The α$\alpha$‐SQG patch problem is illposed in C2,β$C^{2,\beta }$ and W2,p$W^{2,p}$

IF 3.1 1区数学 Q1 MATHEMATICS Communications on Pure and Applied Mathematics Pub Date : 2024-11-16 DOI:10.1002/cpa.22236

Alexander Kiselev, Xiaoyutao Luo

引用次数: 0

Abstract

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

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在 C2,β$C^{2,\beta }$ 和 W2,p$W^{2,p}$ 中，α$\alpha$-SQG 补丁问题存在问题。

我们考虑的是-（表面准地养）SQG 系统的补集问题，其值和分别为二维欧拉方程和 SQG 方程。众所周知，欧拉补集在非端点荷尔德空间以及在Ⅳ空间中都是全局良好的。此外，在合适的正则范围内，除非......，否则每个 Sobolev 空间都具有相同的强失稳性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Communications on Pure and Applied Mathematics 数学-数学

CiteScore

6.70

自引率

3.30%

发文量

审稿时长

>12 weeks

期刊介绍： Communications on Pure and Applied Mathematics (ISSN 0010-3640) is published monthly, one volume per year, by John Wiley & Sons, Inc. © 2019. The journal primarily publishes papers originating at or solicited by the Courant Institute of Mathematical Sciences. It features recent developments in applied mathematics, mathematical physics, and mathematical analysis. The topics include partial differential equations, computer science, and applied mathematics. CPAM is devoted to mathematical contributions to the sciences; both theoretical and applied papers, of original or expository type, are included.

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The α$\alpha$‐SQG patch problem is illposed in C2,β$C^{2,\beta }$ and W2,p$W^{2,p}$ Mean‐field limit of non‐exchangeable systems Semiconvexity estimates for nonlinear integro‐differential equations Issue Information - TOC Issue Information - TOC