Notions of solution and weak-strong uniqueness criteria for the Navier-Stokes equations in Lorentz spaces

TL;DR

This paper extends classical results on the Navier-Stokes equations by establishing energy equality and uniqueness of solutions within Lorentz spaces, broadening the scope from Lebesgue to weak Lebesgue spaces.

Contribution

It generalizes well-known energy and uniqueness criteria for Navier-Stokes solutions to Lorentz spaces, including equivalence of solution notions in these larger spaces.

Findings

Solutions in Lorentz spaces satisfy energy equality.

Uniqueness holds among solutions satisfying the energy inequality.

Generalization of solution notions to Lorentz spaces.

Abstract

For initial data $f\in L^{2}(\mathbb{R}^n)$ ($n\geq 2$), we prove that if $p\in(n,\infty]$, any solution $u\in L_{t}^{\infty}L_{x}^{2}\cap L_{t}^{2}H_{x}^{1}\cap L_{t}^{\frac{2p}{p-n}}L_{x}^{p,\infty}$ to the Navier-Stokes equations satisfies the energy equality, and that such a solution $u$ is unique among all solutions $v\in L_{t}^{\infty}L_{x}^{2}\cap L_{t}^{2}H_{x}^{1}$ satisfying the energy inequality. This extends well-known results due to G. Prodi (1959) and J. Serrin (1963), which treated the Lebesgue space $L_{x}^{p}$ rather than the larger Lorentz (and `weak Lebesgue') space $L_{x}^{p,\infty}$. In doing so, we also prove the equivalence of various notions of solutions in $L_{x}^{p,\infty}$, generalizing in particular a result proved for the Lebesgue setting in Fabes-Jones-Riviere (1972).