The TURBO method for well-posedness of the incompressible Euler equations in Sobolev spaces in any domain

TL;DR

This paper presents the TURBO method, a novel approach for establishing local-in-time solutions to the incompressible Euler equations in Sobolev spaces on arbitrary domains, without modifying the equations.

Contribution

The TURBO method constructs solutions in Sobolev spaces by combining analytic approximation and persistence, applicable to various PDEs without regularization.

Findings

Successfully constructs local solutions in Sobolev spaces on arbitrary domains.

Avoids modification or regularization of the Euler equations.

Applicable to a wide class of PDEs.

Abstract

We introduce a new method for constructing local-in-time solutions the incompressible Euler equations in Sobolev spaces on an arbitrary Sobolev bounded domain. The method is based on construction of an analytic solution in an analytically approximated domain, after which we apply analytic persistence to extend the analytic solution given a priori bounds in Sobolev spaces. The method does not introduce any modification or regularization of the equations themselves and appears applicable to many other PDEs.