On the stabilization of a hyperbolic Stokes system under geometric control condition

Abstract

In this article, we study the stabilization problem for a hyperbolic-type Stokes system posed on a bounded domain. We show that when the damping effects are restricted to a subdomain satisfying the geometric control condition, the energy of the system decays exponentially. The result is a consequence of a new quasi-mode estimate for the Stokes system.

Appendix

We will derive the hyperbolic Stokes system (1.2) from certain limit procedure of Lamé system from elastic theory:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t^2w-\mu \Delta w-(\lambda +\mu )\nabla \text {div} w=0, (t,x)\in [0,T]\times \Omega \\&w(t,.)|_{\partial \Omega }=0 \\&(w(0),\partial _tw(0))=(w_0,z_0)\in (H_0^1(\Omega )\times L^2(\Omega ))^d \end{aligned} \right. \end{aligned}$$

(7.1)

where the solution w(t, x) is vector-valued.

Define $u(t,x):=w(t/\sqrt{\mu },x)$, then we find that

$$\begin{aligned} \partial _t^2 u-\Delta u-\frac{\lambda +\mu }{\mu }\nabla \text {div} u=0. \end{aligned}$$

We let $\epsilon =\frac{\mu }{\mu +\lambda }\ll 1$, in the case that $\lambda \gg \mu >0.$ Thus, we obtain a family of equations

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t^2u_{\epsilon }-\Delta u_{\epsilon }+\nabla p_{\epsilon }=0, (t,x)\in [0,T]\times \Omega \\&u_{\epsilon }(t,.)|_{\partial \Omega }=0 \\&(u_{\epsilon }(0),\partial _tu_{\epsilon }(0))=(u_{0,\epsilon },v_{0,\epsilon })\in (H_0^1(\Omega )\times L^2(\Omega ))^d \end{aligned} \right. \end{aligned}$$

(7.2)

where $p_{\epsilon }=-\frac{1}{\epsilon }\text {div}u_{\epsilon }$ and satisfies $\int _{\Omega }p_{\epsilon }\hbox {d}x=0$.

We make further assumption on the family of initial data $(u_{0,\epsilon },v_{0,\epsilon })$ so that

$$\begin{aligned} \Vert (u_{0,\epsilon },v_{0,\epsilon })-(u_0,v_0)\Vert _{H^1\times L^2}\le C\epsilon \end{aligned}$$

for some divergence free data $(u_0,v_0)\in V\times H$. In particular, we have

$$\begin{aligned} \Vert \mathrm {div }u_{0,\epsilon }\Vert _{L^2(\Omega )}\le C\epsilon . \end{aligned}$$

From the well-posedness of Lame system, we have that $u_{\epsilon }\in C([0,T];H_0^1(\Omega )),\partial _t u_{\epsilon }\in C([0,T];L^2(\Omega ))$, and $p_{\epsilon }\in C([0,T];L^2(\Omega ))$. Moreover, we have the conservation of energy

$$\begin{aligned} E[u_{\epsilon }]=\frac{1}{2}\int _{\Omega }\left( |\partial _t u_{\epsilon }|^2+|\nabla u_{\epsilon }|^2+\epsilon |p_{\epsilon }|^2\right) \hbox {d}x \end{aligned}$$

and therefore

$$\begin{aligned} E[u_{\epsilon }]=\frac{1}{2}\int _{\Omega }\left( |u_{0,\epsilon }|^2+|v_{0,\epsilon }|^2+\frac{1}{\epsilon }|\mathrm {div }u_{0,\epsilon }|^2\right) \hbox {d}x. \end{aligned}$$

From this, we have, up to some subsequence of $(u_{\epsilon },\partial _t u_{\epsilon })$

$$\begin{aligned} \begin{aligned}&\text {div}u_{\epsilon }\rightarrow 0,\text { in } L^{\infty }([0,T];L^2(\Omega )),\\&u_{\epsilon }*\rightharpoonup u, *\text {weakly in }L^{\infty }([0,T];H_0^1(\Omega )),\\&\partial _tu_{\epsilon }*\rightharpoonup \partial _t u, *\text {weakly in }L^{\infty }([0,T];L^2(\Omega )). \end{aligned} \end{aligned}$$

From the uniform bound of $\Vert \partial _t u_{\epsilon }\Vert _{L^{\infty }([0,T];L^2(\Omega ))}$, apply Ascoli theorem, we have that (up to some subsequence)

$$\begin{aligned} u_{\epsilon }\rightarrow u, \text { in } C([0,T];L^2(\Omega )). \end{aligned}$$

Using the equation, we conclude that $\Vert \nabla p_{\epsilon }\Vert _{L^{\infty }([0,T];H^{-1}(\Omega ))}$ is uniformly bounded. Combine with the fact $\int _{\Omega }p_{\epsilon }=0$, we have that $\Vert p_{\epsilon }\Vert _{L^{\infty }([0,T];L^{2}(\Omega ))}$ is uniformly bounded, thus up to some subsequence, we may assume that

$$\begin{aligned} p_{\epsilon }*\rightharpoonup p, *\text {weakly in }L^{\infty }([0,T];L^2(\Omega )). \end{aligned}$$

Now it is not difficult to verify that (u, p) is a weak solution to (1.1). Moreover, p satisfies the zero mean condition

$$\begin{aligned} \int _{\Omega }p\hbox {d}x=0. \end{aligned}$$