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 \((-\Delta )^{\frac{\alpha }{2}}\) ${(-\mathrm{\Delta })}^{\frac{\alpha }{2}}$ with \(\alpha <2\) $\alpha <2$ , converge to a solution of the classical case, with \(-\Delta \) $-\mathrm{\Delta }$ , when \(\alpha \) $\alpha$ goes to 2. Precisely, in the setting of mild solutions, we prove uniform convergence in the \(L^{\infty }_{t,x}\) $L_{t,x}^{\infty }$ -space and derive a precise convergence rate, revealing some phenomenological effects. As a bi-product, we prove strong convergence in the \(L^{p}_{t}L^{q}_{x}\) $L_t^p{L}_x^q$ -space. Finally, our results are also generalized to the coupled setting of the Magnetic-hydrodynamic system.