Euler Equations in Sobolev conormal spaces

TL;DR

This paper proves local existence and uniqueness of solutions to the 3D incompressible Euler equations in Sobolev conormal spaces without requiring normal derivative integrability, expanding understanding of fluid dynamics in complex geometries.

Contribution

It establishes the well-posedness of Euler equations in Sobolev conormal spaces with minimal normal derivative assumptions, a novel approach in fluid dynamics analysis.

Findings

Proves local-in-time existence and uniqueness of solutions.

Handles initial data with Lipschitz regularity and specific conormal derivatives.

No integrability or differentiability assumptions on the normal derivative.

Abstract

We consider the three-dimensional incompressible Euler equations in Sobolev conormal spaces and establish local-in-time existence and uniqueness in the half-space or channel. The initial data is Lipschitz having four square-integrable conormal derivatives and two bounded conormal derivatives. We do not impose any integrability or differentiability assumption for the normal derivative.