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Eigenvalues associated with the vortex patch in 2-D Euler equations. (English) Zbl 1058.76012
Summary: We consider Dirichlet eigenvalue problem associated with a vortex patch for two-dimensional Euler equations. We show that the eigenvalues grow at most doubly exponentially in time. As an application, we derive bounds on the growth of some geometric quantities like the diameter and the inscription radius of the patch. We also discuss the growth of the perimeter of the patch. In particular, we give a double exponential bound on the growth of certain portion of the boundary of the patch.
MSC:
76B03Existence, uniqueness, and regularity theory (fluid mechanics)
76B47Vortex flows
35Q35PDEs in connection with fluid mechanics
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