Monday, May 6, 2013

1305.0773 (Claude Bardos et al.)

Non-uniqueness for the Euler equations: the effect of the boundary    [PDF]

Claude Bardos, László Székelyhidi Jr., Emil Wiedemann
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.
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