Abstract
The equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.
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1 Introduction
We investigate the fluid flow past a rigid body \({\mathcal B}\) that moves through an infinite three-dimensional liquid reservoir with prescribed velocity
with respect to its center of mass \(x_\mathrm {C}\). Here \(t\in \mathbb {R}\) and \(x\in \mathbb {R}^3\) denote time and spatial variable, respectively, \(\xi :=\tfrac{{\mathrm d}}{{\mathrm d}t}x_\mathrm {C}\) is the translation velocity and \(\eta \) the angular velocity of \({\mathcal B}\) with respect to its center of mass. We only consider the case where the angular velocity \(\eta \) is constant, but the translation velocity \(\xi \) may depend on time. In a frame attached to the body, with origin at its center of mass \(x_\mathrm {C}\), the motion of an incompressible Navier–Stokes fluid around \({\mathcal B}\) that adheres to \({\mathcal B}\) at the boundary is described by the equations
see [12, Section 1]. Here \(\Omega :=\mathbb {R}^3\setminus {\overline{{\mathcal B}}}\) is the exterior domain surrounding \({\mathcal B}\), and \(\mathbb {R}\) represents the time axis. The functions \(u:\mathbb {R}\times \Omega \rightarrow \mathbb {R}^3\) and \(\mathfrak {p}:\mathbb {R}\times \Omega \rightarrow \mathbb {R}\) describe velocity and pressure fields of the fluid. The constants \(\rho >0\) and \(\mu >0\) denote density and viscosity, respectively. For the sake of generality, we additionally consider an external body force \(f:\mathbb {R}\times \Omega \rightarrow \mathbb {R}^3\).
In this paper, we investigate a configuration where the rigid body \({\mathcal B}\) translates periodically with some prescribed time period \({\mathcal T}>0\). More precisely, we assume the data
to be \({\mathcal T}\)-time-periodic As the main theorem we show existence of a solution \((u,\mathfrak {p})\) to (1.1) that shares this time periodicity.
We consider a prescribed motion of \({\mathcal B}\) where the axes of translation and rotation do not vary over time and are parallel. Without loss of generality, both are directed along the \(x_1\)-axis such that
for some \({\mathcal T}\)-periodic function \(\alpha :\mathbb {R}\rightarrow \mathbb {R}\) and a constant \(\omega \in \mathbb {R}\). Note that, at least in the case where \(\xi \) is time-independent, this assumption can be made without loss of generality as long as \(\xi \cdot \eta \ne 0\) due to the Mozzi–Chasles theorem.
We assume that the mean translational velocity of the body over one time period is non-zero:
The case of vanishing mean translational velocity shall not be treated here. Not only does the fluid flow exhibit different physical properties when (1.2) is not satisfied, due to the absence of a wake region in this case, but also the mathematical properties of the linearization of (1.1) differ significantly. If (1.2) is satisfied, the linearization of (1.1) is a time-periodic generalized Oseen system, for which we shall establish suitable \(\mathrm {L}^{q}\) estimates in order to show existence of a solution to (1.1). If (1.2) is not satisfied, the linearization of (1.1) is a time-periodic generalized Stokes system, for which similar estimates cannot be derived. In this case, problem (1.1) thus has to be approached in a different way, which has recently been done by Galdi [15].
Since the case \(\eta =0\) was treated in [18], we only consider the case \(\eta \ne 0\) in the following. Observe that then \(\eta \wedge x\cdot \nabla \) is a differential operator with unbounded coefficient. Therefore, the linearization of (1.1) cannot be treated as a lower-order perturbation of the time-periodic Oseen problem, even if \(\eta \) is “small”. In particular, as we will see below, the corresponding resolvent problem also requires an analysis in a different functional setting. This observation reflects the properties of the corresponding stationary problem (see [13, Chapter VIII]), which can be regarded as a special case of the time-periodic problem. In order to find a framework in which the time-periodic generalized Oseen problem is well posed, we employ the idea from [16, 17], where the steady-state problem corresponding to (1.1) was considered, and the rotation term \(\eta \wedge u-\eta \wedge x \cdot \nabla u\) was handled by a change of coordinates into a non-rotating frame. This procedure only yields suitable estimates for time-periodic solutions when the change of coordinates maintains the time periodicity of the involved functions. This is the case if the angular velocity \(\omega \) is an integer multiple of the angular frequency \(2\pi /{\mathcal T}\) of the time-periodic data. For simplicity, we assume
This condition means that during one period the rigid body completes one full revolution. In other words, the rotation and the time-periodic data, which may be regarded as two different sources of time-periodic forcing, have to be compatible.
The equations governing the fluid flow around a rigid body that performs a prescribed rigid motion have been studied by many researchers during the last decades. The first successful attempts of a rigorous mathematical treatment date back to the fundamental works of Oseen [44], Leray [36, 37] and Ladyžhenskaya [34, 35]. The study of time-periodic Navier–Stokes flows was proposed in a short note by Serrin [47], which induced Prodi [45], Yudovich [56] and Prouse [46] to initiate the examination in bounded domains. Through the years, this investigation has been continued and extended to other types of domains and fluid-flow configurations by several authors; see for example [5, 10, 11, 14, 18, 21,22,23, 28,29,30,31, 33, 38,39,43, 49, 51,52,55]. We refer to [19] for a more detailed overview. Concerning in particular time-periodic Navier–Stokes flows around rigid bodies, more specifically the three-dimensional exterior-domain configuration, we emphasize the fundamental work of Yamazaki [55], who introduced a setting of \(\mathrm {L}^{3,\infty }(\Omega )\) spaces to obtain time-periodic solutions in the case \(\xi =\eta =0\). The main estimates in [55] are based on well-known \(\mathrm {L}^{p}\)-\(\mathrm {L}^{q}\) estimates of the Stokes semigroup. If one replaces these estimates with the \(\mathrm {L}^{p}\)-\(\mathrm {L}^{q}\) estimates obtained by Shibata [48] for an Oseen semigroup with rotational effects, the approach in [55] also seems to yield existence of time-periodic solutions to (1.1) in an \(\mathrm {L}^{3,\infty }(\Omega )\) framework in the case of constant non-zero parameters \(\xi \ne 0, \eta \ne 0\). This analysis was recently carried out by Geissert, Hieber and Nguyen [23], who introduced a general semigroup-based approach to show existence of mild solutions to time-periodic problems. Using a recent result by Hishida [27], who established \(\mathrm {L}^{p}\)-\(\mathrm {L}^{q}\) estimates for an evolution operator corresponding to a linearization of the Navier–Stokes equations in the case of time-dependent \(\xi (t)\) and \(\eta (t)\), the approach of Yamazaki [55] even leads to time-periodic solutions in an \(\mathrm {L}^{3,\infty }(\Omega )\) framework for general time-periodic \(\xi (t)\) and \(\eta (t)\). In this general case, Galdi and Silvestre [21] already established the existence of time-periodic solutions in an \(\mathrm {L}^{2}(\Omega )\) setting via a Galerkin approach.
As the main novelty of the present paper, we establish existence of strong solutions to (1.1) in an \(\mathrm {L}^{q}(\Omega )\) setting for a certain range of exponents \(q\in (1,\infty )\). In this setting, better information on the spatial decay of the solutions can be derived compared to the \(\mathrm {L}^{3,\infty }(\Omega )\) and \(\mathrm {L}^{2}(\Omega )\) frameworks described above. Our approach is based on an analysis of the linearization of (1.1) and the associated resolvent problem
for suitable \(s\in \mathbb {R}\) and \(F\in \mathrm {L}^{q}(\Omega )^3\), \(1<q<\infty \). At first glance, it seems reasonable to regard (1.4) as a resolvent problem \((is-A)v=F\) for a closed operator A on the space of solenoidal vector fields in \(\mathrm {L}^{q}(\Omega )^3\). However, the spectral analysis in this setting, which was carried out by Farwig and Neustupa [7, 8], reveals that is, \(s\in \mathbb {R}\), belongs to the spectrum of A when \(s\in \omega \mathbb {Z}\), whereas well-posedness of the time-periodic problem requires invertibility of (1.4) for \(s\in \omega \mathbb {Z}\). Therefore, we propose to investigate the problem in homogeneous Sobolev spaces instead. Although it is merely possible to derive the non-classical resolvent estimate (2.4) in this setting (see Theorem 2.1 below), we are nevertheless able to conclude a suitable solution theory for the linearization of (1.1). To this end, we shall employ a framework of functions with absolutely convergent Fourier series. Finally, a fixed-point argument yields the existence of a solution to the nonlinear problem (1.1) when the data \(f\), \(\xi \) and \(\eta \) are “sufficiently small”.
2 Main Results
In virtue of (1.2) we may assume \(\lambda >0\) without loss of generality, and by (1.3) we have \(\omega =2\pi /{\mathcal T}>0\). To reformulate (1.1) in a non-dimensional way, we let the diameter \(d>0\) of \({\mathcal B}\) serve as a characteristic length scale. We introduce the Reynolds number \(\lambda ':=\lambda \rho d/\mu \), the Taylor number \(\omega ':=\omega \rho d^2/\mu \), and the non-dimensional time and spatial variables \(t'=\omega t\) and \(x'=x/d\). In particular, \(\Omega \) is transformed to \( \Omega ' :=\{x/d\ \vert \ x \in \Omega \}. \) We define \(\alpha '(t'):=\alpha (t)\rho d/\mu \) and the non-dimensional functions
which are time-periodic with period \({\mathcal T}'=2\pi \) and can thus be identified with functions on the torus group \({\mathbb T}=\mathbb {R}/2\pi \mathbb {Z}\) with respect to time. Expressing (1.1) in these new quantities and omitting the primes, we obtain the non-dimensional formulation
Our analysis of (2.1) is based on the study of the linear time-periodic problem
and of the corresponding resolvent problem
for \(k\in \mathbb {Z}\). For the latter we shall derive the following well-posedness result.
Theorem 2.1
Let \(\Omega \subset \mathbb {R}^3\) be an exterior domain of class \(\mathrm {C}^{3}\). Let \(q\in (1,2)\), \(k\in \mathbb {Z}\) and \(\lambda ,\,\omega , \,\theta ,\,B>0\) with \(\lambda ^2\le \theta \omega \le B\). For every \(F\in \mathrm {L}^{q}(\Omega )^3\) there exists a solution \((v,p)\in \mathrm {W}^{2,q}_{\mathrm {loc}}({\overline{\Omega }})^3\times \mathrm {W}^{1,q}_{\mathrm {loc}}({\overline{\Omega }})\) to (2.3) subject to the estimate
for a constant \({C}_{1} ={C}_{1}(\Omega ,q,\lambda ,\omega )>0\) and \(s_1=2q/(2-q)\), \(s_2=4q/(4-q)\). Additionally, if \((w,\mathfrak {q})\) is another solution to (2.3) in the function class defined by the norms on the left-hand side of (2.4), then \(v=w\), and \(p-\mathfrak {q}\) is a constant. Moreover, if \(q\in (1,\frac{3}{2})\), then the constant \({C}_{1}\) can be chosen independently of \(\lambda \) and \(\omega \) such that \({C}_{1} ={C}_{1}(\Omega ,q,\theta ,B)\).
Note that for \(k=0\) we recover the well-known \(\mathrm {L}^{q}\) theory for the corresponding stationary problem; see [13, Theorem VIII.8.1].
In order to transfer estimate (2.4) to the time-periodic setting without losing information on the dependencies of the constant \({C}_{1}\), we work within spaces \(\mathrm {A}({\mathbb T};X)\) of absolutely convergent X-valued Fourier series for suitable Banach spaces X; see (3.1) below. We establish the following solution theory for the time-periodic problem (2.2).
Theorem 2.2
Let \(\Omega \subset \mathbb {R}^3\) be an exterior domain of class \(\mathrm {C}^{3}\). Let \(q\in (1,2)\) and \(\lambda ,\,\omega ,\,\theta ,\,B>0\) with \(\lambda ^2\le \theta \omega \le B\). For every \(f\in \mathrm {A}({\mathbb T};\mathrm {L}^{q}(\Omega ))^3\) there exists a solution \((u,\mathfrak {p})\) to (2.2) subject to the estimate
for the constant \({C}_{1}\) from Theorem 2.1, and \(s_1=2q/(2-q)\), \(s_2=4q/(4-q)\). Additionally, if \((w,\mathfrak {q})\) is another solution to (2.2) in the function class defined by the norms on the left-hand side of (2.5), then \(u=w\) and \(\mathfrak {p}=\mathfrak {q}+\mathfrak {q}_0\) for some (spatially constant) function \(\mathfrak {q}_0:{\mathbb T}\rightarrow \mathbb {R}\).
In Sect. 6, we finally prove the following existence result on solutions to the nonlinear system (2.1).
Theorem 2.3
Let \(\Omega \subset \mathbb {R}^3\) be an exterior domain of class \(\mathrm {C}^{3}\), and let \(q\in \big [\frac{12}{11},\frac{4}{3}\big ]\), \(\rho \in \big (\frac{3q-3}{q},1\big )\) and \(\theta >0\). Then there are constants \(\kappa >0\) and \(\lambda _0>0\) such that for all
there exists \(\varepsilon >0\) such that for all \(f\in \mathrm {A}({\mathbb T};\mathrm {L}^{q}(\Omega ))^3\) and \(\alpha \in \mathrm {A}({\mathbb T};\mathbb {R})\) with \(\frac{{\mathrm d}}{{\mathrm d}t}\alpha \in \mathrm {A}({\mathbb T};\mathbb {R})\) and
there is a solution \((u,\mathfrak {p})\) to (2.1) with
Remark 2.4
The lower bound \(\frac{\lambda ^2}{\theta }\le \omega \) on the angular velocity in (2.6) may seem strange in light of the underlying physics of the problem since from a physical point of view, the limit \(\omega \rightarrow 0\) towards the case of a non-rotating body seems uncritical. The lower bound on \(\omega \) in (2.6) is an artifact of the change of coordinates into the rotating frame of reference employed in the mathematical analysis of the problem, which leads to a priori estimates with constants exhibiting a singular behavior as \(\omega \rightarrow 0\). As a consequence, a lower bound on \(\omega \) is required in Theorem 2.3 to obtain existence of a solution via a fixed-point iteration. A similar observation was made in the investigation of a steady flow past a rotating and translating obstacle carried out in [6]. Therefore, it is not surprising to see the same effect appearing in the more general time-periodic case investigated here.
3 Preliminaries
We use capital letters to denote global constants, while constants in small letters are local to the respective proof. When we want to emphasize that a constant C depends on the quantities \(\alpha ,\beta ,\gamma ,\dots \), we write \(C(\alpha ,\beta ,\gamma ,\dots )\).
We denote points in \({\mathbb T}\times \mathbb {R}^3\) by (t, x), where t and \(x=(x_1,x_2,x_3)\) are referred to as time and spatial variable. The symbol \(\Omega \) always denotes an exterior domain, that is, \(\Omega \subset \mathbb {R}^3\) is connected and the complement of a non-empty compact set. We always assume that the origin is not contained in \(\Omega \).
Inner and outer product of two vectors \(a,b\in \mathbb {R}^3\) are denoted by \(a\cdot b\) and \(a\wedge b\), respectively. For any radius \(R>0\) we set \(\mathrm {B}_R:=\bigl \{x\in \mathbb {R}^3\ \big \vert \ {|x |}<R\bigr \}\), \(\mathrm {B}^R:=\bigl \{x\in \mathbb {R}^3\ \big \vert \ {|x |}>R\bigr \}\), and for a domain \(D\subset \mathbb {R}^3\) we define \(D_R:=D\cap \mathrm {B}_R\) and \(D^R:=D\cap \mathrm {B}^R\).
For \(q\in [1,\infty ]\) and \(k\in \mathbb {N}_0\), the symbols \(\mathrm {L}^{q}(D)\) and \(\mathrm {W}^{k,q}(D)\) denote usual Lebesgue and Sobolev spaces with associated norms \(\Vert \cdot \Vert _{q}=\Vert \cdot \Vert _{q;D}\) and \(\Vert \cdot \Vert _{k,q}=\Vert \cdot \Vert _{k,q;D}\), respectively. Furthermore, \(\mathrm {W}^{1,q}_0(D)\) denotes the subset of functions in \(\mathrm {W}^{1,q}(D)\) with vanishing boundary trace, and \(\mathrm {W}^{-1,q}(D)\) (with norm \(\Vert \cdot \Vert _{-1,q;D}\)) is the dual space of \(\mathrm {W}^{1,q'}_0(D)\) where \(1/q+1/q'=1\) with the usual convention \(1/\infty :=0\). Moreover, \(\mathrm {L}^{2}_{\sigma }(D)\) denotes the set of solenoidal vector fields in \(\mathrm {L}^{2}(D)^3\), that is,
and \({\mathcal P}_\mathrm {H}\) is the corresponding Helmholtz projection that maps \(\mathrm {L}^{2}(D)^3\) onto \(\mathrm {L}^{2}_{\sigma }(D)\).
We always identify \(2\pi \)-periodic functions with functions on the torus group \({\mathbb T}:=\mathbb {R}/2\pi \mathbb {Z}\), which is usually represented by the set \([0,2\pi )\). We consider \({\mathbb T}\) and \(G:={\mathbb T}\times \mathbb {R}^3\) as locally compact abelian groups. The (normalized) Haar measure on \({\mathbb T}\) is given by
and \(G\) is equipped with the corresponding product measure. Recall that the dual group of \({\mathbb T}\) can be identified with \(\widehat{{\mathbb T}}=\mathbb {Z}\) and that of \(G\) with \(\widehat{G}:=\mathbb {Z}\times \mathbb {R}^3\).
For \(H={\mathbb T}\) or \(H=G\), the space \(\mathscr {S}(H)\) is the Schwartz–Bruhat space of generalized Schwartz functions on H, and \(\mathscr {S^\prime }(H)\) denotes the corresponding dual space of tempered distributions; see [1, 4] for precise definitions. The Fourier transform on \({\mathbb T}\) and \(G\) and the respective inverses are given by
and
provided the Lebesgue measure \({\mathrm d}\xi \) is correctly normalized. By duality, \(\mathscr {F}_{\mathbb T}\) and \(\mathscr {F}_G\) extend to homeomorphisms \(\mathscr {F}_{\mathbb T}:\mathscr {S^\prime }({\mathbb T})\rightarrow \mathscr {S^\prime }(\mathbb {Z})\) and \(\mathscr {F}_G:\mathscr {S^\prime }(G)\rightarrow \mathscr {S^\prime }(\widehat{G})\), respectively.
Furthermore, we introduce the Sobolev space
where \(\mathrm {C}^{\infty }_0({\mathbb T}\times \overline{D})\) denotes the space of smooth functions of compact support on \({\mathbb T}\times \overline{D}\) .
Let X be a Banach space. We introduce the projections \({\mathcal P}\) and \({\mathcal P}_\bot \) by
for \(u\in \mathrm {L}^{1}({\mathbb T};X)\). Note that \({\mathcal P}u\in X\) is time-independent, and we have the decomposition \(u={\mathcal P}u+{\mathcal P}_\bot u\) into the steady-state part \({\mathcal P}u\) and the purely periodic part \({\mathcal P}_\bot u\) of \(u\).
Our analysis of the time-periodic problems (2.1) and (2.2) will be carried out within spaces of functions with absolutely convergent Fourier series defined by
Observe that \(\mathrm {A}({\mathbb T};X)\) is the Banach space that coincides with \(\mathscr {F}^{-1}_{\mathbb T}\big [\ell ^{1}(\mathbb {Z};X)\big ]\), which embeds into the X-valued continuous functions on \({\mathbb T}\). It is well known that the scalar-valued space \(\mathrm {A}({\mathbb T};\mathbb {R})\) is an algebra with respect to pointwise multiplication, the so-called Wiener algebra. One can exploit this property to derive estimates in the X-valued case. For example, one readily shows the following correspondences of Hölder’s inequality and interpolation inequalities.
Proposition 3.1
Let \(D\subset \mathbb {R}^n\), \(n\in \mathbb {N}\), be an open set and \(p,q,r\in [1,\infty ]\) such that \(1/p+1/q=1/r\). Moreover, let \(f\in \mathrm {A}({\mathbb T};\mathrm {L}^{p}(D))\) and \(g\in \mathrm {A}({\mathbb T};\mathrm {L}^{q}(D))\). Then \(fg\in \mathrm {A}({\mathbb T};\mathrm {L}^{r}(D))\) and
Proof
By assumption we have \(f=\mathscr {F}^{-1}_{\mathbb T}[(f_k)]\) and \(g=\mathscr {F}^{-1}_{\mathbb T}[(g_k)]\) for elements \((f_k)\in \ell ^{1}(\mathbb {Z};\mathrm {L}^{p}(D))\) and \((g_k)\in \ell ^{1}(\mathbb {Z};\mathrm {L}^{q}(D))\). Then \(fg=\mathscr {F}^{-1}_{\mathbb T}\big [(f_k)*_\mathbb {Z}(g_k)\big ]\) and
where the last estimate is due to Hölder’s inequality. \(\square \)
Proposition 3.2
Let \(D\subset \mathbb {R}^n\), \(n\in \mathbb {N}\), be an open set and \(p,q,r\in [1,\infty ]\) such that \((1-\theta )/p+\theta /q=1/r\) for some \(\theta \in [0,1]\), and let \(f\in \mathrm {A}({\mathbb T};\mathrm {L}^{p}(D))\cap \mathrm {A}({\mathbb T};\mathrm {L}^{q}(D))\). Then \(f\in \mathrm {A}({\mathbb T};\mathrm {L}^{r}(D))\) and
Proof
We have \(f=\mathscr {F}^{-1}_{\mathbb T}[(f_k)]\) for an element \((f_k)\in \ell ^{1}(\mathbb {Z};\mathrm {L}^{p}(D)\cap \mathrm {L}^{q}(D))\). The classical interpolation inequality for Lebesgue spaces yields
where the last estimate follows from Hölder’s inequality on \(\mathbb {Z}\). \(\square \)
4 Embedding Theorem
This section deals with embedding properties of Sobolev spaces of time-periodic functions. The embedding theorem below is a refinement of [18, Theorem 4.1] adapted to the time-scaling employed in (2.1). Clearly, embeddings of the steady-state part \({\mathcal P}u\) are independent of the actual period. Therefore, we only consider the case of purely periodic functions. For the sake of generality, we establish the following theorem in arbitrary dimension \(n\ge 2\).
Theorem 4.1
Let \(n\ge 2\), \(\omega >0\) and \(q\in (1,\infty )\). For \(\alpha \in [0,2]\) with \(\alpha q<2\) and \((2-\alpha ) q <n\) let
and for \(\beta \in [0,1]\) with \(\beta q<2\) and \((1-\beta ) q<n\) let
Then the inequality
holds for all \(u\in {\mathcal P}_\bot \mathrm {W}^{1,2,q}({\mathbb T}\times {\mathbb {R}^n})\) and a constant \({C}_{2}={C}_{2}(n,q,\alpha ,\beta )>0\).
Proof
Since the proof is analogue to [18, Proof of Theorem 4.1], we merely give a brief sketch here. Without restriction we may assume \(u\in \mathscr {S}(G)\). Due to the assumption \(u={\mathcal P}_\bot u\), we have \(\mathscr {F}_G[u]=(1-\delta _\mathbb {Z})\mathscr {F}_G[u]\), where \(\delta _\mathbb {Z}\) is the delta distribution on \(\mathbb {Z}\). Utilizing the Fourier transform, we thus derive the identity
where
Employing the so-called transference principle for Fourier multipliers (see [3, 4]) together with the Marcinkiewicz multiplier theorem, one readily verifies that \(M_\omega \) is an \(\mathrm {L}^{q}(G)\) multiplier for any \(q\in (1,\infty )\) such that
with \({c}_{0}\) independent of \(\omega \). Moreover, when we choose \([-\pi ,\pi )\) as a realization of \({\mathbb T}\), we obtain
for some \(h\in \mathrm {C}^{\infty }({\mathbb T})\); see for example [24, Example 3.1.19]. In particular, this yields \(\gamma _\alpha \in \mathrm {L}^{\frac{1}{1-\alpha /2},\infty }({\mathbb T})\), so that Young’s inequality implies that the mapping \(\varphi \mapsto \gamma _\alpha *\varphi \) extends to a bounded operator \(\mathrm {L}^{q}({\mathbb T})\rightarrow \mathrm {L}^{r_0}({\mathbb T})\). Moreover, it is well known that the mapping \(\varphi \mapsto \mathscr {F}^{-1}_{{\mathbb {R}^n}}\big [{|\xi |}^{\alpha -2}\big ]*\varphi \) extends to a bounded operator \(\mathrm {L}^{q}({\mathbb {R}^n})\rightarrow \mathrm {L}^{p_0}({\mathbb {R}^n})\); see [25, Theorem 6.1.13]. Recalling (4.2), we thus have
where Minkowski’s integral inequality is used in the second estimate. This is the asserted inequality for \(u\). The estimate of \(\nabla u\) follows in the same way. \(\square \)
Remark 4.2
Note that the term on the right-hand side of (4.1) defines a norm equivalent to \(\Vert \cdot \Vert _{1,2,q}\) on \({\mathcal P}_\bot \mathrm {W}^{1,2,q}({\mathbb T}\times \Omega )\) due to Poincaré’s inequality on \({\mathbb T}\).
Remark 4.3
Theorem 4.1 can be generalized to the setting of an exterior domain \(\Omega \subset {\mathbb {R}^n}\) by means of Sobolev extensions. However, to maintain estimate (4.1), one has to construct a specific extension operator that respects the homogeneous second-order Sobolev norm. To this end, one can make use of results from [2].
5 Linear Theory
This section is dedicated to the investigation of the resolvent problem (2.3) and the linear time-periodic problem (2.2). After having shown Theorem 2.1, we establish Theorem 2.2 as an immediate consequence hereof.
5.1 The Whole Space
To study the problems (2.2) and (2.3) in an exterior domain, we first consider the case \(\Omega =\mathbb {R}^3\). In this whole-space setting one can namely change coordinates back to the non-rotating inertial frame and thereby reduce the study of (2.2) to an investigation of the time-periodic Oseen problem without rotation terms, which was analyzed in [18, 32]. In this section, we set
for appropriately fixed q.
Theorem 5.1
Let \(q\in (1,2)\) and \(\lambda ,\,\omega ,\,\theta >0\) with \(\lambda ^2\le \theta \omega \). For every \(f\in \mathrm {L}^{q}({\mathbb T}\times \mathbb {R}^3)^3\) there exists a solution \((u,\mathfrak {p})\in \mathscr {S^\prime }({\mathbb T}\times \mathbb {R}^3)^{3+1}\) to
with \(\partial _tu,\nabla ^2u,\,\nabla \mathfrak {p}\in \mathrm {L}^{q}({\mathbb T}\times \mathbb {R}^3)\). Moreover, there exist constants \({C}_{3}={C}_{3}(q)>0\) and \({C}_{4}={C}_{4}(q,\theta )>0\) such that
Additionally, if \((w,\mathfrak {q})\in \mathscr {S^\prime }({\mathbb T}\times \mathbb {R}^3)^{3+1}\) is another solution to (5.1), then \({\mathcal P}_\bot u={\mathcal P}_\bot w\), and \({\mathcal P}u-{\mathcal P}w\) is a polynomial in each component, and \(\mathfrak {p}-\mathfrak {q}=\mathfrak {p}_0\), where \(\mathfrak {p}_0(t,\cdot )\) is a polynomial for each \(t\in {\mathbb T}\).
Proof
We decompose (5.1) into two problems by splitting 
and 
. For the steady-state part \((u_{\mathrm {s}},\mathfrak {p}_{\mathrm {s}})\) we obtain the system
which is the classical steady-state Oseen problem. The existence of a time-independent solution \((u_{\mathrm {s}},\mathfrak {p}_{\mathrm {s}})\) satisfying estimate (5.2) is well known; see for example [13, Theorem VII.4.1]. The remaining purely periodic part \((u_{\mathrm {p}},\mathfrak {p}_{\mathrm {p}})\) must solve (5.1), but with purely periodic right-hand side \({\mathcal P}_\bot f\). We define
which leads to the system
where \(\widetilde{\lambda }=\lambda \omega ^{-1/2}\). From [32, Theorem 2.1] we conclude the existence of a unique solution \((U,\mathfrak {P})\) that satisfies the estimate
where \({c}_{0}\) is a polynomial in \(\widetilde{\lambda }\) and can thus be bounded uniformly in \(\widetilde{\lambda }\in (0,\sqrt{\theta }]\). Estimate (5.3) with the asserted dependency of the constant \({C}_{4}\) follows after reversing the applied scaling.
The uniqueness statement is readily shown by means of the Fourier transform on \(G={\mathbb T}\times \mathbb {R}^3\). We consider (5.1) with \(f=0\) and apply the divergence operator to (5.1)\(_{1}\). This yields \(\Delta \mathfrak {p}=0\) and thus \({|\xi |}^2\mathscr {F}_{\mathbb {R}^3}[\mathfrak {p}(t,\cdot )]=0\) for all \(t\in {\mathbb T}\). Therefore, we obtain \({{\,\mathrm{supp}\,}}\mathscr {F}_{\mathbb {R}^3}[\mathfrak {p}(t,\cdot )] \subset \{0\}\), so that \(\mathfrak {p}(t,\cdot )\) is a polynomial for all \(t\in {\mathbb T}\). Next we apply the Fourier transform to (5.1)\(_{1}\) to deduce \((i\omega k+{|\xi |}^2-i\xi _1)\mathscr {F}_G[u]+i\xi \mathscr {F}_G[\mathfrak {p}]=0\). Multiplying with the symbol of the Helmholtz projection \(\mathrm {I}-\xi \otimes \xi /{|\xi |}^2\) and utilizing \({{\,\mathrm{div}\,}}u=0\), we obtain \((i\omega k+{|\xi |}^2-i\xi _1)\mathscr {F}_G[u]=0\), which yields \({{\,\mathrm{supp}\,}}\mathscr {F}_G[u]\subset \{(0,0)\}\). Since \({{\mathcal P}_\bot u}=\mathscr {F}^{-1}_G\big [(1-\delta _\mathbb {Z})\mathscr {F}_G[u]\big ]\), it follows that \({\mathcal P}_\bot u=0\), and that each component of \({\mathcal P}u\) is a polynomial. This completes the proof. \(\square \)
Remark 5.2
In the setting of Theorem 5.1 we can write the estimate for the steady-state part \((u_{\mathrm {s}},\mathfrak {p}_{\mathrm {s}})=({\mathcal P}u,{\mathcal P}\mathfrak {p})\) and the purely periodic part \((u_{\mathrm {p}},\mathfrak {p}_{\mathrm {p}})=({\mathcal P}_\bot u,{\mathcal P}_\bot \mathfrak {p})\) in a more condensed way: From the embeddings established in Theorem 4.1 we deduce
Recalling Remark 4.2, we see that (5.2) and (5.3) can be formulated as
for a constant \(C_{6}=C_{6}(q,\theta )\) as long as \(\lambda ^2\le \theta \omega \).
With Theorem 5.1 we now solve the linear problem (2.2) for \(\Omega =\mathbb {R}^3\) and \(f\in \mathrm {L}^{q}({\mathbb T}\times \mathbb {R}^3)^3\).
Theorem 5.3
Let \(q\in (1,2)\) and \(\lambda ,\,\omega , \,\theta >0\) with \(\lambda ^2\le \theta \omega \). For every \(f\in \mathrm {L}^{q}({\mathbb T}\times \mathbb {R}^3)^3\) there exists a solution \((u,\mathfrak {p})\in \mathscr {S^\prime }({\mathbb T}\times \mathbb {R}^3)^{3+1}\) to
with \(\nabla ^2u,\,\partial _1u,\,\nabla \mathfrak {p}\in \mathrm {L}^{q}({\mathbb T}\times \mathbb {R}^3)\). Moreover, there exists a constant \({C}_{7} ={C}_{7}(q,\theta )>0\) such that
Additionally, if \((w,\mathfrak {q})\in \mathscr {S^\prime }({\mathbb T}\times \mathbb {R}^3)^{3+1}\) is another solution to (5.5) with \(w\in \mathrm {L}^{r}({\mathbb T}\times \mathbb {R}^3)\) for some \(r\in [1,\infty )\), then \(u=w\), and \(\mathfrak {p}-\mathfrak {q}=\mathfrak {q}_0\) for some spatially constant function \(\mathfrak {q}_0:{\mathbb T}\rightarrow \mathbb {R}\).
Proof
Let
be the matrix corresponding to the rotation with angular velocity \({{\,\mathrm{e}\,}}_1\). Define
with the new spatial variable \(y=Q(t)x\). Due to
the functions \(u\), \(\mathfrak {p}\) and f satisfy (5.5) if and only if
The assertions in Theorem 5.3 are now a direct consequence of Theorem 5.1 and estimate (5.4). \(\square \)
Remark 5.4
As for the corresponding steady-state problem (see for example [13, Theorem VIII.8.1]), one can extend Theorem 5.3 to the case of an exterior domain \(\Omega \) for \(f\in \mathrm {L}^{q}({\mathbb T}\times \Omega )\), but it is not clear to the authors whether or not the constant in the resulting a priori estimate can then be chosen independently of \(\lambda \) and \(\omega \). Observe that such an independence is obtained in the functional setting of Theorem 2.2 where \(f\in \mathrm {A}({\mathbb T};\mathrm {L}^{q}(\Omega ))\). Since we solve the nonlinear problem (2.1) via a fixed-point iteration which requires \(\lambda \) and \(\omega \) to be chosen sufficiently small, it crucial to obtain an estimate with the constant independent of \(\lambda \) and \(\omega \).
From Theorem 5.3 we can extract a similar result for the resolvent problem (2.3) in the whole space.
Theorem 5.5
Let \(q\in (1,2)\), \(k\in \mathbb {Z}\) and \(\lambda ,\,\omega , \,\theta >0\) with \(\lambda ^2\le \theta \omega \). For every \(F\in \mathrm {L}^{q}(\mathbb {R}^3)^3\) there exists a solution \((v,p)\in \mathscr {S^\prime }(\mathbb {R}^3)^{3+1}\) to
and a constant \({C}_{8} ={C}_{8}(q,\theta )>0\) with
Additionally, if \((w,\mathfrak {q})\in \mathscr {S}(\mathbb {R}^3)^{3+1}\) is another solution to (5.1) with \(w\in \mathrm {L}^{r}(\Omega )\) for some \(r\in [1,\infty )\), then \(v=w\), and \(p-\mathfrak {q}\) is constant.
Proof
First consider a solution \((v,p)\) in the described function class. Then the fields
satisfy (5.5). Therefore, uniqueness of \((v,\nabla p)\) follows from the uniqueness statement in Theorem 5.3. To show existence, let \(F\in \mathrm {L}^{q}(\mathbb {R}^3)\) and define \(f\in \mathrm {L}^{q}({\mathbb T}\times \mathbb {R}^3)\) as above. Theorem 5.3 yields the existence of a pair \((u,\mathfrak {p})\) that solves (5.5). Then the k-th Fourier coefficients \(v(x):=\mathscr {F}_{\mathbb T}[u(\cdot ,x)](k)\) and \(p(x):=\mathscr {F}_{\mathbb T}[\mathfrak {p}(\cdot ,x)](k)\) satisfy (5.7), and estimate (5.8) is a direct consequence of (5.6). \(\square \)
5.2 Uniqueness
Next we show a uniqueness result for the resolvent problem (2.3).
Lemma 5.6
Let \(\lambda \ge 0\), \(\omega >0\), \(k\in \mathbb {Z}\), and let \((v,p)\) be a distributional solution to (2.3) with \(F=0\) and \(\nabla ^2v,\,\partial _1v,\,\nabla p\in \mathrm {L}^{q}(\Omega )\) for some \(q\in (1,\infty )\) and \(v\in \mathrm {L}^{s}(\Omega )\) for some \(s\in (1,\infty )\). Then \(v=0\) and \(p\) is constant.
Proof
We only consider the case \(\lambda >0\) here. The proof for \(\lambda =0\) can be shown in exactly the same way. Fix a radius \(R>0\) such that \(\partial \mathrm {B}_R\subset \Omega \), and define a “cut-off” function \(\chi _0\in \mathrm {C}^{\infty }_0(\mathbb {R}^3)\) with \(\chi _0(x)=1\) for \({|x |}\le 2R\) and \(\chi _0(x)=0\) for \({|x |}\ge 4R\). Set
where \(\mathfrak {B}\) denotes the Bogovskiĭ operator; see for example [13, Section III.3]. Then
with
From the assumptions, we obtain \(v\in \mathrm {W}^{2,q}(\Omega _{4R})\) and \(p\in \mathrm {W}^{1,q}(\Omega _{4R})\). Standard Sobolev embeddings imply \(v, \nabla v, p\in \mathrm {L}^{\frac{3}{2} q}(\Omega _{4R})\). Therefore, we also have \(h\in \mathrm {L}^{r}(\Omega _{4R})\) for all \(1< r\le \frac{3}{2}q\). From well-known regularity results for the Stokes problem in bounded domains (see [13, Theorem IV.6.1]) we obtain \(w\in \mathrm {W}^{2,r}(\Omega _{4R})\) and \(\nabla \mathfrak {q}\in \mathrm {L}^{r}(\Omega _{4R})\). Since \(v=w\) and \(p=\mathfrak {q}\) on \(\Omega _{2R}\), this yields
for all \(1< r\le \frac{3}{2}q\).
Next consider another “cut-off” function \(\chi _1\in \mathrm {C}^{\infty }(\mathbb {R}^3)\) with \(\chi _1(x)=1\) for \({|x |}\ge 2R\) and \(\chi _1(x)=0\) for \({|x |}\le R\). As above, we define
which satisfy the system
with
As above, we see \(f\in \mathrm {L}^{r}(\mathbb {R}^3)\) for all \(1< r\le \frac{3}{2}q\). Since we also have \(u\in \mathrm {L}^{s}(\mathbb {R}^3)\), Theorem 5.5 implies
if additionally \(r<2\). Due to \(v=u\) and \(p=\mathfrak {p}\) on \(\mathrm {B}^{2R}\), we have
for \(1< r\le \frac{3}{2}q\) with \(r<2\).
We combine (5.10) and (5.13) to deduce
for \(1<r\le \frac{3}{2}q\) with \(r<2\). After repeating the above argument a sufficient number of times, we obtain (5.14) for all \(r\in (1,2)\). Since \(v\in \mathrm {L}^{s}(\Omega )\), the Sobolev inequality further yields
In particular, we can employ the divergence theorem to compute
for any \(R>0\) with \(\partial \mathrm {B}_R\subset \Omega \). Passing to the limit \(R\rightarrow \infty \), we obtain
By the above integrability properties, we can further multiply (2.3)\(_{1}\) by \(v\) and integrate over \(\Omega \). By utilizing (5.15) and integration by parts, we conclude
This implies \(\nabla v=0\). The imposed boundary conditions thus yield \(v=0\). Finally, (2.3)\(_{1}\) leads to \(\nabla p=0\), and the proof is complete. \(\square \)
5.3 A Priori Estimate
Next we establish an a priori estimate for the solution to the resolvent problem (2.3).
Lemma 5.7
Let \(q\in (1,2)\), \(k\in \mathbb {Z}\) and \(\lambda ,\,\omega , \,\theta >0\) with \(\lambda ^2\le \theta \omega \). Moreover, let \(F\in \mathrm {L}^{q}(\Omega )\) and \(R>0\) such that \(\partial \mathrm {B}_R\subset \Omega \). Let \((v,p)\in \mathrm {L}^{1}_{\mathrm {loc}}(\Omega )\) with
be a solution to (2.3). Then there exists a constant \({C}_{9} ={C}_{9}(\Omega ,q,\theta ,R)>0\) such that
Proof
Let \(\chi _0\), \(\chi _1\) be the “cut-off” functions from the proof of Lemma 5.6. Define \(w\in \mathrm {W}^{2,q}(\Omega )\) and \(\mathfrak {q}\in \mathrm {W}^{1,q}(\Omega )\) as in (5.9). Then
with
Well-known theory for the Stokes resolvent problem (see for example [9]) yields
In the last estimate we used mapping properties of the Bogovskiĭ operator (see [13, Section III.3]), namely
for \(m\in \mathbb {N}_0\), where
To estimate the last term in (5.18), we introduce the notation
for \(\psi \in \mathrm {C}^{\infty }_0(\mathbb {R}^3)\), and we employ that \({{\,\mathrm{div}\,}}v=0\) in \(\Omega \) and \(v=0\) on \(\partial \Omega \) to deduce the identity
Since Poincaré’s inequality yields
we have
Applying this estimate to the last term in (5.18), we obtain
Next define (\(u,\mathfrak {p})\) as in (5.11), which satisfies the system
with
Theorem 5.5 implies
where we estimated the terms containing the Bogovskiĭ operator as above. Combining this estimate with (5.19), we conclude (5.17). \(\square \)
In the next step we improve estimate (5.17) by showing that the lower-order terms on the right-hand side can be omitted. This leads to the desired estimate (2.4) with the asserted dependencies of the constant \({C}_{1}\).
Lemma 5.8
Let \(q\in (1,2)\), \(k\in \mathbb {Z}\) and \(\lambda ,\,\omega >0\), and let \(F\in \mathrm {L}^{q}(\Omega )\). Let \((v,p)\in \mathrm {L}^{1}_{\mathrm {loc}}(\Omega )\) be a solution to (2.3) in the class (5.16). Then estimate (2.4) holds for a constant \({C}_{1} ={C}_{1}(\Omega ,q,\lambda ,\omega )>0\). If \(q\in (1,\frac{3}{2})\) and \(\lambda ^2\le \theta \omega \le B\), then this constant can be chosen independently of \(\lambda \) and \(\omega \) such that \({C}_{1} ={C}_{1}(\Omega ,q,\theta ,B)\).
Proof
We employ a contradiction argument. At first, consider the case \(q\in (1,\frac{3}{2})\) and assume that (2.4) is not valid for a constant \({C}_{1}={C}_{1}(\Omega ,q,\theta ,B)\). Then there exist sequences of numbers \((\lambda _j)\subset (0,\sqrt{B}]\), \((\omega _j)\subset (0,B/\theta ]\) with \(\lambda _j^2\le \theta \omega _j\), and \((k_j)\subset \mathbb {Z}\), and of functions \((v_j)\), \((p_j)\), \((F_j)\) that satisfy
\(\Vert F_j\Vert _{q}\rightarrow 0\) as \(j\rightarrow \infty \), and
for all \(j\in \mathbb {N}\). Furthermore, without loss of generality, we assume \(\int _{\Omega _R}p_j\,{\mathrm d}x=0\) for \(R>0\) as in Lemma 5.7. Then, \((\lambda _j)\), \((\omega _j)\) and \((k_j)\) contain (improper) convergent subsequences with limits \(\lambda \in [0,\sqrt{B}]\), \(\omega \in [0,B/\theta ]\) and \(k\in \mathbb {Z}\cup \{\pm \infty \}\), respectively, and we have \(\lambda ^2\le \theta \omega \). For simplicity, we identify selected subsequences with the actual sequences. Moreover, (5.20) implies that \(U_j:=(i\omega _j k_jv_j,v_j,p_j)\) is bounded in \(\mathrm {L}^{q}(\Omega _\rho )\times \mathrm {W}^{2,q}(\Omega _\rho )\times \mathrm {W}^{1,q}(\Omega _\rho )\) for any \(\rho >R\). Hence, by a Cantor diagonalization argument, there exists a subsequence that converges weakly in \(\mathrm {L}^{q}(\Omega _\rho )\times \mathrm {W}^{2,q}(\Omega _\rho )\times \mathrm {W}^{1,q}(\Omega _\rho )\) to some \(U:=(w,v,p)\) for each \(\rho >R\). Consequently, passing to the limit \(j\rightarrow \infty \) in (5.21), we obtain
Moreover, by the compact embeddings
we deduce that \(U\) is the strong limit of \((U_j)\) in the topology of \(\mathrm {W}^{-1,q}(\Omega _{4R})\times \mathrm {W}^{1,q}(\Omega _{4R})\times \mathrm {L}^{q}(\Omega _{4R})\). By Lemma 5.7,
Passing to the limit \(j\rightarrow \infty \) in this estimate, we conclude in virtue of (5.20) that
Moreover,
Now we distinguish between several cases:
-
i.
If \(\omega _j k_j\rightarrow s\in \mathbb {R}\) and \(\omega =0\), then \(\lambda =0\) and \(w=isv\), so that (5.22) reduces to a Stokes resolvent problem. If \(s\ne 0\), we also have \(v\in \mathrm {L}^{q}(\Omega )\) and we conclude \(v=\nabla p=0\) from a well-known uniqueness result; see for example [9]. If \(s=0\), we utilize that \(q<\frac{3}{2}\) and \(v_j\in \mathrm {L}^{s_1}(\Omega )\), \(\nabla v_j\in \mathrm {L}^{s_2}(\Omega )\), so that Sobolev’s inequality implies
$$\begin{aligned} \Vert v_j\Vert _{3q/(3-2q)}\le {c}_{0} \Vert \nabla v_j\Vert _{3q/(3-q)}\le {c}_{1}\Vert \nabla ^2v_j\Vert _{q}, \end{aligned}$$and thus \(v\in \mathrm {L}^{3q/(3-2q)}(\Omega )\). Now \(v=\nabla p=0\) follows from classical uniqueness properties of the steady-state Stokes problem, see for example [13, Theorem V.4.6].
-
ii.
If \(\omega _j k_j\rightarrow s\in \mathbb {R}\) and \(\omega \ne 0\) but \(\lambda =0\), then \(k_j\rightarrow k\in \mathbb {Z}\) and \(w=i\omega kv\), so that (5.22) reduces to (2.3) with \(\lambda =0\). As above, we deduce \(v\in \mathrm {L}^{3q/(3-2q)}(\Omega )\). From Lemma 5.6 we conclude \(v=\nabla p=0\).
-
iii.
If \(\omega _j k_j\rightarrow s\in \mathbb {R}\) and \(\omega \ne 0\) and \(\lambda \ne 0\), then \(k_j\rightarrow k\in \mathbb {Z}\) and \(w=i\omega kv\), so that \((v,p)\) satisfies (2.3). Since \(\lambda \ne 0\), it follows from (5.24) that \(v\in \mathrm {L}^{s_1}(\Omega )\). Lemma 5.6 thus implies \(v=\nabla p=0\).
-
iv.
If \(\omega _j {|k_j |}\rightarrow \infty \), we recall (5.20) and estimate
$$\begin{aligned} \omega _j{|k_j |}\Vert v_j\Vert _{q;\Omega _\rho } \le \omega _j\Vert ik_j v_j + {{\,\mathrm{e}\,}}_1\wedge v_j - {{\,\mathrm{e}\,}}_1\wedge x\cdot \nabla v_j \Vert _{q;\Omega _\rho } +{c}_{2}(\rho )\Vert v_j\Vert _{1,q;\Omega _\rho } \le {c}_{3}(\rho ) \end{aligned}$$for any \(\rho >R\). Passing to the limit \(j\rightarrow \infty \), we thus obtain \(v=0\) on \(\Omega _\rho \) for each \(\rho >R\), whence \(v= 0\) on \(\Omega \). Hence, (5.22)\(_{1}\) reduces to \(w+\nabla p=0\). Clearly, we also have \({{\,\mathrm{div}\,}}w=0\) and \(w\big | _{\partial \Omega }=0\), so that \(w+\nabla p=0\) corresponds to the Helmholtz decomposition of 0 in \(\mathrm {L}^{q}(\Omega )\). Since this decomposition is unique, we conclude \(w=\nabla p=0\).
Consequently, all four cases lead to \(w=v=\nabla p=0\), which contradicts (5.23). This completes the proof in the case \(1<q<\frac{3}{2}\).
In the more general case \(q\in (1,2)\), where we do not assert the constant \({C}_{1}\) to be independent of \(\lambda \) and \(\omega \), these parameters remain fixed in the contradiction argument above. Consequently, only the last two cases above have to be considered. The conclusion in both of these cases is valid for all \(q\in (1,2)\), and we thus conclude the lemma. \(\square \)
5.4 Existence
To complete the proof of Theorem 2.1, it remains to show existence of a solution. For this purpose, recall the following property of the Stokes operator.
Lemma 5.9
Let \(D\subset \mathbb {R}^3\) be a bounded domain with \(\mathrm {C}^{3}\)-boundary. Every \(u\in \mathrm {L}^{2}_{\sigma }(D)\cap \mathrm {W}^{1,2}_0(D)\cap \mathrm {W}^{2,2}(D)\) satisfies
for a constant \({C}_{10}={C}_{10}(D)>0\) that does not depend on the “size” of D but solely on its “regularity”. In particular, if \(D=\Omega _R\) for an exterior domain \(\Omega \) with \(\partial \Omega \subset \mathrm {B}_R\), the constant \({C}_{10}\) is independent of R and solely depends on \(\Omega \).
Proof
See [26, Lemma 1]. \(\square \)
We further need the following identity from [20].
Lemma 5.10
Let \(u\in \mathrm {L}^{2}_{\sigma }(\Omega _R)\cap \mathrm {W}^{1,2}_0(\Omega _R)\cap \mathrm {W}^{2,2}(\Omega _R)\) with complex conjugate \(u^*\). Then \( {{\,\mathrm{e}\,}}_1\wedge u - {{\,\mathrm{e}\,}}_1\wedge x\cdot \nabla u \in \mathrm {L}^{2}_{\sigma }(\Omega _R)\) and
Proof
See [20, Lemma 3]. \(\square \)
Existence of a solution to the resolvent problem (2.3) can be shown via a Galerkin approach combined with an “invading domains” technique.
Lemma 5.11
Let \(\Omega \subset \mathbb {R}^3\) be an exterior domain of class \(\mathrm {C}^{3}\). Let \(\lambda ,\,\omega >0\), \(k\in \mathbb {Z}\), and let \(F\in \mathrm {C}^{\infty }_0(\Omega )\). Then there exists a solution \((v,p)\) to (2.3) with
for all \(q\in (1,2)\).
Proof
Let \(R>0\) such that \(\partial \mathrm {B}_R\subset \Omega \), and take \(m\in \mathbb {N}\) with \(m>2R\). Since the Stokes operator in the bounded domain \(\Omega _m\) is a positive self-adjoint invertible operator (see [50, Chapter III, Theorem 2.1.1]), there exist sequences \((\psi _j)_{j\in \mathbb {N}}\) of (real-valued) eigenfunctions and \((\mu _j)_{j\in \mathbb {N}}\subset (0,\infty )\) of eigenvalues, that is,
normalized such that
We show the existence of a function \(u=u^m_n\in X^m_n:={{\,\mathrm{span}\,}}_{\mathbb {C}}\bigl \{\psi _j\ \big \vert \ j=1,\ldots ,n\bigr \}\) satisfying
for all \(j\in \{1,\ldots ,n\}\). Since
for some \(\xi _1,\ldots ,\xi _n\in \mathbb {C}\), this is equivalent to solving the algebraic equation
with \(\xi =(\xi _1,\ldots ,\xi _n) \in \mathbb {C}^n\) and \(M=(M_{\ell j})\in \mathbb {C}^{n\times n}\), \(c=(c_j)\in \mathbb {C}^n\) with
Note that (5.26) is a resolvent problem for the skew-Hermitian matrix M, which is uniquely solvable. Existence of a unique solution \(u=u^m_n\in X^m_n\) to (5.25) thus follows.
Next we need suitable estimates for \(u=u^m_n\). Multiplication of both sides of (5.25) by the complex conjugate coefficient \(\xi _j^*\) and summation over \(j=1,\dots ,n\) yields
Because the integral term on the left-hand side is purely imaginary, taking the real part of this equation leads to the estimate
Recalling the Sobolev inequality \(\Vert u\Vert _{6}\le {c}_{0}\Vert \nabla u\Vert _2\), we obtain
where \({c}_{1} \) is independent of m. If we multiply both sides of (5.25) by \(\mu _j\xi _j^*\) and sum over \(j=1,\dots ,n\), we obtain
Taking real part of both sides and observing that
we conclude, using Hölder’s inequality, the estimate
Using Lemma 5.10, we estimate the remaining integral on the right-hand side to conclude
with \({c}_{2}\) independent of m. Employing the trace inequality [13, Theorem II.4.1] on the domain \(\Omega _{R}\), we further estimate
for small \(\varepsilon >0\). From Lemma 5.9 we deduce
with a constant \({c}_{6} >0\) independent of m. Combining this estimate with (5.28), choosing \(\varepsilon \) sufficiently small and employing estimate (5.27), we arrive at
Using Lemma 5.9 and estimate (5.27) once again and restoring the original notation, we end up with
with \({c}_{9}\) independent of m.
In particular, we see from (5.27), (5.29) and Poincaré’s inequality that \((u^m_n)\) is uniformly bounded in \(\mathrm {W}^{2,2}(\Omega _m)\) and thus contains a subsequence that converges weakly to some function \(v^m\in \mathrm {L}^{2}_{\sigma }(\Omega _m)\cap \mathrm {W}^{1,2}_0(\Omega _m)\cap \mathrm {W}^{2,2}(\Omega _m)\), which obeys the estimate
with \({c}_{10}\) independent of m. Moreover, \(v^m\) satisfies (5.25) for all \(j\in \mathbb {N}\), whence there exists \(p^m\in \mathrm {W}^{1,2}(\Omega _m)\) such that
see [13, Corollary III.5.1]. Since \( {{\,\mathrm{e}\,}}_1\wedge v^m - {{\,\mathrm{e}\,}}_1\wedge x\cdot \nabla v^m \in \mathrm {L}^{2}_{\sigma }(\Omega _m)\) by Lemma 5.10, we deduce from (5.31) and (5.30) the estimate
Combining the estimate above with (5.30), we conclude
with \({c}_{12}\) independent of m.
Now we introduce a sequence of rotationally symmetric “cut-off” functions \((\chi _m)\subset \mathrm {C}^{\infty }_0(\mathbb {R}^3)\) satisfying
and we set \(w^m:=\chi _mv^m\). Then \(w^m\) is an element of \(\mathrm {W}^{2,2}(\Omega )\). Moreover, the rotational symmetry of \(\chi _m\) implies \({{\,\mathrm{e}\,}}_1\wedge x\cdot \nabla \chi _m=0\). Therefore, from (5.32) and the properties of \(\chi _m\), we deduce the estimate
with \({c}_{15}\) independent of m. This implies the existence of a subsequence, still denoted by \((w^m)\), that converges in the sense of distributions to some function \(v\in \mathrm {W}^{2,2}_{\mathrm {loc}}(\Omega )\) that satisfies
Moreover, \(v\big | _{\partial \Omega }=0\). Let \(\varphi \in \mathrm {C}^{\infty }_0(\Omega )\). We choose \(m_0\in \mathbb {N}\) such that \({{\,\mathrm{supp}\,}}\varphi \) is contained in \(\Omega _{m_0/2}\). For \(m\ge m_0\) we have \(w^m=v^m\) on \(\Omega _{m_0/2}\) and thus
by (5.31)\(_{2}\). Passing to the limit \(m\rightarrow \infty \), we conclude \({{\,\mathrm{div}\,}}v=0\). Now let \(\psi \in \mathrm {C}^{\infty }_{0,\sigma }(\Omega )\) and choose \(m_0\) such that \({{\,\mathrm{supp}\,}}\psi \subset \Omega _{m_0/2}\). With the same argument as above, for \(m\ge m_0\) we obtain from (5.31)\(_{1}\) that
Therefore, by passing to the limit \(m\rightarrow \infty \), we see
for all \(\psi \in \mathrm {C}^{\infty }_{0,\sigma }(\Omega )\). Consequently, by the Helmholtz decomposition, there exists a function \(p\) with \(\nabla p\in \mathrm {L}^{2}(\Omega )\) such that \((v,p)\) is a solution to (2.3).
It remains to show that \(v\) and \(p\) belong to the correct function spaces. By Hölder’s inequality, we directly find that
for any \(\rho >R\) and all \(q\in [1,2]\). Repeating the “cut-off” argument from (5.11), we obtain \((u,\mathfrak {p})\) which satisfy (5.12) for some function \(f\in \mathrm {L}^{2}(\mathbb {R}^3)\) with compact support. In particular, this implies \(f\in \mathrm {L}^{q}(\mathbb {R}^3)\) for all \(q\in (1,2)\). Theorem 5.5 yields existence of a solution to (5.12) satisfying (5.8). Since \(u\in \mathrm {L}^{6}(\mathbb {R}^3)\), Theorem 5.5 further ensures that \((u,\mathfrak {p})\) coincides with this solution. We thus have
Since \(v=u\) and \(p=\mathfrak {p}\) on \(\mathrm {B}^{2R}\), the integrability properties above in combination with (5.34) show that \(v\) and \(p\) belong to the correct function spaces. \(\square \)
Combining Lemmas 5.6, 5.8 and 5.11, we can finally complete the proof of Theorem 2.1.
Proof of Theorem 2.1
The uniqueness statement is a direct consequence of Lemma 5.6. Estimate (2.4) has been proved in Lemma 5.8. It thus remains to show existence of a solution for \(F\in \mathrm {L}^{q}(\Omega )\). Consider a sequence \((F_j)\subset \mathrm {C}^{\infty }_0(\Omega )\) that converges to \(F\) in \(\mathrm {L}^{q}(\Omega )\). By Lemma 5.11, for each \(j\in \mathbb {N}\) there exists a solution \((v,p)=(v_j,p_j)\) to (2.3) with \(F=F_j\), which obeys estimate (2.4) by Lemma 5.8. Additionally, this implies that \((v_j,\nabla p_j)\) is a Cauchy sequence in the function space defined by the norm on the left-hand side of (2.4), and thus possesses a limit \((v,\nabla p)\), which satisfies (2.3) and (2.4). \(\square \)
5.5 The Time-Periodic Linear Problem
Proof of Theorem 2.2
An application of the Fourier transform \(\mathscr {F}_{\mathbb T}\) on \({\mathbb T}\) to (2.2) reduces the uniqueness statement to the corresponding uniqueness result for the resolvent problem established in Theorem 2.1. To show existence, consider \(f\in \mathrm {A}({\mathbb T};\mathrm {L}^{q}(\Omega ))\). Then
with \(f_k\in \mathrm {L}^{q}(\Omega )\). Let \((u_k,\mathfrak {p}_k)=(v,p)\) be a solution to the resolvent problem (2.3) with \(F=f_k\) that exists due to Theorem 2.1. We define
By (2.4), \(u\) and \(\mathfrak {p}\) are well defined and satisfy (2.2). We directly conclude estimate (2.5) from estimate (2.4). \(\square \)
6 The Nonlinear Problem
We return to the nonlinear problem (2.1). At first, we reformulate it as a problem with homogeneous boundary conditions. To this end, fix \(R>0\) such that \(\partial \mathrm {B}_R\subset \Omega \). Let \(\varphi \in \mathrm {C}^{\infty }_0(\mathbb {R}^3)\) be a smooth function satisfying \(\varphi (x)=1\) if \({|x |}<R\), and \(\varphi (x)=0\) if \({|x |}>2R\), and define
Then \(U(t,\cdot )\in \mathrm {C}^{\infty }_0(\mathbb {R}^3)^3\) for all \(t\in {\mathbb T}\), \(U\in \mathrm {C}^{1}({\mathbb T}\times \mathbb {R}^3)\), \({{\,\mathrm{div}\,}}U=0\), and a brief calculation shows \(U(t,x)=\alpha (t){{\,\mathrm{e}\,}}_1+\omega {{\,\mathrm{e}\,}}_1\wedge x\) for \((t,x)\in {\mathbb T}\times \partial \Omega \). Now define \(v:=u-U\) and \(p:=\mathfrak {p}\). Then \((u,\mathfrak {p})\) solves (2.1) if and only if \((v,p)\) solves
where
Recall that \({\mathcal P}_\bot \alpha =\alpha -\lambda \). It thus remains to show existence of a solution to the nonlinear system (6.1).
Proof of Theorem 2.3
We define the function space
where \(s_1=2q/(2-q)\), \(s_2=4q/(4-q)\) and
At first, we derive suitable estimates of \({\mathcal N}(v)\). For example, analogously to the proof of Proposition 3.1, we have
Moreover, since \(\frac{2q}{2-q}\le 4\le \frac{3q}{3-2q}\), we can employ estimates (3.2) and (3.3) to obtain
with \(\theta =\frac{12-9q}{2q}\). By the Sobolev inequality we thus deduce
The remaining terms in \({\mathcal N}(v)\) can be estimated in a similar fashion, which results in
Now consider the problem
for given \(v\in {\mathcal X}^q\). Due to estimate (6.2) and Theorem 2.2 there exists a unique velocity field \(w\in {\mathcal X}^q\) and a pressure field \(\mathfrak {q}\) with \(\nabla \mathfrak {q}\in \mathrm {A}^q\) that satisfy (6.3) and the estimate
We thereby obtain a solution map \({\mathcal S}:{\mathcal X}^q\rightarrow {\mathcal X}^q\), \(v\mapsto w\) which is a self-mapping on the ball
provided
Recall that \(\rho \in \big (\frac{3q-3}{q},1\big )\). Choosing \(\delta :=\lambda ^\rho \), one readily verifies that there is a constant \(\kappa >0\) depending on \({c}_{4}\) such the condition above is satisfied with \(\omega \Vert \tfrac{{\mathrm d}}{{\mathrm d}t}\alpha \Vert _{\mathrm {A}({\mathbb T};\mathbb {R})} \le \kappa \lambda ^\rho \), \(\varepsilon =\lambda ^2\) and \(\lambda _0\) sufficiently small. In the same way, one derives the estimate
which ensures that \({\mathcal S}\) is a contraction on \({\mathcal X}^q_\delta \) with a similar choice of parameters. Finally, the contraction mapping principle yields the existence of a fixed point \(v\in {\mathcal X}^q\) of \({\mathcal S}\), and hence of a solution \((v,p)\) to (6.1). Consequently, \((u,\mathfrak {p}):=(v+U,p)\) is a solution to (2.1). \(\square \)
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Eiter, T., Kyed, M. Viscous Flow Around a Rigid Body Performing a Time-periodic Motion. J. Math. Fluid Mech. 23, 28 (2021). https://doi.org/10.1007/s00021-021-00556-4
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DOI: https://doi.org/10.1007/s00021-021-00556-4