Notions of solution and weak-strong uniqueness criteria for the Navier-Stokes equations in 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).