Non-uniqueness for the Euler equations: the effect of the boundary

TL;DR

This paper demonstrates that on bounded domains like an annulus, the Euler equations admit infinitely many weak solutions with the same initial data, highlighting the boundary's role in solution uniqueness and dissipation properties.

Contribution

It shows the existence of multiple admissible weak solutions for the 2D Euler equations on bounded domains using convex integration, emphasizing the boundary's influence on solution behavior.

Findings

Existence of infinitely many weak solutions on an annulus.

Boundary conditions affect solution dissipation.

Hölder continuity near boundary ensures dissipative solutions.

Abstract

We consider rotational initial data for the two-dimensional incompressible Euler equations on an annulus. Using the convex integration framework, we show that there exist infinitely many admissible weak solutions (i.e. such with non-increasing energy) for such initial data. As a consequence, on bounded domains there exist admissible weak solutions which are not dissipative in the sense of P.-L. Lions, as opposed to the case without physical boundaries. Moreover we show that admissible solutions are dissipative provided they are H\"{o}lder continuous near the boundary of the domain.