Regularity In Time Of Holder 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 is an element of (0, 1/2) (the only meaningful range for this result). Assuming that the solution u is an element of L-t(infinity) (C-x(theta)) for some theta is an element of (0, 1) we prove that u is an element of C-t,x(theta), the pressure p is an element of C-t,x(2 theta-) boolean AND C-t(0) (C-x(2 theta)), and the kinetic energy e is an element of C-t(2 theta/1-theta). This result was obtained for the Euler equations in [P. Isett, Regularity in time along the coarse scale flow for the incompressible Euler equations, https://arXiv.org/abs/1307.0565, 2013] with completely different arguments and we believe that our proof, based on a regularization and a commutator estimate, gives a simpler insight into the result.