Full Double Hölder Regularity of the Pressure in Bounded Domains

We consider Hölder continuous weak solutions,, of the incompressible Euler equations on a bounded and simply connected domain. If is of class then the corresponding pressure satisfies in the case, where is the Hölder-Zygmund space, which coincides with the usual Hölder space for $\gamma. This result, together with our previous one in [11] covering the case, yields the full double regularity of the pressure on bounded and sufficiently regular domains. The interior regularity comes from the corresponding estimate for the pressure on the whole space, which in particular extends and improves the known double regularity results (in the absence of a boundary) in the borderline case. The boundary regularity features the use of local normal geodesic coordinates, pseudodifferential calculus and a fine Littlewood-Paley analysis of the modified equation in the new coordinate system. We also discuss the relation between different notions of weak solutions, a step that plays a major role in our approach.