Regularity in time of Hölder solutions of Euler and hypodissipative Navier-Stokes equations

Abstract

In this work we investigate some regularization properties of the incompressible Euler equations and of the fractional Navier-Stokes equations where the dissipative term is given by $(-\hat{I}”{)}^{\hat{I}}\pm$, for a suitable power $\hat{I}\pm \in (0,\frac{1}{2})$ (the only meaningful range for this result). Assuming that the solution $u\in L_t^{\infty }(C^{\hat{I}}{¸}_x)$ for some $\hat{I}¸\in (0,1)$ we prove that $u\in C^{\hat{I}}{¸}_{t,x}$, the pressure $p\in C_{t,x}^{2\hat{I}¸-}$ and the kinetic energy $e\in C_t^{\frac{2\hat{I}¸}{1-\hat{I}¸}}$. This result was obtained for the Euler equations in [Is13] with completely different arguments and we believe that our proof, based on a regularization and a commutator estimate, gives a simpler insight on the result.