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

TL;DR

This paper proves Bernoulli's principle for 3D incompressible Euler equations in Sobolev spaces within bounded Lipschitz domains, utilizing topological and Morse-Sard theorem techniques.

Contribution

It establishes Bernoulli's principle in Sobolev spaces for Euler equations on Lipschitz domains, extending classical results to less regular settings.

Findings

Bernoulli's principle holds in Sobolev spaces for Euler equations.

The proof uses topological properties of level sets and Morse-Sard theorem.

Results apply to bounded Lipschitz domains in -dimensional space.

Abstract

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