On the energy of inviscid singular flows

Abstract

It is known that the energy of a weak solution to the Euler equation is conserved if it is slightly more regular than the Besov space $B^{1/3}_{3,\infty}$. When the singular set of the solution is (or belongs to) a smooth manifold, we derive various $L^p$-space regularity criteria dimensionally equivalent to the critical one. In particular, if the singular set is a hypersurface the energy of $u$ is conserved provided the one sided non-tangential limits to the surface exist and the non-tangential maximal function is $L^3$ integrable, while the maximal function of the pressure is $L^{3/2}$ integrable. The results directly apply to prove energy conservation of the classical vortex sheets in both 2D and 3D at least in those cases where the energy is finite.