Uniqueness of almost periodic-in-time solutions to Navier–Stokes equations in unbounded domains

Abstract

We present a uniqueness theorem for almost periodic-in-time solutions to the Navier–Stokes equations in 3-dimensional unbounded domains. Thus far, uniqueness of almost periodic-in-time solutions to the Navier–Stokes equations in unbounded domain, roughly speaking, is known only for a small almost periodic-in-time solution in \({BC(\mathbb {R};L^{3}_w)}\) within the class of solutions that have sufficiently small \({L^{\infty}(L^{3}_w)}\) -norm. In this paper, we show that a small almost periodic-in-time solution in \({BC(\mathbb {R};L^{3}_w\cap L^{6,2})}\) is unique within the class of all almost periodic-in-time solutions in \({BC(\mathbb {R};L^{3}_w\cap L^{6,2})}\) . The proof of the present uniqueness theorem is based on the method of dual equations.