From non-local to local Navier-Stokes equations

TL;DR

This paper rigorously proves that solutions to fractional Navier-Stokes equations converge to classical solutions as the fractional order approaches 2, with uniform convergence and explicit rates, also extending to MHD systems.

Contribution

It establishes a rigorous framework for the convergence of fractional Navier-Stokes solutions to classical solutions as the fractional parameter approaches 2, including convergence rates and generalization to MHD.

Findings

Uniform convergence in time and space variables.

Explicit convergence rate derived.

Extension to Magnetic-hydrodynamic systems.

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 $(-\Delta)^{\frac{\alpha}{2}}$ with $\alpha<2$, converge to a solution of the classical case, with $-\Delta$, when $\alpha$ 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.