Viscous Flow past a Body Translating by Time-Periodic Motion with Zero Average

TL;DR

This paper rigorously analyzes the Navier-Stokes flow around a body with time-periodic, zero-average motion, demonstrating the existence of a steady streaming phenomenon where oscillatory effects decay with distance, revealing a steady-state behavior far from the body.

Contribution

It provides a rigorous mathematical proof of the steady streaming phenomenon for viscous flow past a body with time-periodic motion and zero average.

Findings

Flow exhibits steady-state characteristics far from the body.

Oscillatory components decay faster than the steady component.

Rigorous proof of steady streaming phenomenon.

Abstract

We study existence, uniqueness, regularity and asymptotic spatial behavior of a Navier-Stokes flow past a body moving by a time-periodic translational motion of period $T$, and with zero average. For example, $\mathscr B$ moves in an oscillating fashion. The flow is also time-periodic with same period $T$. However, sufficiently ``far" from the body, the oscillatory component decays faster than the averaged component, so that the flow shows there a distinctive steady-state character. This provides a rigorous proof of the ``steady streaming" phenomenon.

Equations

$$\begin{array}[]{cc}\vskip 3.0pt plus 1.0pt minus 1.0pt\left.\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt{\partial}_{t}{\mbox{\boldmath$u$}}-\mbox{\boldmath$\xi$}(t)\cdot\nabla{\mbox{\boldmath$u$}}+{\mbox{\boldmath$u$}}\cdot\nabla{\mbox{\boldmath$u$}}=\Delta{\mbox{\boldmath$u$}}-\nabla{p}+{\mbox{\boldmath$b$}}\\ \mbox{\rm div}\,{\mbox{\boldmath$u$}}=0\end{array}\right\}\ \ \mbox{in $\Omega\times(-\infty,\infty)$}\\ {\mbox{\boldmath$u$}}(x,t)=\mbox{\boldmath$\xi$}(t)\,,\ \ (x,t)\in\partial\Omega\times(-\infty,\infty)\,.\end{array}$$

$${\mbox{\boldmath$u$}}(x,t)={\mbox{\boldmath$U$}}(x)+\mbox{\boldmath$\sigma$}(x)+{\mbox{\boldmath$w$}}(x,t).$$

$$\overline{u}:=\frac{1}{T}{\int }_0^{T}u(t)dt.$$

$$\|u\|_{L^{r}(B)}\equiv\left\{\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\big{(}{\displaystyle\int_{0}^{T}}\|u(t)\|_{B}^{r}\big{)}^{\frac{1}{r}}<\infty,\ \ \mbox{if $q\in[1,\infty)\,;$}\\ {\rm ess}\,{{\displaystyle\sup_{{t\in[0,T]}}}}\,\|u(t)\|_{B}<\infty,\ \ \mbox{if $r=\infty.$}\end{array}\right.$$

$$W^{m,r}(B)=\Big{\{}u\in L^{r}(B):\sum_{k=0}^{m}\partial_{t}^{k}u\in L^{r}(B)\Big{\}}\,.$$

$$[]f[{]}_m:=\underset{x\in A}{\sup }|(1+|x{|}^m)f(x)|,[]f[{]}_{\infty ,m}:=\underset{(x,t)\in A\times (0,T)}{\sup }|(1+|x{|}^m)f(x,t)|.$$

$$\begin{array}[]{cc}\vskip 3.0pt plus 1.0pt minus 1.0pt\left.\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt{\partial}_{t}{\mbox{\boldmath$u$}}-\mbox{\boldmath$\xi$}(t)\cdot\nabla{\mbox{\boldmath$u$}}=\Delta{\mbox{\boldmath$u$}}-\nabla{p}+{\mbox{\boldmath$f$}}\\ \mbox{\rm div}\,{\mbox{\boldmath$u$}}=0\end{array}\right\}\ \ \mbox{in $\Omega\times(0,T)$}\\ {\mbox{\boldmath$u$}}(x,t)=\mbox{\boldmath$\xi$}(t)\,,\ \ (x,t)\in\partial\Omega\times[0,T]\,,\end{array}$$

$$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt\widetilde{{\mbox{\boldmath$u$}}}(x,t)=\mbox{\boldmath$\xi$}(t)\,,\ (t,\mbox{\boldmath$x$})\in[0,T]\times\partial\Omega\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\widetilde{{\mbox{\boldmath$u$}}}(x,t)=0\,,\ \mbox{for all $t\in[0,T]$, all $|\mbox{\boldmath$x$}|\geq\rho$, and some $\rho>R_{*}$}\,,\\ \|\widetilde{{\mbox{\boldmath$u$}}}\|_{W^{3,2}(W^{m,q})}\leq C\,\|\mbox{\boldmath$\xi$}\|_{W^{3,2}(0,T)}\,,\end{array}$$

$${\mbox{\boldmath$f$}}=\mbox{\rm div}\,\mbox{\boldmath${\cal F}$}\in W^{2,2}(L^{2})\cap L^{\infty}(D^{1,2})\cap W^{1,2}(D^{2,2}),\ \mbox{with $\mbox{\boldmath${\cal F}$}\in L^{2}(L^{2})$\,,}$$

$${\mbox{\boldmath$u$}}\in W^{2,2}(D^{2,2})\cap W^{1,2}(D^{4,2})\cap L^{\infty}(L^{6})\,,\ \nabla{\mbox{\boldmath$u$}}\in L^{\infty}(W^{2,2})\,,\ \partial_{t}{\mbox{\boldmath$u$}}\in W^{1,\infty}(W^{1,2})\,,$$

$$p\in L^{\infty }(L^6\cap D^{1,2}\cap D^{2,2})\cap W^{1,2}(D^{3,2}).$$

$$\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus 1.0pt\|\partial_{t}{\mbox{\boldmath$u$}}\|_{W^{1,\infty}(W^{1,2})}+\|{\mbox{\boldmath$u$}}\|_{L^{\infty}(L^{6})}+\|\nabla{\mbox{\boldmath$u$}}\|_{L^{\infty}(W^{2,2})}+\|p\|_{L^{\infty}(L^{6})}+\|\nabla p\|_{L^{\infty}(W^{1,2})}\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\hskip 56.9055pt+\|{\mbox{\boldmath$u$}}\|_{W^{2,2}(D^{2,2})}+\|{\mbox{\boldmath$u$}}\|_{W^{1,2}(D^{4,2})}+{\displaystyle\sum_{|k|=3}}\|D^{k}p\|_{W^{1,2}(L^{2})}\\ \hskip 28.45274pt\leq C\,\big{(}\|{\mbox{\boldmath$f$}}\|_{W^{2,2}(L^{2})}+\|{\mbox{\boldmath$f$}}\|_{L^{\infty}(D^{1,2})}+\|{\mbox{\boldmath$f$}}\|_{W^{1,2}(D^{2,2})}+\|\mbox{\boldmath${\cal F}$}\|_{L^{2}(L^{2})}+\|\mbox{\boldmath$\xi$}\|_{W^{3,2}(0,T)}\big{)}\end{array}$$

$$\begin{array}[]{cc}\vskip 3.0pt plus 1.0pt minus 1.0pt\left.\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt{\partial}_{t}{\mbox{\boldmath$v$}}_{k}-\mbox{\boldmath$\xi$}(t)\cdot\nabla{\mbox{\boldmath$v$}}_{k}=\Delta{\mbox{\boldmath$v$}}_{k}-\nabla\widetilde{p}_{k}+{\mbox{\boldmath$f$}}+{\mbox{\boldmath$f$}}_{c}\\ \mbox{\rm div}\,{\mbox{\boldmath$v$}}_{k}=0\end{array}\right\}\ \ \mbox{in $\Omega_{R_{k}}\times(0,T)$}\\ {\mbox{\boldmath$v$}}_{k}(x,t)={\mbox{\boldmath$0$}}\,,\ \ (x,t)\in\partial\Omega_{R_{k}}\times[0,T]\,,\end{array}$$

$${\mbox{\boldmath$f$}}_{c}:=\Delta\widetilde{{\mbox{\boldmath$u$}}}-\partial_{t}\widetilde{{\mbox{\boldmath$u$}}}+\mbox{\boldmath$\xi$}(t)\cdot\nabla\widetilde{{\mbox{\boldmath$u$}}}$$

$$\mbox{$\frac{1}{2}$}{\displaystyle\frac{\mbox{$d$}}{\mbox{$dt$}}}\|{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}+\|\nabla{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}=\langle{\mbox{\boldmath$f$}}+{\mbox{\boldmath$f$}}_{c},{\mbox{\boldmath$v$}}_{k}\rangle\leq c_{0}\left(\|\mbox{\boldmath${\cal F}$}\|_{2}+\|{\mbox{\boldmath$f$}}_{c}\|_{\frac{6}{5}}\right)\|\nabla{\mbox{\boldmath$v$}}_{k}\|_{2}\,,$$

$$\|{\mbox{\boldmath$z$}}\|_{6}\leq\gamma_{0}\,\|\nabla{\mbox{\boldmath$z$}}\|_{2},\ \ \ {\mbox{\boldmath$z$}}\in D_{0}^{1,2}({\mathbb{R}}^{3})\,,$$

$${\displaystyle\frac{\mbox{$d$}}{\mbox{$dt$}}}\|{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}+c_{1R_{k}}\|{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}\leq c_{2}\,\left(\|\mbox{\boldmath${\cal F}$}\|_{2}^{2}+\|{\mbox{\boldmath$f$}}_{c}\|_{\frac{6}{5}}^{2}\right)\,.$$

$$\|\nabla{\mbox{\boldmath$v$}}_{k}\|_{L^{2}(L^{2})}\leq c\,\left(\|\mbox{\boldmath${\cal F}$}\|_{L^{2}(L^{2})}+\|{\mbox{\boldmath$f$}}_{c}\|_{L^{2}(L^{\frac{6}{5}})}\right)\,,$$

$$\|\nabla{\mbox{\boldmath$v$}}_{k}(t_{0})\|_{2}^{2}\leq\,c_{3}\left(\|\mbox{\boldmath${\cal F}$}\|_{L^{2}(L^{2})}^{2}+\|{\mbox{\boldmath$f$}}_{c}\|^{2}_{L^{2}(L^{\frac{6}{5}})}\right)\,.$$

$$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt\mbox{$\frac{1}{2}$}{\displaystyle\frac{\mbox{$d$}}{\mbox{$dt$}}}\|\nabla{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}+\|P\Delta{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}=\langle\mbox{\boldmath$\xi$}\cdot\nabla{\mbox{\boldmath$v$}}_{k},P\Delta{\mbox{\boldmath$v$}}_{k}\rangle+\langle{\mbox{\boldmath$f$}}+{\mbox{\boldmath$f$}}_{c},P\Delta{\mbox{\boldmath$v$}}_{k}(t)\rangle\\ \mbox{$\frac{1}{2}$}{\displaystyle\frac{\mbox{$d$}}{\mbox{$dt$}}}\|\nabla{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}+\|\partial_{t}{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}=\langle\mbox{\boldmath$\xi$}\cdot\nabla{\mbox{\boldmath$v$}}_{k},\partial_{t}{\mbox{\boldmath$v$}}_{k}\rangle+\langle{\mbox{\boldmath$f$}}+{\mbox{\boldmath$f$}}_{c},\partial_{t}{\mbox{\boldmath$v$}}_{k}(t)\rangle\par\end{array}$$

$${\displaystyle\frac{\mbox{$d$}}{\mbox{$dt$}}}\|\nabla{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}+c_{4}(\big{(}\|\partial_{t}{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}+\|P\Delta{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}\big{)}\leq c_{5}\big{(}\|{\mbox{\boldmath$f$}}\|_{2}^{2}+\|{\mbox{\boldmath$f$}}_{c}\|_{2}^{2}+\|\nabla{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}\big{)}\,,$$

$$\|D^{2}{\mbox{\boldmath$z$}}\|_{2,\Omega_{R}}\leq c_{\Omega}\,\left(\|P\Delta{\mbox{\boldmath$z$}}\|_{2,\Omega_{R}}+\|\nabla{\mbox{\boldmath$z$}}\|_{2,\Omega_{R}}\right)\,,\ \ {\mbox{\boldmath$z$}}\in D^{1,2}(\Omega_{R})\cap D^{2,2}(\Omega_{R})\,,$$

$$\begin{array}[]{rl}\vskip 6.0pt plus 2.0pt minus 2.0pt\|{\mbox{\boldmath$v$}}_{k}\|_{L^{\infty}(L^{6})}+\|\nabla{\mbox{\boldmath$v$}}_{k}\|_{L^{\infty}(L^{2})}+&\|\partial_{t}{\mbox{\boldmath$v$}}_{k}\|_{L^{2}(L^{2})}+\|D^{2}{\mbox{\boldmath$v$}}_{k}\|_{L^{2}(L^{2})}\\ \vskip 6.0pt plus 2.0pt minus 2.0pt&\leq c\,\big{(}\|{\mbox{\boldmath$f$}}\|_{L^{2}(L^{2})}+\|\mbox{\boldmath${\cal F}$}\|_{L^{2}(L^{2})}+\|{\mbox{\boldmath$f$}}_{c}\|_{L^{2}(L^{\frac{6}{5}})}\big{)}\\ &\leq C\,\big{(}\|{\mbox{\boldmath$f$}}\|_{L^{2}(L^{2})}+\|\mbox{\boldmath${\cal F}$}\|_{L^{2}(L^{2})}+\|\mbox{\boldmath$\xi$}\|_{W^{2,2}(0,T)}\big{)}\,,\end{array}$$

$$\mbox{$\frac{1}{2}$}{\displaystyle\frac{\mbox{$d$}}{\mbox{$dt$}}}\|\partial_{t}{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}+\|\nabla\partial_{t}{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}=\langle\mbox{\boldmath$\xi$}^{\prime}\cdot\nabla{\mbox{\boldmath$v$}}_{k},\partial_{t}{\mbox{\boldmath$v$}}_{k}\rangle+\langle\partial_{t}{\mbox{\boldmath$f$}}+\partial_{t}{\mbox{\boldmath$f$}}_{c},\partial_{t}{\mbox{\boldmath$v$}}_{k}(t)\rangle\,,$$

$$\begin{array}[]{rl}\vskip 6.0pt plus 2.0pt minus 2.0pt\mbox{$\frac{1}{2}$}{\displaystyle\frac{\mbox{$d$}}{\mbox{$dt$}}}\|\nabla\partial_{t}{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}+\|P&\!\!\Delta\partial_{t}{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}\\ &=\langle\mbox{\boldmath$\xi$}^{\prime}\cdot\nabla{\mbox{\boldmath$v$}}_{k},P\Delta\partial_{t}{\mbox{\boldmath$v$}}_{k}(t)\rangle+\langle\partial_{t}{\mbox{\boldmath$f$}}+\partial_{t}{\mbox{\boldmath$f$}}_{c},P\Delta\partial_{t}{\mbox{\boldmath$v$}}_{k}(t)\rangle\,.\end{array}$$

$$\|\partial_{t}{\mbox{\boldmath$v$}}_{k}(t_{1})\|_{2}\leq c\,\big{(}\|{\mbox{\boldmath$f$}}\|_{L^{2}(L^{2})}+\|\mbox{\boldmath${\cal F}$}\|_{L^{2}(L^{2})}+\|\mbox{\boldmath$\xi$}\|_{W^{2,2}(0,T)}\big{)}\,.$$

$$\|\partial_{t}{\mbox{\boldmath$v$}}_{k}\|_{L^{\infty}(L^{2})}+\|\nabla\partial_{t}{\mbox{\boldmath$v$}}_{k}\|_{L^{2}(L^{2})}\leq C\,\big{(}\|{\mbox{\boldmath$f$}}\|_{W^{1,2}(L^{2})}+\|\mbox{\boldmath${\cal F}$}\|_{L^{2}(L^{2})}+\|\mbox{\boldmath$\xi$}\|_{W^{2,2}(0,T)}\big{)}$$

$$\begin{array}[]{rl}\vskip 6.0pt plus 2.0pt minus 2.0pt\|\nabla\partial_{t}{\mbox{\boldmath$v$}}_{k}\|_{L^{\infty}(L^{2})}+\|&\!\!D^{2}\partial_{t}{\mbox{\boldmath$v$}}_{k}\|_{L^{2}(L^{2})}\\ &\leq C\,\big{(}\|{\mbox{\boldmath$f$}}\|_{W^{1,2}(L^{2})}+\|\mbox{\boldmath${\cal F}$}\|_{L^{2}(L^{2})}+\|\mbox{\boldmath$\xi$}\|_{W^{2,2}(0,T)}\big{)}\,.\end{array}$$

$$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt\|\partial_{t}{\mbox{\boldmath$v$}}_{k}\|_{L^{\infty}(W^{1,2})}+\|{\mbox{\boldmath$v$}}_{k}\|_{L^{\infty}(L^{6})}+\|\nabla{\mbox{\boldmath$v$}}_{k}\|_{L^{\infty}(L^{2})}+\|D^{2}{\mbox{\boldmath$v$}}_{k}\|_{W^{1,2}(L^{2})}\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\hskip 113.81102pt\leq C\,\big{(}\|{\mbox{\boldmath$f$}}\|_{W^{1,2}(L^{2})}+\|\mbox{\boldmath${\cal F}$}\|_{L^{2}(L^{2})}+\|\mbox{\boldmath$\xi$}\|_{W^{2,2}(0,T)}\big{)}\end{array}$$

$$\begin{array}[]{rl}\vskip 6.0pt plus 2.0pt minus 2.0pt\mbox{$\frac{1}{2}$}{\displaystyle\frac{\mbox{$d$}}{\mbox{$dt$}}}\|\nabla\partial_{t}{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}&\!+\|\partial_{t}^{2}{\mbox{\boldmath$v$}}_{k}(t)\|_{2}^{2}\\ &=\langle\mbox{\boldmath$\xi$}^{\prime}\cdot\nabla{\mbox{\boldmath$v$}}_{k}+\mbox{\boldmath$\xi$}\cdot\nabla\partial_{t}{\mbox{\boldmath$v$}}_{k},\partial^{2}_{t}{\mbox{\boldmath$v$}}_{k}\rangle+\langle\partial_{t}{\mbox{\boldmath$f$}}+\partial_{t}{\mbox{\boldmath$f$}}_{c},\partial^{2}_{t}{\mbox{\boldmath$v$}}_{k}\rangle\,.\end{array}$$

Full text

1 Introduction

Consider a body, $\mathscr{B}$, fully immersed in an unbounded Navier-Stokes liquid otherwise at rest, moving by translational motion with velocity \mbox{\boldmath\xi}=\mbox{\boldmath\xi}(t). Suppose $\xi$ is time-periodic with period $T$, and that its average over a period of time, \overline{\mbox{\boldmath\xi}}, is zero. For example, the direction of $\xi$ may be constant, in which case $\mathscr{B}$ oscillates between two fixed configurations. More generally, the center of mass of $\mathscr{B}$ moves periodically along a given closed curve, without $\mathscr{B}$ being able to spin.

The question we would like to address is whether the liquid will execute a corresponding unique time-periodic regular motion, and what will the flow characteristic be at “large” spatial distance away from $\mathscr{B}$.

From the mathematical viewpoint, this question leads us to investigate the same properties for solutions ({\mbox{\boldmathu}},p) to the following set of equations

[TABLE]

Here, $u$ and $p$ are velocity and pressure (1)(1)(1)Divided by the constant density of the liquid. fields of the liquid, respectively, while $\Omega$ is the flow region, namely, the entire space outside $\mathscr{B}$. (2)(2)(2)For simplicity, we set the coefficient of kinematic viscosity to be 1, since its actual value is entirely irrelevant to our aims. Moreover, for completeness and also for allowing the special case \mbox{\boldmath\xi}\equiv{\mbox{\boldmath0}}, we have included a body force {\mbox{\boldmathb}}={\mbox{\boldmathb}}(x,t) which we take to be periodic of the same period $T$.

Despite the very simple formulation, the problem, in its entirety, does not seem to be solvable by the methods currently available, for several reasons that we explain next.

The first contribution to this type of questions –when $\mathscr{B}$ moves in an unbounded viscous liquid in a non-trivial time-periodic fashion(3)(3)(3)It must be emphasized that if $\mathscr{B}$ is kept at rest (\mbox{\boldmath\xi}\equiv{\mbox{\boldmath0}} in our case) or is absent (\mbox{\boldmath\xi}\equiv{\mbox{\boldmath0}} and $\Omega={\mathbb{R}}^{3}$), then problems of existence and uniqueness have been successfully addressed and solved, under different assumptions, by a number of authors; see, e.g., [18, 15, 22, 7, 13] and the review paper [4].– can be found in [6], in the general context where $\mathscr{B}$ is also allowed to rotate. The tool used there is the so called “invading domains” technique, based on the Galerkin method coupled with suitable energy estimates. However, by its own nature, such a method is not capable of furnishing enough information on the spatial behavior of the solution “far” from $\mathscr{B}$. As a consequence, while the existence of weak solutions (for “arbitrary” data) and strong solutions (for data of restricted “size”) can be firmly secured, the question of their uniqueness, which actually requires a certain amount of asymptotic spatial “regularity,” is left open and still remains such. For the same reason, the spatial behavior of these solutions at large distances is still not known.

More recently, two distinct and equally powerful approaches to the study of time-periodic flow past a body have been independently developed by several authors.

The first one [3, 16, 4, 5], consists in splitting the velocity field into its averaged component (over a period), \overline{{\mbox{\boldmathu}}}, and oscillatory one, $w$, with zero average. The crucial property showed in those articles is the validity of maximal $L^{q}$-regularity for the relevant linearized (time-dependent) problem obeyed by $w$. As a result, the authors prove that existence and uniqueness of solutions to the full nonlinear problem (for “small” data), is reduced to show that the steady-state problem satisfied by \overline{{\mbox{\boldmathu}}} is well-posed in appropriate homogeneous Sobolev spaces; see [3, 5]. Now, for the problem treated here, this theory would work fine if \overline{\mbox{\boldmath\xi}}\neq{\mbox{\boldmath0}} (as showed in [3, 5]), thanks to the fact that, in such a case, the linearized steady-state operator is of the Oseen type, for which well-posedness is a classical result [2]. However, our current assumption requires \overline{\mbox{\boldmath\xi}}={\mbox{\boldmath0}}, and then the pertinent linearized operator becomes of the Stokes type, for which well-posedness does not hold [1].

Another, and entirely different line of attack, traces back to the remarkable paper [22]. It is based on a clever duality argument applied to the mild (very weak) formulation of the problem, coupled with sharp time-decay properties ($L^{p}-L^{q}$-estimates) of the evolution operator associated to the relevant linear problem, and of its first spatial derivatives. It must be emphasized that these estimates play a pivotal role for the success of the method. Such an approach, further refined, generalized and improved by several authors [13, 12, 8, 10], is particularly effective, because it allows one to establish existence and uniqueness of mild time-periodic (and almost-periodic) solutions when $\mathscr{B}$ is permitted to translate and also rotate, on condition that both translation and rotation vectors be time independent. However, its extension to the time dependent framework is not at all obvious and probably questionable, since sharp $L^{p}-L^{q}$-estimates in this more general context are not necessarily available. [11, Theorem 2.2 and Remark 2.1].

The method that here we propose and use is based upon a two-fold strategy. Since, eventually, the nonlinear analysis will be carried out by a contraction mapping argument, it is sufficient to develop this strategy for the relevant linear problem $\mathscr{L}\!\mathscr{P}$, say; see (2.1). Thus, in the first place, we establish a number of “energy estimates” that, once combined with the “invading domains” technique of [6], allows us to show existence, uniqueness and corresponding estimates of time-periodic solutions ({\mbox{\boldmathu}},p) to $\mathscr{L}\!\mathscr{P}$ in a very regular function class, provided \mbox{\boldmath\xi}=\mbox{\boldmath\xi}(t) and the “body force” $f$ are sufficiently smooth (see Lemma (6)). Successively, assuming that $f$ possesses suitable spatially asymptotic decay properties, we prove that similar properties must hold also for ({\mbox{\boldmathu}},p). This result –fundamental to the proof of all our main findings– is obtained as follows. By a classical “cut-off” argument applied to $\mathscr{L}\!\mathscr{P}$, we obtain a similar problem, $\mathscr{L}\!\mathscr{P}_{0}$, formulated in the whole space ${\mathbb{R}}^{3}$; see (2.68)–(2.69). Furthermore, with the change of coordinates \mbox{\boldmathx}\to\mbox{\boldmathy}:=\mbox{\boldmathx}-\int_{0}^{t}\mbox{\boldmath\xi}(s)ds, we may absorb the convective term \mbox{\boldmath\xi}\cdot\nabla{\mbox{\boldmathu}} in the time derivative, thus reducing the original system of equations in $\mathscr{L}\!\mathscr{P}_{0}$ to a classical Stokes system; see (2.84). By using the basic properties of the fundamental solution associated to the latter, we then show that all solutions to the corresponding Cauchy problem with vanishing initial data must, along with their first and second spatial derivatives, decay algebraically fast at large spatial distances, uniformly in time, with corresponding estimates; see Lemma 2.3. The decay is, of course, with respect to the $y$-coordinates. However, just thanks to the fact that $\xi$ has zero average, one easily shows that $y$- and $x$-coordinates are “equivalent” at large distances; see (2.82). Moreover, we prove that the solution to the Cauchy problem must tend, as time goes to infinity, to the time-periodic one of problem $\mathscr{L}\!\mathscr{P}_{0}$, which, in turn, for all $x$ away from the boundary, coincides with the solution ({\mbox{\boldmathu}},p) to the original problem $\mathscr{L}\!\mathscr{P}$. This result, combined with the global regularity of ({\mbox{\boldmathu}},p), finally furnishes the desired uniform spatial decay estimates on the whole domain $\Omega$; see Proposition (9).

With such a complete theory for the linear problem, we can then employ the contraction mapping theorem in a ball of a suitable Banach space, $\mathscr{X}$, to extend the result to the fully nonlinear case. In this way, in Theorem 3.3, we show that if the data $\xi$ and $b$ are sufficiently regular and “small in size,” then problem (1.1) possesses one and only one time-periodic solution ({\mbox{\boldmathu}},p) of period $T$ with {\mbox{\boldmathu}}\in\mathscr{X}. In addition, the spatial derivatives of $u$ of order $m=0,1,2$ decay like $|x|^{-m-1}$, uniformly in time. Likewise, $p$ and $\nabla p$ decay as $|x|^{-2}$ and $|x|^{-3}$, respectively, also uniformly in time.

Our approach also allows us to furnish the far–field structure of the solution. More precisely, in Theorem 4.1, we prove that $u$ can be decomposed as

[TABLE]

where $U$ is the velocity field of a specific steady-state problem (see Lemma 4.3), decaying like $|x|^{-1}$, $\sigma$ is also time independent and decays like $|x|^{-1-\alpha}$, for some $\alpha\in(0,1)$, while $w$ is the oscillatory component of $u$, given by subtracting to $u$ its (time) average, and decays faster, like $|x|^{-2}$. The field $U$ is determined up to a (possible other) velocity field, {\mbox{\boldmathU}}_{1}, such that {\mbox{\boldmathU}}-{\mbox{\boldmathU}}_{1} falls like $|x|^{-1-\delta}$, for some $\delta\in(0,1)$. This analysis shows, in particular, the distinctive steady-state behavior of the far field solution, thus providing a rigorous formulation of the steady streaming phenomenon [20, Chapter XV], [19]. In the (less relevant) case \mbox{\boldmath\xi}\equiv{\mbox{\boldmath0}}, we show that $U$ is uniquely determined as the velocity field of a specific Landau solution [14, 13]. Moreover, in this situation, we also prove that the oscillatory component $w$ decays even faster, like $|x|^{-3}$, thus sharpening analogous results of [13].

The outline of the paper is as follows. Section 2 is dedicated to the linear problem obtained from (1.1) by neglecting the nonlinear term. We prove existence, uniqueness and asymptotic behavior of corresponding time-periodic solutions. Successively, in Section 3, we combine this findings with the contraction mapping theorem and prove analogous properties for the full nonlinear problem (1.1), provided $\xi$ and $b$ are sufficiently regular and of restricted “size.” In the final Section 4, we give a detailed analysis of the behavior of our solutions at large spatial distances from $\mathscr{B}$ that shows the peculiar steady-state character of the flow sufficiently “far” from $\mathscr{B}$.

2 Unique Solvability of the Linear Problem

We begin to collect the main notation used throughout. The ball in ${\mathbb{R}}^{3}$ of radius $R>0$ centered at the origin is indicated by $B_{R}$, while $B^{R}$ stands for its complement. $\Omega$ is the complement of the closure of a bounded domain $\Omega_{0}\subset\mathbb{R}^{3}$. We shall assume $\Omega$ of class $C^{4}$,(4)(4)(4)Some of the peripheral results we shall prove require less regularity, but this is irrelevant for our final objective. and take the origin of the coordinate system in $\Omega_{0}$. We indicate by $B_{R_{*}}$ a ball containing the closure of $\Omega_{0}$. For $R\geq R_{*}$, we set $\Omega_{R}=\Omega\cap B_{R}\,,\ \ \Omega^{R}=\Omega\cap{B^{R}}$. Next, for a domain $A\subseteq{\mathbb{R}}^{3}$, by $L^{q}(A)$, $1\leq q\leq\infty,$ $W^{m,q}({A}),$ $W_{0}^{m,q}(A)$, $m\geq 0,$ $(W^{0,q}\equiv W^{0,q}_{0}\equiv L^{q}$), we denote usual Lebesgue and Sobolev classes, with corresponding norms $\|.\|_{q,A}$ and $\|.\|_{m,q,A}$. (5)(5)(5)We shall use the same font style to denote scalar, vector and tensor function spaces. By the letter $P$ we indicate the (Helmholtz) projector from $L^{2}(A)$ onto its subspace constituted by solenoidal (vector) function with vanishing normal component, in distributional sense, at $\partial A$. We also set $\int_{A}u\cdot v=\langle u,v\rangle_{A}$. $D^{m,2}(A)$ stands for the space of (equivalence classes of) functions $u$ such that $\sum_{|k|=m}\|D^{k}u\|_{2,A}<\infty\,.$ Obviously, the latter defines a seminorm in $D^{m,2}(A)$. Also, by $D_{0}^{1,2}(A)$ we denote the completion of $C_{0}^{\infty}(A)$ in the norm $\|\nabla(\cdot)\|_{2}$. In the above notation, the subscript “$A$” will be omitted, unless confusion arises. A function $u:A\times{\mathbb{R}}\mapsto{\mathbb{R}}^{3}$ is $T$-periodic, $T>0$, if $u(\cdot,t+T)=u(\cdot\,t)$, for a.a. $t\in{\mathbb{R}}$, and we shall denote by $\overline{u}$ its average over $[0,T]$, namely,

[TABLE]

Let $B$ be a function space endowed with seminorm $\|\cdot\|_{B}$. For $r=[1,\infty]$, $T>0$, $L^{r}(B)$ is the class of functions $u:(0,T)\rightarrow B$ such that

[TABLE]

Likewise, we put

[TABLE]

Finally, for $A:=\Omega,{\mathbb{R}}^{3}$ and $m\geq 1$, we set

[TABLE]

We now turn to the main objective of this section that consists in showing existence and uniqueness of $T$-periodic solutions, in appropriate function classes, to the following set of linear equations:

[TABLE]

where {\mbox{\boldmathf}}={\mbox{\boldmathf}}(x,t) and \mbox{\boldmath\xi}=\mbox{\boldmath\xi}(t) are suitably prescribed $T$-periodic functions.

To reach this goal, we need a few preparatory lemmas.

Lemma 2.1

Let \mbox{\boldmath\xi}\in W^{2,2}(0,T) be $T$-periodic. There exists a solenoidal, $T$-periodic function \widetilde{{\mbox{\boldmathu}}}\in W^{2,2}(W^{m,q}), $m\in\mathbb{N},$ $q\in[1,\infty],$ such that

[TABLE]

where $C=C(\Omega,m,q)$.

Proof. See [6, Lemma 2.2].

$\square$

Lemma 2.2

Let

[TABLE]

and \mbox{\boldmath\xi}\in W^{3,2}(0,T) be prescribed $T$-periodic functions. Then, there exists at least one $T$-periodic

[TABLE]

solving (2.1) for a corresponding $T$-periodic function

[TABLE]

Moreover, the solution ({\mbox{\boldmathu}},p) satisfies the following estimate

[TABLE]

where $C=C(\Omega,T,\xi_{0})$, for any fixed $\xi_{0}$ such that \|\mbox{\boldmath\xi}\|_{W^{2,2}(0,T)}\leq\xi_{0}. Finally, if \int_{0}^{T}\mbox{\boldmath\xi}(t)dt=0, the solution is also unique in the class (2.2), (2.3).(6)(6)(6)See Footnote (8).

Proof. The proof of existence is obtained by an argument similar to that employed in [6, Sections 3 & 4], that combines the Galerkin method with the “invading domains” procedure. Specifically, we write {\mbox{\boldmathu}}={\mbox{\boldmathv}}+\widetilde{{\mbox{\boldmathu}}}, with \widetilde{{\mbox{\boldmathu}}} given in Lemma 2.1, and begin to consider problem (2.1) along an increasing, unbounded sequence of (bounded) domains $\{\Omega_{R_{k}}\}$ with $\cup_{k\in{\mathbb{N}}}\Omega_{R_{k}}=\Omega$, namely,

[TABLE]

where

[TABLE]

If we formally dot-multiply (2.5)1 by {\mbox{\boldmathv}}_{k} and integrate by parts over $\Omega_{R_{k}}$ we get

[TABLE]

where we have used the assumption on $f$ and the Sobolev inequality

[TABLE]

with $\gamma_{0}$ numerical constant. Employing in (2.6) Cauchy inequality along with Poincarè inequality \|{\mbox{\boldmathv}}_{k}\|_{2}\leq c_{R_{k}}\|\nabla{\mbox{\boldmathv}}_{k}\|_{2} we get, in particular,

[TABLE]

Combining this inequality with Galerkin method one thus shows the existence of a $T$-periodic (distributional) solution {\mbox{\boldmathv}}_{k} to (2.5) with {\mbox{\boldmathv}}_{k}\in L^{\infty}(L^{2}(\Omega_{R_{k}}))\cap L^{2}(D_{0}^{1,2}(\Omega_{R_{k}})) (see [6, Lemma 3.1]). Furthermore,

[TABLE]

where the constant $c$ is independent of $R_{k}$; see [6, Section 3] for technical details. Notice that, by the mean value theorem, from (2.8) it follows that there is $t_{0}\in(0,T)$ such that

[TABLE]

In order to obtain more regular solutions, we need to show uniform (in $k$) estimates for {\mbox{\boldmathv}}_{k} in spaces of higher regularity. For this, we formally dot-multiply (2.5)1 one time by P\Delta{\mbox{\boldmathv}}_{k}, a second time by \partial_{t}{\mbox{\boldmathv}}_{k} and integrate by parts over $\Omega_{R_{k}}$. We thus show

[TABLE]

which, in turn, by Cauchy-Schwarz inequality entails

[TABLE]

with $c_{5}=c_{5}(\xi_{0})$. We now integrate this differential inequality over $[t_{0},t]$, and use the $T$-periodicity property along with (2.9) and the inequality

[TABLE]

where $c_{\Omega}$ depends only on the regularity of $\Omega$ [9, Lemma 1] but not on $R$. One can thus prove that {\mbox{\boldmathv}}_{k}\in W^{1,2}(L^{2}(\Omega_{R_{k}}))\cap L^{\infty}(D_{0}^{1,2}(\Omega_{R_{k}}))\cap L^{2}(D^{2,2}(\Omega_{R_{k}})) and satisfies the uniform bound [6, Lemma 4.1]

[TABLE]

with $C$ independent of $R_{k}$ and where, in the last step, we used Lemma 2.1. Next, we take the time derivative of both sides of (2.5)1, and dot multiply the resulting equation one time by \partial_{t}{\mbox{\boldmathv}}_{k}, a second time by P\Delta\partial_{t}{\mbox{\boldmathv}}_{k} and integrate over $\Omega_{R_{k}}$. We then obtain

[TABLE]

and

[TABLE]

From (2.12) and the mean value theorem we find that there exists at least one $t_{1}\in(0,T)$ such that

[TABLE]

Thus, we integrate (2.13) over $[t_{1},t]$ and use Cauchy-Schwarz inequality, (2.15), (2.12) and the $T$-periodicity of {\mbox{\boldmathv}}_{k}, to show

[TABLE]

Operating in a similar fashion on (2.14), and also employing (2.25) and (2.11), we get

[TABLE]

Therefore, combining (2.12), (2.25), and (2.17) we infer

[TABLE]

where $C$ is independent of $k$. By an entirely similar argument, it is now straightforward to show estimate (2.18) with {\mbox{\boldmathv}}_{k}, $f$ and $\xi$ replaced by \partial_{t}{\mbox{\boldmathv}}_{k}, \partial_{t}{\mbox{\boldmathf}} and \mbox{\boldmath\xi}^{\prime}. To this end, we first differentiate both sides of (2.5)1 with respect to time, dot-multiply the resulting equation by \partial^{2}_{t}{\mbox{\boldmathv}}_{k} and integrate over $\Omega_{R_{k}}$ to get

[TABLE]

Successively, by differentiating two times both sides of (2.5)1 with respect to time and dot-multiplying the resulting equation one time by \partial^{2}_{t}{\mbox{\boldmathv}}_{k}, a second time by P\Delta\partial^{2}_{t}{\mbox{\boldmathv}}_{k}, and integrating over $\Omega$, we show

[TABLE]

and

[TABLE]

Thus, using (2.19)–(2.21) and following exactly the same procedure as the one leading to (2.18), one can prove

[TABLE]

Finally, setting {\mbox{\boldmathF}}_{k}:=\Delta{\mbox{\boldmathv}}_{k}+{\mbox{\boldmathf}}+{\mbox{\boldmathf}}_{c}, from (2.5)1 we get, formally, that $\widetilde{p}_{k}$ obeys for a.a. $t\in[0,T]$ the following Neumann problem(7)(7)(7)Note that \mbox{\boldmath\xi}(t)\cdot\nabla{\mbox{\boldmathv}}_{k}\cdot{\mbox{\boldmathn}}|_{\partial\Omega_{R_{k}}}=0.

[TABLE]

Therefore, multiplying both sides of the first equation by $\widetilde{p}_{k}$ and integrating by parts over $\Omega_{R_{k}}$ we easily establish that the pressure field ${p}_{k}$ associated to {\mbox{\boldmathv}}_{k} satisfies the estimate [6, Lemma 4.3]

[TABLE]

with $c$ independent of $k$. We may now let $R_{k}\to\infty$ and use the uniform estimate (2.18) and Lemma 2.1, to show the existence of a pair ({\mbox{\boldmathu}}:={\mbox{\boldmathv}}+\widetilde{{\mbox{\boldmathu}}},\widetilde{p}), with $u$ $T$-periodic, in the class

[TABLE]

such that

[TABLE]

and which, in addition, solves the original problem (2.1). The proof of this convergence property is entirely analogous to that given in [6, Lemma 3.4 and Section 4], to which we refer for the missing details. We shall now prove the $T$-periodicity of the pressure field. To this end, we notice that, for a.a. $t\in[0,T]$, by [2, Theorem II.6.1], there is a function $p_{0}=p_{0}(t)$ such that ${p}:=\widetilde{p}-p_{0}$ satisfies

[TABLE]

with $c_{0}$ depending only on $\Omega$. Proceeding as in the proof of (2.23), we recognize that ${p}$ must obey (in the sense of distributions) the problem

[TABLE]

with {\mbox{\boldmathG}}:=\Delta{\mbox{\boldmathu}}+\mbox{\boldmath\xi}\cdot\nabla{\mbox{\boldmathu}}-\mbox{\boldmath\xi}^{\prime}+{\mbox{\boldmathf}}. Since ${p}$ satisfies (2.27) and $G$ is $T$-periodic, we may exploit a classical uniqueness result and conclude that ${p}$ can be time-wise extended to the entire line to become $T$-periodic as well. In order to complete the existence part of the lemma, we recall some classical properties of solutions to the Stokes problem:

[TABLE]

In particular, we get that any distributional solution to (2.28) satisfies the following estimate for $m=0,1,2$, [2, Lemma V.4.3]

[TABLE]

Let $h\in L^{2}(\Omega_{R})$ with $\int_{\Omega_{R}}h=0$, and let \mbox{\boldmath\varphi}\in W^{1,2}_{0}(\Omega_{R}) be a solution to the problem \mbox{\rm div}\,\mbox{\boldmath\varphi}=h in $\Omega_{R}$, satisfying \|\mbox{\boldmath\varphi}\|_{1,2}\leq c_{R}\|h\|_{2}. The existence of such a $\varphi$ is well known [2, Theorem III.3.1]. Dot-multiplying both sides of (2.28)1 by $\varphi$ and integrating by parts over $\Omega_{R}$, we get

[TABLE]

From this relation, the properties of $\varphi$ and the arbitrariness of $h$, we deduce that ${\sf p}$, modified by a possible addition of a ($T$-periodic) function of time, must obeys the following inequality

[TABLE]

where, in the last step, we have used Ehrling inequality. As a result, (2.29) furnishes

[TABLE]

We next observe that, for each $t\in[0,T]$, (2.1) can be put in the form (2.28) with

[TABLE]

so that (2.30) leads to

[TABLE]

with $C_{2}=C_{2}(\Omega,m,R,\xi_{0})$. If we take $m=0$ in (2.31) and use (2.26) we then show

[TABLE]

We next take $m=1$ in (2.31) and employ (2.26) and (2.32) to deduce

[TABLE]

Finally, (2.31) with $m=2$ in conjunction with (2.26) and (2.33) furnishes

[TABLE]

We next consider (2.28) with

[TABLE]

and take $m=2$ into (2.30). Again with the help of (2.26), we thus deduce

[TABLE]

In view of (2.32)–(2.35), the proof of the existence property is thus completed. We shall now prove uniqueness. This amounts to show that {\mbox{\boldmathu}}\equiv\nabla p\equiv{\mbox{\boldmath0}} is the only $T$-periodic solution in the class (2.2), (2.3) to the problem (8)(8)(8)As a matter of fact, going into the details of the proof, it is readily seen that uniqueness of a solution in the class (2.2)–(2.3) holds in a much larger class than that defined by (2.2)–(2.3).

[TABLE]

To this end, we begin to split $u$ as

[TABLE]

Since \overline{{\mbox{\boldmathw}}}=0, by Poincaré inequality, Fubini’s theorem and (2.2), we deduce {\mbox{\boldmathw}}\in L^{2}(L^{2}), so that, in particular,

[TABLE]

From classical embedding theorems (e.g. [21, Theorem 2.1]) and (2.38) we deduce

[TABLE]

We next observe that from (2.36) it follows that $p$ obeys the following Neumann problem for a.a. $t\in[0,T]$

[TABLE]

where we used the identity \Delta{\mbox{\boldmathu}}=-\mbox{\rm curl}\,\mbox{\rm curl}\,{\mbox{\boldmathu}}. We may modify $p$ by adding to it a suitable $T$-periodic function of time, in such a way that the redefined pressure field, that we continue to denote by $p$, satisfies (2.27). Thus, on the one hand, by the mean value theorem, (2.3), (2.27) and smoothness properties of harmonic functions we obtain, in particular,

[TABLE]

On the other hand, observing that, by Stokes theorem and (2.40),

[TABLE]

from (2.27) and well-known results on Laplace equation on exterior domains (e.g. [2, Exercise V.3.6]) we find for a.a. $t\in[0,T]$

[TABLE]

where $\mathcal{E}={\cal E}(z)$ is the Laplace fundamental solution. Since

[TABLE]

from (2.41) and (2.42) it follows that

[TABLE]

Let $\psi_{R}=\psi_{R}(x)$ be a smooth cut-off function that is 1 for $|x|\leq 2R$, is 0 for $|x|\geq 3R$ and $|\nabla\psi_{R}|\leq C\,R^{-1}$, with $C$ independent of $R$. Clearly,

[TABLE]

We dot-multiply both sides of (2.36)1 by \psi_{R}{\mbox{\boldmathu}}, and integrate by parts over $\Omega\times(0,T)$. Noticing that {\mbox{\boldmathu}}\in L^{2}(L^{2}(\Omega\rho)), all $\rho\geq R_{*}$, and using $T$-periodicity we thus show

[TABLE]

From Hölder inequality and (2.2)

[TABLE]

which, by (2.45), entails

[TABLE]

Furthermore, employing (2.37) and Fubini’s theorem, we show

[TABLE]

where we have used the assumption \overline{\mbox{\boldmath\xi}}={\mbox{\boldmath0}}. Again by Hölder inequality, and the properties of $\psi_{R}$

[TABLE]

which, by (2.39), implies

[TABLE]

Finally, by using one more time Hölder inequality, we infer

[TABLE]

and so from the latter, (2.39) and (2.2) we obtain

[TABLE]

Uniqueness then follows by letting $R\to\infty$ in (2.46) and using (2.47)–(2.49). The lemma is completely proved.

$\square$

Lemma 2.3

Let $G$ be a second-order tensor field in $\mathbb{R}^{3}\times(0,\infty)$ such that

[TABLE]

Then, the Cauchy problem

[TABLE]

has one and only one solution such that for all $\tau>0$,

[TABLE]

Moreover,

[TABLE]

and the following inequality holds:

[TABLE]

with $C$ a (positive) numerical constant.

Proof. The existence of a unique solution in the class (2.51) is a classical result (e.g. [2, Theorem VIII.4.1]). Moreover, the velocity field $v$ admits the following integral representation

[TABLE]

where \mbox{\boldmath\Gamma}=\mbox{\boldmath\Gamma}(\chi,\rho) is the Oseen fundamental solution to the Stokes problem (see [2, Theorem VIII.4.2]) for which, in particular, the following estimates hold:

[TABLE]

see [2, Lemma VIII.3.3 and Exercise VIII.3.1]. Using (2.53) and (2.54) along with the assumption on $G$, one can then establish the stated pointwise estimate on $v$ [2, Theorem VIII.4.4]. We shall now prove the claimed property for \nabla{\mbox{\boldmathv}}. To this end, let R=\mbox{\frac{1}{2}}|x|>1. From (2.53) we thus deduce

[TABLE]

From (2.54), the assumption, and the fact that $|x-y|\geq R$, $y\in B_{R}$, it follows

[TABLE]

By the same token, again using (2.54), we get

[TABLE]

Furthermore,

[TABLE]

As a result, from a well-known theorem on convolutions [2, Lemma II.9.2] applied to the last integral, we infer

[TABLE]

Finally, if $|x|\leq 2$, from (2.53) and (2.54) we deduce

[TABLE]

The latter, combined with (2.53)–(2.58) thus proves the desired property for \nabla{\mbox{\boldmathv}}. By the same token we get the estimate for D^{2}{\mbox{\boldmathv}}. Actually, from (2.53) we show by a double integration by parts

[TABLE]

Thus, employing (2.54) and [2, Lemma II.9.2], we easily show

[TABLE]

and, likewise,

[TABLE]

Moreover,

[TABLE]

and

[TABLE]

If $|x|\leq 2$, as in the analogous estimate for \nabla{\mbox{\boldmathv}}, we show

[TABLE]

As a result, the claimed estimate for D^{2}{\mbox{\boldmathv}} follows from the latter and (2.59)–(2.63). The estimates for $\phi$ and $\nabla\phi$ are obtained in an entirely similar fashion. In fact, this is a consequence of the following representation, valid for a.a. $t\in[0,T]$,

[TABLE]

and of the fact that $D^{k}{\cal E}(\chi)$ satisfies exactly the same properties as $\Gamma_{k}(\chi)$ in (2.54). We therefore shall omit the proof of these estimates, leaving it to the reader as an exercise. The lemma is proved.

$\square$

We are now in a position to show the main result of this section. Precisely, we have the following.

Proposition 2.1

Let ${\cal F}$ and $\xi$ be prescribed $T$-periodic functions such that

[TABLE]

Then, problem (2.1) has one and only one solution ({\mbox{\boldmathu}},p) in the class (2.2), (2.3), which satisfies the estimate

[TABLE]

In addition, if, for some $\rho\geq R_{\star}$ \nabla\mbox{\boldmath{\cal F}}\in L^{\infty}(L^{\infty}(\Omega_{2\rho})), then \sum_{|k|=0}^{2}[\!]D^{k}{\mbox{\boldmathu}}[\!]_{\infty,|k|+1}, $\sum_{|k|=0}^{1}[\!]p[\!]_{\infty,|k|+2}<\infty$, and we have

[TABLE]

where $C_{i}=C_{i}(\Omega,T,\xi_{0})$, $i=1,2$.(9)(9)(9)Recall that $\xi_{0}$ is defined in Lemma (6).

Proof. We begin to observe that, obviously,

[TABLE]

Therefore, under the given assumptions, the existence and uniqueness of a solution ({\mbox{\boldmathu}},p) in the class (2.2), (2.3) satisfying (2.65) is ensured by Lemma (6). In order to complete the proof of the proposition, it remains to show the pointwise properties of $u$ and $p$, along with the corresponding estimates. To this end, for a fixed $\rho\geq R_{*}$, let $\psi=\psi(x)$ be a smooth “cut-off” function such that $\psi(x)=0$ for $|x|\leq\rho$, $\psi(x)=1$ for $|x|\geq 2\rho$, and set {\mbox{\boldmathw}}:=\psi{\mbox{\boldmathu}}, ${\sf p}:=\psi p$. From (2.1) we thus infer that ({\mbox{\boldmathw}},{\sf p}) obeys the following problem

[TABLE]

where

[TABLE]

We next observe that, by classical embedding theorems,

[TABLE]

Therefore, from the latter and (2.65) we get

[TABLE]

We also notice we have

[TABLE]

with

[TABLE]

where $c=c(\Omega,T,\xi_{0})$. In fact, let

[TABLE]

where, we recall, $\mathcal{E}$ is the Laplace fundamental solution. Clearly, \mbox{\rm div}\,\mbox{\boldmath{\cal H}}={\mbox{\boldmathg}}_{c} and, by (2.69)2, (2.70), $T$-periodicity, and the fact that the support of {\mbox{\boldmathg}}_{c} is contained in $B_{2\rho}$, it follows at once

[TABLE]

Moreover, from (2.43) we find, for a.a. $t\geq 0$

[TABLE]

with $C_{1}=C_{1}(\rho)$. Also, from classical results for convolutions with weakly singular integrals (e.g. [2, Theorem II.11.2]), we have

[TABLE]

with $C_{2}=C_{2}(\rho)$. Consequently, (2.72) follows again from (2.69)2, (2.70), (2.73)–(2.75) and $T$-periodicity. Let

[TABLE]

and write

[TABLE]

Notice that $V$ is $T$-periodic and, as a result, so is {\mbox{\boldmathw}}_{1}. Moreover, using also Sobolev and Calderon-Zygmund theorems and that $h$ has bounded (spatial) support, we easily show that $V$ is in the functional class defined in (2.2). Thus, from (2.68), and taking into account (2.67) we deduce that {\mbox{\boldmathw}}_{1} is a $T$-periodic solution in the class (2.2) to the following problem

[TABLE]

with

[TABLE]

Observe that, by assumption, (2.69)1, (2.72) and $T$-periodicity one has

[TABLE]

We now introduce the following change of coordinates

[TABLE]

Since \int_{0}^{T}\mbox{\boldmath\xi}(t)dt={\mbox{\boldmath0}}, this along with the $T$-periodicity of $\xi$ implies that \mbox{\boldmathx}_{0}(t) is $T$-periodic as well, and also the existence of a constant $M>0$ such that

[TABLE]

In fact, by integrating over $[0,t]$ both sides of the Fourier series for $\xi$:

[TABLE]

we infer at once that \mbox{\boldmathx}_{0} is $T$-periodic. Moreover,

[TABLE]

Notice that from (2.80) and (3.17) it follows that

[TABLE]

Setting

[TABLE]

from (2.80) and (2.77) it follows that ({\mbox{\boldmathW}}_{1},\Pi) satisfies the following Cauchy problem

[TABLE]

In view of (2.80), (2.83) and (2.82), we have

[TABLE]

and, likewise,

[TABLE]

As a result, by (2.79), the tensor field $G$ satisfies the assumptions of Lemma 2.3. Set

[TABLE]

with ({\mbox{\boldmathv}},\phi) solution given in that lemma. From (2.50) and (2.84), we then have that ({\mbox{\boldmathU}},{\sf Q}) satisfies:

[TABLE]

Since both $u$ and $V$ are in the function class defined by (2.2), we have, in particular,

[TABLE]

so that, by classical results on the Cauchy problem for Stokes equations (e.g., [2, Theorem VIII.4.3]) we infer

[TABLE]

which, in turn, by embedding, implies

[TABLE]

From (2.83) and the $T$-periodicity of {\mbox{\boldmathw}}_{1} we have for all $n\in{\mathbb{N}}$

[TABLE]

Thus, setting

[TABLE]

by (2.82) and (2.52) we get

[TABLE]

and, similarly,

[TABLE]

Thus, if we pass to the limit $n\to\infty$ in the relations above and use (2.85), (2.86), (2.90),(2.79), and (2.89) we conclude

[TABLE]

We now recall that {\mbox{\boldmathu}}=(1-\psi){\mbox{\boldmathu}}+{\mbox{\boldmathw}}_{1}, and so the claimed asymptotic property of $u$ follows (2.70) and (2.91). We next observe that from (2.78)1, (2.83)2, (2.87)2 and $T$-periodicity, we get

[TABLE]

Arguing as in the estimate of {\mbox{\boldmathw}}_{1} and taking into account (2.52) and (2.89), from the preceding relation we deduce

[TABLE]

Recalling that h=\nabla\psi\cdot{\mbox{\boldmathu}} (see (2.69)2), we infer $\int_{{\mathbb{R}}^{3}}\partial_{t}h=0$. Therefore, from (2.78)1, also after integrating by parts, we deduce

[TABLE]

By the mean value theorem and (2.43), we infer

[TABLE]

whereas from classical results on convolutions (e.g. [2, Theorem II.11.2])

[TABLE]

Since

[TABLE]

from the latter, (2.92)–(2.95), classical embedding, (2.65), and (2.85), (2.79), we conclude

[TABLE]

Now, as before, we recall that $p=(1-\psi)p+{\sf p}$, so that from (2.35) and (2.96) we prove the desired property for $p$. In an entirely analogous way one can deduce the pointwise estimate for $\nabla p$. The proof of the proposition is therefore completed.

$\square$

3 Unique Solvability of the Nonlinear Problem

We introduce the following function class:

[TABLE]

Clearly, $\mathscr{X}$ becomes a Banach space when endowed with the norm

[TABLE]

Moreover, we set

[TABLE]

with

[TABLE]

The main result of this section reads as follows.

Theorem 3.1

Let \mbox{\boldmath\xi}\in W^{3,2}(0,T) be $T$-periodic with \int_{0}^{T}\mbox{\boldmath\xi}(t)dt={\mbox{\boldmath0}}. Moreover, suppose that {\mbox{\boldmathb}}=\mbox{\rm div}\,{\mbox{\boldmathB}}, where $B$ is a $T$-periodic tensor function such that

[TABLE]

for some fixed $\rho>R_{*}$. Then, setting

[TABLE]

there exists $\varepsilon_{0}>0$ such that if $\mathtt{D}<\varepsilon_{0}\,,$ problem (1.1) has one and only one solution ({\mbox{\boldmathu}},p)\in\mathscr{X}\times\mathscr{P}. Moreover, this solution obeys the following inequality

[TABLE]

Proof. We employ the contraction mapping theorem. To this end, define the map

[TABLE]

with $u$ solving the linear problem

[TABLE]

Set

[TABLE]

where we used the condition $\mbox{\rm div}\,\textbf{{u}}=0$. Clearly,

[TABLE]

Thus, by a straightforward calculation, we show

[TABLE]

Employing in the last term of (3.7) the classical embedding inequality:

[TABLE]

from (3.7) and (3.1) we then conclude

[TABLE]

In a similar fashion, we show

[TABLE]

Again, by classical embedding,

[TABLE]

Therefore, employing (3.8) and (3.11) in (3.10) and taking into account (3.1) we deduce

[TABLE]

Finally, and obviously,

[TABLE]

As a result, from (3.5), (3.9), (3.12) and (3.13) we find that f and F satisfy the assumptions of $f$ and ${\cal F}$, respectively, in Proposition (9), and, in addition,

[TABLE]

Thus, by that proposition and the assumption on the data, we deduce, on the one hand, that \big{(}M(\textbf{{u}}),p)\in\mathscr{X}\times\mathscr{P} –so that, in particular, $M$ is well defined– and, on the other hand, that {\mbox{\boldmathu}}=M(\textbf{{u}}) obeys the estimate:

[TABLE]

Next, suppose $\|\textbf{{u}}\|_{\mathscr{X}}<\delta$. From (3.15) it follows

[TABLE]

from which we infer that if we pick

[TABLE]

we obtain

[TABLE]

Let $\textbf{{u}}_{i}\in\mathscr{X}$ $i=1,2$, and set

[TABLE]

From (3.4) we then get

[TABLE]

Arguing as in the proof of (3.14) we can show

[TABLE]

Consequently, if $\|\textbf{{u}}_{i}\|_{\mathscr{X}}<\delta$, $i=1,2$, from the preceding inequality we find

[TABLE]

and since by (3.16) $2C_{1}\delta<1$, we may conclude that $M$ is a contraction, which ends the proof of existence. Finally, the estimate (3.3) is a consequence of (3.15), (3.17) and the choice of $\delta$ in (3.16).

$\square$

Remark 3.1

In the particular case \mbox{\boldmath\xi}(t)\equiv{\mbox{\boldmath0}}, Theorem 3.3 furnishes (in a better regularity class and with more information about the behavior at infinity) existence results similar to those proved [7, 13].

Remark 3.2

Theorem 3.3 establishes the uniqueness of the solution in the ball of $\mathscr{X}$ of radius $\delta$. However, a more general uniqueness result “in the large” could be actually shown in a sufficiently regular class of solutions (not necessarily “small”), and even in a suitable class of “weak” solutions. In fact, the former could be attained by employing the same “cut-off” procedure used in the proof of Lemma (6), in conjunction with the pointwise asymptotic properties of the solution constructed in Theorem 3.3. As for the latter, one could just follow, step by step, the proof provided in [7, Theorem 5].

4 Asymptotic Spatial Behavior and Steady Streaming

Theorem 3.3 asserts, in particular, that $u$, $p$ and some of their derivatives have a polynomial (spatial) decay rate at large distance from the body $\mathscr{B}$. Objective of this section is to provide a more detailed analysis of this property and show that, “far” from $\mathscr{B}$, the flow velocity field presents a distinctive steady-state character, in spite of being driven by a time-periodic mechanism. This rigorous finding is in agreement with the classical phenomenon of “steady streaming” observed in the motion of a viscous liquid past an oscillating body; see [20, p. 428–432], [19] and the references therein.

To prove the above, we recall the following splitting of $u$ into its averaged and oscillatory components (see (2.37)):

[TABLE]

The following lemma holds.

Lemma 4.1

The oscillatory component $w$ of the solution $u$ of Theorem 3.3 satisfies

[TABLE]

If, in particular, \mbox{\boldmath\xi}(t)\equiv{\mbox{\boldmath0}}, then the faster decay condition is valid:

[TABLE]

Proof. Since \overline{{\mbox{\boldmathw}}}={\mbox{\boldmath0}}, from the Poincaré inequality we get, for all $x\in\Omega$:

[TABLE]

which once combined with an elementary embedding inequality, implies

[TABLE]

Therefore, the claimed properties follow directly from (4.1), (1.1)1 and the pointwise decay estimates established in Theorem 3.3.

$\square$

We next observe that, from (1.1), the averaged component, \overline{{\mbox{\boldmathu}}}, of $u$ and corresponding averaged pressure $\overline{p}$ solve the following boundary-value problem

[TABLE]

where

[TABLE]

Definition 4.1

*Two vector fields {\mbox{\boldmathU}}_{1}, {\mbox{\boldmathU}}_{2}\in L^{\infty}(\Omega) are asymptotically equivalent –and we write {\mbox{\boldmathU}}_{1}\sim{\mbox{\boldmathU}}_{2}– if (i) [\!]{\mbox{\boldmathU}}_{i}[\!]_{1}<\infty, $i=1,2$, and (ii) [\!]{\mbox{\boldmathU}}_{1}-{\mbox{\boldmathU}}_{2}[\!]_{1+\delta}<\infty\,, for some $\delta>0$. *

Lemma 4.2

Let {\mbox{\boldmathG}}\in L^{\infty}({\mathbb{R}}^{3}) with support in $B_{R}$. Further, let $\psi=\psi(|x|)$ be a smooth function that is 0 in $B_{R/2}$ and 1 in $B^{R}$, $R\geq 2R_{*}$. There exists $\varepsilon_{1}>0$ such that if

[TABLE]

then the problem

[TABLE]

has at least one solution ({\mbox{\boldmathU}},P)\in W^{2,2}_{\rm loc}({\mathbb{R}}^{3})\times W^{2,1}_{\rm loc}({\mathbb{R}}^{3}) with [\!]{\mbox{\boldmathU}}[\!]_{1}<\infty, and, moreover,

[TABLE]

Finally, let ({\mbox{\boldmathU}}^{\prime},P^{\prime}) solve the problem

[TABLE]

with {\mbox{\boldmathG}}^{\prime} satisfying the same properties listed for $G$. Then, if

[TABLE]

we have {\mbox{\boldmathU}}\sim{\mbox{\boldmathU}}^{\prime}.

Proof. We begin to notice that, in view of Lemma 4.1, the assumption [\!]{\mbox{\boldmathF}}[\!]_{2}<\infty is meaningful. Let

[TABLE]

Then, clearly \mbox{\rm div}\,{\mbox{\boldmathH}}={\mbox{\boldmathG}}. Furthermore, proceeding as in (2.74), (2.75), we show

[TABLE]

so that, in view of our assumptions, the field {\mbox{\boldmathF}}+{\mbox{\boldmathH}} meets the hypotheses of [2, Lemma X.9.1]. As a result, there is a corresponding solution ({\mbox{\boldmathU}},P) to (4.5) such that

[TABLE]

which, in addition, satisfies [\!]{\mbox{\boldmathU}}[\!]_{1}<\infty, along with the estimate (4.6). Finally, from (4.9) and classical regularity results [2, Theorem X.1.1] we infer ({\mbox{\boldmathU}},P)\in W^{2,2}_{\rm loc}({\mathbb{R}}^{3})\times W^{2,1}_{\rm loc}({\mathbb{R}}^{3}). Next, setting

[TABLE]

from (4.5)–(4.8) we deduce the following integral representation [2, Theorem X.5.2]:

[TABLE]

where $S$ is the Stokes fundamental tensor that, we recall, satisfies the following asymptotic bounds [2, Section IV.2]

[TABLE]

We now regard (4.11) as an integral equation in the unknown $\zeta$. It is simple to show that this equation has a solution, \widehat{\mbox{\boldmath\zeta}}, in the space

[TABLE]

for some $\alpha\in(0,1)$, provided we take $\varepsilon_{1}$ appropriately “small.” Actually, recalling that \mbox{\rm supp}\,({\mbox{\boldmathg}})\subset B_{R} and that both $U$, {\mbox{\boldmathU}}^{\prime} are in $\mathscr{S}_{0}$, from (4.10) and (4.11) we show

[TABLE]

On the other hand, by [2, Lemmas II.9.2, II.11.2] we have

[TABLE]

Thus, using (4.12) and (4.13), by a simple contraction argument it follows that, for a given $\alpha\in(0,1)$, we can choose a corresponding $\varepsilon_{1}$ in (4.4) such that (4.10) has a solution \widehat{\mbox{\boldmath\zeta}}\in\mathscr{S}_{\alpha}. It is also readily proved that \widehat{\mbox{\boldmath\zeta}}=\mbox{\boldmath\zeta}. In fact, setting {\mbox{\boldmathz}}:=\mbox{\boldmath\zeta}-\widehat{{\mbox{\boldmathz}}}, we have

[TABLE]

and so, employing in this relation (4.11), (4.13) with $\alpha=0$, (4.4), and (4.6) we get

[TABLE]

which allows us to conclude {\mbox{\boldmathz}}\equiv{\mbox{\boldmath0}} by taking $\varepsilon_{1}$ sufficiently small. The proof is completed.

$\square$

Remark 4.1

If \mbox{\boldmath\xi}\equiv{\mbox{\boldmath0}}, namely, {\mbox{\boldmathF}}_{2}\equiv{\mbox{\boldmath0}}, then every solution in Lemma 4.2 corresponding to some $G$ that satisfies the assumption of that lemma along with the condition

[TABLE]

is asymptotically equivalent to a specific member of the well-known Landau family of solution [17, 14]. To see this, let $(r,\theta,\phi)$ be a system of polar coordinates, with polar axis oriented in the direction \mbox{\boldmath\beta}/|\mbox{\boldmath\beta}| which, without loss, we take coinciding with the positive $x_{1}-$direction. We recall that the Landau solution corresponding to $\beta$ is a pair ({\mbox{\boldmathU}}^{\beta},P^{\beta}) satisfying

[TABLE]

with $\delta$ Dirac distribution, and defined, for $r>0$, as follows

[TABLE]

where the parameter $A\in(1,\infty)$ is chosen in such a way that

[TABLE]

Since the function on the left-hand side is monotonically decreasing in $A\in(1,\infty)$ and its range coincides with $(0,\infty)$, we deduce that for any given $\beta$ ($>0$) there is one and only one $A$ satisfying (4.16), namely, one and only one Landau solution ({\mbox{\boldmathU}}^{\beta},P^{\beta}). Moreover, observing that $A\to\infty$ as $\beta\to 0$, from (4.15) we also deduce, in particular,

[TABLE]

Now, by following a standard procedure, we regularize ({\mbox{\boldmathU}}^{\beta},P^{\beta}) around $x=0$ by defining \widetilde{{\mbox{\boldmathU}}^{\beta}}:=\psi{\mbox{\boldmathU}}^{\beta}-\textbf{{U}}, $\widetilde{P^{\beta}}:=\psi\,P^{\beta}$, where $\psi=\psi(|x|)$ is the “cut-off” function introduced in Lemma 4.2, while \mbox{\rm div}\,\textbf{{U}}=\nabla\psi\cdot{\mbox{\boldmathU}}^{\beta} in $B_{R}$, $\textbf{{U}}\in C_{0}^{\infty}(B_{R}))$ [14]. It is then readily checked that (\widetilde{{\mbox{\boldmathU}}^{\beta}},\widetilde{P^{\beta}}) is a solution to the following problem

[TABLE]

with \widetilde{{\mbox{\boldmathG}}^{\beta}}\in C_{0}^{\infty}(B_{R}) and such that, by (4.14) and (4.17),

[TABLE]

see [14] for details. Thus, the claimed asymptotic equivalence is a consequence of Lemma (6).

We recall the definition of Cauchy stress:

[TABLE]

with $I$ identity matrix and $\top$ denoting transpose.

Lemma 4.3

Let (\overline{{\mbox{\boldmathu}}},\overline{p}) be the averaged component of the solution ({\mbox{\boldmathu}},p) given in Theorem 3.3, satisfying (4.2), (4.3), and let ({\mbox{\boldmathU}},P) be the solution to (4.5) constructed in Lemma 4.2, corresponding to a vector field

[TABLE]

Then, there is $\varepsilon_{2}>0$ such that, if

[TABLE]

for some $\alpha\in(0,1)$ ($\mathtt{D}$ defined in (3.2)), necessarily \overline{{\mbox{\boldmathu}}}\sim{\mbox{\boldmathU}}.

Proof. Let $\psi=\psi(|x|)$ be the “cut-off” function introduced in Lemma 4.2, and set

[TABLE]

where {\mbox{\boldmathV}}\in W_{0}^{4,2}(B_{R}) satisfies \mbox{\rm div}\,\textsf{{V}}=\nabla\psi\cdot\overline{{\mbox{\boldmathu}}} with

[TABLE]

see [2, Theorem III.3.3]. In view of (4.2), (4.3) and the regularity properties of $u$, we show that ({\mbox{\boldmathv}},q) satisfies the following problem

[TABLE]

where {\mbox{\boldmathG}}_{0}\in L^{\infty}(B_{R}) with \mbox{\rm supp}\,({\mbox{\boldmathG}}_{0})\subset B_{R}. Moreover, also using (4.21), we have

[TABLE]

Observing that

[TABLE]

and that $\psi(R)=1$, integrating both sides of (4.22)1 over $B_{R}$, we get

[TABLE]

where we have used the following properties, consequences of (4.2)1,3:

[TABLE]

Therefore,

[TABLE]

Let \mbox{\boldmath\zeta}:={\mbox{\boldmathv}}-{\mbox{\boldmathU}}. starting with (4.22), (4.5) and proceeding as in the proof of Lemma 4.2 (see (4.10)), we show

[TABLE]

where {\mbox{\boldmathg}}:={\mbox{\boldmathG}}_{0}-{\mbox{\boldmathG}}. Since, by classical trace theorems and Lemma 4.1, one shows

[TABLE]

from the latter, (4.23) and (3.3) we deduce

[TABLE]

Thus, we can argue exactly as in the proof of Lemma 4.2 to show [\!]\mbox{\boldmath\zeta}[\!]_{1+\alpha}<\infty, which completes the proof of the lemma.

$\square$

We are now in a position to show the main result of this section.

Theorem 4.1

Let ({\mbox{\boldmathu}}\equiv\overline{{\mbox{\boldmathu}}}+{\mbox{\boldmathw}},p) be the solution determined in Theorem 3.3. Then, under the assumptions on $\mathtt{D}$ and \overline{{\mbox{\boldmathB}}} of Lemma 4.3, the velocity field $u$ has the following representation

[TABLE]

where ({\mbox{\boldmathU}},P) is the steady-state solution of Lemma 4.3, and for some $\alpha\in(0,1)$,

[TABLE]

The field $U$ is unique up to an asymptotically equivalent velocity field. However, if \mbox{\boldmath\xi}\equiv{\mbox{\boldmath0}}, then $U$ is uniquely determined and coincides with the Landau solution ({\mbox{\boldmathU}}^{\beta},P^{\beta}) where

[TABLE]

Moreover, in such a case, we have a faster decay of the oscillatory component, namely,

[TABLE]

Proof. It is enough to observe that

[TABLE]

and employ Lemma 4.2, Lemma 4.3 and Remark 4.19.

$\square$

Remark 4.2

In the case \mbox{\boldmath\xi}\equiv{\mbox{\boldmath0}}, Theorem 4.1 sharpens an analogous result showed in [13, Theorem 1.2].

Acknowledgement. Work partially supported by NSF grant DMS-1614011.