Elliptic and Parabolic Equations - Abstract

Taniuchi, Yasushi

On the uniqueness of almost periodic-in-time solution to the Navier--Stokes equations in exterior domains

We present a uniqueness theorem for almost periodic-in-time solutions to the Navier-Stokes equations in $3$-dimensional exterior domains $Omega$. It is known that there exists a small almost periodic-in-time solution in $C(R;L^3_w(Omega))$ to the Navier--Stokes equations for a small almost periodic-in-time force. Here $L^n_w(Omega)$ denotes weak $L^n$ space. Thus far, with respect to the uniqueness of almost periodic-in-time solutions to the Navier--Stokes equations in exterior domain, roughly speaking, it has been only known that a small almost periodic-in-time solution in $C(R;L^3_w)$ is unique within the class of solutions which have sufficiently small $L^infty( L^3_w)$-norm, i.e., that if $u$ and $v$ are $L3_w$-solutions for the same force $f$, and if both of them are small, then $u=v$. In this talk, we will show that a small almost periodic-in-time solution in $C(R;L^3_wcap L^infty)$ is unique within the class of all almost periodic-in-time solutions in $C(R;L^3_wcap L^infty)$, i.e., we will show that if $u$ and $v$ are almost periodic-in-time solutions in $C(R;L^3_wcap L^infty)$ for the same force $f$, and if one of them are small, then $u=v$.