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

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 Hölder continuous near the boundary of the domain.

20 pages. Dedicated to the memory of Professor Mark Vishik

References in corpus (2)

• Young Measures Generated by Ideal Incompressible Fluid Flows
• Existence of Weak Solutions for the Incompressible Euler Equations

Cited by in corpus (4)

• Onsager's conjecture and anomalous dissipation on domains with boundary
• Remarks on the emergence of weak Euler solutions in the vanishing viscosity limit
• Remarks on high Reynolds numbers hydrodynamics and the inviscid limit
• Hölder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains