On some regularity results for $\,2-D\,$ Euler equations and linear elliptic b.v. problems

TL;DR

This paper reviews and extends regularity results for 2-D Euler equations and linear elliptic boundary value problems, discussing minimal data assumptions for classical solutions and exploring new findings and open questions.

Contribution

It provides a clear overview of the authors' approach to regularity problems and introduces new results and open problems in the context of elliptic PDEs and Euler equations.

Findings

Regularity conditions for 2-D Euler solutions

Extension of results to general linear elliptic problems

Identification of open problems in the field

Abstract

About thirty years ago we looked for "minimal assumptions" on the data which guarantee that solutions to the $\,2-D\,$ evolution Euler equations in a bounded domain are classical. Classical means here that all the derivatives appearing in the equations and boundary conditions are continuous up to the boundary. Following a well known device, the above problem led us to consider this same regularity problem for the Poisson equation under homogeneous Dirichlet boundary conditions. At this point, one was naturally led to consider the extension of this last problem to more general linear elliptic boundary value problems, and also to try to extend the results to more general data spaces. At that time, some side results in these directions remained unpublished. The first motivation for this note is a clear description of the route followed by us in studying these kind of problems. New results and open problems are also considered.