Uniqueness of mild solutions to the Navier-Stokes equations in $\big(C((0,T];L^d(\mathbb{R}^d))\cap L^\infty((0,T);L^d(\mathbb{R}^d))\big)^d$

Abstract

This paper deals with the uniqueness of mild solutions to the forced or unforced Navier-Stokes equations in the whole space. It is known that the uniqueness of mild solutions to the unforced Navier-Stokes equations holds in $\big(L^{\infty}((0,T);L^d(\mathbb{R}^d))\big)^d$ when $d\geq 4$, and in $\big(C([0,T];L^d(\mathbb{R}^d))\big)^d$ when $d\geq3$. As for the forced Navier-Stokes equations, when $d\geq3$ the uniqueness of mild solutions in $\big(C([0,T];L^{d}(\mathbb{R}^d))\big)^d$ with force $f$ in some Lorentz space is known. In this paper we show that for $d\geq3$, the uniqueness of mild solutions to the forced Navier-Stokes equations in $\big(C((0,T];L^d(\mathbb{R}^d))\cap L^\infty((0,T);L^d(\mathbb{R}^d))\big)^d$ holds when there is a mild solution in $\big(C([0,T];L^d(\mathbb{R}^d))\big)^d$ with the same initial data and force. As a corollary of this result, we establish the uniqueness of mild solutions to the unforced Navier-Stokes equations in $\big(C((0,T];L^3(\mathbb{R}^3))\cap L^\infty((0,T);L^3(\mathbb{R}^3))\big)^3$.