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| The planar Euler equations describe the motion of a 2-D inviscid incompressible fluid, and also arise as a model problem for the study of the barotropic mode (to put it simply, the vertical average) of the Primitive equations of the ocean. It is a result by Yudovich that, in the space-periodic case, there exist a unique weak solution to the Euler system whenever the initial data has bounded vorticity. Relying on a refinement of the sharp $ A_p$ weighted bounds for singular integrals by Buckley, we prove an $L^\infty$ version of Grisvard's shift theorem on domains with corners, and extend the Yudovich theory of weak solutions for the Euler equations to this class of domains. We also discuss analogous results for the barotropic mode of the Primitive equations. This is partly joint work with Roger Temam and Claude Bardos. |
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