A one-point attractor theory for the Navier-Stokes equation on thin domains with no-slip boundary conditions

Abstract:

In an earlier paper related to recent results of Raugel and Sell for periodic boundary conditions, we considered the incompressible Navier-Stokes equations on 3-dimensional thin domains with zero (“no-slip”) boundary conditions and established global regularity results. We extend those results here by developing an attractor theory. We first show that under similar thinness restrictions trajectories of solutions approach each other in $L^4$-norm exponentially. Next, for constant-in-time forcing data $f_1=f_1\left ( x\right ) ,$ we suppose that $f\left ( t\right ) \rightarrow f_1$ in $L^2$ as $t\rightarrow +\infty ,$ and show that if $v$ and $w_1$ solve the equations with forcing data $f$ and $f_1$, respectively, then $\left \| v\left ( t\right ) -w_1\left ( t\right ) \right \| _4\rightarrow 0$ as $t\rightarrow +\infty .$ For similar thinness restrictions we show that the steady-flow equations with forcing data $f_1$ have a unique solution $u_s$. Under both thinness assumptions we then have that all solutions $v\left ( t\right )$ converge to $u_s$ in $L_4$ as $t\rightarrow +\infty$; thus we have a one-point attractor for strong solutions. In fact, we have a one-point attractor for the Leray solutions as well. Moreover, under significantly more relaxed thinness assumptions we are able to show that Leray solutions nonetheless eventually become regular.