Solvability of Euler equations in the fractional Sobolev spaces in a bounded smooth domain

TL;DR

This paper investigates the well-posedness of Euler equations within fractional Sobolev spaces on bounded smooth domains, extending classical results to fractional orders and providing a priori estimates for solutions.

Contribution

It establishes the solvability of Euler equations in fractional Sobolev spaces, using energy and elliptic estimates, for both two and three-dimensional cases.

Findings

Global well-posedness in 2D

Local well-posedness in 3D

A priori estimates in fractional Sobolev spaces

Abstract

Euler equations are the basic system in fluid dynamics describing the motion of incompressible and inviscid ideal fluids. For a bounded smooth domain $\Omega$ in $\mathbb{R}^n$. The well-posedness of Euler equations is well-known in Sobolev spaces $W^{k,p}(\Omega)$ with the integer $k>\frac{n}{p}+1,\, 1<p<\infty$. In this article, we study the well-posedness of Euler equations in fractional Sobolev spaces on a bounded smooth domain. We first give a priori estimates of Euler equations in fractional Hilbert-Sobolev spaces by using the energy method. For the general case of fractional Sobolev spaces, we use the characteristic method together with elliptic estimates to give similar estimates. Finally, using the a priori estimate obtained we give solvability of Euler equations in fractional Sobolev spaces. Similar to the classical case, our result is global in time in the case of two dimensions and local in the three dimensions.