It is known that the energy of a weak solution to the Euler equation isconserved 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 criteriadimensionally equivalent to the critical one. In particular, if the singularset is a hypersurface the energy of $u$ is conserved provided the one sidednon-tangential limits to the surface exist and the non-tangential maximalfunction is $L^3$ integrable, while the maximal function of the pressure is$L^{3/2}$ integrable. The results directly apply to prove energy conservationof the classical vortex sheets in both 2D and 3D at least in those cases wherethe energy is finite.
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机译：众所周知，如果欧拉方程的弱解的能量比贝索夫空间$ B ^ {1/3} _ {3，\ infty} $的规则性稍强，则它的能量是守恒的。当解的奇异集是（或属于）一个光滑流形时，我们得出与临界值等价的各种$ L ^ p $-空间正则性准则。特别是，如果奇异集是一个超曲面，则只要存在该曲面的单侧非切向极限并且非切向最大函数为$ L ^ 3 $可积，而压力的最大函数为$ L ^ {3/2} $可积。该结果直接用于证明至少在能量有限的情况下2D和3D中经典涡旋片的能量守恒。
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