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arXiv · 1305.0773

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

Claude Bardos·László Székelyhidi Jr.·Emil Wiedemann
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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.

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Claude Bardos, László Székelyhidi Jr., Emil Wiedemann. 2013-05-03. Non-uniqueness for the Euler equations: the effect of the boundary. https://doi.org/10.1070/rm2014v069n02abeh004886

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