The α-SQG patch problem is illposed in C<sup>2,β</sup> and W<sup>2,p</sup>

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2024-01-01

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We consider the patch problem for the α-(surface quasi-geostrophic) SQG system with the values α = 0 and (Formula presented.) being the 2D Euler and the SQG equations respectively. It is well-known that the Euler patches are globally wellposed in non-endpoint C2,β Hölder spaces, as well as in W2,p, (Formula presented.) spaces. In stark contrast to the Euler case, we prove that for (Formula presented.), the (Formula presented.) -SQG patch problem is strongly illposed in every (Formula presented.) Hölder space with (Formula presented.). Moreover, in a suitable range of regularity, the same strong illposedness holds for every W2,p Sobolev space unless p = 2.

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https://hdl.handle.net/10161/31790

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https://creativecommons.org/licenses/by-nc/4.0

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10.1002/cpa.22236

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Kiselev, A, and X Luo (2024). The α-SQG patch problem is illposed in C2,β and W2,p. Communications on Pure and Applied Mathematics. 10.1002/cpa.22236 Retrieved from https://hdl.handle.net/10161/31790.

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Kiselev

Alexander A. Kiselev

William T. Laprade Distinguished Professor of Mathematics

My current research interests focus on mathematical fluid mechanics and mathematical biology.
In the past, I have also worked on reaction-diffusion equations and spectral theory of Schredinger operators. 

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