Bernoulli's type Law of Euler Equations on Sobolev Space in ${\mathbb{R}}^3$

Abstract

In this paper, we discuss Bernoulli's principle to the 3-dimensional incompressible Euler equation in a bounded local Lipschitz domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^3$ with a Lipschitz boundary. Using topological properties of the level set and Morse-Sard theorem, we will prove Bernoulli's principle on Sobolev space.

the proof has some error