From non-local to local Navier-Stokes equations

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