The analytic method of constructing local-in-time solutions of the incompressible Euler equations in Sobolev spaces

Abstract

We introduce a new method for constructing local-in-time solutions of the incompressible Euler equations in Sobolev spaces on an arbitrary Sobolev bounded domain. The method is based on a construction of an analytic solution in an analytically approximated domain, after which we apply analytic persistence to extend the analytic solution using 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.