Abstract
We consider Euler equations for an homogeneous incompressible non viscous fluid inside a smooth bounded domain of the plane. For an initial data of smooth vortex patch type, we obtain existence and uniqueness of a solution of the same type, locally in time if the initial patch is tangent to the boundary of the domain, and globally in time if the patch is far away from the boundary. We use pseudo-differential calculus to take care of the boundary condition. For the tangent limit case, we show that the velocity gradient of a vortex patch is Hölder continuous up to the boundary of the patch, using singular integrals. Our method provide also a result for several mutually tangent vortex patches in the plane.
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