Full Double Hölder Regularity of the Pressure in Bounded Domains

We consider H & ouml;lder continuous weak solutions u is an element of C-gamma(Omega), u & sdot;n|(partial derivative Omega)=0, of the incompressible Euler equations on a bounded and simply connected domain Omega subset of R(d )If Omega is of class C2,1 then the corresponding pressure satisfies p is an element of C-& lowast;(2 gamma)(Omega) in the case gamma is an element of(0,12], where C-& lowast;(2 gamma) is the H & ouml;lder-Zygmund space, which coincides with the usual H & ouml;lder space for gamma<12. This result, together with our previous one in [11] covering the case gamma is an element of(12,1), yields the full double regularity of the pressure on bounded and sufficiently regular domains. The interior regularity comes from the corresponding C-& lowast;(2 gamma) estimate for the pressure on the whole space R-d, which in particular extends and improves the known double regularity results (in the absence of a boundary) in the borderline case gamma=1/2. 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 which plays a major role in our approach