Uniqueness of Mild Solutions to the Navier-Stokes Equations in Weak-type \(L^d\) Space

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 \(L^{\infty }(0,T;L^d({\mathbb {R}}^d))\) when \(d\ge 4\), and in \(C([0,T];L^d({\mathbb {R}}^d))\) when \(d\ge 3\). As for the forced Navier-Stokes equations, when \(d\ge 3\) the uniqueness of mild solutions in \(C([0,T];L^{d,\infty }({\mathbb {R}}^d))\) with force f and initial data \(u_{0}\) in appropriate Lorentz spaces is known. In this paper we show that for \(d\ge 3\), the uniqueness of mild solutions to the forced Navier-Stokes equations in \( C((0,T];{\widetilde{L}}^{d,\infty }({\mathbb {R}}^d))\cap L^\beta (0,T;{\widetilde{L}}^{d,\infty }({\mathbb {R}}^d))\) for \(\beta >2d/(d-2)\) holds when there is a mild solution in \(C([0,T];{\widetilde{L}}^{d,\infty }({\mathbb {R}}^d))\) with the same initial data and force. Here \({\widetilde{L}}^{d,\infty }\) is the closure of \({L^{\infty }\cap L^{d,\infty }}\) with respect to \(L^{d,\infty }\) norm.