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 $B_{3,\infty }^{1/3}$. 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.

19 pages

References in corpus (1)

• Energy conservation and Onsager's conjecture for the Euler equations