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

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...

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Published in:arXiv.org 2022-02
Main Authors: Davies, Joseph P, Koch, Gabriel S
Format: Article
Language:English
Subjects:
Fluid flow
Mathematical analysis
Navier-Stokes equations
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Summary: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).
ISSN:2331-8422