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 \((-\Delta )^{\frac{\alpha }{2}}\) 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 the \(L^{\infty }_{t,x}\)-space and derive a precise convergence rate, revealing some phenomenological effects. As a bi-product, we prove strong convergence in the \(L^{p}_{t}L^{q}_{x}\)-space. Finally, our results are also generalized to the coupled setting of the Magnetic-hydrodynamic system.