On Double Hölder regularity of the hydrodynamic pressure in bounded domains

Abstract

We prove that the hydrodynamic pressure p associated to the velocity uin C^theta (Omega ), theta in (0,1), of an inviscid incompressible fluid in a bounded and simply connected domain Omega subset {mathbb {R}}^d with C^{2+} boundary satisfies pin C^{theta }(Omega ) for theta le frac{1}{2} and pin C^{1,2theta -1}(Omega ) for theta >frac{1}{2}. Moreover, when partial Omega in C^{3+}, we prove that an almost double Hölder regularity pin C^{2theta -}(Omega ) holds even for theta <frac{1}{2}. This extends and improves the recent result of Bardos and Titi (Philos Trans R Soc A, 2022) obtained in the planar case to every dimension dge 2 and it also doubles the pressure regularity. Differently from Bardos and Titi (2022), we do not introduce a new boundary condition for the pressure, but instead work with the natural one. In the boundary-free case of the d-dimensional torus, we show that the double regularity of the pressure can be actually achieved under the weaker assumption that the divergence of the velocity is sufficiently regular, thus not necessarily zero.