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Evidence of singularities for a family of contour dynamics equations. (English) Zbl 1135.76315

Summary: In this work, we show evidence of the existence of singularities developing in finite time for a class of contour dynamics equations depending on a parameter \(0 < \alpha \leq 1\). The limiting case \(\alpha\to 0\) corresponds to 2D Euler equations, and \(\alpha = 1\) corresponds to the surface quasi-geostrophic equation. The singularity is point-like, and it is approached in a self-similar manner.

MSC:

76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35A20 Analyticity in context of PDEs
35Q35 PDEs in connection with fluid mechanics
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