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 $(-\Delta)^\alpha$ 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 $p\in C^{2\theta-}_{t,x}\cap C^0_t(C^{2\theta}_x)$, and the kinetic energy $e \in C^{\frac{2\theta}{1-\theta}}_t$. This result was obtained for the Euler equations in [P. Isett, Regularity in time along the coarse scale flow for the incompressible Euler equations, ŭlhttps://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.

Locations

• arXiv (Cornell University) - View - PDF
• SIAM Journal on Mathematical Analysis - View