Asymptotic expansions in a general system of decaying functions for solutions of the Navier–Stokes equations

Abstract

We study the long-time dynamics of the Navier–Stokes equations in the three-dimensional periodic domains with a body force decaying in time. We introduce appropriate systems of decaying functions and corresponding asymptotic expansions in those systems. We prove that if the force has a large-time asymptotic expansion in Gevrey–Sobolev spaces in such a general system, then any Leray–Hopf weak solution admits an asymptotic expansion of the same type. This expansion is uniquely determined by the force, and independent of the solutions. Various applications of the abstract results are provided which particularly include the previously obtained expansions for the solutions in case of power decay, as well as the new expansions in case of the logarithmic and iterated logarithmic decay.

1 Introduction

We study the long-time behavior of viscous, incompressible fluid flows in space ${\mathbb {R}}^3$. First, we recall the Navier–Stokes equations (NSE) that describe the fluid dynamics.

Let $\mathbf {x}\in {\mathbb {R}}^3$ and $t\in {\mathbb {R}}$ denote the space and time variables, respectively. Let the (kinematic) viscosity be denoted by $\nu >0$, the velocity vector field by $\mathbf {u}(\mathbf {x},t)\in {\mathbb {R}}^3$, the pressure by $p(\mathbf {x},t)\in {\mathbb {R}}$, and the body force by $\mathbf f(\mathbf {x},t)\in {\mathbb {R}}^3$. The NSE are

$$\begin{aligned} \begin{aligned}&\frac{\partial \mathbf {u}}{\partial t} + (\mathbf {u}\cdot \nabla )\mathbf {u}-\nu \Delta \mathbf {u}= -\nabla p+{\mathbf {f}} \quad \text {on }{\mathbb {R}}^3\times (0,\infty ),\\&\text {div } \mathbf {u}= 0 \quad \text {on }{\mathbb {R}}^3\times (0,\infty ). \end{aligned} \end{aligned}$$

(1.1)

The initial condition is

$$\begin{aligned} \mathbf {u}(\mathbf {x},0) = \mathbf {u}^0(\mathbf {x}), \end{aligned}$$

(1.2)

where $\mathbf {u}^0(\mathbf {x})$ is a given divergence-free vector field.

We avoid the unbounded domains and the boundary conditions by considering only force ${\mathbf {f}}(\mathbf {x},t)$ and solutions $(\mathbf {u}(\mathbf {x}, t),p(\mathbf {x}, t))$ that are L-periodic for some $L>0$. Hereafter, a function $g(\mathbf {x})$ is said to be L-periodic if

$$\begin{aligned} g(\mathbf {x}+L\mathbf {e}_j)=g(\mathbf {x})\quad \text {for all}\quad \mathbf {x}\in {\mathbb {R}}^3,\ j=1,2,3, \end{aligned}$$

where $\{\mathbf {e}_1,\mathbf {e}_2,\mathbf {e}_3\}$ is the standard basis of ${\mathbb {R}}^3$ and is said to have zero average over the domain $\Omega =(-L/2, L/2)^3$ if

$$\begin{aligned} \int _\Omega g(\mathbf {x})\mathrm{d}\mathbf {x}=0. \end{aligned}$$

By using a particular Galilean transformation, see details in, e.g., [19], we can also assume, for all $t\ge 0$, that $\mathbf f(\mathbf {x},t)$ and $\mathbf {u}(\mathbf {x},t)$, have zero averages over the domain $\Omega$. In light of the Leray–Helmholtz decomposition, and for the sake of convenience, we also assume that $\mathbf f(\mathbf {x},t)$ is divergence-free for all $t\ge 0$.

By rescaling the variables $\mathbf {x}$ and t, we assume throughout, without loss of generality, that $L=2\pi$ and $\nu =1$.

Throughout the paper, we use the following notation

$$\begin{aligned} u(t)=\mathbf {u}(\cdot ,t),\ f(t)={\mathbf {f}}(\cdot ,t),\ u^0=\mathbf {u}^0(\cdot ), \end{aligned}$$

which are function-valued.

In the case of potential force, that is, ${\mathbf {f}}(\mathbf {x},t)=-\nabla \phi (\mathbf {x},t)$, for some scalar function $\phi$, Foias and Saut proved in [15] that any non-trivial, regular solution u(t) in bounded or periodic domains admits an asymptotic expansion (as $t\rightarrow \infty$)

$$\begin{aligned} u(t) \sim \sum _{n=1}^\infty q_n(t)e^{-nt} \end{aligned}$$

(1.3)

in Sobolev spaces $H^m(\Omega )^3$, for all $m\ge 0$. The interested reader is referred to [3, 13] for early results on the solutions’ asymptotic behavior, Foias and Saut [12,13,14,15,16] for associated normalization map and invariant nonlinear manifolds, Foias et al. [6,7,8] for the corresponding Poincaré–Dulac normal form, Foias et al. [4, 5] for their applications to analysis of helicity, statistical solutions, and decaying turbulence. The recent paper [9] is a survey on the subject.

In case of periodic domains, it was then improved in [18] that the expansion (1.3) holds in Gevrey–Sobolev spaces $G_{\alpha ,\sigma }$ for any $\alpha ,\sigma >0$, see definition (2.1) in Sect. 2, which have much stronger norms than those in $H^m(\Omega )^3$. When the force f is not potential, the asymptotic expansion of Leray–Hopf weak solutions is established in [19] for an exponentially decaying force: if the force has an asymptotic expansion

$$\begin{aligned} f(t)\sim \sum _{n=1}^\infty p_n(t)e^{-nt}, \end{aligned}$$

(1.4)

then u(t) has an asymptotic expansion of type (1.3).

The case of power-decaying forces is treated in [1]: if

$$\begin{aligned} f(t)\sim \sum _{n=1}^\infty \phi _n t^{-\gamma _n}, \end{aligned}$$

(1.5)

then all Leray–Hopf weak solutions u(t) admit the same expansion

$$\begin{aligned} u(t)\sim \sum _{n=1}^\infty \xi _n t^{-\mu _n}. \end{aligned}$$

(1.6)

Above $\phi _n$’s and $\xi _n$’s belong to some Gevrey–Sobolev space $G_{\alpha ,\sigma }$. The meanings of the expansions (1.3), (1.4), (1.5), (1.6) are specified precisely in the cited papers.

The current paper aims to develop the results in [1] to cover a very large class of forces. For example, we will prove that if

$$\begin{aligned} f(t)\sim \sum _{n=1}^\infty \phi _n (\ln t)^{-\gamma _n},\text { or , } f(t)\sim \sum _{n=1}^\infty \phi _n (\ln (\ln t))^{-\gamma _n}, \end{aligned}$$

then

$$\begin{aligned} u(t)\sim \sum _{n=1}^\infty \xi _n (\ln t)^{-\mu _n}, \text { or, respectively, } u(t)\sim \sum _{n=1}^\infty \xi _n (\ln (\ln t))^{-\mu _n}. \end{aligned}$$

In fact, we obtain a much more general result which is described very roughly here. Let $(\psi _n)_{n=1}^\infty$ be a sequence of time-decaying functions with $\psi _{n+1}(t)$ decays to zero, as $t\rightarrow \infty$, much faster than $\psi _n(t)$. The functions $\psi _n$’s are assumed to satisfy a certain set of conditions.

Suppose there exist $\alpha \ge 1/2$ and $\sigma \ge 0$ such that

$$\begin{aligned} f(t)\sim \sum _{n=1}^\infty \phi _n\psi _n(t)\quad \text {in } G_{\alpha ,\sigma }. \end{aligned}$$

(1.7)

We will prove that any Leray–Hopf weak solution u(t) will admit an expansion

$$\begin{aligned} u(t)\sim \sum _{n=1}^\infty \xi _n \psi _n(t) \quad \text {in } G_{\alpha ,\sigma }, \end{aligned}$$

(1.8)

where $\xi _n$’s are explicitly determined by $\phi _n$’s. The meaning of the expansions (1.7) and (1.8) is more sophisticated than (1.3)–(1.6), thanks to their generality, and will be made clear later in the paper.

The paper is organized as follows. Section 2 reviews the functional setting for the NSE which is suitable for studying the solutions’ dynamics in time. It also recalls basic inequalities for the Stokes operator and the bilinear form in the NSE. In Sect. 3, we establish the long-time estimates for solutions of both linearized NSE (Sect. 3.1) and the NSE (Sect. 3.2) with very general decaying forces. The results for the linearized NSE play a key role in improving the long-time estimates for the solutions of the NSE. Having their own merits, these estimates are also crucial to the proofs in Sects. 5 and 6. Section 4 introduces the definitions of systems of decaying functions in time and the asymptotic expansions in those systems. In Definition 4.3, we aim to balance between the generality, such as in Definition 4.1, and the technical requirements. Condition 4.4 is particularly emphasized on applications to ordinary and partial differential equations with quadratic or integral power nonlinearity. Condition 4.5 is focused on functions which are larger than the exponential ones. We state and prove elementary properties for these systems and expansions. In Sect. 5, we obtain in Theorem 5.4 the expansions in Gevrey–Sobolev spaces for all Leray–Hopf solutions of the NSE, when a continuum system of decaying functions is available as the expansions’ basis. The result gives precise meanings to the above expansions (1.7) and (1.8). A version of finite sum asymptotic approximations is proved in Theorem 5.6. It is suitable for a force that has limited information about their long-time behavior. In Sect. 6, we study the situation when the discrete system of functions for expansions cannot be embedded directly into a continuum system as in Sect. 5. However, by using a continuum background system, we can still obtain in Theorem 6.3 the asymptotic expansions for solutions of the NSE. An asymptotic approximation result for the discrete system is similarly obtained in Theorem 6.4. Section 7 provides many applications of the abstract results in Sects. 5 and 6. They consist of the recovery of the power decay case previously established in [1], see Sect. 7.1, as well as the new logarithmic and iterated logarithmic decay cases, see Theorem 7.3 and Corollary 7.4. Examples 7.6 and 7.7 demonstrate some asymptotic expansions with trigonometric functions. More complicated expansions are presented in Propositions  7.8 and 7.11, particularly, the latter one is achieved by simply using of the background systems developed in Sect. 6. “Appendix A” contains some criteria for a convergent series of functions to have corresponding asymptotic expansions of the types specified in Sects. 4 and 6.

2 Functional setting and basic facts for the NSE

We recall the standard functional setting for the NSE, see e.g. [2, 10, 21, 22], and some basic inequalities and estimates.

Let $L^2(\Omega )$ and $H^m(\Omega )=W^{m,2}(\Omega )$, for integers $m\ge 0$, denote the standard Lebesgue and Sobolev spaces on $\Omega$. The standard inner product and norm in $L^2(\Omega )^3$ are denoted by $\langle {\cdot ,\cdot }\rangle$ and $|\cdot |$, respectively. (We warn that this notation $|\cdot |$ also denotes the Euclidean norm in ${\mathbb {R}}^n$ and ${{\mathbb {C}}}^n$, for any $n\in {\mathbb {N}}$, but its meaning will be clear based on the context.)

Let ${\mathcal {V}}$ be the set of all $2\pi$-periodic trigonometric polynomial vector fields which are divergence-free and have zero average over $\Omega$. Define

$$\begin{aligned} H, \text { resp. } V\ =\text { closure of }{\mathcal {V}} \text { in } L^2(\Omega )^3, \text { resp. } H^1(\Omega )^3. \end{aligned}$$

Notice that each element of H is divergence-free and has zero average over $\Omega$, and each element of V is $2\pi$-periodic.

We use the following embeddings and identification

$$\begin{aligned} V\subset H=H'\subset V', \end{aligned}$$

where each space is dense in the next one, and the embeddings are compact.

Let ${\mathcal {P}}$ denote the orthogonal (Leray) projection in $L^2(\Omega )^3$ onto H.

The Stokes operator A is a bounded linear mapping from V to its dual space $V'$ defined by

$$\begin{aligned} \langle {A\mathbf {u},\mathbf {v}}\rangle _{V',V}= \langle \!\langle {\mathbf {u},\mathbf {v}}\rangle \!\rangle {\mathop {=}\limits ^\mathrm{def}}\sum _{j=1}^3 \langle { \frac{\partial \mathbf {u}}{\partial x_j} , \frac{\partial \mathbf {v}}{\partial x_j} }\rangle \quad \text {for all } \mathbf {u},\mathbf {v}\in V. \end{aligned}$$

As an unbounded operator on H, the operator A has the domain ${\mathcal {D}}(A)=V\cap H^2(\Omega )^3$, and, under the current consideration of periodicity conditions,

$$\begin{aligned} A\mathbf {u}= - {\mathcal {P}}\Delta \mathbf {u}=-\Delta \mathbf {u}\in H \quad \text {for all}\quad \mathbf {u}\in {\mathcal {D}}(A). \end{aligned}$$

The spectrum of A is known to be

$$\begin{aligned} {\mathfrak {S}}(A)=\{|\mathbf {k}|^2: \ \mathbf {k}\in {\mathbb {Z}}^3, \mathbf {k}\ne {\mathbf {0}}\}, \end{aligned}$$

and each $\lambda \in {\mathfrak {S}}(A)$ is an eigenvalue. Note that ${\mathfrak {S}}(A)\subset {\mathbb {N}}$ and $1\in {\mathfrak {S}}(A)$, hence, the additive semigroup generated by ${\mathfrak {S}}(A)$ is ${\mathbb {N}}$.

For $n\in {\mathfrak {S}}(A)$, we denote by $R_n$ the orthogonal projection in H on the eigenspace of A corresponding to n, and set

$$\begin{aligned} P_n=\sum _{j\in {\mathfrak {S}}(A),j\le n}R_j. \end{aligned}$$

Note that each vector space $P_nH$ is finite dimensional.

For $\alpha ,\sigma \in {\mathbb {R}}$ and $\mathbf {u}=\sum _{\mathbf {k}\ne {\mathbf {0}}} {{\widehat{\mathbf {u}}}}(\mathbf {k})e^{i\mathbf {k}\cdot \mathbf {x}}$, define

$$\begin{aligned} A^{\alpha } \mathbf {u}=\sum _{\mathbf {k}\ne {\mathbf {0}}} |\mathbf {k}|^{2\alpha } {\widehat{\mathbf {u}}}(\mathbf {k})e^{i\mathbf {k}\cdot \mathbf {x}},\quad e^{\sigma A^{1/2}} \mathbf {u}=\sum _{\mathbf {k}\ne {\mathbf {0}}} e^{\sigma |\mathbf {k}|} {\widehat{\mathbf {u}}}(\mathbf {k})e^{i\mathbf {k}\cdot \mathbf {x}}, \end{aligned}$$

and, hence,

$$\begin{aligned} A^\alpha e^{\sigma A^{1/2}} \mathbf {u}=e^{\sigma A^{1/2}}A^\alpha \mathbf {u}=\sum _{\mathbf {k}\ne {\mathbf {0}}} |\mathbf {k}|^{2\alpha }e^{\sigma |\mathbf {k}|} {{\widehat{\mathbf {u}}}}(\mathbf {k})e^{i\mathbf {k}\cdot \mathbf {x}}. \end{aligned}$$

The Gevrey–Sobolev spaces are defined by

$$\begin{aligned} G_{\alpha ,\sigma }={\mathcal {D}}(A^\alpha e^{\sigma A^{1/2}} ){\mathop {=}\limits ^\mathrm{def}}\{ \mathbf {u}\in H: |\mathbf {u}|_{\alpha ,\sigma }{\mathop {=}\limits ^\mathrm{def}}|A^\alpha e^{\sigma A^{1/2}}\mathbf {u}|<\infty \}, \end{aligned}$$

(2.1)

and, in particular, when $\sigma =0$, the domain of the fractional operator $A^\alpha$ is

$$\begin{aligned} {\mathcal {D}}(A^\alpha )=G_{\alpha ,0}=\{ \mathbf {u}\in H: |A^\alpha \mathbf {u}|=|\mathbf {u}|_{\alpha ,0}<\infty \}. \end{aligned}$$

Thanks to the zero-average condition, the norm $|A^{m/2}\mathbf {u}|$ is equivalent to $\Vert \mathbf {u}\Vert _{H^m(\Omega )^3}$ on the space ${\mathcal {D}}(A^{m/2})$ for $m=0,1,2,\ldots$

Note that ${\mathcal {D}}(A^0)=H$, ${\mathcal {D}}(A^{1/2})=V$, and $\Vert \mathbf {u}\Vert {\mathop {=}\limits ^\mathrm{def}}|\nabla \mathbf {u}|$ is equal to $|A^{1/2}\mathbf {u}|$ for $\mathbf {u}\in V$. Also, the norms $|\cdot |_{\alpha ,\sigma }$ are increasing in $\alpha$, $\sigma$, hence, the spaces $G_{\alpha ,\sigma }$ are decreasing in $\alpha$, $\sigma$.

Regarding the nonlinear term in the NSE, a bounded linear map $B:V\times V\rightarrow V'$ is defined by

$$\begin{aligned} \langle {B(\mathbf {u},\mathbf {v}),\mathbf {w}}\rangle _{V',V}=b(\mathbf {u},\mathbf {v},\mathbf {w}){\mathop {=}\limits ^\mathrm{def}}\int _\Omega ((\mathbf {u}\cdot \nabla ) \mathbf {v})\cdot \mathbf {w}\, \mathrm{d}\mathbf {x}, \quad \text {for all}\quad \mathbf {u},\mathbf {v},\mathbf {w}\in V. \end{aligned}$$

In particular,

$$\begin{aligned} B(\mathbf {u},\mathbf {v})={\mathcal {P}}((\mathbf {u}\cdot \nabla ) \mathbf {v}), \quad \text {for all}\quad \mathbf {u},\mathbf {v}\in {\mathcal {D}}(A). \end{aligned}$$

The problems (1.1) and (1.2) can now be rewritten in the functional form as

$$\begin{aligned}&\frac{\mathrm{d}u(t)}{\mathrm{d}t} + Au(t) +B(u(t),u(t))=f(t) \quad \text { in } V' \text { on } (0,\infty ), \end{aligned}$$

(2.2)

$$\begin{aligned}&u(0)=u^0\in H. \end{aligned}$$

(2.3)

(We refer the reader to the books [2, 20,21,22] for more details.)

The next definition makes precise the meaning of weak solutions of (2.2).

Definition 2.1

Let $f\in L^2_{\mathrm{loc}}([0,\infty ),H)$. A Leray–Hopf weak solutionu(t) of (2.2) is a mapping from $[0,\infty )$ to H such that

$$\begin{aligned} u\in C([0,\infty ),H_{\mathrm{w}})\cap L^2_{\mathrm{loc}}([0,\infty ),V),\quad u'\in L^{4/3}_{\mathrm{loc}}([0,\infty ),V'), \end{aligned}$$

and satisfies

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\langle {u(t),v}\rangle +\langle \!\langle {u(t),v}\rangle \!\rangle +b(u(t),u(t),v)=\langle {f(t),v}\rangle \end{aligned}$$

(2.4)

in the distribution sense in $(0,\infty )$, for all $v\in V$, and the energy inequality

$$\begin{aligned} \frac{1}{2}|u(t)|^2+\int _{t_0}^t \Vert u(\tau )\Vert ^2\mathrm{d}\tau \le \frac{1}{2}|u(t_0)|^2+\int _{t_0}^t \langle f(\tau ),u(\tau )\rangle \mathrm{d}\tau \end{aligned}$$

(2.5)

holds for $t_0=0$ and almost all $t_0\in (0,\infty )$, and all $t\ge t_0$. Here, $H_{\mathrm{w}}$ denotes the topological vector space H with the weak topology.

A regular solution is a Leray–Hopf weak solution that belongs to $C([0,\infty ),V)$.

If $t\mapsto u(T+t)$ is a Leray–Hopf weak/regular solution, then we say u is a Leray–Hopf weak/regular solution on $[T,\infty )$.

It is well known that a regular solution is unique among all Leray–Hopf weak solutions.

We denote by ${\mathcal {T}}$ the set of $t_0\in [0,\infty )$ such that (2.5) holds for all $t\ge t_0$. Note that $[0,\infty ){\setminus } {\mathcal {T}}$ has zero measure.

If u(t) is a Leray–Hopf weak solution and $t_0\in {\mathcal {T}}$, then $u(t_0+t)$ is also a Leray–Hopf weak solution. Assume additionally that there exists a regular solution w(t) with $w(0)=u(t_0)$. Then by the uniqueness of w(t), we have $u(t_0+t)=w(t)$ and, hence, u(t) is a regular solution on $[t_0,\infty )$.

We assume throughout the paper that

Assumption 2.2

The function f belongs to $L^\infty _{\mathrm{loc}}([0,\infty ),H)$.

Under Assumption 2.2, for any $u^0\in H$, there exists a Leray–Hopf weak solution u(t) of (2.2) and (2.3), see e.g. [10]. The large-time behavior of u(t) is the focus of our study. More specific conditions on f will be imposed later.

We note that, thanks to Remark 1(e) of [11], the Leray–Hopf weak solutions in Definition 2.1 are the same as the weak solutions defined in [10, Chapter II, section 7]. Hence, according to inequality (A.39) in [10, Chap. II], we have for any such solution u(t) that

$$\begin{aligned} |u(t)|^2\le e^{-t}|u(0)|^2 + \int _0^t e^{-(t-\tau )} |f(\tau )|^2\mathrm{d}\tau \quad \forall t>0. \end{aligned}$$

(2.6)

Below are some inequalities that will be needed in later estimates. First, for any $\sigma ,\alpha >0$, one has

$$\begin{aligned} \max _{x\ge 0} (x^{\alpha }e^{-\sigma x})=d_0(\alpha ,\sigma ){\mathop {=}\limits ^\mathrm{def}}\Big (\frac{\alpha }{e\sigma }\Big )^{\alpha }. \end{aligned}$$

(2.7)

Thanks to (2.7), one can verify, for all $\alpha ,\sigma >0$, that

$$\begin{aligned} |A^\alpha e^{-\sigma A}v|\le & {} d_0(\alpha ,\sigma ) |v|\quad \forall v\in H,\\ |A^\alpha e^{-\sigma A^{1/2}}v|\le & {} d_0(2\alpha ,\sigma ) |v|\quad \forall v\in H,\nonumber \end{aligned}$$

(2.8)

and, consequently,

$$\begin{aligned} |A^\alpha v|=|(A^\alpha e^{-\sigma A^{1/2}})e^{\sigma A^{1/2}}v|\le d_0(2\alpha ,\sigma ) |e^{\sigma A^{1/2}}v|\quad \forall v\in G_{0,\sigma }. \end{aligned}$$

(2.9)

For the bilinear mapping B(u, v), it follows from its boundedness that there exists a constant $K_*>0$ such that

$$\begin{aligned} \Vert B(u,v)\Vert _{V'}\le K_* \Vert u\Vert \, \Vert v\Vert \quad \forall u,v\in V. \end{aligned}$$

(2.10)

The estimate of the Gevrey norms $|B(u,v)|_{0,\sigma }$ was initiated by Foias–Temam [17]. Here we recall an extended and convenient (though not sharp) version from [18, Lemma 2.1].

There exists a constant$K>1$such that if$\sigma \ge 0$and$\alpha \ge 1/2$, then

$$\begin{aligned} |B(u,v)|_{\alpha ,\sigma }\le K^\alpha |u|_{\alpha +1/2,\sigma } \, |v|_{\alpha +1/2,\sigma }\quad \forall u,v\in G_{\alpha +1/2,\sigma }. \end{aligned}$$

(2.11)

Notation

We make clear the meaning of the “big oh” and “small oh” notation. Let f and g be two non-negative functions defined on a neighborhood of infinity (in ${\mathbb {R}}$).

• We write $f(t)={\mathcal {O}}(g(t))$ (implicitly means as $t\rightarrow \infty$) if there exist $T,C>0$, such that $f(t)\le Cg(t)$ for all $t\ge T$.

• We say $f(t){\mathop {=}\limits ^{{\mathcal {O}}}}g(t)$, if $f(t)={\mathcal {O}}(g(t))$ and $g(t)={\mathcal {O}}(f(t))$.

• We write $f(t)=o(g(t))$ (implicitly means as $t\rightarrow \infty$) if for any $\varepsilon >0$, there exist $T_\varepsilon >0$, such that $f(t)\le \varepsilon g(t)$ for all $t\ge T_\varepsilon$.

Let $u(t)={\mathcal {O}}(f(t))$ and $v(t)={\mathcal {O}}(g(t))$. Then clearly $(uv)(t)= {\mathcal {O}}(g(t)g(t))$, which we simply write as

$$\begin{aligned} {\mathcal {O}}(f(t)) {\mathcal {O}}(g(t))={\mathcal {O}}(f(t)g(t)), \text { and particularly, } f(t) {\mathcal {O}}(g(t))={\mathcal {O}}(f(t)g(t)).\nonumber \\ \end{aligned}$$

(2.12)

If $f(t)={\mathcal {O}}(g(t))$, then $(u+v)(t)={\mathcal {O}}(g(t))$, which we write as

$$\begin{aligned} {\mathcal {O}}(f(t))+{\mathcal {O}}(g(t))={\mathcal {O}}(g(t)). \end{aligned}$$

(2.13)

Note that the identities in (2.12) and (2.13) are only for convenience and should be read from left to right.

3 Large-time estimates

This section contains long-time estimates for solutions of the linearized NSE and of the NSE with the force decaying in time.

First, we have a convenient integral estimate which will be used repeatedly later.

Lemma 3.1

Let F be a continuous, decreasing function from $[0,\infty )$ to $[0,\infty )$. For any $\sigma >0$ and $\theta \in (0,1)$, one has

$$\begin{aligned} \int _0^t e^{-\sigma (t-\tau )}F(\tau )\mathrm{d}\tau \le \frac{1}{\sigma }\Big (F(0)e^{-(1-\theta )\sigma t}+F(\theta t)\Big )\quad \forall t\ge 0. \end{aligned}$$

(3.1)

Proof

We follow the proof of [1, Lemma 2.2]. We split

$$\begin{aligned} \int _0^ t e^{-\sigma (t- \tau )} F(\tau ) \mathrm{d}\tau = I_1+I_2, \end{aligned}$$

where $I_1$ is the integral from 0 to $\theta t$, and $I_2$ is the integral from $\theta t$ to t. Using the monotonicity of F, we estimate

$$\begin{aligned} I_1&\le F(0) \int _0^{\theta t} e^{-\sigma (t-\tau )} \mathrm{d}\tau \le F(0) \frac{e^{-(1-\theta )\sigma t}}{\sigma },\\ I_2&\le F(\theta t)\int _{\theta t}^ t e^{-\sigma (t- \tau )} \mathrm{d}\tau \le F(\theta t) \frac{1}{\sigma }. \end{aligned}$$

Thus, we obtain (3.1). $\square$

3.1 The linearized NSE

We establish an explicit estimate for solutions of the linearized NSE in terms of the decaying force.

Theorem 3.2

Given $\alpha ,\sigma \ge 0$, let $\xi \in G_{\alpha ,\sigma }$, and f be a function from $(0,\infty )$ to $G_{\alpha ,\sigma }$ that satisfies

$$\begin{aligned} |f(t)|_{\alpha ,\sigma }\le MF(t) \quad \text {a.e. in }(0,\infty ), \end{aligned}$$

(3.2)

where M is a positive constant, and F is a continuous, decreasing function from $[0,\infty )$ to $[0,\infty )$.

Let $w_0\in G_{\alpha ,\sigma }$. Suppose $w\in C([0,\infty ),H_\mathrm{w})\cap L^1_{\mathrm{loc}}([0,\infty ),V)$, with $w'\in L^1_\mathrm{loc}([0,\infty ),V')$, is a weak solution of

$$\begin{aligned} w'=-Aw+\xi +f \text { in }V'\text { on }(0,\infty ), \quad w(0)=w_0, \end{aligned}$$

i.e., it holds, for all $v\in V$, that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\langle {w,v}\rangle =-\langle \!\langle {w,v}\rangle \!\rangle +\langle {\xi +f,v}\rangle \text { in the distribution sense on }(0,\infty ). \end{aligned}$$

Then the following statements hold true.

• (i) $w(t)\in G_{\alpha +1-\varepsilon ,\sigma }$ for all $\varepsilon \in (0,1)$ and $t>0$.

• (ii) For any numbers $a,a_0\in (0,1)$ with $a+a_0<1$ and any $\varepsilon \in (0,1)$, there exists a positive constant C depending on $a_0$, a, $\varepsilon$, M, F(0), $|\xi |_{\alpha ,\sigma }$ and $|w_0|_{\alpha ,\sigma }$ such that

  $$\begin{aligned} |w(t)-A^{-1}\xi |_{\alpha +1-\varepsilon ,\sigma }\le C\big (e^{-a_0 t}+F(a t)\big ) \quad \forall t \ge 1. \end{aligned}$$

  (3.3)

• (iii) Assume, in addition, that

  • There exist $k_0>0$ and $D_1>0$ such that

    $$\begin{aligned} e^{-k_0 t}\le D_1 F(t)\quad \forall t\ge 0, \text { and } \end{aligned}$$

    (3.4)

  • For any $a\in (0,1)$, there exists $D_2=D_{2,a}>0$ such that

    $$\begin{aligned} F(at)\le D_2 F(t)\quad \forall t\ge 0. \end{aligned}$$

    (3.5)

  Then there exists $C>0$ such that

  $$\begin{aligned} |w(t)-A^{-1}\xi |_{\alpha +1-\varepsilon ,\sigma }\le CF(t) \quad \forall t \ge 1. \end{aligned}$$

  (3.6)

Proof

• (i) This regularity result is the same as [1, Lemma 2.4(i)] in which we set $f:=\xi +f$.

• (ii) First, we state and prove a more technical version of (3.3).

  For any$\varepsilon ,\delta ,\theta ,\theta '\in (0,1)$, there exists$C>0$depending on$\varepsilon$, $\delta$, $\theta$, M, F(0), $|\xi |_{\alpha ,\sigma }$and$|w_0|_{\alpha ,\sigma }$such that

  $$\begin{aligned} |w(t)-A^{-1}\xi |_{\alpha +1-\varepsilon ,\sigma }\le C\big (e^{-(1-\theta ')\theta \delta t}+F(\theta '\theta t)\big ) \quad \forall t \ge 1. \end{aligned}$$

  (3.7)

Proof of (3.7) We follow the proof of [1, Lemma 2.3].

• (a) Let $N\in {\mathfrak {S}}(A)$. We recall formula (2.19) of [1]:

  $$\begin{aligned} P_N\big (w(t)-A^{-1}\xi \big )=e^{-tA}P_N w_0-A^{-1}e^{-tA}P_N \xi +\int _0^t e^{-(t-\tau )A}P_N f(\tau )\mathrm{d}\tau \quad \forall t\ge 0.\nonumber \\ \end{aligned}$$

  (3.8)

(This formula was derived by using the equation of $P_N w$ and the variation of constant formula.)

Let $\varepsilon \in (0,1)$. Applying $A^{1-\varepsilon }$ to both sides of (3.8), and estimating the $|\cdot |_{\alpha ,\sigma }$ norm of the resulting quantities, we obtain

$$\begin{aligned} \begin{aligned} |P_N(w(t)-A^{-1}\xi )|_{\alpha +1-\varepsilon ,\sigma }&\le |A^{1-\varepsilon }e^{-tA}w_0|_{\alpha ,\sigma }+|A^{-\varepsilon }e^{-tA}\xi |_{\alpha ,\sigma }\\&\quad +\int _0^t |e^{-(t-\tau )A}A^{1-\varepsilon } f(\tau )|_{\alpha ,\sigma }\mathrm{d}\tau . \end{aligned} \end{aligned}$$

(3.9)

Let $\theta ,\delta \in (0,1)$ and $t\ge 1$. We estimate each term on the right-hand side of (3.9) separately.

• (b) Rewriting the first term on the right-hand side of (3.9) and applying (2.8) yield

  $$\begin{aligned} |A^{1-\varepsilon }e^{-tA}w_0|_{\alpha ,\sigma }&= |A^{1-\varepsilon }e^{-(1-\delta )tA} (e^{-\delta t A} w_0)|_{\alpha ,\sigma } \le \Big [ \frac{1-\varepsilon }{e(1-\delta )t}\Big ]^{1-\varepsilon }|e^{-\delta tA} w_0|_{\alpha ,\sigma }\\&\le \Big [ \frac{1-\varepsilon }{e(1-\delta )}\Big ]^{1-\varepsilon } e^{-\delta t} |w_0|_{\alpha ,\sigma }. \end{aligned}$$

The second term on the right-hand side of (3.9) can be easily estimated by

$$\begin{aligned} |A^{-\varepsilon }e^{-tA}\xi |_{\alpha ,\sigma } \le | e^{-tA} \xi |_{\alpha ,\sigma } \le e^{-t} |\xi |_{\alpha ,\sigma }. \end{aligned}$$

• (c) Dealing with the last integral in (3.9), we split it into two integrals

  $$\begin{aligned} \int _0^t |e^{-(t-\tau )A}A^{1-\varepsilon }f(\tau )|_{\alpha ,\sigma }\mathrm{d}\tau =I_1+I_2, \end{aligned}$$

where $I_1$ is the integral from 0 to $\theta t$, and $I_2$ from $\theta t$ to t.

For $I_1$, we have

$$\begin{aligned} I_1&= \int _0^{\theta t} \Big |e^{-(t-\tau )(1-\delta )A}\Big ( e^{-(t-\tau )\delta A} A^{1-\varepsilon } f(\tau )\Big )\Big |_{\alpha ,\sigma }\mathrm{d}\tau \\&\le \int _0^{\theta t} \Big |e^{-(1-\theta )t(1-\delta ) A}A^{1-\varepsilon } \Big (e^{-(t-\tau )\delta A}f(\tau )\Big )\Big |_{\alpha ,\sigma }\mathrm{d}\tau . \end{aligned}$$

Utilizing (2.8) and then using hypothesis (3.2), we obtain

$$\begin{aligned} I_1&\le \int _0^{\theta t} \Big [\frac{1-\varepsilon }{e(1-\theta )(1-\delta )t}\Big ]^{1-\varepsilon }|e^{-(t-\tau )\delta A}f(\tau )|_{\alpha ,\sigma }\mathrm{d}\tau \\&\le \Big [\frac{1-\varepsilon }{e(1-\theta )(1-\delta )t}\Big ]^{1-\varepsilon } \int _0^{\theta t} e^{-(t-\tau )\delta } MF(\tau )\mathrm{d}\tau .\\&\le M\Big [\frac{1-\varepsilon }{e(1-\theta )(1-\delta )}\Big ]^{1-\varepsilon } \int _0^{\theta t} e^{-\delta (\theta t-\tau )} F(\tau )\mathrm{d}\tau . \end{aligned}$$

Let $\theta '\in (0,1)$. Then by Lemma 3.1,

$$\begin{aligned} I_1&\le M \Big [\frac{1-\varepsilon }{e(1-\theta )(1-\delta )}\Big ]^{1-\varepsilon } \frac{1}{\delta }\big (F(0) e^{-(1-\theta ')\delta \theta t}+F(\theta '\theta t)\big ). \end{aligned}$$

For $I_2$, we apply (2.8) and use (3.2) to have

$$\begin{aligned} I_2&= \int _{\theta t}^t |e^{-(t-\tau )\delta A} (e^{-(t-\tau )(1-\delta )A}A^{1-\varepsilon }f(\tau ))|_{\alpha ,\sigma }\mathrm{d}\tau \\&\le \int _{\theta t}^t e^{-(t-\tau )\delta } |e^{-(t-\tau )(1-\delta )A}A^{1-\varepsilon }f(\tau )|_{\alpha ,\sigma }\mathrm{d}\tau \\&\le \int _{\theta t}^t e^{-\delta (t-\tau )} \Big [\frac{1-\varepsilon }{e(1-\delta )(t-\tau )}\Big ]^{1-\varepsilon } |f(\tau )|_{\alpha ,\sigma }\mathrm{d}\tau \\&\le \Big [\frac{1-\varepsilon }{e(1-\delta )}\Big ]^{1-\varepsilon } M F(\theta t) \int _{\theta t}^t \frac{e^{-\delta (t-\tau )}}{(t-\tau )^{1-\varepsilon }}\mathrm{d}\tau . \end{aligned}$$

We estimate the last integral by

$$\begin{aligned} \int _{\theta t}^t \frac{e^{-\delta (t-\tau )}}{(t-\tau )^{1-\varepsilon }}\mathrm{d}\tau&=\int _0^{(1-\theta )t} z^{\varepsilon -1}e^{-\delta z}\mathrm{d}z \le \int _0^{1-\theta } z^{\varepsilon -1}\mathrm{d}z + (1-\theta )^{\varepsilon -1}\int _{1-\theta }^{(1-\theta )t} e^{-\delta z}\mathrm{d}z\\&\le \frac{(1-\theta )^\varepsilon }{\varepsilon }+ \frac{(1-\theta )^{\varepsilon -1}}{\delta }e^{-\delta (1-\theta )}. \end{aligned}$$

• (d) Combining the above calculations, we obtain

  $$\begin{aligned} |P_N\big ( w(t)-A^{-1}\xi \big )|_{\alpha +1-\varepsilon ,\sigma }\le C\big (e^{-(1-\theta ')\theta \delta t}+F(\theta '\theta t)\big ) \quad \forall t \ge 1, \end{aligned}$$

  (3.10)

with constant C independent of N. By passing $N\rightarrow \infty$ in (3.10), and the fact $A^{-1}\xi \in G_{\alpha +1-\varepsilon ,\sigma }$, we obtain $w(t)\in G_{\alpha +1-\varepsilon ,\sigma }$ together with the estimate (3.7).

Proof of (3.3) Take $\theta \in (a+a_0,1)$, and set $\delta =a_0/(\theta -a)$ and $\theta '=a/\theta$. Then $\theta -a>a_0>0$, which gives $\delta ,\theta '\in (0,1)$. It is also clear that $\theta \theta '=a$ and $(1-\theta ')\theta \delta =(\theta -a)\delta =a_0$. Therefore, with these values of $\theta ,\theta ',\delta$, inequality (3.3) follows (3.7).

(iii) Note, by the monotonicity of F, that the property (3.5) in fact holds true for all $a>0$, with $D_{2,a}=1$ for all $a\ge 1$. Then we have

$$\begin{aligned} \begin{aligned} e^{-a_0 t}+F(a t)&=e^{-k_0 \cdot a_0 t/k_0}+F(a t)\\&\le D_1 F(a_0 t/k_0) +F(at)\le (D_1 D_{2,a_0/k_0}+D_{2,a})F(t). \end{aligned} \end{aligned}$$

(3.11)

Combining (3.11) with (3.3), we obtain inequality (3.6). The proof is complete. $\square$

3.2 The NSE

This subsection aims at establishing the large-time estimates for weak solutions of the NSE. First, we obtain a result for small initial data and force.

Theorem 3.3

Let F be a continuous, decreasing, non-negative function on $[0,\infty )$. Given $\alpha \ge 1/2$ and numbers $\theta _0,\theta \in (0,1)$ such that $\theta _0+\theta <1$. Then there exist positive numbers $c_k=c_k(\alpha ,\theta _0,\theta ,F)$, for $k=0,1,2,3$, such that the following holds true. If

$$\begin{aligned} |A^\alpha u^0|&\le c_0, \end{aligned}$$

(3.12)

$$\begin{aligned} |f(t)|_{\alpha -1/2,\sigma }&\le c_1F(t)\quad \text {a.e. in } (0,\infty )\text { for some } \sigma \ge 0, \end{aligned}$$

(3.13)

then there exists a unique regular solution u(t) of (2.2) and (2.3), which also belongs to $C([0,\infty ),{\mathcal {D}}(A^\alpha ))$ and satisfies, for all $t\ge 8\sigma (1-\theta )/(1-\theta -\theta _0)$,

$$\begin{aligned} |u(t)|_{\alpha ,\sigma }&\le c_2 (e^{-2\theta _0 t}+F^2(\theta t))^{1/2}, \end{aligned}$$

(3.14)

$$\begin{aligned} \int _t^{t+1} |u(\tau )|_{\alpha +1/2,\sigma }^2\mathrm{d}\tau&\le c_3^2 (e^{-2\theta _0 t}+F^2(\theta t)). \end{aligned}$$

(3.15)

Proof

The proof follows [1, Theorem 3.1]. The calculations below are formal, but can be made rigorous by using the Galerkin approximations and then pass to the limit.

Let $\theta _*=\theta _0/(1-\theta )\in (\theta _0,1)$ and denote $t_*=8\sigma /(1-\theta _*)=8\sigma (1-\theta )/(1-\theta -\theta _0)$.

• (a) For $\sigma >0$, let $\varphi$ be a $C^\infty$-function on ${\mathbb {R}}$ such that $\varphi ((-\infty ,0])=\{0\}$, $\varphi ([t_*,\infty ))=\{\sigma \}$, and $0< \varphi '< 2\sigma /t_*$ on $(0,t_*)$. We derive from (2.2) that

  $$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}(A^{\alpha }e^{\varphi (t) A^{1/2}}u)&=\varphi '(t)A^{1/2}A^{\alpha }e^{\varphi (t) A^{1/2}} u+A^{\alpha }e^{\varphi (t) A^{1/2}}\frac{\mathrm{d}u}{\mathrm{d}t} \nonumber \\&=\varphi '(t)A^{\alpha +1/2}e^{\varphi (t) A^{1/2}} u+A^{\alpha }e^{\varphi (t) A^{1/2}}(-Au-B(u,u)+f). \end{aligned}$$

  (3.16)

By taking the inner product in H of (3.16) with $A^{\alpha }e^{\varphi (t) A^{1/2}}u(t)$, we obtain

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}|u|_{\alpha ,\varphi (t)}^2 + |A^{1/2}u|_{\alpha ,\varphi (t)}^2 =\varphi '(t)\langle A^{\alpha +1/2}e^{\varphi (t) A^{1/2}}u,A^{\alpha }e^{\varphi (t) A^{1/2}} u\rangle \\&\quad -\langle A^{\alpha }e^{\varphi (t) A^{1/2}}B(u,u),A^{\alpha }e^{\varphi (t) A^{1/2}}u\rangle + \langle A^{\alpha -1/2}e^{\varphi (t) A^{1/2}}f,A^{\alpha +1/2}e^{\varphi (t) A^{1/2}}u\rangle . \end{aligned}$$

Using the Cauchy–Schwarz inequality, and estimating the second term on the right-hand side by (2.11), we get

$$\begin{aligned} \begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}|u|_{\alpha ,\varphi (t)}^2 + |A^{1/2}u|_{\alpha ,\varphi (t)}^2&\le \varphi '(t) |A^{1/2}u|_{\alpha ,\varphi (t)}^2\\&\quad +K^\alpha |A^{1/2}u|_{\alpha ,\varphi (t)}^2 |u|_{\alpha ,\varphi (t)} + |f(t)|_{\alpha -1/2,\varphi (t)}|A^{1/2}u|_{\alpha ,\varphi (t)}. \end{aligned} \end{aligned}$$

(3.17)

Using the bound of $\varphi '(t)$ and applying Cauchy’s inequality to the last term gives

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}|u|_{\alpha ,\varphi (t)}^2 + |A^{1/2}u|_{\alpha ,\varphi (t)}^2 \le \frac{2\sigma }{t_*} |A^{1/2}u|_{\alpha ,\varphi (t)}^2 \\&\quad +K^\alpha |u|_{\alpha ,\varphi (t)} |A^{1/2}u|_{\alpha ,\varphi (t)}^2 +\frac{2\sigma }{t_*}|A^{1/2}u|_{\alpha ,\varphi (t)}^2+ \frac{t_*}{8\sigma }|f(t)|_{\alpha -1/2,\varphi (t)}^2, \end{aligned}$$

which, together with the fact $\varphi (t)\le \sigma$, implies

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}|u|_{\alpha ,\varphi (t)}^2 + \Big (1-\frac{4\sigma }{t_*} -K^\alpha |u|_{\alpha ,\varphi (t)}\Big )|A^{1/2}u|_{\alpha ,\varphi (t)}^2 \le \frac{t_*}{8\sigma }|f(t)|_{\alpha -1/2,\sigma }^2. \end{aligned}$$

Thus,

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}|A^\alpha u|^2+\Big (1-\frac{1-\theta _*}{2}-K^\alpha |A^\alpha u|\Big )|A^{\alpha +1/2}u|^2\le \frac{1}{1-\theta ^*}|A^{\alpha -1/2}f|^2. \end{aligned}$$

(3.18)

• (b) For $\sigma =0$, let $\varphi \equiv 0$ on ${\mathbb {R}}$. Then the first term in (3.17) vanishes. Applying Cauchy’s inequality to the last term of (3.17):

  $$\begin{aligned} |f(t)|_{\alpha -1/2,\varphi (t)}|A^{1/2}u|_{\alpha ,\varphi (t)}\le \frac{1-\theta _*}{2} |A^{1/2}u|_{\alpha ,\varphi (t)}^2 + \frac{1}{1-\theta _*} |f(t)|_{\alpha -1/2,\varphi (t)}^2, \end{aligned}$$

  we obtain the same inequality (3.18).

• (c) In the calculations below, we use the following constants

  $$\begin{aligned} c_*&=c_*(\alpha ,\theta _0,\theta ,F)= \frac{1-\theta _*}{4K^\alpha },&\gamma =\gamma (F)=\frac{1}{F(0)+1}\in (0,1],\\ c_0&=c_0(\alpha ,\theta _0,\theta ,F)=\gamma c_*,&c_1=c_1(\alpha ,\theta _0,\theta ,F)=\gamma ^2 c_* (\theta _*(1-\theta _*))^{1/2},\\ c_2&=c_2(\alpha ,\theta _0,\theta ,F)=\sqrt{2} \gamma c_*,&c_3=c_3(\alpha ,\theta _0,\theta ,F)=(1+\theta _*^{-1})^{1/2}\gamma c_* . \end{aligned}$$

At the initial time, we have

$$\begin{aligned} |u(0)|_{\alpha ,\varphi (0)}=|A^\alpha u^0|<2c_0\le 2c_*. \end{aligned}$$

Let $T\in (0,\infty )$. Assume that

$$\begin{aligned} |u(t)|_{\alpha ,\varphi (t)}\le 2c_*\quad \forall t\in [0,T). \end{aligned}$$

(3.19)

This and the definition of $c_*$ give

$$\begin{aligned} K^\alpha |u(t)|_{\alpha ,\varphi (t)} \le 2c_* K^\alpha =(1-\theta _*)/2 \quad \forall t\in [0,T). \end{aligned}$$

(3.20)

For $t\in (0,T)$, we have from (3.18) for both $\sigma >0$ and $\sigma =0$, and (3.20) that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}|u|_{\alpha ,\varphi (t)}^2 + 2\theta _*|A^{1/2}u|_{\alpha ,\varphi (t)}^2 \le 2(1-\theta _*)^{-1}|f(t)|_{\alpha -1/2,\sigma }^2. \end{aligned}$$

(3.21)

Applying Gronwall’s inequality in (3.21) and using (3.12), (3.13) yield, for all $t\in (0,T)$, that

$$\begin{aligned} |u(t)|_{\alpha ,\varphi (t)}^2&\le e^{-2\theta _*t}|u^0|_{\alpha ,0}^2+2(1-\theta _*)^{-1}\int _0^t e^{-2\theta _*(t-\tau )}|f(\tau )|_{\alpha -1/2,\sigma }^2 \mathrm{d}\tau \\&\le e^{-2\theta _0 t}c_0^2+2(1-\theta _*)^{-1}c_1^2\int _0^t e^{-2\theta _*(t-\tau )}F^2(\tau ) \mathrm{d}\tau . \end{aligned}$$

For the last integral, applying (3.1) with $F:=F^2$, $\sigma :=2\theta _*$ and noting that $(1-\theta )\theta _*=\theta _0$ give

$$\begin{aligned} \int _0^t e^{-2\theta _* (t-\tau )}F^2(\tau )\mathrm{d}\tau \le \frac{1}{2\theta _*}\Big (F^2(0)e^{-2\theta _0 t}+F^2(\theta t)\Big )\quad \forall t\ge 0. \end{aligned}$$

Then we obtain

$$\begin{aligned} |u(t)|_{\alpha ,\varphi (t)}^2&\le c_0^2 e^{-2\theta _0 t} + [\theta _*(1-\theta _*)]^{-1}c_1^2\big ( \gamma ^{-2}e^{-2\theta _0 t}+ F^2(\theta t)\big )\\&\le 2 c_*^2 \gamma ^2(e^{-2\theta _0 t} + F^2(\theta t)). \end{aligned}$$

This implies

$$\begin{aligned} |u(t)|_{\alpha ,\varphi (t)} \le \sqrt{2} c_* \gamma (e^{-2\theta _0 t}+F^2(\theta t))^{1/2} \quad \forall t\in [0,T). \end{aligned}$$

(3.22)

Letting $t\rightarrow T^-$ in (3.22) and using the monotonicity of F give

$$\begin{aligned} \lim _{t\rightarrow T^-}|u(t)|_{\alpha ,\varphi (t)} \le \sqrt{2} c_* \gamma (1+F^2(0))^{1/2} < 2c_*. \end{aligned}$$

(3.23)

By the standard contradiction argument, it follows (3.19) and (3.23) that the inequalities (3.19) and (3.22), in fact, hold true for $T=\infty$. Then, due to the fact $\varphi (t)=\sigma$ for all $t\ge t_*$, inequality (3.22) implies (3.14).

• (d) For $t\ge t_*$, by integrating (3.21) from t to $t+1$, and using estimates (3.14), (3.13), we obtain

  $$\begin{aligned} \int _t^{t+1} |A^{1/2}u(\tau )|_{\alpha ,\sigma }^2\mathrm{d}\tau&\le \frac{1}{2\theta _*}| u(t)|_{\alpha ,\sigma }^2+[\theta _*(1-\theta _*)]^{-1}c_1^2 \int _t^{t+1} F^2(\tau )\mathrm{d}\tau \\&\le c_*^2 \gamma ^2 \theta _*^{-1}(e^{-2\theta _0 t}+F^2(\theta t))+ c_*^2 \gamma ^2 F^2(\theta t). \end{aligned}$$

Then inequality (3.15) follows. The proof is complete. $\square$

Theorem 3.4

Let F be a continuous, decreasing, non-negative function on $[0,\infty )$ that satisfies

$$\begin{aligned} \lim _{t\rightarrow \infty } F(t)=0. \end{aligned}$$

(3.24)

Suppose there exist $\sigma \ge 0$, $\alpha \ge 1/2$ such that

$$\begin{aligned} |f(t)|_{\alpha ,\sigma }={\mathcal {O}}(F(t)). \end{aligned}$$

(3.25)

Let u(t) be a Leray–Hopf weak solution of (2.2). Then there exists ${\hat{T}}>0$ such that u(t) is a regular solution of (2.2) on $[{\hat{T}},\infty )$, and for any $\varepsilon ,\lambda \in (0,1)$, and $a_0,a,\theta _0,\theta \in (0,1)$ with $a_0+a<1$, $\theta _0+\theta <1$, there exists $C>0$ such that

$$\begin{aligned} |u({{\hat{T}}} + t)|_{\alpha +1-\varepsilon ,\sigma } \le C(e^{-a_0 t}+e^{-2\theta _0 a t}+F^{2\lambda }(\theta a t)+F(at))\quad \forall t\ge 0. \end{aligned}$$

(3.26)

If, in addition, F satisfies (3.4) and (3.5), then

$$\begin{aligned} |u({\hat{T}}+t)|_{\alpha +1-\varepsilon ,\sigma } \le CF(t)\quad \forall t\ge 0. \end{aligned}$$

(3.27)

Proof

By (3.25), there exist $T_1>0$ and $C_1>0$ such that

$$\begin{aligned} |f(t)|_{\alpha ,\sigma }\le C_1 F(t)\quad t\ge T_1. \end{aligned}$$

(3.28)

We claim the following fact which is weaker than the desired estimate (3.27).

ClaimFor any$\lambda \in (0,1)$, and$\theta ,\theta _0\in (0,1)$with$\theta +\theta _0<1$, there exists${\hat{T}}\ge T_1$such thatu(t) is a regular solution of (2.2) on$[{\hat{T}},\infty )$, and one has for all$t\ge 0$that

$$\begin{aligned} |u({\hat{T}}+t)|_{\alpha +1/2,\sigma } \le C(e^{-2\theta _0 t}+ F^{2\lambda }(\theta t))^{1/2}, \end{aligned}$$

(3.29)

for some positive constantC.

Accepting this Claim at the moment, we prove (3.27). Rewrite the NSE (2.2) as the linearized NSE:

$$\begin{aligned} u_t+Au={{\tilde{f}}}(t){\mathop {=}\limits ^\mathrm{def}}-B(u(t),u(t))+f(t) . \end{aligned}$$

(3.30)

Then from (3.29) and (2.11) we obtain for t large,

$$\begin{aligned} |B(u({{\hat{T}}}+t),u({{\hat{T}}}+t))|_{\alpha ,\sigma } \le K^\alpha |u({{\hat{T}}}+t)|_{\alpha +1/2,\sigma }^2 \le C_2(e^{-2\theta _0 t}+F^{2\lambda }(\theta t)) \end{aligned}$$

for some positive constant $C_2$. From this and (3.28), we have, for $t\ge 0$,

$$\begin{aligned} |{{\tilde{f}}}({{\hat{T}}}+t)|_{\alpha ,\sigma }\le C_2(e^{-2\theta _0 t}+F^{2\lambda }(\theta t))+C_1 F({{\hat{T}}}+t )\le C_3 {{\tilde{F}}}(t), \end{aligned}$$

(3.31)

where $C_3=C_1+C_2$ and ${{\tilde{F}}}(t)=e^{-2\theta _0 t}+F^{2\lambda }(\theta t)+F(t)$.

By (3.30) and (3.31), we apply part (iii) of Theorem 3.2 with $w(t):=u({{\hat{T}}}+t)$, $f(t):={{\tilde{f}}}({{\hat{T}}}+t)$, $\xi =0$, $M=C_3$, $F(t):={{\tilde{F}}}(t)$ to obtain from (3.3), with $t:=t+1$, that

$$\begin{aligned} |u({{\hat{T}}} + t + 1)|_{\alpha +1-\varepsilon ,\sigma }&\le C_4(e^{-a_0(t+1)}+{{\tilde{F}}}(a(t+1)))\\&\le C_4(e^{-a_0 t}+e^{-2\theta _0 a t}+F^{2\lambda }(\theta a t)+F(at)). \end{aligned}$$

for all $t \ge 0$ and some constant $C_4>0$. By re-denoting ${{\hat{T}}}:={{\hat{T}}}+1$, we obtain (3.26).

Now, assume (3.4) and (3.5). Taking $\lambda =1/2$, $a_0=\theta _0\in (0,1/2)$ and $a=1/2$, we obtain from (3.26)

$$\begin{aligned} |u({\hat{T}}+t)|_{\alpha +1-\varepsilon ,\sigma } \le C(e^{-\theta _0 t}+F(\theta t/2))\quad \forall t\ge 0. \end{aligned}$$

(3.32)

Similarly to proving (3.11), we obtain inequality (3.27) from (3.32).

The rest of this proof is to prove the Claim.

• (a) By Assumption 2.2, there exists $C_0>0$ such that

  $$\begin{aligned} |f(t)|\le C_0, \quad \text {a.e. in } (0,T_1). \end{aligned}$$

  (3.33)

On the one hand, using (2.6), (3.28), (3.33) we have, for all $t\ge T_1$, that

$$\begin{aligned} |u(t)|^2&\le e^{-t}|u_0|^2 + C_0^2 \int _0^{T_1} e^{-(t-\tau )}\mathrm{d}\tau + C_1^2 \int _{T_1}^t e^{-(t-\tau )}F^2(\tau )\mathrm{d}\tau \\&\le e^{-t}|u_0|^2 + C_0^2 e^{-t}e^{T_1} + C_1^2 \int _0^t e^{-(t-\tau )}F^2(\tau )\mathrm{d}\tau . \end{aligned}$$

To estimate the last integral, we apply inequality (3.1) with $\sigma :=1$, $\theta :=1/2$, $F:=F^2$, hence, obtain

$$\begin{aligned} |u(t)|^2\le e^{-t}(|u_0|^2+C_0^2e^{T_1}) + C_1^2 (F^2(0)e^{-t/2}+F^2(t/2))\quad \forall t\ge T_1. \end{aligned}$$

(3.34)

On the other hand, we estimate in (2.5)

$$\begin{aligned} |\langle {f(\tau ),u(\tau )}\rangle |\le \frac{1}{2} |u(\tau )|^2+\frac{1}{2} |f(\tau )|^2 \le \frac{1}{2} \Vert u(\tau )\Vert ^2+\frac{1}{2} |f(\tau )|^2. \end{aligned}$$

Hence, we obtain

$$\begin{aligned} |u(t)|^2+\int _{t_0}^t \Vert u(\tau )\Vert ^2\mathrm{d}\tau \le |u(t_0)|^2+\int _{t_0}^t |f(\tau )|^2\ \mathrm{d}\tau \end{aligned}$$

(3.35)

for all $t_0\in {\mathcal {T}}$ and $t\ge t_0$.

Let $t_0\in {\mathcal {T}}\cap [T_1,\infty )$. Setting $t=t_0+1$ in (3.35), using (3.34) to estimate $|u(t_0)|^2$, and (3.28) to estimate $|f(\tau )|$, we derive

$$\begin{aligned} \int _{t_0}^{t_0+1}\Vert u(\tau )\Vert ^2\mathrm{d}\tau \le e^{-t_0}(|u_0|^2+C_0^2 e^{T_1}) + C_1^2 (F^2(0)e^{-t_0/2}+F^2(t_0/2)) + C_1^2 F^2(t_0), \end{aligned}$$

thus,

$$\begin{aligned} \int _{t_0}^{t_0+1}\Vert u(\tau )\Vert ^2\mathrm{d}\tau \le e^{-t_0/2}(|u_0|^2 +C_0^2 e^{T_1}+ C_1^2 F^2(0)) + 2C_1^2F^2(t_0/2). \end{aligned}$$

(3.36)

Let $t\ge T_1$ be arbitrary now. There exists a sequence $\{t_n\}_{n=1}^\infty$ in ${\mathcal {T}}\cap (T_1,\infty )$ such that $\lim _{n\rightarrow \infty }t_n=t$. Then (3.36) holds for $t_0=t_n$, and letting $n\rightarrow \infty$ gives

$$\begin{aligned} \int _t^{t+1}\Vert u(\tau )\Vert ^2\mathrm{d}\tau\le & {} M_t{\mathop {=}\limits ^\mathrm{def}}e^{-t/2}(|u_0|^2 + C_0^2 e^{T_1}+C_1^2F^2(0))\nonumber \\&+2C_1^2F^2(t/2)\quad \forall t\ge T_1. \end{aligned}$$

(3.37)

Note that the quantity $M_t$ in (3.37) is decreasing in t, and goes to zero as t tends to infinity.

• (b) Consider $\sigma >0$. Let $\lambda \in (0,1)$. For $T>0$, we write

  $$\begin{aligned} F(t+T)=F^{1-\lambda }(t+T)F^{\lambda }(t+T)\le F(T)^{1-\lambda }F^\lambda (t). \end{aligned}$$

  (3.38)

Choose $T_2>T_1$ such that

$$\begin{aligned} M_{T_2}< c_0(1/2,F^\lambda )/2\text { and } F(T_2)^{1-\lambda }\le c_1(1/2,F^\lambda )/C_1. \end{aligned}$$

By applying inequality (3.37) to $t=T_2$, there exists $t_0\in {\mathcal {T}}\cap (T_2,T_2+1)$ such that

$$\begin{aligned} |A^{1/2}u(t_0)|\le 2M_{t_0}\le 2 M_{T_2} <c_0(1/2,F^\lambda ). \end{aligned}$$

Moreover, for $t\ge 0$, by (3.28) and (3.38),

$$\begin{aligned} |f(t_0+t)|_{0,\sigma }&\le C_1 F(t_0+t)\le C_1 F(t_0)^{1-\lambda }F^\lambda (t)\le C_1 F(T_2)^{1-\lambda }F^\lambda (t)\nonumber \\&\le c_1(1/2,F^\lambda ) F^\lambda (t). \end{aligned}$$

(3.39)

Applying Theorem 3.3 to the unique regular solution $u(t):= u(t_0+t)$, force $f(t):= f(t_0+t)$ with parameters $\alpha =1/2$ and $F(t):=F^\lambda (t)$, we obtain from (3.14) that

$$\begin{aligned} |u(t_0+t)|_{1/2,\sigma }\le c_2(1/2,F^\lambda ) (e^{-\theta _0 t}+F^\lambda (\theta t))\quad \forall t\ge t_*, \end{aligned}$$

(3.40)

where $t_*$ is a non-negative number. Then by (2.9), we have for all $t\ge t_*$ that

$$\begin{aligned} |A^{\alpha +1/2} u(t_0+t)| \le d_0(2\alpha +1,\sigma ) |e^{\sigma A^{1/2}} u(t_0+t)| \le d_0(2\alpha +1,\sigma ) |u(t_0+t)|_{1/2,\sigma }, \end{aligned}$$

and, hence, thanks to (3.40),

$$\begin{aligned} \lim _{t\rightarrow \infty } |A^{\alpha +1/2} u(t_0+t)|=0. \end{aligned}$$

(3.41)

Using (3.41), and similar to (3.39) with the norm $|\cdot |_{\alpha ,\sigma }$ replacing $|\cdot |_{0,\sigma }$, we deduce that there is $T\in {\mathcal {T}}\cap (t_0+t_*,\infty )$ so that

$$\begin{aligned} |A^{\alpha +1/2} u(T)|&\le c_0(\alpha +1/2,F^\lambda ), \end{aligned}$$

(3.42)

$$\begin{aligned} |f(T+t)|_{\alpha ,\sigma }&\le c_1(\alpha +1/2,F^\lambda )F^\lambda (t) \quad \forall t\ge 0. \end{aligned}$$

(3.43)

• (c) We will establish (3.42) and (3.43) when $\sigma =0$. First, we observe the following: if $j\in {\mathbb {N}}$ such that $j\le 2\alpha +1$ and

  $$\begin{aligned} \lim _{t\rightarrow \infty }\int _t^{t+1}|A^{j/2}u(\tau )|^2\mathrm{d}\tau =0, \end{aligned}$$

  (3.44)

then

$$\begin{aligned} \lim _{t\rightarrow \infty }\int _t^{t+1}|A^{(j+1)/2}u(\tau )|^2\mathrm{d}\tau =0. \end{aligned}$$

(3.45)

Indeed, since $(j-1)/2 \le \alpha$, and thanks to (3.25), we have

$$\begin{aligned} |A^\frac{j-1}{2}f(t)|={\mathcal {O}}(F(t)). \end{aligned}$$

(3.46)

By (3.44) and (3.46), we obtain, similar to (3.42) and (3.43) that there exists $T_3\in {\mathcal {T}}\cap [T_1,\infty )$ so that

$$\begin{aligned} |A^{j/2} u(T_3)|\le & {} c_0(j/2,F^\lambda ), \\ |A^{j/2-1/2}f(T_3+t)|\le & {} c_1(j/2,F^\lambda )F^\lambda (t) \quad \forall t\ge 0. \end{aligned}$$

Applying Theorem 3.3 to $u(t):=u(T_3+t)$, $f(t):=f(T_3+\cdot )$, $F(t):=F^\lambda (t)$, $\alpha :=j/2$, $\sigma :=0$, we obtain from (3.15) that

$$\begin{aligned} \int _t^{t+1}|A^{(j+1)/2}u(\tau )|^2\mathrm{d}\tau&=\int _{t-T_3}^{t+1-T_3}|A^{(j+1)/2}u(T_3+\tau )|^2\mathrm{d}\tau \\&={\mathcal {O}}(e^{-2\theta _0(t-T_3)}+F^{2\lambda }(\theta (t-T_3))), \end{aligned}$$

which proves (3.45), thanks to (3.24).

Now, let m be a non-negative integer such that $2\alpha \le m<2\alpha +1$.

Note that $m\ge 1$, and, because of (3.37), condition (3.44) holds true for $j=1$. Hence we obtain (3.45) with $j=1$, which is (3.44) for $j=2$. We apply the arguments recursively for $j=1,2,\ldots ,m$, and obtain, when $j=m$, from (3.45) that

$$\begin{aligned} \lim _{t\rightarrow \infty } \int _t^{t+1}|A^{(m+1)/2}u(\tau )|^2\mathrm{d}\tau =0. \end{aligned}$$

Since $\alpha \le m/2$, it follows that

$$\begin{aligned} \lim _{t\rightarrow \infty }\int _t^{t+1}|A^{\alpha +1/2}u(\tau )|^2\mathrm{d}\tau =0. \end{aligned}$$

(3.47)

By (3.47), (3.28), (3.38) and (3.24), there exists $T\in {\mathcal {T}}\cap [T_1,\infty )$ so that (3.42) and (3.43) similarly hold true (for this case of $\sigma =0$.)

• (d) With $T\in {\mathcal {T}}\cap [T_1,\infty )$ in (b) and (c), we apply Theorem 3.3 to the unique regular solution $u(t):=u(T+t)$, $f(t):=f(T+t)$, $F(t):=F^\lambda (t)$, $\alpha :=\alpha +1/2$, and obtain that there is $t_*\ge 0$ such that, following (3.14) with $t:=t+t_*$,

  $$\begin{aligned} |u(T+t_* +t)|_{\alpha +1/2,\sigma }&\le c_2(\alpha +1/2,\theta _0,\theta ,F^\lambda )\Big (e^{-2\theta _0(t_*+ t)}+ F^{2\lambda }(\theta (t_*+t))\Big )^{1/2}\\&\le C(e^{-2\theta _0 t}+ F^{2\lambda }(\theta t))^{1/2} \end{aligned}$$

for all $t\ge 0$. By setting ${{\hat{T}}} =T+t_*$, this estimate implies (3.29). The proof is complete. $\square$

Remark 3.5

Because the constants $c_0$ and $c_1$ in (3.42), (3.43) can be small, we could not prove (3.29) for $\lambda =1$ directly. Rather, we use (3.30) and the estimate (3.6) in Theorem 3.2 for the linearized NSE to improve (3.29) to (3.27).

4 General asymptotic expansions

Now, we introduce a very general definition of an asymptotic expansion in a normed space with respect to a system of time-decaying functions.

Definition 4.1

Let $(\psi _n)_{n=1}^\infty$ be a sequence of non-negative functions defined on $[T_*,\infty )$ for some $T_*\in {\mathbb {R}}$ that satisfies the following two conditions:

• (a) For each $n\in {\mathbb {N}}$,

  $$\begin{aligned} \lim _{t\rightarrow \infty }\psi _n(t)=0. \end{aligned}$$

  (4.1)

• (b) For $n>m$,

  $$\begin{aligned} \psi _n(t)=o(\psi _m(t)). \end{aligned}$$

  (4.2)

Let $(X, \Vert \cdot \Vert )$ be a normed space, and g be a function from $[T_*,\infty )$ to X. We say g has an asymptotic expansion (implicitly as $t\rightarrow \infty$)

$$\begin{aligned} g(t)\sim \sum _{n=1}^\infty \xi _n \psi _n(t) \text { in }X, \end{aligned}$$

(4.3)

where $\xi _n\in X$ for all $n\in {\mathbb {N}}$, if, for any $N\in {\mathbb {N}}$,

$$\begin{aligned} \left\| g(t)-\sum _{n=1}^N \xi _n\psi _n(t) \right\| =o(\psi _N(t)). \end{aligned}$$

(4.4)

Obviously, if $g(t)=\sum _{n=1}^N \xi _n \psi _n(t)$ for some $N\in {\mathbb {N}}$, then $g(t)\sim \sum _{n=1}^\infty \xi _n \psi _n(t)$ where $\xi _n=0$ for $n>N$. In case of the infinite sum, the convergent series

$$\begin{aligned} g(t)=\sum _{n=1}^\infty \xi _n \psi _n(t) \end{aligned}$$

(4.5)

does not necessarily imply the expansion (4.3). We refer to “Appendix A” for some criteria for both (4.3) and (4.5) to hold, with the infinite sum not reduced to a finite one.

Note that the expansion (4.3) does not determine the function g. Indeed, if $h:[T_*,\infty )\rightarrow X$ is a function that satisfies $\Vert h(t)\Vert =o(\psi _n(t))$ for all $n\in {\mathbb {N}}$, then both g and $g+h$ have the same expansion on the right-hand side of (4.3). The converse is considered in the next proposition.

Proposition 4.2

Let $(\psi _n)_{n=1}^\infty$, $(X,\Vert \cdot \Vert )$ and g be as in Definition 4.1. Suppose, for each $n\in {\mathbb {N}}$, that the function $\psi _n$ is not identically zero on $[T,\infty )$ for all $T\ge T_*$. Then the asymptotic expansion (4.3), if exists, is unique.

Proof

Suppose g(t) has two expansions

$$\begin{aligned} g(t)\sim \sum _{n=1}^\infty \phi _n \psi _n(t)\text { and }g(t)\sim \sum _{n=1}^\infty \xi _n \psi _n(t). \end{aligned}$$

(4.6)

We will prove by induction that $\phi _n=\xi _n$ for all $n\in {\mathbb {N}}$.

One has from the triangle inequality and each expansion in (4.6) that

$$\begin{aligned} \Vert (\phi _1-\xi _1)\psi _1(t)\Vert \le \Vert \phi _1\psi _1(t)-g(t)\Vert +\Vert g(t)-\xi _1\psi _1(t)\Vert =o(\psi _1(t)). \end{aligned}$$

(4.7)

Since $\psi _1$ is asymptotically non-trivial, then one can verify from (4.7) that $\phi _1=\xi _1$.

Let $N\in {\mathbb {N}}$ and assume $\phi _n=\xi _n$ for $n=1,2,\ldots ,N$. Then

$$\begin{aligned}&\Vert (\phi _{N+1}-\xi _{N+1})\psi _{N+1}(t)\Vert =\left\| \sum _{n=1}^{N+1}(\phi _n-\xi _n)\psi _n(t)\right\| \\&\le \left\| \sum _{n=1}^{N+1}\phi _n\psi _n(t)-g(t)\right\| +\left\| g(t)-\sum _{n=1}^{N+1}\xi _n\psi _n(t)\right\| =o(\psi _{N+1}(t)). \end{aligned}$$

Hence, $\phi _{N+1}=\xi _{N+1}$. By the induction principle, $\phi _n=\xi _n$ for all $n\in {\mathbb {N}}$. $\square$

Note that the rate of convergence, as $t\rightarrow \infty$, in (4.4), in fact, can be related to the next term $\psi _{N+1}(t)$. Indeed,

$$\begin{aligned} \left\| g(t)-\sum _{n=1}^N \xi _n\psi _n(t)\right\| \le \left\| g(t)-\sum _{n=1}^{N+1} \xi _n\psi _n(t)\right\| +\psi _{N+1}(t)\Vert \xi _{N+1}\Vert . \end{aligned}$$

Hence, we can replace equivalently (4.4) by

$$\begin{aligned} \left\| g(t)-\sum _{n=1}^N \xi _n\psi _n(t)\right\| =O(\psi _{N+1}(t)). \end{aligned}$$

(4.8)

The equivalence of (4.4) and (4.8) is essentially due to the infinite sum in (4.3). If the sum is finite, this is no more the case. Moreover, for general $\psi _n$’s, the relation (4.2) is not informative enough to work with.

These prompt us to have the following more specific definition.

Definition 4.3

Let $\Psi =(\psi _\lambda )_{\lambda >0}$ be a system of functions that satisfies the following two conditions.

• (a) There exists $T_*\ge 0$ such that, for each $\lambda >0$, $\psi _\lambda$ is a positive function defined on $[T_*,\infty )$, and

  $$\begin{aligned} \lim _{t\rightarrow \infty }\psi _\lambda (t)=0. \end{aligned}$$

  (4.9)

• (b) For any $\lambda >\mu$, there exists $\eta >0$ such that

  $$\begin{aligned} \psi _\lambda (t)={\mathcal {O}}(\psi _\mu (t)\psi _\eta (t)). \end{aligned}$$

  (4.10)

Let $(X, \Vert \cdot \Vert )$ be a real normed space, and g be a function from $(0,\infty )$ to X. The function g is said to have the asymptotic expansion

$$\begin{aligned} g(t) {\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^\infty \xi _n \psi _{\lambda _n}(t) \text { in } X, \end{aligned}$$

(4.11)

where $\xi _n\in X$ for all $n\in {\mathbb {N}}$, and $(\lambda _n)_{n=1}^\infty$ is a strictly increasing, divergent sequence of positive numbers, if it holds, for any $N\ge 1$, that there exists $\varepsilon >0$ such that

$$\begin{aligned} \left\| g(t)- \sum _{n=1}^N \xi _n \psi _{\lambda _n}(t)\right\| ={\mathcal {O}}(\psi _{\lambda _N}(t)\psi _\varepsilon (t)). \end{aligned}$$

(4.12)

We have the following remarks on Definition 4.3.

• (a) If $\lambda >\mu$, it follows (4.10) and (4.9) that

  $$\begin{aligned} \psi _\lambda (t)=o(\psi _\mu (t)). \end{aligned}$$

  (4.13)

• (b) If a function g has an expansion (4.11), then $g(t) \sim \sum _{n=1}^\infty \xi _n \psi _{\lambda _n}(t)$ in X in the sense of Definition 4.1.

• (c) Thanks to (b) and Proposition 4.2, the $\xi _n$’s in (4.11) are unique. Similarly, following the proof of Proposition 4.2, we also have the uniqueness of $\xi _n$’s in (4.16).

• (d) The main difference between Definition 4.1 and Definition 4.3 is the specific decaying rate $\psi _\eta (t)$ on the right-hand side of (4.10), in contrast with the non-specific one in (4.2). In the proofs, this crucially allows comparisons and estimates for different quantities.

We have the following special cases for the expansion (4.11).

• (i) Assume (4.11). If there exists $N\in {\mathbb {N}}$, such that

  $$\begin{aligned} \xi _n=0\text { for all }n>N, \end{aligned}$$

  (4.14)

  then it holds for all $\lambda >0$ that

  $$\begin{aligned} \left\| g(t)- \sum _{n=1}^N \xi _n \psi _{\lambda _n}(t)\right\| ={\mathcal {O}}(\psi _\lambda (t)). \end{aligned}$$

  (4.15)

• (ii) Assume there exist $N\in {\mathbb {N}}$, $\xi _n\in X$ for $1\le n\le N$, and $\lambda _n$’s, for $1\le n\le N$, are positive numbers, strictly increasing in n such that (4.15) holds for all $\lambda >0$. We extend $\xi _n\in X$ for $1\le n\le N$ to a sequence $(\xi _n)_{n=1}^\infty$ with (4.14), and extend $\lambda _n$ for $1\le n\le N$ to any sequence $(\lambda _n)_{n=1}^\infty$ that is a strictly increasing and divergent. Then one can verify that (4.11) holds true.

  Therefore, we say in cases (i) and (ii) that the function g has the asymptotic expansion

  $$\begin{aligned} g(t) {\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^N \xi _n \psi _{\lambda _n}(t) \text { in } X. \end{aligned}$$

  (4.16)

• (iii) If $\xi _n=0$ for all $n\in N$ in (4.11), then

  $$\begin{aligned} \Vert g(t)\Vert ={\mathcal {O}}(\psi _\lambda (t)) \end{aligned}$$

  (4.17)

  for all $\lambda >0$.

• (iv) Assume (4.17) holds for all $\lambda >0$. Let $\xi _n=0$ for all $n\in {\mathbb {N}}$. Let $(\lambda _n)_{n=1}^\infty$ be any strictly increasing, divergent sequence of positive numbers. Then we have (4.11).

  Therefore, we say in cases (iii) and (iv) that the function g has the asymptotic expansion

  $$\begin{aligned} g(t) {\mathop {\sim }\limits ^{\Psi }}0 \text { in } X. \end{aligned}$$

• (v) For $N=0$, we conveniently set the sum $\sum _{n=1}^N \xi _n \psi _{\lambda _n}(t)$ to be zero in (4.16), and see that the condition (4.15) is, in fact, (4.17). Thus the expression

  $$\begin{aligned} g(t) {\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^0 \xi _n \psi _{\lambda _n}(t)\quad \text { will mean }\quad g(t) {\mathop {\sim }\limits ^{\Psi }}0. \end{aligned}$$

• (vi) If a function g has an asymptotic expansion

  $$\begin{aligned} g(t) {\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^N \xi _n \psi _{\lambda _n}(t) \text { in } X, \end{aligned}$$

  for $N\in {\mathbb {N}}\cup \{0,\infty \}$, then by remark (c) above, this expansion is unique for g.

For solutions of ODEs or PDEs, the linear and nonlinear structures of the equations will impose more conditions on the system. We consider below the ones that are appropriate to our current study of the NSE.

Condition 4.4

The system $\Psi =(\psi _\lambda )_{\lambda >0}$ satisfies (a) and (b) in Definition 4.3 and the following.

• (i) For any $\lambda ,\mu >0$, there exist $\gamma > \max \{\lambda ,\mu \}$ and a nonzero constant $d_{\lambda ,\mu }$ such that

  $$\begin{aligned} \psi _\lambda \psi _\mu =d_{\lambda ,\mu }\psi _\gamma . \end{aligned}$$

  (4.18)

• (ii) For each $\lambda >0$, the function $\psi _\lambda$ is continuous and differentiable on $[T_*,\infty )$, and its derivative $\psi _\lambda '$ has an expansion in the sense of Definition 4.3

  $$\begin{aligned} \psi _\lambda '(t){\mathop {\sim }\limits ^{\Psi }}\sum _{k=1}^{N_\lambda } c_{\lambda ,k}\psi _{\lambda ^\vee (k)}(t) \text { in }{\mathbb {R}}, \end{aligned}$$

  (4.19)

  where $N_\lambda \in {\mathbb {N}}\cup \{0,\infty \}$, all $c_{\lambda ,k}$ are constants, all $\lambda ^\vee (k)>\lambda$, and, for each $\lambda >0$, $\lambda ^\vee (k)$’s are strictly increasing in k.

The following remarks on Condition 4.4 are in order.

• (a) By (4.13), the numbers $\gamma$ and $d_{\lambda ,\mu }$ in (4.18) are unique.

• (b) By (4.18), we have the reverse of (4.10) in the following sense:

  $$\begin{aligned} \psi _\mu (t)\psi _\eta (t)={\mathcal {O}}(\psi _ \lambda (t))\text { for some }\lambda >\mu . \end{aligned}$$

  (4.20)

• (c) Thanks to (4.10) and (4.20), the condition (4.12) is equivalent to

  $$\begin{aligned} \left\| g(t)- \sum _{n=1}^N \xi _n \psi _{\lambda _n}(t)\right\| ={\mathcal {O}}(\psi _\lambda (t)), \end{aligned}$$

  for some $\lambda >\lambda _N$.

• (d) Expansion (4.19) of $\psi _\lambda '$, in all cases of $N_\lambda$, and property (4.10) imply that there exists $\eta >0$ such that

  $$\begin{aligned} |\psi _{\lambda }'(t)| = {\mathcal {O}}(\psi _{\lambda }(t)\psi _{\eta }(t)). \end{aligned}$$

  (4.21)

• (e) If, instead of (4.19), $\psi _\lambda '=c_\lambda \psi _\lambda$ for all $\lambda$, then $\psi _\lambda$’s are exponential functions. This case was studied in [19]. For other examples of (4.19), see Sects. 7.3 and 7.4.

Notation

Denote $\gamma$ in (4.18) by $\lambda \wedge \mu$, which is uniquely determined thanks to remark (a) above.

For the current study, we focus on decaying functions that are larger than the exponentially decaying ones; hence, we impose more specific conditions.

Condition 4.5

The system $\Psi =(\psi _\lambda )_{\lambda >0}$ satisfies (a), (b) of Definition 4.3, and the following.

• (i) For each $\lambda >0$, the function $\psi _\lambda$ is decreasing (in t).

• (ii) If $\lambda ,\alpha >0$, then

  $$\begin{aligned} e^{-\alpha t}=o(\psi _\lambda (t)). \end{aligned}$$

  (4.22)

• (iii) For any number $a\in (0,1)$,

  $$\begin{aligned} \psi _\lambda (at)={\mathcal {O}}(\psi _\lambda (t)). \end{aligned}$$

  (4.23)

The followings are direct consequences of Condition 4.5.

• (a) By (4.22), for any $\alpha ,\lambda >0$, there exists a positive constant $C_{\alpha ,\lambda }$ such that

  $$\begin{aligned} e^{-\alpha (t+T_*)}\le C_{\alpha ,\lambda } \psi _\lambda (t+T_*)\quad \forall t\ge 0, \end{aligned}$$

  hence, by denoting $D_1(\lambda ,\alpha )=e^{\alpha T_*} C_{\lambda ,\alpha }$, we have

  $$\begin{aligned} e^{-\alpha t}\le D_1(\lambda ,\alpha ) \psi _\lambda (t+T_*). \end{aligned}$$

  (4.24)

• (b) For $a\in (0,1)$, we have from (i) that $\psi _\lambda (t)={\mathcal {O}}(\psi _\lambda (at))$. Thus, the condition (4.23), in fact, is equivalent to

  $$\begin{aligned} \psi _\lambda (at){\mathop {=}\limits ^{{\mathcal {O}}}}\psi _\lambda (t). \end{aligned}$$

• (c) Property (4.23) and the decrease of $\psi _\lambda (t)$ in t imply, for $a\in (0,1)$, that

  $$\begin{aligned} \psi _\lambda (at)\le D_2(a,\lambda ) \psi _\lambda (t)\quad \forall t\ge T_*/a, \end{aligned}$$

  where $D_2(a,\lambda )$ is a constant in $[1,\infty )$. Consequently, for $a\in (0,1)$ and $t\ge 0$,

  $$\begin{aligned} \psi _\lambda (at+T_*)=\psi _\lambda (a(t+T_*/a))\le D_2(a,\lambda ) \psi _\lambda (t+T_*/a) \le D_2(a,\lambda ) \psi _\lambda (t+T_*).\nonumber \\ \end{aligned}$$

  (4.25)

  Then by the decrease of $\psi _\lambda$ in t, we have

  $$\begin{aligned} \psi _\lambda (at+T_*)\le D_2(a,\lambda ) \psi _\lambda (t)\quad \forall t\ge T_*. \end{aligned}$$

  (4.26)

  In particular, for any $T\ge 0$ and $t\ge 2(T_*+T)$, we have $t-T\ge t/2+T_*$, then by (4.26),

  $$\begin{aligned} \psi _\lambda (t-T)\le \psi _\lambda (t/2+T_*)\le D_3(\lambda ) \psi _\lambda (t), \text { where } D_3(\lambda )=D_2(1/2,\lambda ). \end{aligned}$$

  Combining this with the boundedness of $\psi _\lambda (t-T)/\psi _\lambda (t)$ for small $t\in [T_*+T,2(T_*+T)]$, we obtain

  $$\begin{aligned} \psi _\lambda (t-T)\le D_4(\lambda ,T) \psi _\lambda (t)\quad \forall t\ge T_*+T, \end{aligned}$$

  which yields

  $$\begin{aligned} \psi _\lambda (t)\le D_4(\lambda ,T) \psi _\lambda (t+T)\quad \forall t\ge T_*, \end{aligned}$$

  for some positive constant $D_4(\lambda ,T)$. Consequently, for any $T\in {\mathbb {R}}$,

  $$\begin{aligned} \psi _\lambda (t){\mathop {=}\limits ^{{\mathcal {O}}}}\psi _\lambda (t+T). \end{aligned}$$

  (4.27)

In applications to the NSE, suppose that the force f has an expansion containing the terms $\psi _{\gamma _n}$’s for some numeric sequence $(\gamma _n)_{n=1}^\infty$. Then the operators in the NSE require that a solution u(t), in case of having an expansion itself, may need many more terms in addition to $\psi _{\gamma _n}$’s. We describe below a general principle to find those other terms.

For any $x\in (0,\infty )$, define the set

$$\begin{aligned} G_x={\left\{ \begin{array}{ll} \emptyset ,&{}\text { if } N_x=0,\\ \{x^\vee (k):1\le k\le N_x\}, &{} \text { if }N_x\in {\mathbb {N}},\\ \{x^\vee (k):k\in {\mathbb {N}}\},&{}\text { if }N_x=\infty . \end{array}\right. } \end{aligned}$$

A non-empty subset S of $(0,\infty )$ is said to preserve the operation $\vee$ if

$$\begin{aligned} \forall x\in S: G_x\subset S. \end{aligned}$$

(4.28)

Similarly, S is said to preserve the operation $\wedge$ if

$$\begin{aligned} \forall x,y\in S: x\wedge y\in S. \end{aligned}$$

(4.29)

Lemma 4.6

Let S be any non-empty subset of $(0,\infty )$.

• (i) There exists a smallest set $S_*\subset (0,\infty )$ that contains S, and preserves the operations $\vee$ and $\wedge$.

• (ii) In fact, $S_*=S^*$, where $S^*$ is constructed explicitly in (4.30) below.

Proof

• (i) For any non-empty subset M of $(0,\infty )$, we denote

  $$\begin{aligned} M^{\wedge }&= \{x\wedge y:x,y\in M\},\\ M^{\vee }&=\bigcup _{x\in M} G_x. \end{aligned}$$

  • (a) Let $S_0=S$. We define recursively the sets $S_n$, for $n\in {\mathbb {N}}$, by

    $$\begin{aligned} S_{2k+1}=S_{2k} \cup S_{2k}^{\vee }\text { and } S_{2k+2}=S_{2k+1} \cup S_{2k+1}^{\wedge }\text { for }k\ge 0. \end{aligned}$$

Define

$$\begin{aligned} S^*=\bigcup _{n=0}^\infty S_{n}. \end{aligned}$$

(4.30)

We obviously have

$$\begin{aligned} S_{2k}\subset S_{2k+1}\subset S_{2k+2}\subset S_{2k+3}\quad \forall k\ge 0. \end{aligned}$$

(4.31)

It follows (4.31) that $(S_n)$, $(S_{2n})$ and $(S_{2n+1})$ are increasing sequences, and, hence,

$$\begin{aligned} S^*=\bigcup _{k=0}^\infty S_{2k}=\bigcup _{k=0}^\infty S_{2k+1}. \end{aligned}$$

(4.32)

Clearly, $S=S_0\subset S^*$. Next, we prove $S^*$ preserves the operations $\vee$ and $\wedge$.

Let $x\in S^*$. By (4.32), $x\in S_{2k}$ for some $k\ge 0$. Then obviously by definition $G_x \subset S_{2k}^{\vee } \subset S_{2k+1}\subset S^*$. Thus, $G_x\subset S^*$.

For $x,y\in S^*$, then by (4.32), $x\in S_{2k+1}$, $y\in S_{2m+1}$ for some $k,m\ge 0$. Assume $k\ge m$, then $y\in S_{2k+1}$ by (4.31). This implies $x\wedge y\in S_{2k+2}\subset S^*$.

• (b) Let ${\mathcal {C}}$ be the collection of sets M that contain S and preserve the operations $\vee$ and $\wedge$. Because $S^*\in {\mathcal {C}}$, then the collection ${\mathcal {C}}$ is non-empty.

Let $S_*$ be the intersections of all the elements in ${\mathcal {C}}$. Then $S_* \subset S^*$. Let $M \in {\mathcal {C}}$, properties (4.28) and (4.29) for $S:=M$ clearly imply

$$\begin{aligned} M^{\vee } \subset M \text { and }M^{\wedge } \subset M. \end{aligned}$$

(4.33)

Thus,

$$\begin{aligned} (S_*)^\vee \subset M^{\vee } \subset M \quad \text { and } (S_*)^{\wedge }\subset M^{\wedge } \subset M. \end{aligned}$$

It follows that

$$\begin{aligned} (S_*)^\vee \subset \bigcap _{M\in C}M=S_* \quad \text { and } (S_*)^{\wedge }\subset \bigcap _{M\in C}M=S_*. \end{aligned}$$

Therefore, $S_*\in {\mathcal {C}}$. By its definition, $S_*$ is the smallest set in ${\mathcal {C}}$.

• (ii) We prove $S_*=S^*$. It suffices to show $S^* \subset S_*$.

Let M be an arbitrary element in ${\mathcal {C}}$. We shall show that $S^* \subset M$. First, we see that $S_0 \subset M$. By (4.33),

$$\begin{aligned} S_1=S_0 \cup S_0^{\vee } \subset M \cup M^{\vee }=M, \end{aligned}$$

and then,

$$\begin{aligned} S_2=S_1 \cup S_1^{\wedge } \subset M \cup M^{\wedge }=M. \end{aligned}$$

By induction, we can prove similarly that $S_k \subset M$ for all k. Therefore $S^*=\bigcup _{k=0}^\infty S_{k} \subset M$.

Then $S^*\subset \bigcap _{M\in {\mathcal {C}}} M=S_*$. This completes the proof of the lemma. $\square$

Notation

We denote the set $S_*$ in Lemma 4.6 by ${\mathcal {G}}_\Psi (S)$.

5 Asymptotic expansions in a continuum system

Let $\Psi =(\psi _\lambda )_{\lambda >0}$ be a system of functions that satisfies both Conditions 4.4 and 4.5.

Assumption 5.1

Suppose there exist real numbers $\sigma \ge 0$, $\alpha \ge 1/2$, a strictly increasing, divergent sequence of positive numbers $(\gamma _n)_{n=1}^\infty$ and a sequence $({{\tilde{\phi }}}_n)_{n=1}^\infty$ in $G_{\alpha ,\sigma }$ such that, in the sense of Definition 4.3,

$$\begin{aligned} f(t){\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^\infty {{\tilde{\phi }}}_n \psi _{\gamma _n}(t) \text { in }G_{\alpha ,\sigma }. \end{aligned}$$

(5.1)

Note from (5.1) that f(t) belongs to $G_{\alpha ,\sigma }$ for all t sufficiently large.

Let u(t) be a Leray–Hopf weak solution of the NSE. We search for an asymptotic expansion of u(t) in the form

$$\begin{aligned} u(t){\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^\infty \xi _n \psi _{\lambda _n}(t). \end{aligned}$$

(5.2)

Formally substituting expansion (5.2) into the NSE (2.2), we find that the indices $\lambda _n$’s naturally take values in the set

$$\begin{aligned} {\mathcal {G}}_\Psi (\{\gamma _n:n\in {\mathbb {N}}\}). \end{aligned}$$

(5.3)

However, the expansion (5.2) only agrees with (4.11) in Definition 4.3 if the set in (5.3) does not have a finite cluster point. Therefore, we impose one more condition.

Assumption 5.2

There exists a set $S_*$ that contains $\{\gamma _n:n\in {\mathbb {N}}\}$, preserves the operations $\vee$ and $\wedge$, and can be ordered so that

$$\begin{aligned} S_*=\{\lambda _n:n\in {\mathbb {N}}\}, \text { where }\lambda _n\text {'s are strictly increasing to infinity.} \end{aligned}$$

(5.4)

We usually choose $S_*$ in Assumption 5.2 to be (5.3), but this is not the only choice.

Under Assumption 5.2, we can show that the expansion (5.1) implies

$$\begin{aligned} f(t){\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^{\infty } \phi _n \psi _{\lambda _n}(t)\quad \text {in }G_{\alpha ,\sigma }\quad \text {as } t\rightarrow \infty , \end{aligned}$$

(5.5)

where the sequence $(\phi _n)_{n=1}^\infty$ in $G_{\alpha ,\sigma }$ is defined by $\phi _n={{\tilde{\phi }}}_k$ if there exists $k\ge 1$ such that $\lambda _n=\gamma _k$, and $\phi _n=0$ otherwise. Note in the former case that such an index k, when exists, is unique.

Remark 5.3

In case the set $S=\{\gamma _n:n\in {\mathbb {N}}\}$ itself preserves the operations $\vee$ and $\wedge$, then $S={\mathcal {G}}_{\Psi }(S)$. Hence, Assumption 5.2 is met with $S_*=S$ and (5.5) holds with $\lambda _n=\gamma _n$, $\phi _n={{\tilde{\phi }}}_n$, i.e., expansion (5.5) is just the original (5.1).

Our first main result on the expansion of the Leray–Hopf weak solutions is the following.

Theorem 5.4

Let Assumptions 5.1 and 5.2 hold true, and let f have the asymptotic expansion (5.5). Then any Leray–Hopf weak solution u(t) of (2.2) has the asymptotic expansion

$$\begin{aligned} u(t){\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^\infty \xi _n \psi _{\lambda _n}(t)\quad \text {in }G_{\alpha +1-\rho ,\sigma }\text { for all } \rho \in (0,1), \end{aligned}$$

(5.6)

where $\xi _n$’s are defined recursively by

$$\begin{aligned} \xi _1&=A^{-1}\phi _1, \end{aligned}$$

(5.7)

$$\begin{aligned} \xi _n&=A^{-1}\Big (\phi _n - \chi _n - \sum _{{\mathop {\lambda _k\wedge \lambda _m=\lambda _n}\limits ^{1\le k,m\le n-1,}}}d_{\lambda _k,\lambda _m}B(\xi _k,\xi _m)\Big ) \quad \text {for } n \ge 2, \end{aligned}$$

(5.8)

where

$$\begin{aligned} \chi _n= {\left\{ \begin{array}{ll} \sum _{{\mathop {\lambda _p^\vee (k)=\lambda _n}\limits ^{(p,k)\in [1,n-1]\times {\mathbb {N}}:}}}c_{\lambda _p,k} \xi _p , &{}\quad \text {if }\ \exists p\in [1, n-1],k\in {\mathbb {N}}:\lambda _p^\vee (k)=\lambda _n,\\ 0,&{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

(5.9)

Proof

The proof is divided into parts A, B, $\ldots$, steps 1,2, and substeps (a), (b), $\ldots$

A. Notation. For $n\in {\mathbb {N}}$, denote

$$\begin{aligned}&F_n(t)=\phi _n \psi _{\lambda _n}(t),\quad {{\bar{F}}}_n(t)=\sum _{j=1}^n F_j(t),\quad \text {and}\quad {{\tilde{F}}}_n(t)=f(t)-{{\bar{F}}}_n(t),\\&u_n(t)=\xi _n \psi _{\lambda _n}(t),\quad {{\bar{u}}}_n(t)=\sum _{j=1}^n u_j(t),\quad \text {and}\quad v_n=u(t)-{{\bar{u}}}_n(t). \end{aligned}$$

According to the expansion (5.5) and Definition 4.3, we can assume that

$$\begin{aligned} |{{\tilde{F}}}_N(t)|_{\alpha ,\sigma }={\mathcal {O}}(\psi _{\lambda _N}(t)\psi _{\delta _N}(t)) , \end{aligned}$$

(5.10)

for any $N\in {\mathbb {N}}$, with some $\delta _N>0$.

B. We observe that

$$\begin{aligned} \xi _n\in G_{\alpha +1,\sigma }\quad \forall n \ge 1. \end{aligned}$$

(5.11)

The proof of (5.11) is by induction and is the same as in [1, Lemma 4.2].

By (5.11), we have

$$\begin{aligned} |{{\bar{u}}}_n(t)|_{\alpha +1,\sigma }={\mathcal {O}}(\psi _{\lambda _1}(t))\quad \forall n\in {\mathbb {N}}. \end{aligned}$$

(5.12)

C. As a preparation, we need to establish the large-time decay for u(t) first. Letting $N=1$ in (5.10) gives $|f(t)- \phi _1 \psi _{\lambda _1}(t)|_{\alpha ,\sigma }={\mathcal {O}}(\psi _{\lambda _1}(t)\psi _{\delta _1}(t) ),$ which implies

$$\begin{aligned} |f(t)|_{\alpha ,\sigma }={\mathcal {O}}(\psi _{\lambda _1}(t) )={\mathcal {O}}(\psi _{\lambda _1}(t+T_*)). \end{aligned}$$

The last relation is due to (4.27).

Let $F(t)=\psi _{\lambda _1}(t+T_*)$. Then $|f(t)|_{\alpha ,\sigma }={\mathcal {O}}(F(t))$, and, by (4.24) and (4.25), the function F satisfies (3.4) and (3.5). We now apply Theorem 3.4 with $\varepsilon =1/2$. Then there exists time ${\hat{T}}>0$ and a constant $C>0$ such that u(t) is a regular solution of (2.2) on $[{\hat{T}},\infty )$, and

$$\begin{aligned} |u({\hat{T}}+t)|_{\alpha +1/2,\sigma } \le C\psi _{\lambda _1}(t+T_*) \quad \forall t\ge 0. \end{aligned}$$

(5.13)

It follows (2.11) and (5.13) that

$$\begin{aligned} |B(u({\hat{T}}+t),u({\hat{T}}+t))|_{\alpha ,\sigma }\le C|u({\hat{T}}+t)|_{\alpha +1/2,\sigma }^2 \le C\psi _{\lambda _1}^2(t+T_*) \quad \forall t\ge 0. \end{aligned}$$

(5.14)

D. It suffices to prove, for any $N\in {\mathbb {N}}$, that there exists a number $\varepsilon _N>0$ such that

$$\begin{aligned} |v_N(t)|_{\alpha ,\sigma } ={\mathcal {O}}(\psi _{\lambda _N}(t)\psi _{\varepsilon _N}(t) ). \end{aligned}$$

(5.15)

We will prove (5.15) by induction in N. In calculations below, all differential equations hold in $V'$-valued distribution sense on $(T,\infty )$ for any $T>0$, which is similar to (2.4). One can easily verify them by using (2.10), and the facts $u\in L^2_{\mathrm{loc}}([0,\infty ),V)$ and $u'\in L^1_\mathrm{loc}([0,\infty ),V')$ in Definition 2.1.

Step 1:$N=1$ Define $w_1(t)=\psi _{\lambda _1}^{-1}(t) u(t)$.

• (a) Equation for $w_1(t)$. We have

  $$\begin{aligned} w_1'(t)&=\psi _{\lambda _1}^{-1}(t) u'(t) - \psi _{\lambda _1}^{-2}(t) \psi _{\lambda _1}'(t)u(t)\\&=\psi _{\lambda _1}^{-1}(t) \big (-Au(t)-B(u,u)+ \phi _1 \psi _{\lambda _1}(t)+{{\tilde{F}}}_1(t)\big ) - \psi _{\lambda _1}^{-2}(t) \psi _{\lambda _1}'(t)u(t). \end{aligned}$$

Thus,

$$\begin{aligned} w_1'(t) + A w_1(t) = \phi _1 + H_1(t),\quad t>T_*, \end{aligned}$$

(5.16)

where

$$\begin{aligned} H_1(t)=\psi _{\lambda _1}^{-1} [ {{\tilde{F}}}_1(t) - B(u(t),u(t))]- \psi _{\lambda _1}' \psi _{\lambda _1}^{-2}u(t). \end{aligned}$$

• (b) Estimation of $|H_1(t)|_{\alpha ,\sigma }$. By estimates (5.13), (5.14), and the relations in (4.27), we have

  $$\begin{aligned} |u(t)|_{\alpha +1/2,\sigma }= & {} {\mathcal {O}}(\psi _{\lambda _1}(t) ). \end{aligned}$$

  (5.17)
  $$\begin{aligned} |B(u(t),u(t)))|_{\alpha ,\sigma }= & {} {\mathcal {O}}(\psi ^2_{\lambda _1}(t) ). \end{aligned}$$

  (5.18)

By (5.10), (5.17), (5.18), and properties (4.21), (4.27), there exist $T_0\ge T_*$, $\eta _1>0$, and $D_0>0$ such that for $t\ge 0$,

$$\begin{aligned} \psi _{\lambda _1}^{-1}(T_0+t) |{{\tilde{F}}}_1(T_0+t)|_{\alpha ,\sigma }&\le D_0 \psi _{\lambda _1}^{-1}(T_0+t) \psi _{\lambda _1}(T_0+t)\psi _{\delta _1}(T_0+t)\\&\le D_0 \psi _{\delta _1}(T_0+t),\\ \psi _{\lambda _1}^{-1}(T_0+t)|B(u(T_0+t),u(T_0+t))|_{\alpha ,\sigma }&\le D_0 \psi _{\lambda _1}^{-1}(T_0+t)\psi _{\lambda _1}^2(T_0+t)\\&\le D_0\psi _{\lambda _1}(T_0+t), \end{aligned}$$

and

$$\begin{aligned}&\psi _{\lambda _1}'(T_0+t) \psi _{\lambda _1}^{-2}(T_0+t)|u(T_0+t)|_{\alpha +1/2,\sigma }\\&\quad \le D_0 \psi _{\lambda _1}(T_0+t) \psi _{\eta _1}(T_0+t) \psi _{\lambda _1}^{-2}(T_0+t) \psi _{\lambda _1}(T_0+t)\\&\quad \le D_0\psi _{\eta _1}(T_0+t). \end{aligned}$$

Let $\varepsilon _1=\min \{\delta _1,\eta _1,\lambda _1\}$. Then

$$\begin{aligned} |H_1(T_0+t)|_{\alpha ,\sigma } \le 3D_0 \psi _{\varepsilon _1}(T_0+t)\quad \forall t\ge 0. \end{aligned}$$

• (c) We apply Theorem 3.2(iii) to Eq. (5.16) in $G_{\alpha ,\sigma }$ with $w(t):=w_1(T_0+t)$, $f(t):=H_1(T_0+t)$, $F(t):=\psi _{\varepsilon _1}(T_0+t)$ and $\xi =\phi _1$. We obtain from (3.6) that

  $$\begin{aligned} |w_1(T_0+t)-A^{-1}\phi _1 |_{\alpha +1-\rho ,\sigma }= {\mathcal {O}} (\psi _{\varepsilon _1}(T_0+t)) \end{aligned}$$

for any $\rho \in (0,1)$, which yields

$$\begin{aligned} |w_1(t)-A^{-1}\phi _1 |_{\alpha +1-\rho ,\sigma }= {\mathcal {O}} (\psi _{\varepsilon _1}(t)). \end{aligned}$$

Multiplying this equation by $\psi _{\lambda _1} (t)$ gives

$$\begin{aligned} |u(t)-\xi _1\psi _{\lambda _1} (t)|_{\alpha +1-\rho ,\sigma }= {\mathcal {O}} (\psi _{\lambda _1}(t)\psi _{\varepsilon _1}(t)) \quad \forall \rho \in (0,1). \end{aligned}$$

This proves that (5.15) holds for $N=1$.

Step 2: Induction step Let $N \ge 1$ be an integer and assume there exists $\varepsilon _N>0$ such that

$$\begin{aligned} |v_N(t)|_{\alpha +1-\rho ,\sigma }={\mathcal {O}} (\psi _{\lambda _N}(t)\psi _{\varepsilon _N}(t))\quad \forall \rho \in (0,1). \end{aligned}$$

(5.19)

(a) We will find an equation for $v_N$ which is suitable to study its asymptotic behavior. First, we have the preliminary calculations.

${\underline{Rewriting\,\,u'.}}$ By the NSE,

$$\begin{aligned} u'&=-Au-B(u,u)+f(t)\\&=-Av_N - A{{\bar{u}}}_N -B({{\bar{u}}}_N+v_N,{{\bar{u}}}_N+v_N)+{{\bar{F}}}_N+F_{N+1} +{{\tilde{F}}}_{N+1}\\&=-Av_N- A{{\bar{u}}}_N+{{\bar{F}}}_N-B({{\bar{u}}}_N,{{\bar{u}}}_N)+ \phi _{N+1}\psi _{\lambda _{N+1}}(t) +h_{N+1,1}, \end{aligned}$$

where

$$\begin{aligned} h_{N+1,1}=-B({{\bar{u}}}_N,v_N)-B(v_N,{{\bar{u}}}_N)-B(v_N,v_N)+{\tilde{F}}_{N+1}. \end{aligned}$$

On the one hand,

$$\begin{aligned} -A{{\bar{u}}}_N+{{\bar{F}}}_N= - \sum _{n=1}^N \psi _{\lambda _n}(t)\Big ( A\xi _n-\phi _n\Big ). \end{aligned}$$

On the other hand,

$$\begin{aligned} B({{\bar{u}}}_N,{{\bar{u}}}_N)&=\sum _{ m,j=1}^N \psi _{\lambda _m}(t)\psi _{\lambda _j}(t)B(\xi _m,\xi _j)\\&=\sum _{n=1}^{N}\psi _{\lambda _n}(t)\left( \sum _{{\mathop {\lambda _m\wedge \lambda _j=\lambda _n}\limits ^{1\le m,j\le N,}}}d_{\lambda _m,\lambda _j}B(\xi _m,\xi _j)\right) \\&\quad +\psi _{\lambda _{N+1}}(t) \sum _{{\mathop {\lambda _m\wedge \lambda _j=\lambda _{N+1}}\limits ^{1\le m,j\le N,}}}d_{\lambda _m,\lambda _j}B(\xi _m,\xi _j) +h_{N+1,2}, \end{aligned}$$

where

$$\begin{aligned} h_{N+1,2}=\sum _{{\mathop {\lambda _m\wedge \lambda _j\ge \lambda _{N+2}}\limits ^{1\le m,j\le N,}}} \psi _{\lambda _m}(t)\psi _{\lambda _j}(t) B(\xi _m,\xi _j). \end{aligned}$$

Then we obtain the equation

$$\begin{aligned} \begin{aligned} u'&=-Av_N - \sum _{n=1}^N \psi _{\lambda _n}(t)\left( A\xi _n-\phi _n+ \sum _{{\mathop {\lambda _m\wedge \lambda _j=\lambda _n}\limits ^{1\le m,j\le N,}}}d_{\lambda _m,\lambda _j}B(\xi _m,\xi _j)\right) \\&\quad -\psi _{\lambda _{N+1}}(t) \left( \sum _{{\mathop {\lambda _m\wedge \lambda _j=\lambda _{N+1}}\limits ^{1\le m,j\le N,}}}d_{\lambda _m,\lambda _j}B(\xi _m,\xi _j)- \phi _{N+1}\right) -h_{N+1,2} +h_{N+1,1}. \end{aligned} \end{aligned}$$

(5.20)

In calculations below to the end of this proof, $\varepsilon$ denotes a generic positive index used for function $\psi _\varepsilon (t)$.

Utilizing (5.10), (5.12) and (5.19), we estimate, with the use of the short-hand notation $\psi _\lambda =\psi _\lambda (t)$,

$$\begin{aligned} |h_{N+1,1}(t)|_{\alpha ,\sigma }&={\mathcal {O}}(\psi _{\lambda _1}){\mathcal {O}}(\psi _{\lambda _N}\psi _{\varepsilon _N}) + {\mathcal {O}}(\psi _{\lambda _N}\psi _{\varepsilon _N}) {\mathcal {O}}(\psi _{\lambda _1})\\&\quad + {\mathcal {O}}(\psi _{\lambda _N}\psi _{\varepsilon _N}) {\mathcal {O}}(\psi _{\lambda _N}\psi _{\varepsilon _N}) + {\mathcal {O}}(\psi _{\lambda _{N+1}}\psi _{\delta _{N+1}})\\&= {\mathcal {O}}(\psi _{\lambda _1}\psi _{\lambda _N}\psi _{\varepsilon _N}) + {\mathcal {O}}(\psi _{\lambda _{N+1}}\psi _{\delta _{N+1}})\\&= {\mathcal {O}}(\psi _{\lambda _1\wedge \lambda _N}\psi _{\varepsilon _N}) + {\mathcal {O}}(\psi _{\lambda _{N+1}}\psi _{\delta _{N+1}}). \end{aligned}$$

Since $\lambda _1\wedge \lambda _N\ge \lambda _{N+1}$, we have

$$\begin{aligned} |h_{N+1,1}(t)|_{\alpha ,\sigma } = {\mathcal {O}}(\psi _{\lambda _{N+1}}(t)\psi _{\varepsilon }(t)). \end{aligned}$$

It is also clear that

$$\begin{aligned} |h_{N+1,2}(t)|_{\alpha ,\sigma }&\le \sum _{{\mathop {\lambda _m\wedge \lambda _j\ge \lambda _{N+2}}\limits ^{1\le m,j\le N,}}} |d_{\lambda _m,\lambda _j}|\psi _{\lambda _m\wedge \lambda _j}(t) |B(\xi _m,\xi _j)|_{\alpha ,\sigma }\\&= {\mathcal {O}}(\psi _{\lambda _{N+2}}(t))= {\mathcal {O}}(\psi _{\lambda _{N+1}}(t)\psi _{\varepsilon }(t)). \end{aligned}$$

${\underline{Rewriting\,\,{{\bar{u}}}_N'.}}$ We have

$$\begin{aligned} {{\bar{u}}}_N'&=\sum _{p=1}^N \psi _{\lambda _p}'\xi _p = \sum _{p=1}^{N}\xi _p\left( \sum _{k=1}^{{{\widetilde{N}}}_p} c_{\lambda _p,k}\psi _{\lambda _p^\vee (k)} + \psi _{\lambda _p}'-\sum _{k=1}^{{{\widetilde{N}}}_p} c_{\lambda _p,k}\psi _{\lambda _p^\vee (k)}\right) , \end{aligned}$$

where ${{\widetilde{N}}}_p$ is the largest integer k such that

$$\begin{aligned} \lambda _p^\vee (k)\le \lambda _{N+1}. \end{aligned}$$

Then we obtain

$$\begin{aligned} {{\bar{u}}}_N'=\sum _{n=1}^{N} \psi _{\lambda _n}\chi _n+\psi _{\lambda _{N+1}}\chi _{N+1} +h_{N+1,3}, \end{aligned}$$

(5.21)

where

$$\begin{aligned} h_{N+1,3} =\sum _{p=1}^{N}\xi _p\left( \psi _{\lambda _p}'-\sum _{k=1}^{{{\widetilde{N}}}_p} c_{\lambda _p,k}\psi _{\lambda _p^\vee (k)}\right) . \end{aligned}$$

Note from (4.19) and the definition of ${{\widetilde{N}}}_p$ that

$$\begin{aligned} |\psi _{\lambda _p}'(t)-\sum _{k=1}^{{{\widetilde{N}}}_p} c_{\lambda _p,k}\psi _{\lambda _p^\vee (k)}(t)|={\mathcal {O}}(\psi _\lambda (t)),\quad \text {some } \lambda >\lambda _{N+1}. \end{aligned}$$

Together with (4.10), we have

$$\begin{aligned} |h_{N+1,3}(t)|_{\alpha ,\sigma } ={\mathcal {O}}(\psi _{\lambda _{N+1}}(t)\psi _{\varepsilon }(t)). \end{aligned}$$

${\underline{Equation\,\,for\,\,v_N.}}$ Combining (5.20) and (5.21) yields

$$\begin{aligned} v_N'&=u'-{{\bar{u}}}_N'\\&=-Av_N- \sum _{n=1}^N \psi _{\lambda _n}(t)\left( A\xi _n+\sum _{{\mathop {\lambda _m\wedge \lambda _j=\lambda _n}\limits ^{1\le m,j\le N,}}}d_{\lambda _m,\lambda _j}B(\xi _m,\xi _j)-\phi _n + \chi _n\right) \\&\quad +\psi _{\lambda _{N+1}}(t)\left( - \sum _{{\mathop {\lambda _m\wedge \lambda _j=\lambda _{N+1}}\limits ^{1\le m,j\le N,}}}d_{\lambda _m,\lambda _j}B(\xi _m,\xi _j)+\phi _{N+1}-\chi _{N+1}\right) +h_{N+1,4}(t), \end{aligned}$$

where

$$\begin{aligned} h_{N+1,4}=h_{N+1,1}-h_{N+1,2}-h_{N+1,3}. \end{aligned}$$

Note, for $1\le n\le N+1$, that

$$\begin{aligned} \sum _{{\mathop {\lambda _m\wedge \lambda _j=\lambda _n}\limits ^{1\le m,j\le N,}}}B(\xi _m,\xi _j)=\sum _{{\mathop {\lambda _m\wedge \lambda _j=\lambda _n}\limits ^{1\le m,j \le n-1,}}}B(\xi _m,\xi _j). \end{aligned}$$

Therefore, one has, for $1\le n\le N$,

$$\begin{aligned} A\xi _n+\sum _{{\mathop {\lambda _m\wedge \lambda _j=\lambda _n}\limits ^{1\le m,j \le N,}}}d_{\lambda _m,\lambda _j}B(\xi _m,\xi _j)-\phi _n+\chi _n =0, \end{aligned}$$

and

$$\begin{aligned} -\sum _{{\mathop {\lambda _m\wedge \lambda _j=\lambda _{N+1}}\limits ^{1\le m,j \le N,}}}d_{\lambda _m,\lambda _j}B(\xi _m,\xi _j)+\phi _{N+1}-\chi _{N+1}=A\xi _{N+1}. \end{aligned}$$

These yield

$$\begin{aligned} v_N'=-Av_N + \psi _{\lambda _{N+1}}(t)A\xi _{N+1}+h_{N+1,4}(t). \end{aligned}$$

(5.22)

• (b) Estimation of $v_N(t)$. In Eq. (5.22), we have

  $$\begin{aligned} |h_{N+1,4}(t)|_{\alpha ,\sigma }\le |h_{N+1,1}|_{\alpha ,\sigma }+|h_{N+1,2}|_{\alpha ,\sigma }+|h_{N+1,3}|_{\alpha ,\sigma }={\mathcal {O}}(\psi _{\lambda _{N+1}}(t)\psi _{\varepsilon }(t)).\nonumber \\ \end{aligned}$$

  (5.23)

One has from (5.23) that

$$\begin{aligned} |\psi _{\lambda _{N+1}}(t)A\xi _{N+1}+h_{N+1,4}(t)|_{\alpha ,\sigma } \le \psi _{\lambda _{N+1}}(t)|A\xi _{N+1}|_{\alpha ,\sigma }+|h_{N+1,4}(t)|_{\alpha ,\sigma } ={\mathcal {O}}(\psi _{\lambda _{N+1}}(t)). \end{aligned}$$

Similar to part (c) of Step 1, we apply Theorem 3.2(iii) to the linearized NSE (5.22) with $w(t)=v_N(T_0+t)$, $\xi =0$, $f(t)=\psi _{\lambda _{N+1}}(T_0+t)A\xi _{N+1}+h_{N+1,4}(T_0+t)$, and $F(t)=\psi _{\lambda _{N+1}}(T_0+t)$, where $T_0\ge T_*$ is an appropriate, sufficient large time. We have from (3.6), for any $\rho \in (0,1)$, that

$$\begin{aligned} |v_N(t)|_{\alpha +1-\rho ,\sigma }={\mathcal {O}}(\psi _{\lambda _{N+1}}(t)). \end{aligned}$$

(5.24)

• (c) We will improve the precision of decay in (5.24). Define $w_{N+1}(t)=\psi _{\lambda _{N+1}}(t)^{-1} v_N(t)$ for $t\ge T_*$. We have

  $$\begin{aligned} w_{N+1}'=\psi _{\lambda _{N+1}}^{-1}(t)v_N' +\psi _{\lambda _{N+1}}'(t)\psi _{\lambda _{N+1}}^{-2}(t) v_N, \end{aligned}$$

which, thanks to (5.22), yields

$$\begin{aligned} w_{N+1}'=-Aw_{N+1}+A\xi _{N+1}+H_{N+1}(t), \end{aligned}$$

(5.25)

where $H_{N+1}(t) = \psi _{\lambda _{N+1}}^{-1}(t)h_{N+1,4}(t)+ \psi _{\lambda _{N+1}}'(t)\psi _{\lambda _{N+1}}^{-2}(t) v_N(t).$

We estimate $H_{N+1}(t)$ next. By (5.23),

$$\begin{aligned} |\psi _{\lambda _{N+1}}^{-1}(t)h_{N+1,4}(t)|_{\alpha ,\sigma }=\psi _{\lambda _{N+1}}^{-1}(t){\mathcal {O}}(\psi _{\lambda _{N+1}}(t)\psi _{\varepsilon }(t))={\mathcal {O}}(\psi _{\varepsilon }(t)). \end{aligned}$$

For the second term, we use (4.21) and (5.24) to obtain

$$\begin{aligned} |\psi _{\lambda _{N+1}}'(t)\psi _{\lambda _{N+1}}^{-2}(t) v_N(t)|_{\alpha ,\sigma } =\psi _{\lambda _{N+1}}^{-2}(t){\mathcal {O}} (\psi _{ \lambda _{N+1}}(t)\psi _\varepsilon (t)) {\mathcal {O}}(\psi _{\lambda _{N+1}}(t)) ={\mathcal {O}}(\psi _\varepsilon (t)). \end{aligned}$$

Hence, there exists $\varepsilon _{N+1}>0$ such that

$$\begin{aligned} |H_{N+1}(t)|_{\alpha ,\sigma } ={\mathcal {O}}(\psi _{\varepsilon _{N+1}}(t)). \end{aligned}$$

• (d) Note from (5.11) that $A\xi _{N+1}\in G_{\alpha ,\sigma } \subset G_{\alpha -\frac{1}{2},\sigma }$. Again, by applying Theorem 3.2(iii) to equation (5.25) with $w(t):=w_{N+1}(T_1+t)$, $\xi :=A\xi _{N+1}$, $f(t):=H_{N+1}(t+T_1)$, $F(t):=\psi _{\varepsilon _{N+1}}(t+T_1)$ for some $T_1\ge T_*$ sufficiently large, we obtain from (3.6), for any $\rho \in (0,1)$, that

  $$\begin{aligned} |w_{N+1}(T_1+t)-A^{-1}(A\xi _{N+1})|_{\alpha +1-\rho ,\sigma } \le C \psi _{\varepsilon _{N+1}}(t+T_1)\quad \forall t\ge 1. \end{aligned}$$

Thus, $|w_{N+1}(t)-\xi _{N+1}|_{\alpha +1-\rho ,\sigma }= {\mathcal {O}}(\psi _{\varepsilon _{N+1}}(t)).$ Multiplying this equation by $\psi _{\lambda _{N+1}}(t)$ yields

$$\begin{aligned} |v_N(t)- \xi _{N+1}\psi _{\lambda _{N+1}}(t)|_{\alpha +1 -\rho ,\sigma }={\mathcal {O}}(\psi _{\lambda _{N+1}}(t)\psi _{\varepsilon _{N+1}}(t)). \end{aligned}$$

Since the left-hand side of this equation is $|v_{N+1}(t)|_{\alpha +1 -\rho ,\sigma }$, it proves that the statement (5.15) holds true for $N:=N+1$.

Conclusion By the induction principle, we have (5.15) holds true for all $N\in {\mathbb {N}}$. Our proof is complete. $\square$

In Theorem 5.4, both force f and solution u have infinite sum expansions which means that they can be approximated by infinitely many terms $\psi _\lambda$’s as $\lambda \rightarrow \infty$. The case of finite sum approximations can be treated similarly. We briefly discuss the idea and result here.

Assumption 5.5

Suppose there exist numbers $\sigma \ge 0$, $\alpha \ge 1/2$, an integer $N_0\ge 1$, strictly increasing, positive numbers $\gamma _n$ and functions ${{\tilde{\phi }}}_n\in G_{\alpha ,\sigma }$ for $1\le n\le N_0$ such that

$$\begin{aligned} \left| f(t) - \sum _{n=1}^{N_0} {{\tilde{\phi }}}_n \psi _{\gamma _n}(t)\right| _{\alpha ,\sigma } ={\mathcal {O}}(\psi _\lambda (t))\text { for some } \lambda >\gamma _{N_0}. \end{aligned}$$

(5.26)

Assume further that there exists a set $S_\infty$ that contains $\{\gamma _n:1\le n\le N_0\}$ and preserves the operations $\vee$ and $\wedge$, so that the set $S_*{\mathop {=}\limits ^\mathrm{def}}S_\infty \cap [\gamma _1,\gamma _{N_0}]$ is finite.

In applications, we often choose $S_\infty ={\mathcal {G}}_{\Psi }(\{\gamma _n:1\le n\le N_0\})$, but it can be more general than this.

We rewrite $S_*=\{\lambda _n: 1 \le n\le N_*\}$ for some integer $N_*\ge N_0$, where $\lambda _n$’s are strictly increasing. Note that $\lambda _{N_*}=\gamma _{N_0}$. Then from (5.26) we have

$$\begin{aligned} \Big |f(t) - \sum _{n=1}^{N_*} \phi _n \psi _{\lambda _n}(t)\Big |_{\alpha ,\sigma } ={\mathcal {O}}(\psi _\lambda (t))\text { for some } \lambda >\lambda _{N_*}, \end{aligned}$$

(5.27)

where $\phi _n\in G_{\alpha ,\sigma }$ for all $1\le n\le N_*$.

Theorem 5.6

Let Assumption 5.5 hold true, and let f have the asymptotic approximation (5.27). Let $\xi _n$ be defined by (5.7) and (5.8) for $1\le n\le N_*$. For any Leray–Hopf weak solution u(t) of (2.2), it holds that

$$\begin{aligned} \left| u(t) - \sum _{n=1}^{N_*} \xi _n \psi _{\lambda _n}(t)\right| _{\alpha ,\sigma } ={\mathcal {O}}(\psi _\lambda (t))\text { for some } \lambda >\lambda _{N_*}. \end{aligned}$$

Proof

The proof of Theorem 5.6 is the same as that of Theorem 5.4 except that we only use finite induction to establish (5.15) for $1\le N\le N_*$. $\square$

6 Asymptotic expansions in a discrete system with a continuum background

In this section, we investigate the case that the system of functions $(\psi _n)_{n=1}^\infty$ in Definition 4.1 cannot be mapped directly to a system $(\psi _{\lambda _n})_{n=1}^\infty$ to be embedded into a continuum system $(\psi _\lambda )_{\lambda >0}$. Hence, Definition 4.3 will not apply. However, we consider below the case when each $\psi _n$ is of the same decaying order, when $t\rightarrow \infty$, as $\varphi _{\lambda _n}$ with the functions $\varphi _{\lambda _n}$’s being part of a continuum system $(\varphi _\lambda )_{\lambda >0}$.

Definition 6.1

Let $\Psi =(\psi _n)_{n=1}^\infty$ be a sequence of positive functions defined on $[T_*,\infty )$ for some $T_*\in {\mathbb {R}}$, and $\Phi =(\varphi _{\lambda })_{\lambda >0}$ be a continuum system as in Definition 4.3 such that there exists a strictly increasing, divergent sequence $(\lambda _n)_{n=1}^\infty$ of positive numbers such that

$$\begin{aligned} \psi _n(t){\mathop {=}\limits ^{{\mathcal {O}}}}\varphi _{\lambda _n}(t)\quad \text {for all }n\in {\mathbb {N}}. \end{aligned}$$

(6.1)

Let $(X, \Vert \cdot \Vert )$ be a normed space, and g be a function from $(0,\infty )$ to X. We define the asymptotic expansions

$$\begin{aligned} g(t) \underset{\Phi }{\sim } \sum _{n=1}^\infty \xi _n \psi _n(t),\quad g(t) \underset{\Phi }{\sim } \sum _{n=1}^N \xi _n \psi _n(t) \text { with }N\in {\mathbb {N}},\quad g(t) \underset{\Phi }{\sim } 0 \text { in } X, \end{aligned}$$

(6.2)

in the same way as Definition 4.3 and the special cases (i)–(iv) below it, where we replace $\psi _{\lambda _n}$ with $\psi _n$, replace (4.12) with

$$\begin{aligned} \left\| g(t)- \sum _{n=1}^N \xi _n \psi _n(t)\right\| ={\mathcal {O}}(\psi _N(t)\varphi _\varepsilon (t)), \end{aligned}$$

(6.3)

replace (4.15) with

$$\begin{aligned} \left\| g(t)- \sum _{n=1}^N \xi _n \psi _n(t)\right\| ={\mathcal {O}}(\varphi _\lambda (t)), \end{aligned}$$

(6.4)

replace (4.17) with

$$\begin{aligned} \Vert g(t)\Vert ={\mathcal {O}}(\varphi _\lambda (t)). \end{aligned}$$

(6.5)

We refer to $\Phi$ as a background system of $(\psi _n)_{n=1}^\infty$. We will write expansions in (6.2) as

$$\begin{aligned} g(t) \underset{\Phi }{\sim } \sum _{n=1}^N \xi _n \psi _n(t) \text { in } X,\text { for }N=\infty ,\,N\in {\mathbb {N}},\text { and }N=0,\text { respectively.} \end{aligned}$$

The following remarks on Definition 6.1 are in order.

• (a) It follows (6.1) immediately that property (4.1) holds true. Moreover, for any $n>m$, there exists $\eta >0$ such that

  $$\begin{aligned} \psi _n(t)={\mathcal {O}}(\psi _m(t) \varphi _\eta (t)). \end{aligned}$$

  (6.6)

  where $\eta$ is a number such that $\varphi _{\lambda _n}(t)={\mathcal {O}}(\varphi _{\lambda _m}(t)\varphi _\eta (t))$, thanks to property (4.10) for the system $\Phi$. Thus, property (4.2) is also true. Therefore, Definition 4.1 for the asymptotic expansions $g(t)\sim \sum _{n=1}^\infty \xi _n \psi _n(t)$ in X still applies.

  Obviously, if $g(t) \underset{\Phi }{\sim } \sum _{n=1}^N \xi _n \psi _n(t)$ then $g(t)\sim \sum _{n=1}^N \xi _n \psi _n(t)$ in the sense of Definition 4.1. Then, thanks to Proposition 4.2, the uniqueness of the latter expansion implies the uniqueness of the former one.

• (b) We can equivalently replace ${\mathcal {O}}(\psi _N(t)\varphi _\varepsilon (t))$ in (6.3) with ${\mathcal {O}}(\varphi _{\lambda _N}(t)\varphi _\varepsilon (t))$, or ${\mathcal {O}}(\varphi _\lambda (t))$ for some $\lambda >\lambda _N$.

• (c) For a given sequence $(\psi _n)_{n=1}^\infty$, there may be different background systems. However, the asymptotic expansion of a function g as defined in (6.2), thanks to remark (a), is unique disregarding the choice of the background system $\Phi$.

• (d) Let $\Phi =(\varphi _\lambda )_{\lambda >0}$ and $\Theta =(\vartheta _\lambda )_{\lambda >0}$ be two systems as in Definition 4.3. If there exists a strictly increasing bijection $\mu$ from $(0,\infty )$ to $(0,\infty )$ such that $\varphi _\lambda (t){\mathop {=}\limits ^{{\mathcal {O}}}}\vartheta _{\mu (\lambda )}(t)$ for all $\lambda >0$, then

  $$\begin{aligned} g(t) \underset{\Phi }{\sim } \sum _{n=1}^N \xi _n \psi _n(t) \quad \text { if and only if }\quad g(t) \underset{\Theta }{\sim } \sum _{n=1}^N \xi _n \psi _n(t) . \end{aligned}$$

• (e) Let $\Psi =(\psi _\lambda )_{\lambda >0}$ and $\Phi =(\varphi _\lambda )_{\lambda >0}$ satisfy (a) and (b) of Definition 4.3. Suppose there exists a strictly increasing bijection $\mu$ from $(0,\infty )$ to $(0,\infty )$ such that

  $$\begin{aligned} \psi _\lambda (t){\mathop {=}\limits ^{{\mathcal {O}}}}\varphi _{\mu (\lambda )}(t)\text { for all }\lambda >0. \end{aligned}$$

  (6.7)

  Let X, $(\xi _n)_{n=1}^\infty$ and $(\lambda _n)_{n=1}^\infty$ be as in Definition 4.3. Set ${{\tilde{\psi }}}_n=\psi _{\lambda _n}$ for all $n\in {\mathbb {N}}$. If g is a function from $(0,\infty )$ to X, then

  $$\begin{aligned} g(t){\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^\infty \xi _n\psi _{\lambda _n}(t)\text { if and only if } g(t)\underset{\Phi }{\sim } \sum _{n=1}^\infty \xi _n{{\tilde{\psi }}}_n(t). \end{aligned}$$

  (6.8)

  For simplicity, we will write the last expansion as

  $$\begin{aligned} g(t)\underset{\Phi }{\sim } \sum _{n=1}^\infty \xi _n\psi _{\lambda _n}(t). \end{aligned}$$

  (6.9)

While the functions $\psi _n$’s, with discrete index n, are the actual functions presented in the expansions in (6.2), the functions $\varphi _\lambda$, with the continuum index $\lambda$, provide specific rates in comparison (6.6) and remainder estimates (6.5), (6.4), (6.3). The fact that $\lambda$ has the range $(0,\infty )$ gives $\varphi _\lambda$ the flexibility in many comparisons and estimates, while the structure of the expansions is maintained by $\psi _n$’s. Note also that $\psi _n$’s are not required to be decreasing anymore.

Assumption 6.2

For the rest of this section, we assume that $\Psi =(\psi _n)_{n=1}^\infty$ and $\Phi =(\varphi _\lambda )_{\lambda >0}$ are a pair of systems as in Definition 6.1 that further satisfy

• (i) For any $m,n\in {\mathbb {N}}$, there exist a natural number $k > \max \{m,n\}$ and a nonzero constant $d_{m,n}$ such that

  $$\begin{aligned} \psi _m\psi _n=d_{m,n}\psi _k. \end{aligned}$$

  (6.10)

• (ii) For each $n\in {\mathbb {N}}$, $\psi _n$ is continuous and differentiable on $[T_*,\infty )$, and $\psi _n'$ has an expansion in the sense of Definition 6.1

  $$\begin{aligned} \psi _n'(t)\underset{\Phi }{\sim } \sum _{k=n+1}^{N_n} c_{n,k}\psi _{k}(t) \text { in }{\mathbb {R}}, \end{aligned}$$

  where $N_n\in {\mathbb {N}}\cup \{0,\infty \}$, all $c_{n,k}$ are constants.

• (iii) The system $\Phi =(\varphi _\lambda )_{\lambda >0}$ satisfies Condition 4.5.

Notation

We denote the unique number k in (6.10) by $m\wedge n$.

We obtain the asymptotic expansions of type (6.2) for the NSE.

Theorem 6.3

Suppose there exist $\alpha \ge 1/2$, $\sigma \ge 0$, and $\phi _n\in G_{\alpha ,\sigma }$ for all $n\in {\mathbb {N}}$ such that

$$\begin{aligned} f(t)\underset{\Phi }{\sim } \sum _{n=1}^\infty \phi _n \psi _n(t)\text { in }G_{\alpha ,\sigma }. \end{aligned}$$

Then any Leray–Hopf weak solution u(t) of (2.2) has the asymptotic expansion

$$\begin{aligned} u(t)\underset{\Phi }{\sim } \sum _{n=1}^\infty \xi _n \psi _n(t)\quad \text {in }G_{\alpha +1-\rho ,\sigma } \text { for all } \rho \in (0,1), \end{aligned}$$

(6.11)

where

$$\begin{aligned} \xi _1=A^{-1}\phi _1,\quad \xi _n=A^{-1}\left( \phi _n - \chi _n - \sum _{{\mathop {k\wedge m=n}\limits ^{1\le k,m\le n-1,}}} d_{k,m}B(\xi _k,\xi _m)\right) \text { for }n \ge 2, \end{aligned}$$

(6.12)

with $\chi _n=\sum _{p=1}^{n-1}c_{p,n} \xi _p$.

Proof

We follow the proof of Theorem 5.4 and make the following replacements:

• $\psi _{\lambda _n}$ is replaced with $\psi _n$ for all $n\in {\mathbb {N}}$, and

• $\psi _\sharp$ is replaced with $\varphi _\sharp$ whenever the subscript symbol $\sharp$ is $\delta _1$, $\varepsilon$, $\varepsilon _1$, $\eta _1$, $\varepsilon _{N}$, $\varepsilon _{N+1}$, $\delta _{N+1}$.

It results in the expansion (6.11) as desired. $\square$

For finite sum asymptotic approximations in a discrete system, we obtain the following counter part of Theorem 5.6.

Theorem 6.4

Suppose there exist numbers $\sigma \ge 0$, $\alpha \ge 1/2$, $N_*\in {\mathbb {N}}$, and functions $\phi _n\in G_{\alpha ,\sigma }$ for $1\le n\le N_*$ such that

$$\begin{aligned} \left| f(t) - \sum _{n=1}^{N_*} \phi _n \psi _n(t)\right| _{\alpha ,\sigma } ={\mathcal {O}}(\varphi _\lambda (t))\text { for some } \lambda >\lambda _{N_*}. \end{aligned}$$

Let $\xi _n$ be defined by (6.12) for $1\le n\le N_*$. Then any Leray–Hopf weak solution u(t) satisfies

$$\begin{aligned} \left| u(t) - \sum _{n=1}^{N_*} \xi _n \psi _n(t)\right| _{\alpha ,\sigma } ={\mathcal {O}}(\varphi _\lambda (t))\text { for some } \lambda >\lambda _{N_*}. \end{aligned}$$

Proof

The proof of Theorem 6.4 is the same as that of Theorem 5.6 with the use of replacements in the proof of Theorem 6.3. $\square$

7 Applications

We will apply results in Sects. 5 and 6 to obtain specific expansions for solutions of the NSE corresponding to different types of forces. We focus on the infinite expansions, hence, show only applications of Theorems 5.4 and 6.3. Their counterparts using the finite asymptotic approximations in Theorems 5.6 and 6.4 can be similarly obtained. However, they will not be presented here, for the sake of avoiding repetitions and keeping the paper concise.

First, we discuss a very frequently used type of systems of functions for long-time asymptotic expansions.

Definition 7.1

A P-system is a system $\Psi =(\psi _\lambda )_{\lambda >0}$, with $\psi _\lambda =\varphi ^\lambda$, where $\varphi$ is a positive function defined on $[T_*,\infty )$ for some $T_*\ge 0$, and $\varphi (t)\rightarrow 0$ as $t\rightarrow \infty$.

Property (P)Clearly, a P-system$\Psi$satisfies (a) and (b) in Definition 4.3with$\eta =\lambda -\mu$, and (i) in Condition4.4with$d_{\lambda ,\mu }=1$and$\gamma =\lambda \wedge \mu =\lambda +\mu$.

In this case, a set $S\subset (0,\infty )$ preserves the operation $\wedge$, see (4.29), if and only if it preserves the addition, i.e., $x+y\in S$ whenever $x,y\in S$.

In Sects. 7.1–7.4, let $\sigma \ge 0$ and $\alpha \ge 1/2$ be given numbers, $(\gamma _n)_{n=1}^\infty$ be a strictly increasing, divergent sequence of positive numbers, and $({{\tilde{\phi }}}_n)_{n=1}^\infty$ be a sequence in $G_{\alpha ,\sigma }$.

7.1 The system of power-decaying functions

We quickly demonstrate how to apply Theorem 5.4 to recover one of the main theorems in [1] on the expansions in the system of power-decaying functions.

Let $\Psi =(t^{-\lambda })_{\lambda >0}$ which is a P-system.

• (i) By Property (P), $\Psi$ satisfies (a) and (b) of Definition 4.3.

• (ii) By Property (P), $\Psi$ satisfies (i) of Condition 4.4. In addition, it satisfies (ii) of Condition 4.4 with

  $$\begin{aligned} N_\lambda =1, \quad c_{\lambda ,1}=-\lambda ,\quad \lambda ^\vee (1)=\lambda +1\text { for all }\lambda >0. \end{aligned}$$

  (7.1)

  Thus, $\Psi$ meets Condition 4.4.

• (iii) Elementary calculations show $\Psi$ meets Condition 4.5.

Therefore, $\Psi$ satisfies the conditions set from the beginning of Sect. 5.

Note from (7.1) that a set $S\subset (0,\infty )$ preserves the operation $\vee$, see (4.28), if and only if it preserves the increments by 1, i.e., $x+1\in S$ whenever $x\in S$.

We assume the force has an expansion

$$\begin{aligned} f(t) {\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^\infty {{\tilde{\phi }}}_n t^{-\gamma _n} \quad \text { in }G_{\alpha ,\sigma }. \end{aligned}$$

(7.2)

Let

$$\begin{aligned} S_*=\left\{ \sum _{j=1}^p \gamma _{n_j} +k : \ p,n_1,n_2,\ldots ,n_p\in {\mathbb {N}},\ k\in {\mathbb {N}}\cup \{0\}\right\} . \end{aligned}$$

(7.3)

Clearly, the set $S_*$ in (7.3) satisfies Assumption 5.2. We assume (5.4), and rewrite (7.2) as

$$\begin{aligned} f(t){\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^{\infty } \phi _n t^{-\lambda _n}\quad \text {in }G_{\alpha ,\sigma }, \end{aligned}$$

where the sequence $(\phi _n)_{n=1}^\infty \subset G_{\alpha ,\sigma }$ is defined as in (5.5). Then Theorem 5.4 implies that any Leray–Hopf weak solution u(t) of the NSE (2.2) has the asymptotic expansion

$$\begin{aligned} u(t) {\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^\infty \xi _n t^{-\lambda _n}\quad \text {in }G_{\alpha +1-\rho ,\sigma }\text { for all } \rho \in (0,1), \end{aligned}$$

where

$$\begin{aligned} \xi _1=A^{-1}\phi _1,\quad \xi _n=A^{-1}\left( \phi _n + \chi _n - \sum _{{\mathop {\lambda _k+\lambda _m=\lambda _n}\limits ^{1\le k,m\le n-1,}}}B(\xi _k,\xi _m)\right) \quad \text {for } n\ge 2, \end{aligned}$$

with $\chi _n=\lambda _p \xi _p$ if there exists an integer $p\in [1, n-1]$ such that $\lambda _p +1= \lambda _n$, and $\chi _n=0$ otherwise.

We have recovered Theorem 4.3 in [1] as a consequence of Theorem 5.4.

7.2 Systems of iterated logarithmic, decaying functions

We consider the case when the force decays as logarithmic or iterated logarithmic functions.

For $k,m \in {\mathbb {N}}$, let

$$\begin{aligned} L_k(t)= \underbrace{\ln (\ln (\cdots \ln (t)))}_{k\text {-times}}\quad \text {and}\quad {\mathcal {L}}_m(t)= (L_1(t),L_2(t), \ldots , L_m(t)). \end{aligned}$$

Let $Q_0: {\mathbb {R}}^m \rightarrow {\mathbb {R}}$ be a polynomial in m variables:

$$\begin{aligned} Q_0(z)=\sum _\alpha c_\alpha z^\alpha \text { for }z\in {\mathbb {R}}^m, \end{aligned}$$

(7.4)

where the sum is taken over finitely many multi-index $\alpha =(\alpha _1,\alpha _2,\ldots ,\alpha _m)$, and $c_\alpha$’s are (real) constants. We use the lexicographic order for the multi-indices in (7.4).

We assume that $Q_0(z)$ has positive degree and positive leading coefficient. Denote by $\alpha _*=(\alpha _{*1},\alpha _{*2},\ldots ,\alpha _{*m})$ the largest multi-index (with the lexicographic order) in (7.4) for which $c_{\alpha _*}\ne 0$. Then we have $|\alpha _*|\ge 1$ and $c_{\alpha _*}>0$.

Let $Q_1$ be a polynomial in one variable of positive degree with positive leading coefficient. Denote the degree of $Q_1$ by $d\ge 1$, and the leading coefficient by $a_d>0$.

Given a number $\beta >0$, we define

$$\begin{aligned} \omega (t)= (Q_0 \circ {\mathcal {L}}_m \circ Q_1)(t^\beta ))\text { with } t\in {\mathbb {R}}. \end{aligned}$$

(7.5)

One can see that there exists $T_*>0$ such that $\omega$ is a positive function defined on $[T_*,\infty )$ and $\omega (t)\rightarrow \infty$ as $t\rightarrow \infty$.

Let $\psi _\lambda (t)= \omega (t)^{-\lambda }$ for $\lambda >0$, and let $\Psi$ be the P-system $(\psi _\lambda )_{\lambda >0}$.

Lemma 7.2

If $\lambda >0$, then

$$\begin{aligned} \lim _{t\rightarrow \infty } \psi _\lambda (t) {\mathcal {L}}_m( t)^{\lambda \alpha _*}=(c_{\alpha _*}(\beta d)^{\alpha _{*1}})^{-\lambda }. \end{aligned}$$

(7.6)

Proof

First, if $k<j$, then $L_j(t)=o(L_k(t))$. With the lexicographic order, we have

$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{ (Q_0\circ {\mathcal {L}}_m)(t)}{ c_{\alpha _*} {\mathcal {L}}_m(t)^{\alpha _*}}=1, \text { which implies } \lim _{t\rightarrow \infty } \frac{ \omega (t)}{ {\mathcal {L}}_m(Q_1(t^\beta ))^{\alpha _*}}=c_{\alpha _*}>0. \end{aligned}$$

Moreover,

$$\begin{aligned} \lim _{t\rightarrow \infty } \frac{ {\mathcal {L}}_m(Q_1(t^\beta ))^{\alpha _*}}{ {\mathcal {L}}_m(a_d t^{\beta d})^{\alpha _*}}=1. \end{aligned}$$

By the properties of the logarithmic function, one has, for any $a,r>0$, that

$$\begin{aligned} \lim _{t\rightarrow \infty } \frac{L_k(at^r)}{L_k(t)}={\left\{ \begin{array}{ll} r,&{} \quad \text { for } k=1,\\ 1,&{}\quad \text {for } k>1. \end{array}\right. } \end{aligned}$$

(7.7)

Combining these gives

$$\begin{aligned}&\lim _{t\rightarrow \infty }\frac{ \omega (t)}{{\mathcal {L}}_m( t)^{\alpha _*}}\\&\quad =\lim _{t\rightarrow \infty } \frac{ \omega (t)}{ {\mathcal {L}}_m(Q_1(t^\beta ))^{\alpha _*}}\cdot \lim _{t\rightarrow \infty } \frac{ {\mathcal {L}}_m(Q_1(t^\beta ))^{\alpha _*}}{ {\mathcal {L}}_m(a_d t^{\beta d})^{\alpha _*}}\cdot \lim _{t\rightarrow \infty } \frac{{\mathcal {L}}_m(a_d t^{\beta d})^{\alpha _*}}{{\mathcal {L}}_m(t)^{\alpha _*}} =c_{\alpha _*} (\beta d)^{\alpha _{*1}}. \end{aligned}$$

Thus, (7.6) follows. $\square$

As a consequence of (7.6), we have

$$\begin{aligned} \psi _\lambda (t){\mathop {=}\limits ^{{\mathcal {O}}}}{\mathcal {L}}_m( t)^{-\lambda \alpha _*}. \end{aligned}$$

(7.8)

In particular, if $\alpha _*= p_0 e_k$ for some $p_0\in {\mathbb {N}}$, where $e_k$ is the k-th unit vector of the canonical basis of ${\mathbb {R}}^m$, then

$$\begin{aligned} \psi _\lambda (t){\mathop {=}\limits ^{{\mathcal {O}}}}L_k(t)^{-p_0 \lambda }. \end{aligned}$$

(7.9)

We verify Conditions 4.4 and  4.5 for the P-system $\Psi$.

Verification of Condition 4.4 Because of Property (P) for $\Psi$, we only need to check (ii) of Condition 4.4. By Chain Rule,

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}({\mathcal {L}}_m(t)^\alpha )&=\sum _{k=1}^m \alpha _k {\mathcal {L}}_m(t)^{\alpha -e_k}\frac{\mathrm{d}}{\mathrm{d}t}L_k(t) =\sum _{k=1}^m \alpha _k {\mathcal {L}}_m(t)^{\alpha -e_k}\frac{1}{t \Pi _{p=1}^{k-1}L_p(t)}\\&=\frac{1}{t}\sum _{k=1}^m \alpha _k {\mathcal {L}}_m(t)^{\alpha -e_1-e_2-\cdots e_k}. \end{aligned}$$

Then,

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}({\mathcal {L}}_m(Q_1(t^\beta ))^\alpha )&= \frac{\beta t^{\beta -1} Q_1'(t^\beta )}{Q_1(t^\beta )}\sum _{k=1}^m \alpha _k {\mathcal {L}}_m(Q_1(t^\beta ))^{\alpha -e_1-e_2-\cdots -e_k}. \end{aligned}$$

We estimate

$$\begin{aligned} \left| \frac{\mathrm{d}}{\mathrm{d}t}({\mathcal {L}}_m(Q_1(t^\beta ))^\alpha )\right| ={\mathcal {O}}(t^{-1}{\mathcal {L}}_m(Q_1(t^\beta ))^{\alpha _*}) ={\mathcal {O}}(t^{-1}{\mathcal {L}}_m(t))^{\alpha _*}). \end{aligned}$$

Hence,

$$\begin{aligned} |\omega '(t)| ={\mathcal {O}}(t^{-1}{\mathcal {L}}_m(t))^{\alpha _*}). \end{aligned}$$

Since $\psi _\lambda '(t)=-\lambda \omega (t)^{-\lambda -1} \omega '(t)$, we combine these with Lemma 7.2 to obtain

$$\begin{aligned} |\psi _\lambda '(t)| = {\mathcal {O}} \left( \frac{{\mathcal {L}}_m(t)^{\alpha _*}}{t \cdot {\mathcal {L}}_m(t)^{(1+\lambda )\alpha _*}}\right) = {\mathcal {O}} (t^{-1}). \end{aligned}$$

This implies

$$\begin{aligned} |\psi _\lambda '(t)| = {\mathcal {O}} \left( {\mathcal {L}}_m(t)^{-\mu \alpha _*}\right) ={\mathcal {O}}(\psi _\mu (t)), \quad \forall \mu >0. \end{aligned}$$

Therefore, by definition,

$$\begin{aligned} \psi _\lambda '(t) {\mathop {\sim }\limits ^{\Psi }}0\text { for all }\lambda >0. \end{aligned}$$

(7.10)

Thus, $\Psi$ satisfies (ii) of Condition 4.4 with $N_\lambda =0$ for all $\lambda >0$.

Verification of Condition 4.5 Thanks to (7.6) and (7.7), the requirements (ii) and (iii) are met. For (i), using the above calculations we find

$$\begin{aligned} \omega '(t)&= \frac{\beta t^{\beta -1} Q_1'(t^\beta )}{Q_1(t^\beta )}\sum _{\alpha =(\alpha _1,\alpha _2,\ldots ,\alpha _m)} c_\alpha \sum _{k=1}^m \alpha _k {\mathcal {L}}_m(Q_1(t^\beta ))^{\alpha -e_1-e_2-\cdots e_k}. \end{aligned}$$

Let $\gamma _*$ be the largest multi-index among $\alpha -e_1-e_2-\cdots -e_k$ with nonzero $c_\alpha \alpha _k$.

Then $\gamma _*=\alpha _*-e_1-e_2-\cdots -e_k$ where k is the smallest index for which the component $\alpha _{*k}\ge 1$. Hence the corresponding coefficient is $c_{\alpha _*}\alpha _{*k}>0$. Note also that $Q_1'(t^\beta )>0$ for large t. We conclude, for sufficiently large t, that $\omega '(t)>0$, and hence $\psi _\lambda '(t)<0$ .

Now, we assume the force has the following expansion

$$\begin{aligned} f(t) {\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^\infty {{\tilde{\phi }}}_n \omega (t)^{-\gamma _n}= \sum _{n=1}^\infty {{\tilde{\phi }}}_n \psi _{\gamma _n}(t) \text { in }G_{\alpha ,\sigma }. \end{aligned}$$

Let $S_*= \{ \sum _{j=1}^p \gamma _{n_j} : p,n_1,n_2,\ldots ,n_p\in {\mathbb {N}}\}$. Then, again, this set $S_*$ satisfies Assumption 5.2; hence, we can assume (5.4) and the expansion (5.5), which reads as

$$\begin{aligned} f(t){\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^{\infty } \phi _n \omega (t)^{-\lambda _n}\quad \text {in }G_{\alpha ,\sigma }, \end{aligned}$$

(7.11)

for a sequence $(\phi _n)_{n=1}^\infty$ in $G_{\alpha ,\sigma }$.

Theorem 7.3

Let $\omega$ be defined by (7.5) and assume the expansion (7.11).

• (i) Then any Leray–Hopf weak solution u(t) of the NSE (2.2) has the asymptotic expansion

  $$\begin{aligned} u(t){\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^\infty \xi _n \omega (t)^{-\lambda _n}\quad \text {in }G_{\alpha +1-\rho ,\sigma }\text { for all } \rho \in (0,1), \end{aligned}$$

  (7.12)

  where

  $$\begin{aligned} \xi _1=A^{-1}\phi _1,\quad \xi _n=A^{-1}\left( \phi _n - \sum _{{\mathop {\lambda _k+\lambda _m=\lambda _n}\limits ^{1\le k,m\le n-1,}}}B(\xi _k,\xi _m)\right) \quad \text {for } n\ge 2. \end{aligned}$$

  (7.13)

• (ii) By defining $\varphi _\lambda (t)={\mathcal {L}}_m(t)^{-\lambda \alpha _*}$ and $\Phi =(\varphi _\lambda )_{\lambda >0}$, we can equivalently replace ${\mathop {\sim }\limits ^{\Psi }}$ with $\underset{\Phi }{\sim }$ in (7.11) and (7.12), in the sense of (6.9).

  In particular, if $\alpha _*$ is co-linear with the k-th unit vector $e_k$ of the canonical basis of ${\mathbb {R}}^m$, then this replacement still holds true for $\Phi =(L_k(t)^{-\lambda })_{\lambda >0}$.

Proof

• (i) Applying Theorem 5.4 while noting that $\chi _n =0$ in (5.9) for all $n \in {\mathbb {N}}$ due to the fact (7.10), we deduce (7.12) from (5.6).

• (ii) Let $\varphi _\lambda (t)={\mathcal {L}}_m(t)^{-\lambda \alpha _*}$ and $\Phi =(\varphi _\lambda )_{\lambda >0}$. Note by (7.8) that $\psi _\lambda (t){\mathop {=}\limits ^{{\mathcal {O}}}}\varphi _\lambda (t)$ which implies (6.7) with $\mu (\lambda )=\lambda$. Then the replacement is valid thanks to (6.8) and (6.9) in remark (e) after Definition 6.1.

  Now, consider the case when $\alpha _*=p_0 e_k$ for some $p_0\in {\mathbb {N}}$. Let $\varphi _{\lambda }(t)=L_k(t)^{-\lambda }$ and, again, $\Phi =(\varphi _\lambda )_{\lambda >0}$. By (7.9), we have

  $$\begin{aligned} \psi _\lambda (t){\mathop {=}\limits ^{{\mathcal {O}}}}\varphi _{p_0\lambda }(t)=\varphi _{\mu (\lambda )}(t),\quad \text {where }\mu (\lambda )=p_0\lambda . \end{aligned}$$

Hence, the replacement is valid again by the same (6.8) and (6.9). $\square$

Corollary 7.4

Given $m\in {\mathbb {N}}$, define $\Psi =(L_m(t)^{-\lambda })_{\lambda >0}$. Suppose $(\lambda _n)_{n=1}^\infty$ is a strictly increasing, divergent sequence of positive numbers such that the set $\{\lambda _n:n\in {\mathbb {N}}\}$ preserves the addition. If

$$\begin{aligned} f(t){\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^{\infty } \phi _n L_m(t)^{-\lambda _n}\quad \text {in }G_{\alpha ,\sigma }, \end{aligned}$$

then any Leray–Hopf weak solution u(t) of the NSE (2.2) admits the same asymptotic expansion

$$\begin{aligned} u(t){\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^{\infty } \xi _n L_m(t)^{-\lambda _n}\quad \text {in }G_{\alpha +1-\rho ,\sigma }\text { for all } \rho \in (0,1), \end{aligned}$$

where $\xi _n$’s are defined by (7.13).

Proof

We choose $Q_0(z_1,z_2,\ldots ,z_m)=z_m$, $Q_1(t)=t$ and $\beta =1$. Notice, in this case, that (7.5) becomes $\omega (t)=L_m(t)$. Then the result in this corollary follows Theorem 7.3(i) and Remark 5.3. $\square$

Example 7.5

Thanks to Theorem 7.3, we can have asymptotic expansions of many different types. We illustrate it with just two here. Let $(\lambda _n)_{n=1}^\infty$ be a strictly increasing, divergent sequence of positive numbers that preserves the addition. By Remark 5.3, we can use, from the beginning, expansion (7.11) for the force. We will also use the replacements indicated in Theorem 7.3(ii).

• (a) When $m=5$, $Q_0(z_1,z_2,\ldots ,z_5)=3z_1^2 z_3-2 z_2 z_5^4$, $Q_1(t)=t$, and $\beta =1$, if

  $$\begin{aligned} f(t)\underset{\Phi }{\sim } \sum _{n=1}^\infty \phi _n \big [3(\ln t)^2 L_3(t) - 2 L_2(t)L_5(t)^4\big ]^{-\lambda _n} \quad \text {in }G_{\alpha ,\sigma }, \end{aligned}$$

  where $\Phi =\big ((\ln t)^{-2\lambda }L_3(t)^{-\lambda }\big )_{\lambda >0}$, then

  $$\begin{aligned} u(t)\underset{\Phi }{\sim } \sum _{n=1}^\infty \xi _n \big [3(\ln t)^2 L_3(t) - 2 L_2(t)L_5(t)^4\big ]^{-\lambda _n}\quad \text {in }G_{\alpha +1-\rho ,\sigma }\text { for all } \rho \in (0,1). \end{aligned}$$

• (b) When $m=7$, $Q_0(z_1,z_2,\ldots ,z_7)=4z_2-z_7^5+3$, $Q_1(t)=t^3-3t+1$, and $\beta =1/2$, if

  $$\begin{aligned} f(t)\underset{\Phi }{\sim } \sum _{n=1}^\infty \phi _n \big [4 L_2(t^{3/2}-3 t^{1/2}+1) - L_7(t^{3/2}-3t^{1/2}+1)^5 +3\big ]^{-\lambda _n} \quad \text {in }G_{\alpha ,\sigma }, \end{aligned}$$

  where $\Phi =(L_2(t)^{-\lambda })_{\lambda >0}$, then

  $$\begin{aligned}&u(t)\underset{\Phi }{\sim } \sum _{n=1}^\infty \xi _n \big [4 L_2(t^{3/2}-3 t^{1/2}+1) - L_7(t^{3/2}-3t^{1/2}+1)^5 +3\big ]^{-\lambda _n}\\&\quad \text {in }G_{\alpha +1-\rho ,\sigma }\text { for all } \rho \in (0,1). \end{aligned}$$

Example 7.6

Given $m \in {\mathbb {N}}$. Consider the system

$$\begin{aligned} \Psi =(\psi _\lambda )_{\lambda>0}=\left( \left[ \sin (L_m^{-1}(t))\right] ^{\lambda } \right) _{\lambda >0}. \end{aligned}$$

Clearly, $\sin (L_m^{-1}(t))$ is a positive function defined on $[T_*,\infty )$ for some $T_* > 0$ and $\sin (L_m^{-1}(t))\rightarrow 0$ as $t\rightarrow \infty$. Thus $\Psi$ is a P-system.

Since $\sin ^{\lambda }(x)$ is increasing in x on $(0,\pi /2)$, it implies that $\left[ \sin (L_m^{-1}(t))\right] ^{\lambda }$ is decreasing on $[T_1,\infty )$ for some sufficiently large number $T_1$. Noting that

$$\begin{aligned} \frac{2x}{\pi } \le \sin (x) \le x, \quad \text {for } 0 \le x \le \pi /2, \end{aligned}$$

and $0 < L_m^{-1}(t) \le 1$ for large t, one has $\sin (L_m^{-1}(t)) {\mathop {=}\limits ^{{\mathcal {O}}}}L_m^{-1}(t)$.

This and Lemma 7.2 yield

$$\begin{aligned} \left[ \sin (L_m^{-1}(at))\right] ^{\lambda } {\mathop {=}\limits ^{{\mathcal {O}}}}\left[ L_m^{-1}(at)\right] ^{\lambda } {\mathop {=}\limits ^{{\mathcal {O}}}}\left[ L_m^{-1}(t)\right] ^{\lambda } {\mathop {=}\limits ^{{\mathcal {O}}}}\left[ \sin (L_m^{-1}(t))\right] ^{\lambda }. \end{aligned}$$

Clearly,

$$\begin{aligned} \lim _{ t \rightarrow \infty } \frac{e^{-\alpha t }}{\psi _{\lambda }(t)}= \lim _{ t \rightarrow \infty } e^{-\alpha t }L_m(t)^{\lambda }= 0. \end{aligned}$$

Therefore, $\Psi$ satisfies Condition 4.5. We write

$$\begin{aligned} \psi '_\lambda (t)= -\lambda \left( \sin ( L_m^{-1}(t) \right) ^{\lambda -1}\cos ( L_m^{-1}(t) ) \frac{1}{t \left( \Pi _{p=1}^{m-1}L_p(t) \right) L_m(t)^2}, \end{aligned}$$

and estimate

$$\begin{aligned} |\psi '_\lambda (t)|={\mathcal {O}} \left( t^{-1} (\sin ( L_m^{-1}(t))^{\lambda -1}\right) ={\mathcal {O}} \left( t^{-1} L_m(t)^{-\lambda +1}\right) ={\mathcal {O}} \left( L_m(t)^{-\mu } \right) = {\mathcal {O}} (\psi _{\mu }(t)) \end{aligned}$$

for all $\mu > 0$. Then $\Psi$ satisfies (ii) of Condition 4.4 with $N_\lambda =0$ for all $\lambda >0$, and thus, all parts of Condition 4.4 due to Property (P). Let $(\lambda _n)_{n=1}^\infty$ be as in Example 7.5. By Theorem 5.4, if

$$\begin{aligned} f(t){\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^{\infty } \phi _n \left[ \sin (L_m^{-1}(t))\right] ^{\lambda _n} \quad \text {in }G_{\alpha ,\sigma }, \end{aligned}$$

then

$$\begin{aligned} u(t){\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^{\infty } \xi _n \left[ \sin (L_m^{-1}(t))\right] ^{\lambda _n} \quad \text {in }G_{\alpha +1-\rho ,\sigma }\text { for all } \rho \in (0,1), \end{aligned}$$

with $\xi _n$’s being (7.13).

Example 7.7

Given $m \in {\mathbb {N}}$. Consider the system

$$\begin{aligned} \Psi =(\psi _\lambda )_{\lambda>0}=\left( \left[ \tan (L_m^{-1}(t))\right] ^{\lambda } \right) _{\lambda >0}. \end{aligned}$$

Similar to Example 7.6, using the fact that

$$\begin{aligned} x \le \tan (x) \le 2x, \quad \text {for } 0 \le x \le 1/2, \end{aligned}$$

one can verify that $\Psi$ satisfies Condition 4.5 and Condition 4.4 with $N_\lambda =0$ for all $\lambda >0$.

Again, let $(\lambda _n)_{n=1}^\infty$ be as in Example 7.5. We obtain that if

$$\begin{aligned} f(t){\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^{\infty } \phi _n \left[ \tan (L_m^{-1}(t))\right] ^{\lambda _n} \quad \text {in }G_{\alpha ,\sigma }, \end{aligned}$$

then

$$\begin{aligned} u(t){\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^{\infty } \xi _n \left[ \tan (L_m^{-1}(t))\right] ^{\lambda _n} \quad \text {in }G_{\alpha +1-\rho ,\sigma }\text { for all } \rho \in (0,1), \end{aligned}$$

with $\xi _n$’s from (7.13).

7.3 A system with infinite expansions for the derivatives

In the previous two subsections, the expansion (4.19) is zero or a finite sum. In this subsection, we demonstrate the case when each expansion (4.19) is an infinite sum.

Consider a particular P-system $\Psi =(\psi _\lambda )_{\lambda >0}$ with $\psi _\lambda =(\sqrt{t}+1)^{-\lambda }$. Let $\lambda >0$. We see, for any $t>0$, that

$$\begin{aligned} \psi _\lambda '(t)&=-\lambda (\sqrt{t}+1)^{-\lambda -1 }\frac{1}{2}\frac{1}{\sqrt{t}} =-\frac{\lambda }{2} (\sqrt{t}+1)^{-\lambda -1 }\frac{1}{\sqrt{t}+1}\cdot \frac{1}{1-\frac{1}{\sqrt{t}+1}}\\&=-\frac{\lambda }{2} (\sqrt{t}+1)^{-\lambda -1 }\sum _{k=1}^\infty \frac{1}{(\sqrt{t}+1)^k} =\sum _{k=1}^\infty -\frac{\lambda }{2} (\sqrt{t}+1)^{-\lambda -k-1 }. \end{aligned}$$

Applying Lemma A.1 to $X={\mathbb {R}}$, $\varphi (t)=(\sqrt{t}+1)^{-1}$, $\lambda _n=\lambda +n+1$, $M=2$, $\xi _n=-\lambda /2$, $c_0=\lambda /2$, $\kappa =1$, we deduce that the derivative $\psi _\lambda '(t)$, in fact, has the expansion

$$\begin{aligned} \psi _\lambda '(t) {\mathop {\sim }\limits ^{\Psi }}\sum _{k=1}^\infty -\frac{\lambda }{2} (\sqrt{t}+1)^{-\lambda -k-1 }. \end{aligned}$$

Thus, we have expansion (4.19) with

$$\begin{aligned} N_\lambda =\infty ,\text { and}\quad c_{\lambda ,k}=-\lambda /2,\quad \lambda ^\vee (k)=\lambda +1+k\text { for all }k\in {\mathbb {N}}. \end{aligned}$$

With this, Property (P), and some elementary estimates, we can verify that Conditions 4.4 and 4.5 are met.

We assume the force has the expansion

$$\begin{aligned} f(t){\mathop {\sim }\limits ^{\Psi }}\, \sum _{n=1}^\infty {{\tilde{\phi }}}_n \psi _{\gamma _n}(t) =\sum _{n=1}^\infty {{\tilde{\phi }}}_n (\sqrt{t}+1)^{-\gamma _n} \text { in }G_{\alpha ,\sigma }. \end{aligned}$$

Define

$$\begin{aligned} S_*=\Big \{ (\sum _{j=1}^p \gamma _{n_j}) + k: p,n_1,n_2,\ldots ,n_p\in {\mathbb {N}}, \ k=0 \text { or }k\in {\mathbb {N}}\cap [2,\infty )\Big \}. \end{aligned}$$

Then the set $S_*$ satisfies Assumption 5.2, and we can assume (5.4) and the expansion

$$\begin{aligned} f(t) {\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^{\infty } \phi _n (\sqrt{t}+1)^{-\lambda _n }\quad \text {in }G_{\alpha ,\sigma }, \end{aligned}$$

(7.14)

for a sequence $(\phi _n)_{n=1}^\infty$ in $G_{\alpha ,\sigma }$. We apply Theorem 5.4 and obtain the following.

Proposition 7.8

Assume (7.14). Then any Leray–Hopf weak solution u(t) of the NSE (2.2) admits the asymptotic expansion

$$\begin{aligned} u(t) {\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^\infty \xi _n (\sqrt{t}+1)^{-\lambda _n}\quad \text {in }G_{\alpha +1-\rho ,\sigma }\text { for all } \rho \in (0,1), \end{aligned}$$

where

$$\begin{aligned} \xi _1=A^{-1}\phi _1,\quad \xi _n&=A^{-1}\left( \phi _n + \frac{1}{2}\sum _{ p \in {\mathcal {Z}}_n} \lambda _p \xi _p - \sum _{{\mathop {\lambda _k+\lambda _m=\lambda _n}\limits ^{1\le k,m\le n-1,}}}B(\xi _k,\xi _m)\right) \quad \text {for } n\ge 2, \end{aligned}$$

with ${\mathcal {Z}}_n= \{ p \in {\mathbb {N}}\cap [1, n-1]: \exists k\in {\mathbb {N}}, \lambda _p+1+k=\lambda _n\}.$

7.4 Expansions using a background system

In this subsection, we present a scenario for which the use of the background systems in Sect. 6 is essential. To motivate our more general force f later, we consider a simple case first. Let $\gamma \in (0,1)$, $\beta _0>0$, and

$$\begin{aligned} f(t)= \frac{\phi }{(t^\gamma +1)^{\beta _0}}\quad \text {with } \phi \in G_{\alpha ,\sigma }. \end{aligned}$$

We expect a solution u(t) of the NSE (2.2) to have an asymptotic expansion containing at least $(t^\gamma +1)^{-\beta }$ for some $\beta >0$. (Here, the structure of f(t) is maintained without being converted to a different form such as (7.2).) The derivative term $u_t$ in the NSE will contain $\frac{\mathrm{d}}{\mathrm{d}t}(t^\gamma +1)^{-\beta }$, which is

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}(t^\gamma +1)^{-\beta }&=\frac{ -\gamma \beta }{(t^\gamma +1)^{ \beta +1}t^{1-\gamma } } =\frac{ -\gamma \beta }{(t^\gamma +1)^{ \beta +1}(t^{1-\gamma }+1)(1-\frac{1}{t^{1-\gamma }+1}) }\\&=\frac{ -\gamma \beta }{(t^\gamma +1)^{ \beta +1} } \sum _{k=1}^\infty \frac{1}{(t^{1-\gamma }+1)^k}, \end{aligned}$$

thus,

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}(t^\gamma +1)^{-\beta } = \sum _{k=1}^\infty \frac{ -\gamma \beta }{(t^\gamma +1)^{ \beta +1} (t^{1-\gamma }+1)^k}. \end{aligned}$$

(7.15)

Thanks to the term Au in the NSE, (7.15) in turn suggests that a possible asymptotic expansion of u(t) may have to include infinitely many terms $(t^\gamma +1)^{-\lambda }(t^{1-\gamma }+1)^{-\mu }$.

Because of this, we now consider a function $\psi (t)=(t^\gamma +1)^{-\lambda }(t^{1-\gamma }+1)^{-\mu }$. Taking the derivative by the product rule gives

$$\begin{aligned} \psi '(t)=\Big (\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{(t^\gamma +1)^\lambda }\Big )\frac{1}{(t^{1-\gamma }+1)^\mu } +\frac{1}{(t^\gamma +1)^\lambda }\Big (\frac{\mathrm{d}}{\mathrm{d}t}\frac{1}{(t^{1-\gamma }+1)^\mu }\Big ). \end{aligned}$$

Using (7.15) with $\beta :=\lambda$ for the first derivative, and with $\gamma :=1-\gamma$, $\beta :=\mu$ for second derivative, we obtain

$$\begin{aligned} \psi '(t) = \sum _{k=1}^\infty \frac{ -\gamma \lambda }{(t^\gamma +1)^{ \lambda +1} (t^{1-\gamma }+1)^{\mu +k}} + \sum _{k=1}^\infty \frac{ -(1-\gamma ) \mu }{(t^{1-\gamma }+1)^{ \mu +1} (t^\gamma +1)^{\lambda +k}}. \end{aligned}$$

(7.16)

Observe that the sums in (7.16) involve the functions of the same form as $\psi$, but with different powers. Also, the equality can be converted, under proper conditions, to an asymptotic expansion with the background system $\Phi =(t^{-\lambda })_{\lambda >0}$.

Fixing a background system Let us fix the P-system $\Phi =(\varphi _\lambda )_{\lambda >0}$, where $\varphi _\lambda (t)=t^{-\lambda }$. By (i)–(iii) in Sect. 7.1, we see that $\Phi$ satisfies Condition (iii) of Assumption 6.2.

From the above observation, we consider a force f having the following general expansion

$$\begin{aligned} f(t) \underset{\Phi }{\sim } \sum _{n=1}^{\infty } {{\tilde{\phi }}}_n {{\tilde{\psi }}}_n(t)=\sum _{n=1}^{\infty } \frac{{{\tilde{\phi }}}_n}{(t^\gamma +1)^{{{\tilde{\alpha }}}_n}(t^{1-\gamma }+1)^{{{\tilde{\beta }}}_n} } \quad \text { in }G_{\alpha ,\sigma }, \end{aligned}$$

(7.17)

where ${{\tilde{\psi }}}_n(t)= (t^\gamma +1)^{-{{\tilde{\alpha }}}_n}(t^{1-\gamma }+1)^{-{{\tilde{\beta }}}_n}$, $\gamma$ is a constant in the interval (0, 1), $({{\tilde{\alpha }}}_n)_{n=1}^\infty$ and $({{\tilde{\beta }}}_n)_{n=1}^\infty$ are sequences of non-negative numbers such that

$$\begin{aligned} {{\tilde{\lambda }}}_n{\mathop {=}\limits ^\mathrm{def}}\gamma {{\tilde{\alpha }}}_n+(1-\gamma ){{\tilde{\beta }}}_n\text { is positive, strictly increasing (in }n\text {) to infinity}. \end{aligned}$$

(7.18)

Note that the expansion in (7.17) is understood in the sense of Definition 6.1 with ${{\tilde{\Psi }}}=({{\tilde{\psi }}}_n)_{n=1}^\infty$ replacing $\Psi$, and ${{\tilde{\lambda }}}_n$ replacing $\lambda _n$.

A simple example of (7.17) is a finite sum $f(t)= \sum _{n=1}^N {{\tilde{\phi }}}_n {{\tilde{\psi }}}_n(t)$ for some $N\in {\mathbb {N}}$. For more complicated cases of infinite sums, see Corollary A.3.

Assumption 7.9

The number $\gamma$ is irrational, while numbers ${{\tilde{\alpha }}}_n$ and ${{\tilde{\beta }}}_n$ are rational for all $n\in {\mathbb {N}}$.

Define the sets

$$\begin{aligned} S_*&=\left\{ \gamma \left( \sum _{j=1}^p {{\tilde{\alpha }}}_{n_j} + k\right) + (1-\gamma )\left( \sum _{j=1}^p {{\tilde{\beta }}}_{n_j} + \ell \right) : \right. \nonumber \\&\qquad \left. p,n_1,n_2,\ldots ,n_p\in {\mathbb {N}},\ (k,\ell )\in {\mathbb {N}}^2\cup (0,0)\right\} ,\\ E_1&=\left\{ \sum _{j=1}^p {{\tilde{\alpha }}}_{n_j} + k: p,n_1,n_2,\ldots ,n_p\in {\mathbb {N}},\ k\in {\mathbb {N}}\cup \{0\}\right\} , \nonumber \\ E_2&=\left\{ \sum _{j=1}^q {{\tilde{\beta }}}_{k_j} + \ell :q,k_1,k_2,\ldots ,k_q\in {\mathbb {N}},\ \ell \in {\mathbb {N}}\cup \{0\}\right\} .\nonumber \end{aligned}$$

(7.19)

Note that

$$\begin{aligned} \{{{\tilde{\lambda }}}_n:n\in {\mathbb {N}}\}\subset S_*\subset E^*{\mathop {=}\limits ^\mathrm{def}}\{ \gamma \alpha +(1-\gamma )\beta : \alpha \in E_1,\ \beta \in E_2 \}. \end{aligned}$$

(7.20)

We see that $S_*$ and $E^*$ preserve the addition.

Lemma 7.10

For each $\mu \in E^*$, there exists a unique pair $(\alpha ,\beta )\in E_1\times E_2$ such that

$$\begin{aligned} \mu =\gamma \alpha +(1-\gamma )\beta . \end{aligned}$$

(7.21)

Proof

The existence of the decomposition (7.21) comes directly from (7.20). We prove the uniqueness now. Let $\mu \in E^*$ and suppose there are $(\alpha ,\beta ),(\alpha ',\beta ')\in E_1\times E_2$ such that

$$\begin{aligned} \mu =\gamma \alpha +(1-\gamma )\beta =\gamma \alpha '+(1-\gamma )\beta '. \end{aligned}$$

Then $\gamma (\alpha -\alpha ')=-(1-\gamma )(\beta -\beta ').$ Note from this relation that $\alpha =\alpha '$ if and only if $\beta =\beta '$.

Consider the case $\alpha \ne \alpha '$ and $\beta \ne \beta '$. Then

$$\begin{aligned} \frac{\gamma }{1-\gamma }=-\frac{\beta -\beta '}{\alpha -\alpha '}. \end{aligned}$$

(7.22)

We have $\alpha -\alpha '=\sum _{j=1}^p \pm {{\tilde{\alpha }}}_{n_j} + k\ne 0$, and $\beta -\beta '=\sum _{j=1}^q \pm {{\tilde{\beta }}}_{k_j} + \ell \ne 0$, for some $k,\ell \in {\mathbb {Z}}$, and some particular $+/-$ signs in the two sums.

Since $\gamma$ is irrational, so is $\gamma /(1-\gamma )$, while the right-hand side of (7.22) is rational, which yields a contradiction. Therefore, we can only have $\alpha =\alpha '$ and $\beta =\beta '$. $\square$

We rewrite $S_*$ as

$$\begin{aligned} S_*=\left\{ \sum _{j=1}^p {{\tilde{\lambda }}}_{n_j} + \gamma k+ (1-\gamma )\ell : p,n_1,n_2,\ldots ,n_p\in {\mathbb {N}},\ (k,\ell )\in {\mathbb {N}}^2\cup (0,0)\right\} .\nonumber \\ \end{aligned}$$

(7.23)

Since ${{\tilde{\lambda }}}_n\rightarrow \infty$, it follows (7.23) that we can order $S_*$ to be a sequence $(\lambda _n)_{n=1}^\infty$ as in (5.4).

Note that $\lambda _n\rightarrow \infty$, hence consequently, $\alpha _n+\beta _n\rightarrow \infty$.

The discrete system for expansions Let $\Psi =(\psi _n)_{n=1}^\infty$, where

$$\begin{aligned} \psi _n(t)=(t^\gamma +1)^{-\alpha _n}(t^{1-\gamma }+1)^{-\beta _n}, \end{aligned}$$

with $(\alpha _n,\beta _n)$, thanks to Lemma 7.10, being the unique pair in $E_1\times E_2$ such that

$$\begin{aligned} \lambda _n=\gamma \alpha _n+(1-\gamma )\beta _n. \end{aligned}$$

(7.24)

Clearly, $\psi _n(t){\mathop {=}\limits ^{{\mathcal {O}}}}t^{-\lambda _n}=\varphi _{\lambda _n}(t)$. Hence, $\Psi$ and $\Phi$ satisfy condition (6.1) in Definition 6.1.

We still need to verify the remaining Conditions (i) and (ii) of Assumption 6.2.

Verification of Condition (i) For $m,n \in {\mathbb {N}}$, we have

$$\begin{aligned} \psi _m(t)= & {} (t^\gamma +1)^{-\alpha _m}(t^{1-\gamma }+1)^{-\beta _m},\quad \lambda _m=\gamma \alpha _m+(1-\gamma )\beta _m\in S_*, \end{aligned}$$

(7.25)

$$\begin{aligned} \psi _n(t)= & {} (t^\gamma +1)^{-\alpha _n}(t^{1-\gamma }+1)^{-\beta _n},\quad \lambda _n=\gamma \alpha _n+(1-\gamma )\beta _n\in S_*, \end{aligned}$$

(7.26)

where $\alpha _m,\alpha _n\in E_1$ and $\beta _m,\beta _n\in E_2$. Then

$$\begin{aligned} (\psi _m\psi _n)(t)=(t^\gamma +1)^{-\alpha _m-\alpha _m}(t^{1-\gamma }+1)^{-\beta _n-\beta _n}. \end{aligned}$$

Since $S_*$ preserves the addition we have $\lambda _m+\lambda _n\in S_*$, hence there exists k such that

$$\begin{aligned} \lambda _k=\lambda _m+\lambda _n. \end{aligned}$$

(7.27)

By (7.27), (7.25) and (7.26), we have $\lambda _k=\gamma (\alpha _m+\alpha _n)+(1-\gamma )(\beta _m+\beta _n)$. Because $\alpha _m+\alpha _n\in E_1$, $\beta _m+\beta _n\in E_2$, and by the uniqueness of the decomposition of $\lambda _k$, we deduce $\alpha _m+\alpha _n=\alpha _k$ and $\beta _n+\beta _m=\beta _k$. Therefore,

$$\begin{aligned} (\psi _m\psi _n)(t)=(t^\gamma +1)^{-\alpha _k}(t^{1-\gamma }+1)^{-\beta _k}=\psi _k(t). \end{aligned}$$

This proves that (6.10) of Assumption 6.2 holds true with $d_{m,n}=1$ and $k=m\wedge n\in {\mathbb {N}}$ satisfying (7.27).

Thus, Condition (i) of Assumption 6.2 is met.

Verification of Condition (ii) Using (7.16), we have, for $n\in {\mathbb {N}}$, $t>0$,

$$\begin{aligned} \psi _n'(t)&= \sum _{k=1}^\infty \frac{ -\gamma \alpha _n}{(t^\gamma +1)^{ \alpha _n +1} (t^{1-\gamma }+1)^{\beta _n+k}} + \sum _{k=1}^\infty \frac{ -(1-\gamma ) \beta _n}{(t^\gamma +1)^{\alpha _n+k}(t^{(1-\gamma }+1)^{ \beta _n +1} }\\&=g_{n,1}(t)+g_{n,2}(t), \end{aligned}$$

where

$$\begin{aligned} g_{n,1}(t)= & {} \sum _{k=1}^\infty \frac{ -\gamma \alpha _n}{(t^\gamma +1)^{ \alpha _n +1} (t^{1-\gamma }+1)^{\beta _n+k}},\\ g_{n,2}(t)= & {} \sum _{k=1}^\infty \frac{ -(1-\gamma ) \beta _n}{(t^\gamma +1)^{\alpha _n+k}(t^{(1-\gamma }+1)^{ \beta _n +1} }. \end{aligned}$$

Let $n\in {\mathbb {N}}$ be fixed momentarily. We apply Lemma A.2 to

$$\begin{aligned} \varphi (t)= & {} t^{-1},\ {{\bar{\psi }}}_k(t)=(t^\gamma +1)^{-\alpha _n-1} (t^{1-\gamma }+1)^{-\beta _n-k},\ T_*=1, \\ X= & {} {\mathbb {R}},\ {{\bar{\xi }}}_k=-\gamma \alpha _n,\ {{\bar{\nu }}}_k=\gamma (\alpha _n+1)+(1-\gamma )(\beta _n+k). \end{aligned}$$

Note, for $t\ge 1$, that

$$\begin{aligned} 2^{-\alpha _n-\beta _n-k-1}t^{-{{\bar{\nu }}}_k}=(2t^\gamma )^{-\alpha _n-1} (2t^{1-\gamma })^{-\beta _n-k} \le {{\bar{\psi }}}_k(t)\le t^{-{{\bar{\nu }}}_k}, \end{aligned}$$

which yields

$$\begin{aligned} D_k^{-1}\varphi (t)^{{{\bar{\nu }}}_k} \le {{\bar{\psi }}}_k(t)\le D_k \varphi (t)^{{{\bar{\nu }}}_k}, \end{aligned}$$

where $D_k= 2^{\alpha _n+\beta _n+k+1}$. Taking $c_0=\gamma \alpha _n$, $\kappa =1$, and $M=3^{1/(1-\gamma )}$, we have

$$\begin{aligned} |\xi _k|\le c_0 \kappa ^{{{\bar{\nu }}}_k},\quad \sum _{k=1}^\infty D_k M^{-{{\bar{\nu }}}_k}<\infty . \end{aligned}$$

Since $({{\bar{\nu }}}_k)_{k=1}^\infty$ is already strictly increasing, it is its own strictly increasing re-arrangement.

Then, by Lemma A.2,

$$\begin{aligned} g_{n,1}(t)\underset{\Phi }{\sim } \sum _{k=1}^\infty {{\bar{\xi }}}_k{{\bar{\psi }}}_k(t)= \sum _{k=1}^\infty \frac{ -\gamma \alpha _n}{(t^\gamma +1)^{ \alpha _n +1} (t^{1-\gamma }+1)^{\beta _n+k}}. \end{aligned}$$

(7.28)

Now, with ${{\hat{\xi }}}_k=-(1-\gamma ) \beta _n$ replacing $\xi _k$, ${{\hat{\psi }}}_k(t)=(t^\gamma +1)^{-\alpha _n-k}(t^{1-\gamma }+1)^{-\beta _n -1}$ replacing ${{\bar{\psi }}}_k(t)$, and ${{\hat{\nu }}}_k=\gamma (\alpha _n+k)+(1-\gamma )(\beta _n+1)$ replacing ${{\bar{\nu }}}_k$, we similarly obtain

$$\begin{aligned} g_{n,2}(t)\underset{\Phi }{\sim }\sum _{k=1}^\infty {{\hat{\xi }}}_k{{\hat{\psi }}}_k(t) =\sum _{k=1}^\infty \frac{ -(1-\gamma ) \beta _n}{(t^\gamma +1)^{\alpha _n+k}(t^{1-\gamma }+1)^{ \beta _n +1} }. \end{aligned}$$

(7.29)

Since ${{\bar{\nu }}}_k$ and ${{\hat{\nu }}}_k$ belong to $S_*$, all functions ${{\bar{\psi }}}_k(t)$ in (7.28), and ${{\hat{\psi }}}_k(t)$ in (7.29) belong to the collection $\{\psi _n:n\in {\mathbb {N}}\}$. Then we can rewrite

$$\begin{aligned} g_{n,i}(t)\underset{\Phi }{\sim } \sum _{k=1}^\infty {{\tilde{c}}}_{n,i,k}\psi _k(t),\quad \text {for } i=1,2, \end{aligned}$$

for some constants ${{\tilde{c}}}_{n,i,k}$. Therefore,

$$\begin{aligned} \psi _n'(t)=g_{n,1}(t)+g_{n,2}(t)\underset{\Phi }{\sim } -\sum _{k=1}^\infty c_{n,k}\psi _k(t), \end{aligned}$$

(7.30)

where

$$\begin{aligned} c_{n,k}=-({{\tilde{c}}}_{n,1,k}+{{\tilde{c}}}_{n,2,k}) =\gamma \sum _j\alpha _j +(1-\gamma )\sum _\ell \beta _\ell , \end{aligned}$$

(7.31)

with $\alpha _j+1=\alpha _k$, $\beta _j+p=\beta _k$, and $\alpha _\ell +q=\alpha _k$, $\beta _\ell +1=\beta _k$ for some $p,q\in {\mathbb {N}}\cup \{0\}$.

Note that these pairs (j, p) and $(\ell ,q)$ are only finitely many. Indeed, since

$$\begin{aligned} \alpha _j+\beta _j+p+1=\alpha _k+\beta _k\text { and }\alpha _j+\beta _j\rightarrow \infty , \end{aligned}$$

we have, for each fixed k, there are only finitely many j and p. The same arguments apply to $(\ell ,q)$. Therefore, the sums in (7.31) are only finite ones.

With (7.30), Condition (ii) of Assumption 6.2 is met.

Conclusion on Assumption 6.2 We have checked that the systems $\Psi$ and $\Phi$ satisfy Assumption 6.2.

We now return to the expansion (7.17) for the force. Since ${{\tilde{\lambda }}}_n\in S_*$, we have that each function $(t^\gamma +1)^{-{{\tilde{\alpha }}}_n}(t^{1-\gamma }+1)^{-{{\tilde{\beta }}}_n}$ in the sum in (7.17) belongs to $\{\psi _n:n\in {\mathbb {N}}\}$. Hence, we can rewrite (7.17) as

$$\begin{aligned} f(t)\underset{\Phi }{\sim } \sum _{n=1}^\infty \phi _n \psi _n(t) =\sum _{n=1}^{\infty } \frac{ \phi _n}{(t^\gamma +1)^{ \alpha _n}(t^{1-\gamma }+1)^{ \beta _n} } \quad \text {in }G_{\alpha ,\sigma }, \end{aligned}$$

(7.32)

for some sequence $(\phi _n)_{n=1}^\infty$ in $G_{\alpha ,\sigma }$.

By applying Theorem 6.3, we obtain the following result.

Proposition 7.11

Assume (7.32). Then any Leray–Hopf weak solution u(t) of the NSE (2.2) admits the asymptotic expansion

$$\begin{aligned} u(t) \underset{\Phi }{\sim } \sum _{n=1}^{\infty } \xi _n\psi _n(t)=\sum _{n=1}^{\infty } \frac{ \xi _n}{(t^\gamma +1)^{ \alpha _n}(t^{1-\gamma }+1)^{ \beta _n} } \quad \text {in }G_{\alpha +1-\rho ,\sigma }\text { for all } \rho \in (0,1), \end{aligned}$$

where

$$\begin{aligned} \xi _1=A^{-1}\phi _1,\quad \xi _n=A^{-1}\left( \phi _n + \sum _{ p=1}^{n-1} c_{p,n} \xi _p - \sum _{{\mathop {\lambda _k+\lambda _m=\lambda _n}\limits ^{1\le k,m\le n-1,}}}B(\xi _k,\xi _m)\right) \quad \text {for } n\ge 2, \end{aligned}$$

with $c_{p,n}$ being defined in (7.31).

Remark 7.12

This is a counterpart of Remark 5.3, but applied to the expansion (7.17).

Assume $({{\tilde{\alpha }}}_n)_{n=1}^\infty$ and $({{\tilde{\beta }}}_n)_{n=1}^\infty$ are sequences of non-negative numbers such that

• (a) Property (7.18) holds true,

• (b) For each $n\in {\mathbb {N}}$, the right-hand side of (7.18) is the unique decomposition among $\gamma {{\tilde{\alpha }}}_k+(1-\gamma ) {{\tilde{\beta }}}_j$ for $k,j\in {\mathbb {N}}$,

• (c) Each set $E_{*,1}=\{{{\tilde{\alpha }}}_n:n\in {\mathbb {N}}\}$, $E_{*,2}=\{{{\tilde{\beta }}}_n:n\in {\mathbb {N}}\}$ preserves the addition, and the increments by 1,

• (d) The set $E_*{\mathop {=}\limits ^\mathrm{def}}\{{{\tilde{\lambda }}}_n:n\in {\mathbb {N}}\}$ is equal to ${{\bar{E}}}{\mathop {=}\limits ^\mathrm{def}}\{\gamma {{\tilde{\alpha }}}_k+(1-\gamma ) {{\tilde{\beta }}}_j:k,j\in {\mathbb {N}}\}$.

Define $E = \Big \{\gamma ({{\tilde{\alpha }}}_n+k)+(1-\gamma )({{\tilde{\beta }}}_n+j):n\in {\mathbb {N}},\ k,j\in {\mathbb {N}}\cup \{0\}\Big \}$.

By the preservation of $E_{*,1}$ and $E_{*,2}$ in (c), we have ${{\tilde{\alpha }}}_n+k\in E_{*,1}$ and ${{\tilde{\beta }}}_n+k\in E_{*,2}$. Then $E_*\subset E\subset {{\bar{E}}}=E_*$, which implies $E_*=E$.

Let $S_*$ be defined by (7.19). We have $E_*\subset S_*\subset E=E_*$. Hence, $S_*=E_*$, which, by (7.18), is already ordered by $({{\tilde{\lambda }}}_n)_{n=1}^\infty$. By this and (5.4), (7.24), we have $\lambda _n={{\tilde{\lambda }}}_n$, $\alpha _n={{\tilde{\alpha }}}_n$, $\beta _n={{\tilde{\beta }}}_n$, and $\psi _n={{\tilde{\psi }}}_n$. Therefore, the expansion (7.32) is the original (7.17).

A Appendix

One way to generate an infinite expansion of the type (4.3) is to start with a function as a convergent series in (4.5). We give a criterion for such a conversion.

Lemma A.1

Let $\varphi$ be a positive function defined on $[T_*,\infty )$ for some $T_*\ge 0$, and $\varphi (t)\rightarrow 0$ as $t\rightarrow \infty$.

Let $(\lambda _n)_{n=1}^\infty$ be a strictly increasing, divergent sequence of positive numbers and there exists a number $M>1$ such that

$$\begin{aligned} \sum _{n=1}^\infty M^{-\lambda _n}<\infty . \end{aligned}$$

(A.1)

Let $(X, \Vert \cdot \Vert )$ be a Banach space and let $(\xi _n)_{n=1}^\infty \subset X$ satisfy

$$\begin{aligned} \Vert \xi _n\Vert \le c_0\kappa ^{\lambda _n} \quad \forall n\in {\mathbb {N}}, \end{aligned}$$

(A.2)

for some positive constants $c_0$ and $\kappa$.

Then the series $\sum _{n=1}^\infty \xi _n \varphi (t)^{\lambda _n}$ converges absolutely and uniformly to a function g(t) on $[T_0,\infty )$ for some $T_0 \ge T_*$, and g has the expansion

$$\begin{aligned} g(t){\mathop {\sim }\limits ^{\Psi }}\sum _{n=1}^\infty \xi _n \varphi (t)^{\lambda _n} \text { in } X,\text { where }\Psi =(\varphi ^\lambda )_{\lambda >0}. \end{aligned}$$

(A.3)

Proof

Since $\varphi (t) \rightarrow 0$ as $t\rightarrow \infty$, there exists $T_0\ge T_*$ such that for all $t \ge T_0$,

$$\begin{aligned} \varphi (t) \le \frac{1}{M\kappa }. \end{aligned}$$

(A.4)

Combining (A.2) and (A.4) yields, for all $n \in {\mathbb {N}}$,

$$\begin{aligned} \sup _{[T_0,\infty )}\Vert \xi _n \varphi (t)^{\lambda _n}\Vert \le c_0 M^{-\lambda _n}. \end{aligned}$$

This and (A.1) imply that $\sum _{n=1}^\infty \xi _n \varphi (t)^{\lambda _n}$ converges absolutely and uniformly on $[T_0,\infty )$, with $g(t)=\sum _{n=1}^\infty \xi _n \varphi (t)^{\lambda _n}$ being its limit function.

It remains to prove the expansion (A.3). We note, by the convergence of the series $\sum _{n=1}^\infty \xi _n \varphi (T_0)^{\lambda _n}$, that

$$\begin{aligned} \sup _{n\in {\mathbb {N}}} \Vert \xi _n\Vert \varphi (T_0)^{\lambda _n} = c_1<\infty , \end{aligned}$$

which implies

$$\begin{aligned} \Vert \xi _n\Vert \le \frac{c_1}{\varphi (T_0)^{\lambda _n}}\quad \forall n\in {\mathbb {N}}. \end{aligned}$$

(A.5)

Again, since $\varphi (t) \rightarrow 0$ as $t\rightarrow \infty$, there exists $T_1\ge T_0$ such that for all $t \ge T_1$,

$$\begin{aligned} \frac{\varphi (t)}{\varphi (T_0)} \le \frac{1}{M}. \end{aligned}$$

(A.6)

Using (A.5) and (A.6), we estimate, for all $t \ge T_1$,

$$\begin{aligned} \left\| g(t)-\sum _{n=1}^N \xi _n \varphi (t)^{\lambda _n}\right\|&=\left\| \sum _{n=N+1}^\infty \xi _n \varphi (t)^{\lambda _n}\right\| \\&\le c_1 \sum _{n=N+1}^\infty \frac{ \varphi (t)^{\lambda _n}}{\varphi (T_0)^{\lambda _n}} = c_1 \,\frac{ \varphi (t)^{\lambda _{N+1}}}{\varphi (T_0)^{\lambda _{N+1}}} \sum _{n=N+1}^\infty \left( \frac{ \varphi (t)}{\varphi (T_0)}\right) ^{\lambda _n-\lambda _{N+1}}\\&\le c_1 \,\frac{ \varphi (t)^{\lambda _{N+1}}}{\varphi (T_0)^{\lambda _{N+1}}} \sum _{n=N+1}^\infty M^{-\lambda _n+\lambda _{N+1}} \le C_N\varphi (t)^{\lambda _{N+1}}, \end{aligned}$$

where $C_N=c_1 (M/\varphi (T_0))^{\lambda _{N+1}} \sum _{n=1}^\infty M^{-\lambda _n}<\infty$. Therefore, we obtain the expansion (A.3), according to Definition 4.3 with $\psi _\lambda =\varphi ^\lambda$. $\square$

We extend Lemma A.1 to cover the expansions with a background system such as those in Sect. 6.

Lemma A.2

Let $\varphi (t)$ and $\psi _n(t)$, for $n\in {\mathbb {N}}$, be positive functions defined on $[T_*,\infty )$ for some $T_*\ge 0$ that tend to zero as $t\rightarrow \infty$. Assume, for each $n\in {\mathbb {N}}$, there exist a numbers $D_n\ge 1$ such that

$$\begin{aligned} D_n^{-1}\varphi (t)^{\lambda _n}\le \psi _n(t) \le D_n\varphi (t)^{\lambda _n}\quad \forall t\ge T_*, \end{aligned}$$

(A.7)

where $(\lambda _n)_{n=1}^\infty$ is a sequence of positive numbers and $\lambda _n\rightarrow \infty$ as $n\rightarrow \infty$. Assume further that there exists $M>0$ such that

$$\begin{aligned} \sum _{n=1}^\infty D_n M^{-\lambda _n}<\infty . \end{aligned}$$

(A.8)

Let $(X,\Vert \cdot \Vert )$ be a Banach space, and $(\xi _n)_{n=1}^\infty$ be a sequence in X such that (A.2) holds.

(i) Then the series $\sum _{n=1}^\infty \xi _n \psi _n(t)$ converges absolutely and uniformly on $[T_0,\infty )$ for some $T_0\ge T_*$. Define

$$\begin{aligned} f(t)=\sum _{n=1}^\infty \xi _n \psi _n(t)\in X \quad \forall t\ge T_0. \end{aligned}$$

(A.9)

(ii) Assume the mapping $n\mapsto \lambda _n$ is one-to-one. Let $(\mu _n)_{n=1}^\infty$ be the strictly increasing re-arrangement of $(\lambda _n)_{n=1}^\infty$. Define $\psi _n^*=\psi _k$ and $\xi _n^*=\xi _k$ with $\mu _n=\lambda _k$. Let $\Phi =(\varphi (t)^\lambda )_{\lambda >0}$. Then

$$\begin{aligned} f(t)\underset{\Phi }{\sim }\sum _{n=1}^\infty \xi _n^* \psi _n^*(t). \end{aligned}$$

(A.10)

Proof

• (i) There is $T_0\ge T_*$ such that $\varphi (t)\le 1/\kappa M$ for all $t\ge T_0$. Then for all $t\ge T_0$, by (A.2), (A.7), and (A.8),

  $$\begin{aligned} \sum _{n=1}^\infty \Vert \xi _n\Vert \psi _n(t)&\le \sum _{n=1}^\infty \Vert \xi _n\Vert D_n\varphi (t)^{\lambda _n} \le \sum _{n=1}^\infty c_0 D_n M^{-\lambda _n}<\infty . \end{aligned}$$

  Therefore, we obtain the absolute and uniform convergence on $[T_0,\infty )$.

• (ii) Let $D^*_n=D_k$ with $\mu _n=\lambda _k$. After the re-arrangement we still have

  $$\begin{aligned} \sum _{n=1}^\infty D_n^* M^{-\mu _n}=\sum _{k=1}^\infty D_k M^{-\lambda _k}<\infty , \end{aligned}$$

  (A.11)

and, because of the absolute convergence,

$$\begin{aligned} f(t)=\sum _{n=1}^\infty \xi _n^* \psi _n^*(t) \text { for }t\ge T_0. \end{aligned}$$

(A.12)

For each $n\in {\mathbb {N}}$, let $k\in {\mathbb {N}}$ such that $\mu _n=\lambda _k$, then we have

$$\begin{aligned} (D_n^*)^{-1}\varphi ^{\mu _n}=D_k^{-1}\varphi ^{\lambda _k}\le \psi _n^*=\psi _k\le D_k\varphi ^{\lambda _k}=D^*_n\varphi ^{\mu _n}. \end{aligned}$$

The convergence of $f(T_0)$ in (A.12) implies that there exists $c_1>0$ such that

$$\begin{aligned} \Vert \xi _n^*\Vert \le c_1 \psi _n^*(T_0)^{-1}\le c_1 D_n^* \varphi (T_0)^{-\mu _n}\quad \forall n\in {\mathbb {N}}. \end{aligned}$$

Let $T_1\ge T_0$ such that $\varphi (t)/\varphi (T_0)\le 1/M^2$ for all $t\ge T_1$. Then, for $t\ge T_1$,

$$\begin{aligned} \left\| f(t)-\sum _{n=1}^N \xi _n^* \psi _n^*(t)\right\|&=\left\| \sum _{n=N+1}^\infty \xi _n^* \psi _n^*(t)\right\| \le \sum _{n=N+1}^\infty c_1 D_n^* \varphi (T_0)^{-\mu _n}\cdot D_n^*\varphi (t)^{\mu _n}\\&\le c_1 (\varphi (t)/\varphi (T_0))^{\mu _{N+1}}\sum _{n=N+1}^\infty (D_n^*)^2(\varphi (t)/\varphi (T_0))^{\mu _n-\mu _{N+1}}\\&\le c_1 (\varphi (t)/\varphi (T_0))^{\mu _{N+1}}\sum _{n=N+1}^\infty (D_n^*)^2M^{-2\mu _n+2\mu _{N+1}}\\&\le c_1 (M^2\varphi (t)/\varphi (T_0))^{\mu _{N+1}}\left( \sum _{n=N+1}^\infty D_n^* M^{-\mu _n}\right) ^2 \le C\varphi (t)^{\mu _{N+1}}, \end{aligned}$$

where thanks to (A.11), C is a positive number. Since $\mu _{N+1}>\mu _N$, we obtain (A.10). $\square$

We emphasize that the sum in (A.9) must be re-arranged to have a meaningful expansion as in (A.10). In particular cases, Lemma A.2 is used to obtain expansions when $\psi _n$ is generated by two functions with two different sequences of powers.

Corollary A.3

Suppose $\zeta (t)$, $\vartheta (t)$ and $\varphi (t)$ are three positive functions defined on $[T_*,\infty )$ for some $T_*\ge 0$ that tend to zero as $t\rightarrow \infty$, and there exist numbers $D\ge 1$, $s_1,s_2>0$ such that

$$\begin{aligned} D^{-1}\varphi (t)^{s_1}\le \zeta (t)\le D\varphi (t)^{s_1},\quad D^{-1}\varphi (t)^{s_2}\le \vartheta (t)\le D\varphi (t)^{s_2} \quad \forall t\ge T_*. \end{aligned}$$

(A.13)

Let $(\alpha _n)_{n=1}^\infty$ and $(\beta _n)_{n=1}^\infty$ be two sequences of non-negative numbers such that $\alpha _n+\beta _n\rightarrow \infty$.

Define $\lambda _n=s_1 \alpha _n+s_2\beta _n$ for $n\in {\mathbb {N}}$. Let $(X,\Vert \cdot \Vert )$ be a Banach space, and $(\xi _n)_{n=1}^\infty$ be a sequence in X. Assume (A.1) and (A.2).

(i) Then the series $\sum _{n=1}^\infty \xi _n \zeta (t)^{\alpha _n}\vartheta (t)^{\beta _n}$ converges absolutely and uniformly on $[T_0,\infty )$ for some $T_0\ge T_*$. Define

$$\begin{aligned} f(t)=\sum _{n=1}^\infty \xi _n \zeta (t)^{\alpha _n}\vartheta (t)^{\beta _n} \quad \forall t\ge T_0. \end{aligned}$$

(ii) Suppose the mapping $n\mapsto \lambda _n$ is one-to-one. Let $(\mu _n)_{n=1}^\infty$ be the strictly increasing re-arrangement of $(\lambda _n)_{n=1}^\infty$. Define $\psi _n^*=\zeta ^{\alpha _k}\vartheta ^{\beta _k}$ and $\xi _n^*=\xi _k$ with $\mu _n=\lambda _k$.

Let $\Phi =(\varphi (t)^{-\lambda })_{\lambda >0}$. Then

$$\begin{aligned} f(t)\underset{\Phi }{\sim }\sum _{n=1}^\infty \xi _n^* \psi _n^*(t). \end{aligned}$$

Proof

Let $\psi _n(t)=\zeta (t)^{\alpha _n}\vartheta (t)^{\beta _n}$ and $D_n=D^{\alpha _n+\beta _n}$. Thanks to (A.13), we have

$$\begin{aligned} D_n^{-1}\varphi ^{\lambda _n}=D^{-(\alpha _n+\beta _n)}\varphi ^{\lambda _n}\le \psi _n\le D^{\alpha _n+\beta _n}\varphi ^{\lambda _n}=D_n\varphi ^{\lambda _n}. \end{aligned}$$

Note that $\lambda _n\rightarrow \infty$. Denote $s=\max \{1/s_1,1/s_2\}$. Then

$$\begin{aligned} \sum _{n=1}^\infty D_n (M D^s)^{-\lambda _n}= \sum _{n=1}^\infty M^{-\lambda _n} D^{(1- s_1 s)\alpha _n+(1- s_2 s )\beta _n} \le \sum _{n=1}^\infty M^{-\lambda _n} <\infty . \end{aligned}$$

Hence, (A.8) holds true with $M:=MD^s$. Applying Lemma A.2, we obtain the desired statements in (i) and (ii). $\square$