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

2021/11/08 by Joseph P. Davies, Gabriel S. Koch, Davies, Joseph P. +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2111.04350

openalex publication_date 2021/11/08 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28

Abstract

For initial data f∈ L²(ℝⁿ) (n≥ 2), we prove that if p∈(n,∞], any solution u∈ L_t^∞L_x²∩ L_t²H_x¹∩ L_t^(2p)/(p-n)L_x^p,∞ to the Navier-Stokes equations satisfies the energy equality, and that such a solution u is unique among all solutions v∈ L_t^∞L_x²∩ L_t²H_x¹ 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,∞. In doing so, we also prove the equivalence of various notions of solutions in L_x^p,∞, generalizing in particular a result proved for the Lebesgue setting in Fabes-Jones-Riviere (1972).

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