Abstract
We study global well-posedness for the inhomogeneous Navier–Stokes equations on \({\mathbb {R}}^n\), \(n\ge 2\), with initial velocity in endpoint critical Besov spaces \(B^{-1+n/q}_{q,\infty }({\mathbb {R}}^n)\), \(n\le q<2n\), and merely bounded initial density with a positive lower bound. First, we consider a multiplication property of \(L^\infty \)-functions in some Bessel potential and Besov spaces. Based on it and on maximal \(L^\infty _\gamma \)-regularity of the Stokes operator in little Nicolskii spaces, we show solvability for the momentum equations with fixed bounded density. Finally, proof for existence of a solution to the inhomogeneous Navier–Stokes equations is done via an iterative scheme when \(B^{-1+n/q}_{q,\infty }\)-norm of initial velocity and relative variation of initial density are small, while uniqueness of a solution is proved via a Lagrangian approach when initial velocity belongs to \(B^{-1+n/q}_{q,\infty }({\mathbb {R}}^n)\cap B^{-1+n/q}_{r,\infty }({\mathbb {R}}^n)\) for slightly larger \(r>q\).
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1 Introduction and Main Result
Consider the inhomogeneous Navier–Stokes equations
where \(0<T\le \infty \), \(n\ge 2\), \(\mu \) is the dynamic viscosity, and \(\rho \), u, P are density, velocity and pressure of an incompressible flow, respectively. The system (1.1) describes the motion of viscous incompressible flows with variable density and its global well-posedness has been widely studied, (cf. [1,2,3, 7,8,9,10,11,12,13,14,15,16,17, 19] and the references therein).
Writing \(a:=\frac{{\bar{\rho }}-\rho }{\rho }\) with a suitable constant \({\bar{\rho }}>0\), the system (1.1) is reduced to the following equivalent system:
where and in what follows
The equations of (1.2) are invariant under the scaling
and hence it is important to show the existence of solutions to (1.2) in critical spaces, i.e., the function spaces with norms invariant under the scaling (1.3).
In [7], Danchin proved that if \((a_0, u_0)\) belongs to the critical spaces \(({\dot{B}}^{n/2}_{2,r}({{\mathbb {R}}}^n)\cap L^\infty ({{\mathbb {R}}}^n))\times {\dot{B}}^{n/2-1}_{2,1}({{\mathbb {R}}}^n)\) (\(r\in [1,\infty ]\) for \(n=3\) and \(r=1\) for \(n=2\)) and \(\Vert a_0\Vert _{{\dot{B}}^{n/2}_{2,\infty }}+\Vert u_0\Vert _{{\dot{B}}^{n/2-1}_{2,1}}\) is small enough, then (1.2) has a unique solution \((\rho ,u)\) such that
This result was generalized by Abidi [1] to the case where the spaces for (a, u) are extended to \({\dot{B}}^{n/q}_{q,1}({{\mathbb {R}}}^n)\times {\dot{B}}^{-1+n/q}_{q,1}({{\mathbb {R}}}^n)\), \(1<q<2n\), showing existence for \(1<q<2n\) and uniqueness for \(1<q\le n\); the gap in the uniqueness for \(n<q<2n\) was filled by Danchin and Mucha [10] via Lagrangian approach. Abidi, Gui and Zhang in [2] proved global well-posedness of (1.1) without smallness of initial density variation \(a_0\in B^{3/2}_{2,1}({{\mathbb {R}}}^3)\) for \(u_0\in {\dot{B}}^{1/2}_{2,1}({{\mathbb {R}}}^3)\) whose norm is small depending on \(\Vert a_0\Vert _{B^{3/2}_{2,1}({{\mathbb {R}}}^3)}\); the result was extended in [3] to the case \((a_0,u_0)\in B^{3/q}_{q,1}({{\mathbb {R}}}^3)\times {\dot{B}}^{-1+3/p}_{p,1}({{\mathbb {R}}}^3)\), \(q\in [1,2]\), \(p\in [3,4]\), \(\frac{1}{p}+\frac{1}{q}>\frac{5}{6}\), \(\frac{1}{q}-\frac{1}{p}\le \frac{1}{3}\). Recently, in [23], global well-posedness of (1.1) with \((a_0,u_0)\in B^{3/q}_{q,1}({{\mathbb {R}}}^3)\times {\dot{B}}^{-1+3/p}_{p,1}({{\mathbb {R}}}^3)\), \(1<q\le p<6\), \(\frac{1}{p}+\frac{1}{q}>\frac{1}{2}\), \(\frac{1}{q}-\frac{1}{p}\le \frac{1}{3}\), was proved under the condition that \(\Vert a_0\Vert _{BMO}\) and \(\Vert u_0\Vert _{{\dot{B}}^{-1+3/p}_{p,1}({{\mathbb {R}}}^3)}\) are small enough. Here, we note that consideration of the system (1.1) or (1.2) with initial velocity in a type of critical Besov spaces \({\dot{B}}^s_{q,1}\) is mathematically ideal for showing both existence and uniqueness of a solution in the same class of functions, since in this case the velocity can be found in \(L^1(0,T; C_{Lip}({{\mathbb {R}}}^n))\), where \(C_{Lip}({{\mathbb {R}}}^n)\) denotes the set of all Lipschitz continuous functions on \({{\mathbb {R}}}^n\), thanks to the maximal \(L^1\)-regularity property of the Stokes operator in \({\dot{B}}^s_{q,1}\)-type Besov spaces.
On the other hand, it is a common interest to study (1.2) (equivalently (1.1)) when the velocity does not belong to \(L^1(0,T; C_{Lip}({\mathbb {R}}^n))\) and initial density is merely bounded, cf. [12, 14]. Huang et al. [14] proved existence of a global solution to (1.2) under a smallness condition of \(a_0\in L^\infty ({{\mathbb {R}}}^n)\) and \(u_0\in {\dot{B}}^{-1+n/q}_{q,r}({{\mathbb {R}}}^n)\), \(q\in (1,n), r\in (1,\infty )\), and uniqueness of such solution under a slightly higher regularity assumption on initial velocity \(u_0\); this result is extended to the half-space setting by Danchin and Zhang [12]. The main ideas of [14] and [12] are to employ, for existence, maximal \(L^p\)-regularity for the Stokes operator in Lebesgue spaces and, for uniqueness, a Lagrangian approach which was exploited in [10].
Global well-posedness for initial boundary value problem corresponding to (1.1) on bounded domains \(\Omega \subset {{\mathbb {R}}}^n\) with \(\rho _0\in L^\infty (\Omega )\) was considered in [15] and [11] for some regular and small initial velocity in Sobolev and Besov spaces, while in [19] for small piecewise constant \(a_0\) and \(u_0\in B_{q,\infty }^0(\Omega )\), \(q\ge n\ge 2\).
In this paper, we prove global well-posedness of (1.1) for \(u_0\in B^{-1+n/q}_{q,\infty }({{\mathbb {R}}}^n)\), \(n\le q<2n\), and \(\rho _0\in L^\infty ({{\mathbb {R}}}^n)\) positive away from 0 provided the corresponding norms of initial velocity and initial density variation is suitably small. For uniqueness of a solution we require a slightly higher regularity of the initial velocity.
Before introducing the main result of the paper, we need to give some notations. For a linear normed space X the notation \(X'\) stands for the dual space of X. We always denote the conjugate number of \(q\in (1,\infty )\) by \(q'\), i.e. \(q'=q/(q-1)\). Let \([\cdot ,\cdot ]_\theta \), \((\cdot ,\cdot )_{\theta ,r}\) and \((\cdot ,\cdot )^0_{\theta ,\infty }\) for \(\theta \in (0,1)\), \(1\le r\le \infty \) be complex, real and continuous interpolation functors, respectively, see [5, 21] for real and complex interpolation functors, and see e.g. [4], §§2.4.4, §2.5, [21] §§1.11.2, page 69 for continuous interpolation functors. We use standard notation \(L^q, H^s_q, B^s_{q,r}\) for Lebesgue spaces, Bessel potential spaces and Besov spaces, respectively, without distinguishing whether or not it is the space of scalar-valued functions or vector-valued functions. For \(q\in (1,\infty )\) and \(s\in {{\mathbb {R}}}\) let \(b^{s}_{q,\infty }({{\mathbb {R}}}^n)\) be the little Nicolskii space defined by the completion of \(H^s_q({{\mathbb {R}}}^n)\) in \(B^s_{q,\infty }({{\mathbb {R}}}^n)\).
Given \(\gamma \in (0,1]\), \(0<T\le \infty \) and Banach space X, let
The space of all pointwise multipliers in Y is denoted by \({{\mathcal {M}}}(Y)\), i.e.,
The characteristic function for a subset G of \({{\mathbb {R}}}^n\) is denoted by \(\chi _G\). We denote the tensor product of two tensors a, b by \(a\otimes b\) and by \(A:B=\sum _{i,j}a_{ij}b_{ij}\) for two matrices \(A=(a_{ij})_{1\le i,j\le n}\) and \(B=(b_{ij})_{1\le i,j\le n}\).
Definition 1.1
Let \(2\le n\le q<2n\), \(0<T\le \infty \) and let \(s\in (0,1)\). We say that a pair of functions \((\rho , u)\) is a solution to (1.1) if it satisfies the followings:
-
(i)
$$\begin{aligned} \rho \in L^\infty (0,T; L^\infty ({{\mathbb {R}}}^n)),\; u\in L^\infty _{loc}([0,T), B^{1+n/q-s}_{q,\infty }({{\mathbb {R}}}^n)),\quad \text {div}\,u=0. \end{aligned}$$(1.4)
-
(ii)
Two identities
$$\begin{aligned} \int _0^T\int _{{{\mathbb {R}}}^n}(\rho \psi _t+\rho u\cdot \nabla \psi )\,dxdt +\int _{{{\mathbb {R}}}^n} \rho _0\psi (0,\cdot )\,dx=0,\; \forall \psi \in C_0^1([0,T)\times {{\mathbb {R}}}^n), \end{aligned}$$(1.5)and
$$\begin{aligned}&\displaystyle \int _0^T\int _{{{\mathbb {R}}}^n}[\rho u\cdot \varphi _t+\mu u\cdot \Delta \varphi +\rho u\otimes u:\nabla \varphi ]\,dxdt +\int _{{{\mathbb {R}}}^n} \rho _0u_0\cdot \varphi (0,\cdot )\,dx=0,\nonumber \\&\quad \forall \varphi \in C_0^\infty ([0,T)\times {{\mathbb {R}}}^n)^n\; (\text {div}\,\varphi =0), \end{aligned}$$(1.6)hold true.
Note that if \((\rho , u)\) satisfies (1.4), the integrals of (1.5) and (1.6) make a sense due to \(u\in L^\infty _{loc}([0,T)\times {{\mathbb {R}}}^n)\) by Sobolev embedding. If \((\rho , u)\) is a solution to (1.1) in the sense of Definition 1.1, it follows by a standard argument using De-Rham’s lemma that there is a distribution P, associated pressure, such that \(\rho , u\) and P satisfy the equations of (1.1) in the sense of distribution and the initial conditions in (1.1) are satisfied in a weak sense. Hence, if necessary in the below, the triple \((\rho , u, \nabla P)\) will also be called a solution to (1.1).
The main result of the paper is stated as follows:
Theorem 1.1
Let \(0<T\le \infty \) and let \( 2\le n\le q<2 n\). Let
with positive constants \(\rho _{0i},i=1,2\), \(u_0\in B^{-1+n/q}_{q,\infty }({{\mathbb {R}}}^n)\cap B^{-1+n/q}_{r,\infty }({{\mathbb {R}}}^n)\), \(r\ge q\), and \(\text {div}\,u_0=0\). Then, for any \(s\in (0,\min \{\frac{n-1}{q},\frac{2n}{q}-1\})\) there are some constants \(\delta _i=\delta _i(q,n,s)>0, i=1,2,\) independent of T, such that if
then (1.1) has a solution \((\rho ,u,\nabla P)\) satisfying (1.4)–(1.6) and, in addition,
The solution \((\rho ,u,\nabla P)\) is unique in the class of functions satisfying (1.4) and (1.8) with \(r>q\) and \(s<\frac{n}{q}-\frac{n}{r}\).
Remark 1.2
By Theorem 1.1, in particular, we have an existence result when \(u_0\in B^{-1+n/q}_{q,\infty }({{\mathbb {R}}}^n)\), while uniqueness result when \(u_0\in B^{-1+n/q+\varepsilon }_{q,\infty }({{\mathbb {R}}}^n)\) for any small \(\varepsilon >0\).
The proof of Theorem 1.1 has two aspects, i.e., existence part and uniqueness part for a solution to the system (1.1).
The existence of a solution is proved via an iterative scheme for the momentum equations with fixed density and the transfer equation with fixed velocity. First, we need to consider solvability for momentum equations, that is,
with \(a\equiv \frac{{\bar{\rho }}}{\rho }-1, {\bar{\rho }}\in [\rho _0,\rho _1],\) fixed; to this end, we rely on maximal \(L^\infty _\gamma \)-regularity of the Stokes operator. It is known that the unsteady Stokes system
with \(v_0\in B^{\alpha +\gamma }_{q,\infty }({{\mathbb {R}}}^n)\) and \(f\in L^\infty _\gamma (0,T; b^{2\alpha }_{q,\infty }({{\mathbb {R}}}^n))\) for \(\gamma \in (0,1)\), \(\alpha \in (-\frac{1}{2},1)\), \(2\alpha -1/q\notin {{\mathbb {Z}}}\), \(\alpha +\gamma <1\), has a unique solution such that \(\nabla ^2 v,\nabla p\in L^\infty _\gamma (0,T; b^{2\alpha }_{q,\infty }({{\mathbb {R}}}^n))\), see [18]. In order to apply this property to solvability for (1.9) using a fixed point argument, the term \(a(\nu \Delta u -\nabla p)\) should belong to \(L^\infty _\gamma (0,T; b^{2\alpha }_{q,\infty }({{\mathbb {R}}}^n))\). For this reason, we prove a multiplication property of \(L^\infty \)-functions in little Nicolskii spaces \(b^{\tau }_{q,\infty }, \tau \in (-1+1/q,1/q),\) (also in corresponding Bessel potential and Besov spaces), which is of independent significance. Then, applying the above maximal \(L^\infty _\gamma \)-regularity result for the Stokes operator with \(\gamma =s/2,\alpha =-1/2+n/(2q)-s/2\) for sufficiently small \(s>0\), we obtain solvability for (1.9). Finally, the existence of a solution to (1.1) is proved via an iterative scheme where the density transport equation involves the velocity as a regularization of the solution to (1.9) since it does not belong to \(L^1(0,T; C_{Lip}({{\mathbb {R}}}^n))\), generally.
The uniqueness part of Theorem 1.1 is proved by Lagrangian approach similarly as in [10, 12] and [14], but Theorem 3.3 on the unsteady Stokes system with nonzero divergence and Lemma 5.1 on pointwise multiplication in Besov spaces are essentially used.
The remaining part of the paper is organized as follows. In Sect. 2 we show a multiplication property of \(L^\infty \)-functions in Bessel potential, Besov and little Nicolskii spaces. Section 3 is devoted to solvability for momentum equations involved in (1.1) with fixed variable density. The proof for the existence part of Theorem 1.1 is given in Sect. 4, while the uniqueness part in Sect. 5.
Throughout the paper, we denote the estimate constants appearing in inequalities by the same symbol c or C as long as no confusion arises.
2 A Multiplication Property of \(L^\infty \)-Functions
Lemma 2.1
Let \(\Omega \) be a domain of \({{\mathbb {R}}}^n\), \(n\in {{\mathbb {N}}}\), and let \(1<q<\infty \) and \(s\in (-1+1/q,1/q)\). Then, for any open subset \(\Omega '\) of \(\Omega \)
where the space \(H^{s}_{q}(\Omega )\) is endowed with the norm of the complex interpolation space \([H^{-1}_{q}(\Omega ), H^1_{q,0}(\Omega )]_{\frac{s+1}{2}}\).
Proof
First, suppose that \(s\in (0,1/q)\). Define the operator \(E_0: L^q(\Omega ')\mapsto L^q(\Omega )\) by \(E_0f:= {\tilde{f}}\), where \({\tilde{f}}\) is the extension of f by zero on \(\Omega \setminus \Omega '\). Then, obviously,
where and in what follows, \(H^1_{q,0}(G)\) for \(1<q<\infty \), denotes the closures of \(C^\infty _0(G)\) in \(H^1_q(G)\). Hence, by complex interpolation \([L^q,H^1_{q,0}]_s=H_q^s\) for \(s\in (0,1/q)\), we have
where note that the complex interpolation functor is an exact interpolation functor, cf. [5].
On the other hand, for \(r_{\Omega '}\) being the restriction operator onto \(\Omega '\subset \Omega \), we have
Therefore, since \(\chi _{\Omega '}f=E_0r_{\Omega '}f\), we get
The assertion of the lemma for the case \(s\in (-1+1/q,0)\) follows by duality argument using the assertion for \(s\in (0,1/q)\).
Finally, the assertion for the case \(s=0\) directly follows by interpolation. \(\square \)
Lemma 2.2
Let \(\Omega \) be a domain of \({{\mathbb {R}}}^n\), \(n\in {{\mathbb {N}}}\), and let \(M\subset L^\infty (\Omega )\) be the set of all finite or countable summations \(g(x)=\sum _{k}a_k\chi _{\Omega _k}(x)\) such that \(a_k\in {{\mathbb {R}}}\) and \(\Omega _k\) ’s are disjoint Lipschitz subdomains of \(\Omega \) .
Proof
Denote the Lebesgue measure by m. It is known that for every \(f\in (L^\infty (\Omega ))'\) there is a unique \(m-\)absolutely continuous finitely additive set function \(\mu _f\) defined on m-measurable subsets of \(\Omega \) such that
see [22], p.118.
Now, let \(f\in (L^\infty (\Omega ))'\) be such that \(\langle f, g\rangle _{(L^\infty (\Omega ))', L^\infty (\Omega )}=0\) for all \(g\in M\). Note that for the union \(\Pi =\bigcup _j Q_j\) of any finite or countable number of disjoint (open) cubes \(Q_j\subset \Omega \) which are parallel to the coordinate axis of \({{\mathbb {R}}}^n\) one has \(\chi _{\Pi }\in M\) and hence
Moreover, since \(\mu _f\) is \(m-\)absolutely continuous, for every \(\varepsilon >0\) there is \(\delta =\delta (\varepsilon )>0\) such that
Let G be any \(m-\)measurable subset of \(\Omega \). Then, there is a finite or countable number of disjoint cubes \(Q_j\subset \Omega \) such that
holds for \(\Pi =\cup _j Q_j\), where “\(\ominus \)” denotes symmetric difference of two sets. In fact, if \(m(G)<\infty \), then, by the definition of the Lebesgue measure, (2.5) obviously holds with some finite disjoint cubes \(Q_j\). Let \(m(G)=\infty \). Then, by \(\sigma \)-finiteness of Lebesgue measure, G can be expressed as a union \(G=\cup _{k=1}^\infty G_k\) of m-measurable sets \(G_k\) with \(m(G_k)<\infty \), where without loss of generality \(G_k\)’s can be regarded as disjoint with each other. For each \(G_k\) one can find \(\Pi _k\subset \Omega \) which is the union of a finite number of disjoint cubes and satisfies \(m(G_k\,\ominus \,\Pi _k)<2^{-k}\delta \). Then, \(\Pi \equiv \cup _{k=1}^\infty \Pi _k\) satisfies (2.5), where \(\Pi _k\)’s also can be regarded as disjoint with each other.
Thus we have \(\mu _f(G)=0\), since \(\varepsilon >0\) is arbitrarily taken.
Therefore, by (2.2) we have \( \langle f, g\rangle _{(L^\infty (\Omega ))', L^\infty (\Omega )}=0\) for all \(g\in L^\infty (\Omega )\), and hence \(f=0\). Thus M is dense in \(L^\infty (\Omega )\). \(\square \)
Proposition 2.3
Let \(\Omega \) be a domain of \({{\mathbb {R}}}^n,n\in {{\mathbb {N}}}\), and let \(q\in (1,\infty ), r\in [1,\infty ]\) and \(s\in (-1+1/q,1/q)\). Then, every \(f\in L^\infty (\Omega )\) is a pointwise multiplier of \(H^s_{q}(\Omega ), B^s_{q,r}(\Omega )\) and \(b^s_{q,\infty }(\Omega )\), respectively, and
where \(Y=H^s_{q}(\Omega ), B^s_{q,r}(\Omega )\) and \(b^s_{q,\infty }(\Omega )\) and the constant \(C>0\) depends only on q, r, s and \(\Omega \).
Proof
Let \(f\in L^\infty (\Omega )\) and \(f_k=\sum _i a_{k_i}\chi _{\Omega _{k_i}}, k=1,2,\ldots ,\) be a sequence converging to f in \(L^\infty (\Omega )\) as \(k\rightarrow \infty \), where summation is finite or countable, \(a_{k_i}\in {{\mathbb {R}}}\), and \(\Omega _{k_i}\subset \Omega \) are disjoint Lipschitz subdomains. Note that
Let the operator \(E_0: L^q(\cup _i\Omega _{k_i})\mapsto L^q(\Omega )\) by \(E_0f:= {\tilde{f}}\), where \({\tilde{f}}\) is the extension of f by zero on \(\Omega \setminus \cup _i\Omega _{k_i}\). We have
On the other hand, it follows that if \(\varphi \in H^1_{q,0}(\cup _{i}\Omega _{k_i})\), then
In fact, since \(\Omega _{k_i}\)’s are disjoint domains, we have
Moreover,
where \(E_{i0}\) is an operator of extension by 0 from \(\Omega _{ki}\) onto \(\Omega \); since \(\Omega _{k_i}\) is a Lipschitz domain, we have
Hence, if \(\cup _i\Omega _{k_i}\) is finite union, then, obviously, \(E_0\varphi \in H^1_{q,0}(\Omega )\) and, by (2.9) and (2.11), we have
If \(\cup _i\Omega _{k_i}\) is infinite countable union, then, in view of (2.9),
Hence, it follows by (2.10) that \(E_0\varphi \in H^1_{q,0}(\Omega )\) and
Thus, (2.8) is proved.
For all \( v\in H^1_{q,0}(\cup _i\Omega _{k_i})\), it follows that \(f_k v=\sum _i a_{k_i}\chi _{\Omega _{k_i}}v\in H^1_{q,0}(\cup _{i}\Omega _{k_i})\) and
Hence, by (2.8),
Now, for the moment let us assume that the space \(H^s_q\) for \(s\in (0,1/q)\) is endowed with norm of the complex interpolation space \([H_q^{-1}, H^1_{q,0}]_{\frac{s+1}{2}}\). Then, since the complex interpolation functor is an exact interpolation functor, cf. [5], it follows by (2.7) and (2.12) that
On the other hand, denoting the restriction operator onto \(\cup _i\Omega _{k_i}\) by R, we have
Hence, we get from (2.13), (2.14) that
and, in particular,
Moreover, by (2.15), we have
Thus \(\{f_k\}\) becomes a Cauchy sequence in \({{\mathcal {M}}}(H^s_{q}(\Omega ))\) as well, and \(f_k\) converges to some \({\tilde{f}}\in {\mathcal M}(H^s_{q}(\Omega ))\) as \(k\rightarrow \infty \). Then we have \(f={\tilde{f}}\) since \(f_k\) converges to both f and \({\tilde{f}}\) in the Banach space \({{\mathcal {M}}}(H^s_{q}(\Omega ))+L^\infty (\Omega )\) as \(k\rightarrow \infty \), where note that both the space \({{\mathcal {M}}}(H^s_{q}(\Omega ))\) and \(L^\infty (\Omega )\) are contained in the distribution spaces \({\mathcal D}'(\Omega )\) and hence the sum of the two spaces may be defined. Thus, we have \(L^\infty (\Omega )\hookrightarrow {{\mathcal {M}}}(H^s_{q}(\Omega ))\) and, by (2.15),
yielding
Thus, without the assumption that \(H^s_q(\Omega )\) is endowed with the norm of the complex interpolation space \([H_q^{-1}(\Omega ), H^1_{q,0}(\Omega )]_{\frac{s+1}{2}}\), one generally has
It then directly follows from (2.16) by standard duality argument that
Thus, the assertion of the lemma is proved for \(Y=H^s_q(\Omega )\).
The remaining assertions for \(Y=B^s_{q,r}(\Omega ), 1\le r\le \infty ,\) and \(b^s_{q,\infty }(\Omega )\) follows directly from (2.16) and (2.17) by standard argument using real and continuous interpolation, respectively.
The proof of the lemma is complete. \(\square \)
Remark 2.4
From the proof of Proposition 2.3, it is easy to see that if the Banach spaces \(Y=H^s_{q}(\Omega ), B^s_{q,r}(\Omega )\) and \(b^s_{q,\infty }(\Omega )\) for \(s\in (-1+1/q,1/q)\) are endowed with the norm of the interpolation spaces between \(H^{-1}_q(\Omega )\) and \(H^{1}_{q,0}(\Omega )\) by the complex interpolation functor, real and continuous interpolation functors by K-method, which are exact, respectively, (cf. [5]), then the constant C in (2.6) may be regarded as 1.
3 Solvability for Momentum Equations with Fixed Density
Let \(\rho \in L^\infty (0,T;L^\infty (\Omega ))\) be fixed such that \(\rho _{01}\le \rho (t,x)\le \rho _{02}\) with some positive constants \(\rho _{01},\rho _{02}\) and let \(\frac{1}{\rho }=\frac{1}{{\bar{\rho }}}(1+a)\) with a constant \({\bar{\rho }}\in [\rho _{01}, \rho _{02}]\) and \(a=a(t,x)\in L^\infty ((0,T)\times {{\mathbb {R}}}^n)\). Then the momentum equations in (1.1) may be formally written as
Beforehand, we consider the unsteady Stokes equations with nonzero divergence.
3.1 Unsteady Stokes Equations
Consider the Cauchy problem for the unsteady Stokes equations
where \(n\ge 2\) and \(0<T\le \infty \).
Lemma 3.1
There is a unique solution operator \({{\mathcal {R}}}\) for the problem
such that \(u={{\mathcal {R}}}g\) for \(g\in {{\mathcal {S}}'}\) becomes a solution to (3.3) in the sense of distribution and for \(q\in (1, \infty )\) and \(\alpha \in {{\mathbb {R}}}\)
where the constant \(c>0\) is independent of \(q,\nu \) and \(\alpha \), and \(F_q^\alpha \in \{H^\alpha _q, B^\alpha _{q,r} (1\le r\le \infty ), b^\alpha _{q,\infty }\}\).
Proof
It is formally checked that \(u=\Delta ^{-1}\nabla g\) and \(\nabla p=\nu \nabla g\) uniquely solve (3.3). Defining \({{\mathcal {R}}}\) in the space of tempered distributions \({{\mathcal {S}}'}({{\mathbb {R}}}^n)\) by
then we have
with constant \(c>0\) independent of q, \(\nu \) and \(\alpha \), which follows directly by the classical Fourier multiplier theorem. In particular, from (3.5) we have
Thus, (3.4) follows from (3.6) by real and continuous interpolation in view of
for \(\alpha =(1-\theta )\alpha _1+\theta \alpha _2, \alpha _1, \alpha _2\in {{\mathbb {R}}},\theta \in (0,1)\) and \(r\in [1,\infty ]\).
The proof is complete. \(\square \)
Lemma 3.2
Let \(0<T\le \infty \), \(\gamma \in (0,1)\) and \(\alpha \le 0\). If \(w_t,\nabla ^2 w\in L^\infty _\gamma (0,T; b^{2\alpha }_{q,\infty }({{\mathbb {R}}}^n)\), \(1<q<\infty \), then \(w\in L^\infty _{1-\theta +\gamma }(0,T; X)\) for all \(\theta \in (\gamma ,1)\) and the estimate
holds true, where \(X:={\dot{B}}^{2\theta }_{q,1}({{\mathbb {R}}}^n)+{\dot{B}}^{2(\alpha +\theta )}_{q,1}({{\mathbb {R}}}^n)\). More precisely, it holds
where \(w_1:=({\hat{w}}\chi _{|\xi |\le 1})^{\vee },\,\,w_2:=({\hat{w}}\chi _{|\xi |> 1})^{\vee }\), and \({\hat{\varphi }}\) and \({\varphi }^\vee \), respectively, denote the Fourier and inverse Fourier transform of \(\varphi \) and \(\chi _{|\xi |\le 1}\), \(\chi _{|\xi |> 1}\) denote the characteristic functions of the sets \(\{\xi : |\xi |\le 1\}\), \(\{\xi : |\xi |> 1\}\), respectively.
Proof
First, consider the case \(\alpha <0\). Then, by the assumption of w, we have
and
Using these inequalities, we get by [21], Theorem 1.8.2 that
and that
Then, by interpolation of (3.8) and the first relation of (3.7), we get for almost all \(t\in (0,T)\) that
yielding
On the other hand, by interpolation of (3.9) and the second relation of (3.7), we get for almost all \(t\in (0,T)\) that
yielding
Since \(w:=w_1+w_2\), the proof of the theorem is complete. \(\square \)
Theorem 3.3
Let \(q\in (1,\infty )\), \(\gamma \in (0,1)\), \(0<T\le \infty \), \(\alpha \in (-\frac{1}{2},0]\), \(2\alpha -1/q\notin {{\mathbb {Z}}}\), and \(\alpha +\gamma <1\). Let \(u_0\in B^{2(\alpha +\gamma )}_{q,\infty }({{\mathbb {R}}}^n), \text {div}\,u_0=0\) and \(f\in L^\infty _\gamma (0,T;b^{2\alpha }_{q,\infty }({{\mathbb {R}}}^n))\) and \(\nabla g\in L^\infty _{\gamma }(0,T;b^{2\alpha }_{q,\infty }({{\mathbb {R}}}^n))\), \(g=\text {div}\,R\) with some distribution \(R=R(t,\cdot )\) such that \(R_t\in L^\infty _{\gamma }(0,T;b^{2\alpha }_{q,\infty }({{\mathbb {R}}}^n))\) and \(R(0)=0\). Then the problem (3.2) has a unique solution \((u,\nabla p)\) such that
with constant \(c>0\) depending only on q, n, s and independent of T.
In addition, the solution u to (3.2) satisfies \(u\in L^\infty _{1-\theta +\gamma }(0,T; X)\), where
and the estimate
with constant \(c=c(q,n,s)>0\) independent of T and \(\nu \).
Remark 3.4
If \(T<\infty \), then it easily follows that
Hence, if \(T<\infty \) is assumed in Theorem 3.3, then \(u\in L^\infty _{1-\theta +\gamma }(0,T; B^{2(\alpha +\theta )}_{q,1}({{\mathbb {R}}}^n))\) and
holds.
Proof of Theorem 3.3
First, assume that \(\nu =1\). Let \(w(t)={{\mathcal {R}}}g(t)=\Delta ^{-1}\nabla g(t), t\in (0,T)\), where \({{\mathcal {R}}}\) is the solution operator for (3.3) given by Lemma 3.1. Then, \(\text {div}\,w(t)=g(t)\) for all \(t\in (0,T)\). It is easily checked that
Moreover, since \(w_t={{\mathcal {R}}}g_t={{\mathcal {R}}}\text {div}\,R_t\) and \(R_t\in L^\infty _{\gamma }(0,T;b^{2\alpha }_{q,\infty }({{\mathbb {R}}}^n))\), we get by Lemma 3.1 that
Now, with the introduction of the new unknown \(U=u-w\), the problem (3.2) is reduced to a divergence-free problem, that is,
where \(F:=f-w_t+\Delta w\in L^\infty _{\gamma }(0,T; b^{2\alpha }_{q,\infty }({{\mathbb {R}}}^n))\). It follows by [18], Corollary 4.13 (i) that the problem (3.15) has a unique solution U satisfying
which yields by (3.13) and (3.14) that for \(u:=U+w\)
Then, the estimate (3.11) with \(\nu =1\) follows directly by Lemma 3.2 and by the above proved fact.
Finally, assuming \(\nu \ne 1\), let us prove (3.10) and (3.11). Notice that the rescaling transform
reduces the system (3.2) with \(\nu \ne 1\) to the case with \(\nu =1\) and that \({\tilde{g}}_t(t,x)=\text {div}\,{\tilde{R}}(t,x)\), where \({\tilde{R}}(t,x):=\frac{1}{\nu ^2} R(\frac{t}{\nu },x)\). Then we have
with \(c=c(q,n,\alpha ,\gamma ,\theta )\) independent of \(\nu \) and T. Here we have
and
Hence, we get (3.10) and (3.11) for the general case \(\nu \ne 1\).
The proof of the theorem is complete. \(\square \)
3.2 Momentum Equations with Fixed Variable Density
Let us consider the existence for the momentum equations with fixed density. We need the following lemma for the proof of Theorem 3.6 which is the main result of this section.
Lemma 3.5
Let \(\Omega \) be a domain of \({{\mathbb {R}}}^n,n\in {{\mathbb {N}}}\), \( n\le q<2n\), \(s\in (0,\frac{2n}{q}-1)\) and \(q\le r\). Then,
-
(i)
\(\forall \xi \in (1-\frac{n}{q},\min \{-s+\frac{n}{q},1-\frac{n}{q}+\frac{n}{r}\})\);
$$\begin{aligned}&{\dot{H}}^{\xi -1+n/q}_{q}(\Omega )\cdot {\dot{H}}^{-\xi -s+n/q}_{r}(\Omega ) \hookrightarrow H^{-s-1+n/q}_{r}(\Omega ),\nonumber \\&{\dot{H}}^{\xi -1+n/q}_{r}(\Omega )\cdot {\dot{H}}^{-\xi -s+n/q}_{q}(\Omega ) \hookrightarrow H^{-s-1+n/q}_{r}(\Omega ). \end{aligned}$$(3.17) -
(ii)
\(\forall \xi \in (1-\frac{n}{q},\min \{-s+\frac{n}{q},1-\frac{n}{q}+\frac{n}{r}\})\), \(\forall \eta \in (0,1-\xi )\), \(\forall \zeta \in (0,\xi +s)\);
$$\begin{aligned}&\big ({\dot{H}}^{\xi -1+n/q+\eta }_{q}+{\dot{H}}^{\xi -1+n/q}_{q}\big )\cdot \big ({\dot{H}}^{-\xi -s+n/q+\zeta }_{r}+{\dot{H}}^{-\xi -s+n/q}_{r}\big ) \hookrightarrow H^{-s-1+n/q}_{r},\nonumber \\&\big ({\dot{H}}^{\xi -1+n/q+\eta }_{r}+{\dot{H}}^{\xi -1+n/q}_{r}\big )\cdot \big ({\dot{H}}^{-\xi -s+n/q+\zeta }_{q}+{\dot{H}}^{-\xi -s+n/q}_{q}\big ) \hookrightarrow H^{-s-1+n/q}_{r}. \end{aligned}$$(3.18)
Proof
In the proof, we omit the symbol \(\Omega \) for the domain. Under the assumption on q, r, s and \(\xi \), we have \(0<\xi -1+n/q<n/q\) and \(0<-\xi -s+n/q<n/r\). Hence, we have
where
Note that \(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=1\). Hence, the first relation of (3.17) follows by Hölder’s inequality. In the same way, we get the second relation of (3.17).
Then, (3.18) is an easy consequence of (3.17). Indeed, we have
where, by (3.17),
and, with \(p_1,p_2, p_3\) in (3.19) and some positive numbers \(\varepsilon _i,i=1\sim 4,\)
proving the first relation of (3.18). The second relation of (3.18) can be proved in the same way.
Thus the lemma is proved. \(\square \)
Theorem 3.6
Let \(n\ge 2\), \(0<T\le \infty \). Let \(n\le q<2 n\), \(s\in (0,\min \{\frac{n-1}{q},\frac{2n}{q}-1\})\) and \(u_0\in B^{-1+n/q}_{q,\infty }({{\mathbb {R}}}^n)\cap B^{-1+n/q}_{r,\infty }({{\mathbb {R}}}^n)\), \(q\le r\), \(\text {div}\,u_0=0\) and \(a\in L^\infty (0,T;L^\infty ({{\mathbb {R}}}^n))\). Then, there are some constants \(\delta _i=\delta _i(q,r,n,s)>0, i=1,2,\) and \(M=M(q,r,n,s)>0\) independent of T such that if
then the nonlinear problem (3.1) has a solution \((u,\nabla p)\) satisfying
In addition, \( u\in L^\infty _{1-\theta +s/2}(0,T;Y),\;\forall \theta \in (\frac{s}{2},1),\) where
and the estimate
holds true. The solution \((u,\nabla p)\) is unique in the class of functions satisfying (3.21) with sufficiently small \(M>0\) on the right-hand side.
Proof
The proof is based on a linearization and a fixed point argument.
Consider the linear system (3.2) with arbitrarily fixed \(f\in L^\infty _{s/2}(0,T;b^{-1+n/q-s}_{q,\infty }({{\mathbb {R}}}^n)\cap b^{-1+n/q-s}_{r,\infty }({{\mathbb {R}}}^n))\) and \(u_0\in B^{-1+n/q}_{q,\infty }({{\mathbb {R}}}^n)\cap B^{-1+n/q}_{r,\infty }({{\mathbb {R}}}^n)\) with \(q\le r\). Then, by Theorem 3.3 with \(\alpha =\frac{1}{2}(-1+\frac{n}{q}-s)\) the system (3.2) has a unique solution \((u,\nabla p)\) such that
Moreover, \(u\in L^\infty _{1-\theta +s/2}(0,T; Y_\theta )\) holds for all \(\theta \in (\frac{s}{2},1)\) and \(Y_\theta \) given by (3.20), and the estimate
holds with constant \({\tilde{C}}>0\) depending on \(q,r,n,\theta \) and s. Note that \(r=q\) is allowed in (3.22). Now, fix \(\theta _1\in \big (\frac{1}{2}(s+1-\frac{n}{q}),\min \{\frac{n}{2q},\frac{1}{2} (s+1-\frac{n}{q}+\frac{n}{r})\}\big )\). Then
satisfy the assumption of Lemma 3.5. Let \( \theta _2:=1-\frac{\xi }{2}\in (\frac{s}{2},1)\). Then,
Let
Then, for \(f_1,f_2\in {{\mathcal {X}}}\) and almost all \(t\in (0,T)\) we get, using (2.7) and the first relation of (3.17) of Lemma 3.5 with \(\xi ,\eta \) and \(\zeta \) of (3.23) and Theorem 3.3 that
where we used that \(\theta _1+\theta _2=1+s/2\). Interchanging the role of \(f_1\) and \(f_2\) and using the second relation of (3.17), we can get the counterpart inequality of (3.24) where the role of \(f_1\) and \(f_2\) are interchanged. Thus,
holds with \(c=c(q,r,n,s)\).
Moreover, using Proposition 2.3 in view of
due to \(0<s<\frac{n-1}{q}\) and \(q\ge n\), we have
Now, define the mapping \(\Phi \) from \({{\mathcal {X}}}\) to itself by
where \((u_f,\nabla p_f)\) is the unique solution to the system (3.2) corresponding to \(u_0\) and f. If we show that \(\Phi \) has a fixed point \({\tilde{f}}\in {{\mathcal {X}}}\), then \((u_{{\tilde{f}}},\nabla p_{{\tilde{f}}})\) is obviously a solution to the system (3.1). Let
Obviously, \(G_K\) is a closed subset of \({\mathcal {X}}\). If \(f\in G_{K}\), then, by (3.25) with \(r=q\) and (3.27),
holds, where \(C_1=C_1(q,r,n,s)\).
On the other hand, if \(f_1,f_2\in G_{K}\), then
Note that \((u_{f_1}-u_{f_2}, \nabla (p_{f_1}- p_{f_2}))\) is a solution to (3.2) with zero initial value and \(g\equiv 0\) and right-hand side \({f_1}-{f_2}\). Hence, by (3.25), the first part (I) on the right-hand side of (3.29) is estimated as
and, by Proposition 2.3 and (3.10) of Theorem 3.3, the part (II) is estimated as
in view of (3.26), where \(C_2=C_2(q,r,n,s)>0\). Finally, we have
Now, in view of (3.28) and (3.30), consider the inequality
where \(C_0:=\max \{C_1,C_2\}\). By elementary calculations, it follows that, if
then for any
with
the inequality (3.31) holds true. In other words, if (3.32) and (3.33) are satisfied, then \(\Phi (G_{K})\subset G_{K}\) and \(\Phi : G_{K}\mapsto G_{K}\) is a contraction mapping. Thus, by the Banach fixed point theorem \(\Phi \) has a unique fixed point \({\tilde{f}}\) in \(G_K\), and \(u=u_{{\tilde{f}}}\) is a solution to (3.1).
Tracking the above arguments, we can infer that the constant \(C_0=C_0(q,r,n,s)\) in (3.31) and hence the constants \(k_{1,2}\) in (3.34) are continuously dependent on s. Denote \(k_{1,2}\) in (3.34) by \(k_{1,2}(s)\). Then, one can choose sufficiently small \(\alpha =\alpha (q,r,n,s)>0\) so that \(\max \{k_1(s-\alpha ), k_1(s+\alpha )\}<\min \{k_2(s-\alpha ), k_2(s+\alpha )\}\). Let
where \({{\mathcal {X}}}_\pm :=L^\infty _{(s-\alpha )/2}(0,T; b^{-s-1+n/q \pm \alpha }_{q,\infty }({{\mathbb {R}}}^n)\cap b^{-s-1+n/q\pm \alpha }_{r,\infty }({{\mathbb {R}}}^n))\) and the norm in \({\mathcal {{{\tilde{X}}}}}\) is given as maximum of the two norms. We construct the mapping \(\Psi : {\mathcal {\tilde{X}}}\mapsto {\mathcal {{{\tilde{X}}}}}\) by
where \((u_f,\nabla p_f)\) is the unique solution to (3.2) corresponding to \(u_0\) and \(f\in {\mathcal {{{\tilde{X}}}}}\). Then, it is obvious from the argument of the proof of (i) (see (3.30)–(3.33)) that \(\Psi \) maps
into \(B_K\), (note that \(B_K\) is closed in \({\mathcal {\tilde{X}}}\)) and becomes a contraction mapping on \(B_K\) provided that
see (3.32), and that
see (3.33). Hence, \(\Psi \) has a unique fixed point \({\bar{f}}\) and \((u_{{\bar{f}}},\nabla p_{{\bar{f}}})\) becomes a solution to (3.1). Note that \(K<\frac{\nu ^{1+s/2}}{2C_0}\). It follows from (3.22), (3.32) and (3.33) that the solution u satisfies
with \(M\equiv \frac{5{\tilde{C}}}{8C_0}\).
Thus, (3.21) follows from (3.35) by the real interpolation relation \((\cdot ,\cdot )_{1/2,1}\). Here, we recall the fact that
for Banach interpolation couple \((X_1,X_2)\) and that
which follows by [5], Theorem 3.4.2 (d), Theorem 6.4.5 (1).
Let \(K'\equiv \frac{K}{3}\) and the norm of solution \((u, \nabla p)\) in (3.21) is bounded by \(K'\). Then
Hence, in view of uniqueness of the fixed point of \(\Phi \) in \(B_K\), \((u,\nabla p)\) must be the only solution satisfying the inequality (3.21) with \(K'\) on the right-hand side.
Thus, the theorem is proved. \(\square \)
4 Proof of the Main Result: Existence Part
Let \(2\le n\le q<2n\), \(0<T\le \infty \) and \(s\in (0,\min \{\frac{n-1}{q},\frac{2n}{q}-1\})\). Suppose that
In order to prove existence of solutions to the system (1.1), we construct an iterative scheme. Let \(\eta _m\in C^\infty ({{\mathbb {R}}}^n), m\in {{\mathbb {N}}},\) be mollifiers such that
For \(m=1,2,\ldots \), let us construct an iterative scheme for (1.1) as
where \(\nu :=\frac{\mu }{{\bar{\rho }}}\) with constant \({\bar{\rho }}\in [\rho _{01},\rho _{02}]\) fixed, \(u^{(0)}\equiv 0\) and \(a_m(t,x):=\frac{{\bar{\rho }}}{\rho _m(t,x)}-1\), \(u^{(m)}=\eta _m\star u_m\), \(m\in {{\mathbb {N}}}\), where “\(\star \)” is used to denote the convolution.
Remark 4.1
For \(f\in H^1_q({{\mathbb {R}}}^n)\) (or \(H^{-1}_q({{\mathbb {R}}}^n)\)), \(q\in (1,\infty )\), it holds \(\Vert \eta _m\star f-f\Vert _{H^1_q({{\mathbb {R}}}^n)}\rightarrow 0\) (or \(\Vert \eta _m\star f-f\Vert _{H^{-1}_q({{\mathbb {R}}}^n)}\rightarrow 0\) ) as \(m\rightarrow \infty \). Let \({\mathcal F}^s_{q}\in \{b_{q,\infty }^s, B_{q,r}^s, 1\le r\le \infty \}\). Then it follows by interpolation that
Moreover, by the Banach-Steinhaus theorem, it follows that
with \(C=C(q,s)>0\) independent of \(m\in {{\mathbb {N}}}\).
Furthermore, if \(u_m\in L^\infty _{s/2}(0,T; b^{2-s}_{q,\infty }({{\mathbb {R}}}^n))\) for some \(q\ge n\) and \(s\in (0,2)\), then \(u^{(m)}(t)\in C^\infty ({{\mathbb {R}}}^n)\) for almost all \(t\in (0,T),\) and, in particular,
We have the following lemma.
Lemma 4.2
Let \(n\ge 2\), \(n\le q<2n\) and \(0<T\le \infty \). Suppose that (4.1) holds. Then for any \(s\in (0, \min \{\frac{n-1}{q}, \frac{2n}{q}-1\})\) there are some constants \(\delta _i=\delta _i(q,r,n,s)>0, i=1,2,\) and \(M=M(q,r,n,s)>0\) independent of \(m\in {{\mathbb {N}}}\) with the following property: If
the iterative system (4.2) has a solution \(\{(\rho _m,u_m, \nabla p_m):m\in {{\mathbb {N}}}\}\) such that
and
and the estimates
hold true.
Proof
By [6], Theorem II.3, it follows that when \(v\in L^1(0,T; W^{1,\infty }({{\mathbb {R}}}^n))\), \(\text {div}\,v=0\), and \(\rho _0\in L^\infty ({{\mathbb {R}}}^n)\), the transport equation
has a unique solution \(\rho \in L^\infty (0,T;L^\infty ({{\mathbb {R}}}^n))\) such that \(\Vert \rho \Vert _{L^\infty (0,T;L^\infty ({{\mathbb {R}}}^n))}=\Vert \rho _0\Vert _{L^\infty ({{\mathbb {R}}}^n)}\).
This fact, together with Theorem 3.6, yields the assertions of the lemma. \(\square \)
Remark 4.3
Let \(n\ge 2\), \(0<T \le \infty \), \(n\le q\le r\) and let \(s\in (0,1)\). Let \(v\in L^\infty _{1-\theta +s/2}(0,T; Y_\theta )\) for all \(\theta \in (s/2,1)\), where \(Y_\theta \) is given by (3.20). Then, if we take \(\theta =\theta _1\) as
we get for any finite \(T'\le T\) and bounded domain \(G\subset {{\mathbb {R}}}^n\) that
and
On the other hand, if we take \(\theta =\theta _2\in \big (\frac{s+1}{2}, \frac{s}{2}+\frac{2}{3}\big )\), then for any finite \(T'\le T\) and bounded domain \(G\subset {{\mathbb {R}}}^n\) we have
Remark 4.4
Let \(2\le n\le q< 2n\), \(s\in (0,\min \{\frac{n-1}{q},\frac{2n}{q}-1\})\). Let \(\{(\rho _m,u_m, \nabla p_m):m\in {{\mathbb {N}}}\}\) be solutions to the iterative system (4.2) whose existence is guaranteed by Lemma 4.2. Then,
and so is for \(u^{(m)}\). Therefore, if
then by Proposition 2.3 we get for \(v=u_m\) or \(u^{(m)}\) that
for almost all \(t\in (0,T)\). Therefore, if (4.10) holds, then
Proof of Theorem 1.1: Existence part
Let \(s\in (0, \min \{\frac{n-1}{q},\frac{2n}{q}-1\})\) and let \(\{(\rho _m, u_m, \nabla p_m):m\in {{\mathbb {N}}}\}\) be the solutions to the iterative system (4.2), the existence of which is given by Lemma 4.2. By Lemma 4.2\(\{u_m\}\) is bounded in \(L^\infty _{1-\theta +s/2}(0,T; Y_\theta )\) for all \(\theta \in (s/2,1)\), and \(\{u_{mt},\nabla p_m\}\), \(\{\rho _m\}\) are bounded in \(L^\infty _{s/2}(0,T; b^{-s-1+n/q}_{q,\infty }({{\mathbb {R}}}^n))\) and \(L^\infty ((0,T)\times {{\mathbb {R}}}^n)\), respectively. Note that
Therefore, it follows by standard arguments that \(\{\rho _m, u_m, \nabla p_m\}\) has a subsequence (denote it by the same symbols) such that
for some u, \(\rho \) and distribution P. Moreover, it follows from (4.6) and (4.7) that
for all \(\theta _1\in (s/2,1)\) satisfying (4.5), finite \(T'\le T\) and bounded \(G\subset {{\mathbb {R}}}^n\).
We shall show that \((\rho , u,\nabla P)\) is a solution to (1.1) satisfying the assertion of Theorem 1.1.
By (4.12), obviously, \((\rho , u, \nabla P)\) satisfies (1.4) and (1.8).
Next, in order to prove (1.6), note that by (4.3), (4.4) the sequence \(\{u_{mt}\}\) weakly converges in \(L^{\alpha }(0,T; H^{-s-1+n/q}_q(G))\) for some \(\alpha >1\). Therefore, in view of (4.13) and compact embedding \({\dot{H}}^{2\theta _1-s-1+n/q}_{q}(G)\hookrightarrow \hookrightarrow L^{2n}(G)\), we get by a compactness theorem ([20], Ch.3, Theorem 2.1) that
for any finite \(T'\le T\) and bounded G as \(m\rightarrow \infty \). Moreover, we get from (4.8) that \(\{\nabla u_{m}\}\) is bounded in \(L^{3/2}(0,T'; L^{(2n)/(2n-1)}(G))\).
Rewriting the second equation of the system (4.2), we have
which is equivalent to
in view of
Let \(\varphi \in C_0^\infty ([0,T)\times {{\mathbb {R}}}^n)^n\), \(\text {div}\,\varphi =0\) be given arbitrarily such that \(\mathrm{supp}\,\varphi \subset [0,T')\times G\) with finite \(T'>0\) and bounded \(G\subset {{\mathbb {R}}}^n\). Since each term of (4.15) belongs to \(L^1(0,T; B^{-s-1+n/q}_{q,1}({{\mathbb {R}}}^n))\) by (4.3) and \((b^{-s-1+n/q}_{q,\infty }({{\mathbb {R}}}^n))'=B^{s+1-n/q}_{q',1}({{\mathbb {R}}}^n)\), we can test (4.15) with \(\varphi \) to obtain
By (4.9) we have
with \(\tau =s-\theta +3/2-n/q\) being \(s/2<\tau <1\), which holds true if (4.10) is satisfied. Hence, if \((\tau -s/2)p'<1\), i.e., if \(0<1/p<\theta -s/2+n/q-1/2\) and (4.10) is satisfied, then
where note that \(-2\theta +s+2-n/q>0\). On the other hand, by (4.11), we have \(\rho _{mt}\varphi \in L^p(0,T;H^{2\theta -s-2+n/q}_{q}({{\mathbb {R}}}^n))\) for \(\theta \) satisfying (4.10) and \(\theta -s/2<1/p<\infty \). Therefore, if
and (4.10) is satisfied, then we have
Moreover, for p and \(\theta \) satisfying (4.16) and (4.10) we get by (4.11) that
Therefore, we have
For the estimate of the last term on the right-hand side of (4.17) we need the following lemma.
Lemma 4.5
Let \(T'\le T\) be finite and \(G\subset {{\mathbb {R}}}^n\) be a bounded domain. Let p and \(\theta \) satisfy (4.16) and put
Then \(R_m\) tends to 0 as \(m\rightarrow \infty \).
Proof
By Remark 4.3 and Lemma 4.2 we have
Then, since \(\Vert \nabla (u_{m}\cdot \varphi )\Vert _{L^{3/2}(0,T';L^{2n/(2n-1)}(G))}\) is bounded with respect to \(m\in {{\mathbb {N}}}\), see (4.8), the proof of the lemma will be complete if we show that
Note that
where the first term on the right-hand side tends to 0 as \(m\rightarrow \infty \) due to (4.14).
In order to show
we write
Here, \(L^{3}(0,T';L^{2n}(G))\)-norm of \(\eta _{m-1}\star u-u\) obviously tends to 0 as \(m\rightarrow \infty \). The \({L^{3}(0,T';L^{2n}(G))}\)-norm of \(\eta _{m-1}\star (u_{m-1}-u)\) goes to zero as \(m\rightarrow \infty \) thanks to (4.14) and the fact that \(\Vert \eta _{m}\star \,\,\cdot \Vert _{{{\mathcal {L}}}(L^{3}(0,T';L^{2n}(G)))}\) is uniformly bounded with respect to \(m\in {{\mathbb {N}}}\). Thus, (4.19) and, consequently, (4.18) are proved. The proof of the lemma comes to end. \(\square \)
Let us continue the proof of existence part of Theorem 1.1. In (4.17), we get easily that
as \(m\rightarrow \infty \) due to \(*\)-weak convergence \(\rho _{m} \rightharpoonup \rho \) in \(L^\infty (0,T;L^\infty ({{\mathbb {R}}}^n))\) and strong convergence \(u_{m} \rightarrow u\) in \(L^{3}(0,T'; L^{2n}(G))\), see (4.12) and (4.14). By (4.14) we have \(u_{m}\otimes u_{m}\rightarrow u\otimes u\) in \(L^{1}((0,T')\times G)\) as \(m\rightarrow \infty \) and therefore,
Thus, letting \(m\rightarrow \infty \) in (4.17), it follows that \((\rho ,u)\) satisfies (1.6).
Finally, in order to show (1.5), test the first equation of (4.2) with arbitrary \(\psi \in C_0^1([0,T)\times {{\mathbb {R}}}^n)\) to get
Obviously,
Moreover, in view of (4.19) and (4.12), we have
as \(m\rightarrow \infty \). Therefore, \((\rho ,u)\) satisfies (1.5) in the limiting case \(m\rightarrow \infty \) in (4.20).
Thus, the proof of existence part of Theorem 1.1 is completed. \(\square \)
5 Proof of the Main Result: Uniqueness Part
Lemma 5.1
Let \(\Omega \) be a Lipschitz domain of \({{\mathbb {R}}}^n\), \(n\ge 2\), and let \(n\le q<r<2n\) and \(s\in (0,\frac{n}{q}-\frac{n}{r})\). Then the following statements hold:
-
(i)
\(\forall f\in b^{-s+n/q}_{r,\infty }(\Omega ), \varphi \in B^{s+1-n/q}_{r',1}(\Omega )\);
$$\begin{aligned} \Vert f\varphi \Vert _{B^{s+1-n/q}_{r',1}(\Omega )}\lesssim \Vert f\Vert _{b^{-s+n/q}_{r,\infty }(\Omega )} \Vert \varphi \Vert _{B^{s+1-n/q}_{r',1}(\Omega )}. \end{aligned}$$ -
(ii)
\(\forall f\in b^{-s+n/q}_{r,\infty }(\Omega ), g\in b^{-s-1+n/q}_{r,\infty }(\Omega )\);
$$\begin{aligned} \Vert f g\Vert _{b^{-s-1+n/q}_{r,\infty }(\Omega )}\le l_1\Vert f\Vert _{b^{-s+n/q}_{r,\infty }(\Omega )}\Vert g\Vert _{b^{-s-1+n/q}_{r,\infty }(\Omega )}. \end{aligned}$$ -
(iii)
\(\forall f,g\in b^{-s+n/q}_{r,\infty }(\Omega )\);
$$\begin{aligned} \Vert f g\Vert _{b^{-s+n/q}_{r,\infty }(\Omega )}\le l_2\Vert f\Vert _{b^{-s+n/q}_{r,\infty }(\Omega )}\Vert g\Vert _{b^{-s+n/q}_{r,\infty }(\Omega )}. \end{aligned}$$ -
(iv)
\(\forall f\in {\dot{B}}^{-s+n/q}_{r,\infty }(\Omega ), g\in B^{-1+n/q}_{r,\infty }(\Omega )\);
$$\begin{aligned} \Vert f g\Vert _{B^{-s-1+n/q}_{r,\infty }(\Omega )} \lesssim \Vert f\Vert _{{\dot{B}}^{-s+n/r}_{r,\infty }(\Omega )}\Vert g\Vert _{B^{-1+n/q}_{r,\infty }(\Omega )}. \end{aligned}$$
Proof
– Proof of (i): It is clear from \(r>n\) and Sobolev and Hölder inequality that
Moreover, we have
where \(\zeta :=(\frac{1}{r}-\frac{s+1-n/q}{n})^{-1}(-s+\frac{n}{q})>1\) holds true thanks to \(s<n/q-n/r\) and \(r<2n\).
From (5.1) and (5.2) we get by bilinear real interpolation that
cf. [5, 21]. Hence, in view of
we have
which implies the assertion (i) in view of \(b^{-s+n/q}_{r,\infty }(\Omega )\hookrightarrow B^{n/r}_{r,1}(\Omega )\subset {\dot{B}}^{n/r}_{r,1}(\Omega )\) due to \(-s+n/q>n/r\).
– Proof of (ii): Using duality argument in view of \((B^{-s-1+n/q}_{r',1}(\Omega ))'=B^{s+1-n/q}_{r,\infty }(\Omega )\) due to \(0<s+1-n/q<1/r'\), it follows directly from (i) that
for all \(f\in b^{-s+n/q}_{r,\infty }(\Omega )\) and \(g\in B^{-s-1+n/q}_{r,\infty }(\Omega )\). Then, the assertion (ii) follows by a density argument.
–Proof of (iii): The assertion (iii) can be proved exactly in the same way as for the assertion (i). In fact, using
where \(\frac{s+1-n/q}{\sigma }+\frac{-s+n/q}{n}=\frac{1}{r}\), we get by bilinear continuous interpolation \((\cdot ,\cdot )^0_{-s+n/q,\infty }\) that
which yields the assertion (iii).
–Proof of (iv): Note that \(s+1-n/q-n/r<0\). It then easily follows by Sobolev embedding theorem that
which together with (5.3) yields by real interpolation \((\cdot ,\cdot )_{{\tilde{\theta }},1}\) with \({\tilde{\theta }}=\frac{r}{n}(-s+\frac{n}{r})\) that
Hence, by duality, the assertion (iv) is proved. \(\square \)
Proof of Theorem 1.1: Uniqueness part
The uniqueness proof relies on a Lagrangian coordinates approach following the idea of [12].
First let us recall some facts concerning Lagrangian coordinates. Let u be a vector field such that
and let X(t, y) be the (unique) solution to the ordinary differential system:
The unique solution X to (5.5), that is,
determines a unique continuous semiflow, i.e., \(t\rightarrow X(t,y)\) for each y is continuous and \(X(0,\cdot )=\text {Id}\), \(X(t+s,y)=X(t, X(s,y))\) for \(t,s>0\). Then, Eulerian coordinates x and Lagrangian coordinates y are related by
Note that \(W^{1,\infty }({{\mathbb {R}}}^n)\subset C_{Lip}({{\mathbb {R}}}^n)\). Moreover, if \(\text {div}\,u=0\), then \(X(t,\cdot )\) for each \(t>0\) is a \(C^1\)-diffeomorphism and measure preserving due to the Jacobian \(|D_y X(t,y)|=1\). Let \(Y(t,\cdot )\) be the inverse mapping of \(X(t,\cdot )\), then
where and in what follows we use the notation \((\nabla u)_{i,j}=(\partial _iu^j)_{1\le i,j\le n}, Du=(\nabla u)^T\). Let \(v(t,y):=u(t, X(t,y))\). Then,
see [10], ”Appendix”. In view of (5.7), we use the notation
Now, let \((\rho , u,\nabla P)\) be a solution to (1.1) satisfying (1.4) and (1.8) with \(r>q\) and \(s\in (0, \frac{n}{q}-\frac{n}{r})\). Then, it follows from \(u_t\in L^\infty _{s/2}(0,T; b^{-s-1+n/q}_{r,\infty }({{\mathbb {R}}}^n))\) and \(u_0\in B^{-1+n/q}_{r,\infty }({{\mathbb {R}}}^n)\subset b^{-s-1+n/q}_{r,\infty }({{\mathbb {R}}}^n)\) that \(u\in L^\infty _{s/2}(0,T'; b^{-s-1+n/q}_{r,\infty }({{\mathbb {R}}}^n))\) for any finite \(T'\le T\), which together with \(\nabla ^2 u\in L^\infty _{s/2}(0,T; b^{-s-1+n/q}_{r,\infty }({{\mathbb {R}}}^n))\) yields that \(u\in L^\infty _{s/2}(0,T'; b^{-s+1+n/q}_{r,\infty }({{\mathbb {R}}}^n))\). Therefore, we have
by Sobolev embedding \(b^{-s+1+n/q}_{r,\infty }({{\mathbb {R}}}^n)\subset W^{1,\infty }({{\mathbb {R}}}^n)\) in view of \(-s+1+n/q-n/r>1\), and thus u satisfies (5.4).
For \(a(t,x):=\frac{{\bar{\rho }}}{\rho (t,x)}-1\), \((t,x)\in (0,T)\times {{\mathbb {R}}}^n\), and
where X(t, y) is given by (5.6), we get from (1.5) that \(b_t=0\) in the sense of distribution. In fact, given any \({\tilde{\psi }}\in C^1_0((0,T)\times {{\mathbb {R}}}^n)\) and \(\psi (t,x):={\tilde{\psi }}(t,Y(t,x))\), we have \({\tilde{\psi }}(t,y)=\psi (t, X(t,y))\) and \(\psi \in C^1_0((0,T)\times {{\mathbb {R}}}^n)\). Hence, we have
Therefore, \(b(t,y)\equiv b(0,y)\equiv a_0(y)\) for each \(y\in {{\mathbb {R}}}^n\).
Assuming \(\mu =1\) without loss of generality, \(\{v,Q\}\) solves the system:
Now, in order to prove uniqueness of solutions, let \((\rho _i, u_i, \nabla P_i)\), \(i=1,2\), be two solutions to (1.1) which satisfy (1.4) and (1.8) with \(r>q\) and \(s\in (0,\frac{n}{q}-\frac{n}{r})\). Note that \(r<2n\) may be assumed without loss of generality in view of the initial assumption \(q <2n\) and \(u_0\in B^{-1+n/q}_{q,\infty }({{\mathbb {R}}}^n)\cap B^{-1+n/q}_{r,\infty }({{\mathbb {R}}}^n)\subset B^{-1+n/q}_{{\tilde{r}},\infty }({{\mathbb {R}}}^n)\), \(\forall {\tilde{r}}\in (q,r)\).
For \(i=1,2\), let \(X_i\) be the semiflow corresponding to \(u_i\) (see (5.6)) and let \((b_i, v_i, Q_i)\) be the corresponding density perturbation, velocity and pressure in the Lagrangian coordinates. Then, for \(\delta v=v_1-v_2, \delta Q=Q_1-Q_2\) we get from (5.9) that
where
Note that \(\delta R(0)=0\). Therefore, by Theorem 3.3 with \(q\equiv r\), \(\gamma \equiv s/2, \alpha \equiv -s-1+n/q\), we have
with constant \(K>0\) independent of t and \(\nu \). In particular, if \(t\le 1\), it follows from (5.12) that
see Remark 3.4.
From now on, let us get estimate of the right-hand side of (5.12). Thanks to (5.8), there is some \(m_0>0\) such that, if \(0\le t_1<t_2<1, t_2-t_1<m_0\),
where \(l:=\max \{l_1, l_2\}\) with constants \(l_1\), \(l_2\) appearing in the estimates of Lemma 5.1. Throughout the proof, we assume that \(0<t<m_0\).
Note that
due to \(-1+1/r<-s-1+n/q<1/r\) and by Proposition 2.3. Hence, \(a_0(y)=\frac{{\bar{\rho }}-\rho _0(y)}{\rho _0(y)}\) implies that there is some \(\delta _1>0\) independent of \(\nu \) such that if
then
where K is the constant appearing in (5.12).
On the one hand, we have
where
Hence one has
with
By elementary calculation using the first relation of (5.14) we have
with a generic constant C. Hence, it follows from (5.15) and \(v\in L^\infty _{s/2}(0,T'; b^{-s+1+n/q}_{r,\infty }({{\mathbb {R}}}^n))\) for any finite \(T'\le T\) that
with \(c=c(q,r,s)>0\).
By definition of \(C_i\) and the second relation of (5.14) we have
and, in particular, by Lemma 5.1 (iii)
Then, using (5.24) and Lemma 5.1 (iii), we have
with \(c=c(q,r,s)>0\).
Next, let us get estimate of \(\Vert \nabla \delta g\Vert _{L^\infty _{s/2}(0,t;b^{-s-1+n/q}_{r,\infty }({{\mathbb {R}}}^n))}\). From (5.11) one has
where the right-hand side can be estimated as below.
Since \(DA_i=\sum _{k\ge 1}(-1)^kkC_i^{k-1}DC_i\) due to (5.18), we get by Lemma 5.1 (ii) and (5.23) that
and
with \(c=c(q,r,s)\). Furthermore, by (5.18), Lemma 5.1 (iii) and (5.23) we have
and, consequently,
with \(c=c(q,r,s)\).
Note that
where
Then, we get by Lemma 5.1 (ii), (5.23) and the second relation of (5.14) that
with \(c=c(q,r,s)\). Hence, by Lemma 5.1 (ii),
with \(c=c(q,r,s)\). Then, by Lemma 5.1 (ii) we have
where the constant c depends on q, r, s and \(\Vert v_1,v_2\Vert _{L^\infty _{s/2}(0,t;b^{-s+1+n/q}_{r,\infty })}\).
By (5.25) and Lemma 5.1 (ii) we have
where the constant c depends on q, r, s and \(\Vert v_1,v_2\Vert _{L^\infty _{s/2}(0,t;b^{-s+1+n/q}_{r,\infty })}\).
Thus, from (5.26), (5.28), (5.30), (5.31) and (5.32) we get
with some \(\eta _1(t)\) such that \(\eta _1(t)\rightarrow 0\) as \(t\rightarrow +0\).
Following the same procedure as the derivation of (5.33), we can get estimate
under the condition (5.16), where \(c>0\) depends on q, r, s, \(\Vert v_1,v_2\Vert _{L^\infty _{s/2}(0,t;b^{-s+1+n/q}_{r,\infty }({{\mathbb {R}}}^n))}\) and \(\Vert \nabla Q_1\Vert _{L^\infty _{s/2}(0,t;b^{-s-1+n/q}_{r,\infty }({{\mathbb {R}}}^n))}\); we omit the details here.
On the other hand, we have
Here, by Lemma 5.1 (ii) we have
Note that, by Lemma 5.1 (ii), (iii), (5.27) and (5.14),
and, by Lemma 5.1 (iii),
in view of \(A_2^T(t) A_2(t)-\text {Id}= \int _0^t (Dv_2+Dv_2^T)\,d\tau + \int _0^tDv_2^T\,d\tau \cdot \int _0^tDv_2\,d\tau \). Therefore,
Furthermore, using \(A_2^T A_2-A_1^T A_1= A_2^T \delta A+ \delta A^T A_1\), Lemma 5.1 (ii), (iii), (5.25) and the second relation of (5.14), we have
Thus, from (5.35)–(5.37) we have
which together with (5.34) yields
with some \(\eta _2(t)\) such that \(\eta _2(t)\rightarrow 0\) as \(t\rightarrow +0\).
Finally, let us get estimate of \(\Vert \delta R_t\Vert \). Recall that \(\delta R=(\text {Id}-A_2)\delta v-\delta A v_1\) and hence \((\delta R)_t=-A_{2t}\delta v+(\text {Id}-A_2)\delta v_t-\delta A_t v_1-\delta A v_{1t}\). Since we have
see (5.18), it follows by (5.23) and Lemma 5.1 (ii) that
where we used that
On the other hand, using (5.29), we have
It is easy to see that \(\delta A_t= -A_1^2 D\delta v-\delta A(A_1+A_2)Dv_2\). By (5.23) and (5.25), we have
Moreover,
where we used that \(B^{2\theta -s-2+n/q}_{r,1}\cdot B^{-1+n/q}_{r,\infty } \hookrightarrow B^{-s-1+n/q}_{r,\infty }\) for \(\theta =1-\frac{1}{2}(\frac{n}{q}-\frac{n}{r})\) thanks to Lemma 5.1 (iv).
Finally, in view of the expression \(\delta A(t)=h_1(t)\int _0^t D\delta v\,d\tau \) and (5.25), we get by Lemma 5.1 (ii) that
Thus, from (5.39)–(5.42) it follows that
with some \(\eta _3(t)\) converging to 0 as \(t\rightarrow +0\).
Summarizing, we can conclude from (5.13), (5.17), (5.33), (5.38) and (5.43) that \(\delta v(t)=0, \delta Q(t)=0\) for all \(t\in (0,T_1)\) with some \(T_1>0\). Then, by standard continuation argument, it can be shown that \(\delta v(t)=0, \delta Q(t)=0\) for all \(t\in (0,T)\).
Now, the proof of the uniqueness part of Theorem 1.1 comes to end. \(\square \)
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Ri, MH. Global Well-Posedness for Inhomogeneous Navier–Stokes Equations in Endpoint Critical Besov Spaces. J. Math. Fluid Mech. 23, 16 (2021). https://doi.org/10.1007/s00021-020-00532-4
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DOI: https://doi.org/10.1007/s00021-020-00532-4