Abstract.
We consider the Dirichlet eigenvalue problem associated with the vortex patch in 2-D 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 of the growth of certain portion of the boundary of the patch.
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Kim, N. Eigenvalues associated with the vortex patch in 2-D Euler equations. Math. Ann. 330, 747–758 (2004). https://doi.org/10.1007/s00208-004-0568-4
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DOI: https://doi.org/10.1007/s00208-004-0568-4