Uniqueness of mild solutions to the Navier-Stokes equations in weak-type L^d space

2024/02/02 by Zhirun Zhan, Zhan, Zhirun · 1 citation
Mathematics · Engineering · #Navier-Stokes equation solutions #Advanced Mathematical Physics Problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2402.01174

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^∞(0,T;L^d(ℝ^d)) when d≥ 4, and in C([0,T];L^d(ℝ^d)) when d≥3. As for the forced Navier-Stokes equations, when d≥3 the uniqueness of mild solutions in C([0,T];L^d,∞(ℝ^d)) with force f and initial data u₀ in some proper Lorentz spaces is known. In this paper we show that for d≥3, the uniqueness of mild solutions to the forced Navier-Stokes equations in C((0,T];\widetildeL^d,∞(ℝ^d))∩ L^β(0,T;\widetildeL^d,∞(ℝ^d)) for β>2d/(d-2) holds when there is a mild solution in C([0,T];\widetildeL^d,∞(ℝ^d)) with the same initial data and force. Here \widetildeL^d,∞ is the closure of L^∞∩ L^d,∞ with respect to L^d,∞ norm.

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