Geodesic
Non-uniqueness for the Euler equations: the~effect of the boundary
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 69 (2014) no. 2, pp. 189-207

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Rotational initial data is considered for the two-dimensional incompressible Euler equations on an annulus. With use of the convex integration framework it is shown that there exist infinitely many admissible weak solutions (that is, 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 Lions, as opposed to the case without physical boundaries. Moreover, it is shown that admissible solutions are dissipative if they are Hölder continuous near the boundary of the domain. Bibliography: 34 titles.
Keywords: non-uniqueness, wild solutions, boundary effects, convex integration, rotational flows.
Mots-clés : Euler equations, dissipative solutions, inviscid limit 
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     author = {C. Bardos and L. Sz\'ekelyhidi and Jr. and E. Wiedemann},
     title = {Non-uniqueness for the {Euler} equations: the~effect of the boundary},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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C. Bardos; L. Székelyhidi; Jr.; E. Wiedemann. Non-uniqueness for the Euler equations: the~effect of the boundary. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 69 (2014) no. 2, pp. 189-207. http://geodesic.mathdoc.fr/item/RM_2014_69_2_a0/