Hoelder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains

Abstract

We consider the three-dimensional incompressible Euler equations on a bounded domain with C3 boundary. We prove that if the velocity field u 2 C0;�( ) with � > 0 (where we are omitting the time dependence), it follows that the pressure p 2 C0;�( ). In order to prove this result we use a local parametrisation of the boundary and a very weak formulation of the boundary condition for the pressure, as was introduced in [C. Bardos and E.S. Titi, Philos. Trans. Royal Soc. A, 380 (2022), 20210073]. Moreover, we provide an example illustrating the necessity of this new very weak formulation of the boundary condition for the pressure. This result is of importance for the proof of the �rst half of the Onsager Conjecture, the su�cient conditions for energy conservation of weak solutions to the three-dimensional incompressible Euler equations in bounded domains.