On the energy of inviscid singular flows

TL;DR

This paper investigates conditions under which energy is conserved in weak solutions to the Euler equations with singularities, especially on smooth manifolds like vortex sheets, extending known regularity criteria.

Contribution

It derives new $L^p$ regularity criteria for energy conservation when singular sets are smooth manifolds, including hypersurfaces, with specific integrability conditions.

Findings

Energy conservation holds if the non-tangential limits exist and the maximal functions are integrable.

Results apply to classical vortex sheets in 2D and 3D with finite energy.

Provides criteria linking geometric singularities to energy conservation in inviscid 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.