From non-local to local Navier-Stokes equations

2023/09/25 by Jarrin, Oscar, Loachamin, Geremy
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.13784

Abstract

Inspired by some experimental (numerical) works on fractional diffusion PDEs, we develop a rigorous framework to prove that solutions to the fractional Navier-Stokes equations, which involve the fractional Laplacian operator (-Δ)^\fracα2 with α<2, converge to a solution of the classical case, with -Δ, when α goes to 2. Precisely, in the setting of mild solutions, we prove uniform convergence in both the time and spatial variables and derive a precise convergence rate, revealing some phenomenological effects. Finally, our results are also generalized to the coupled setting of the Magnetic-hydrodynamic (MHD) system.

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