Regularity in time of H\"older solutions of Euler and hypodissipative Navier-Stokes equations

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 $(-\Delta)^\alpha$, for a suitable power $\alpha \in (0,\frac{1}{2})$ (the only meaningful range for this result). Assuming that the solution $u \in L^\infty _t(C^\theta_x)$ for some $\theta \in (0,1)$ we prove that $u \in C^\theta_{t,x}$, the pressure $p\in C^{2\theta-}_{t,x}$ and the kinetic energy $e \in C^{\frac{2\theta}{1-\theta}}_t$. 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.