论广义 SQG方程的局部自相似膨胀

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations Pub Date : 2024-09-23 DOI:10.1016/j.jde.2024.09.025

Anne Bronzi , Ricardo Guimarães , Cecilia Mondaini

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引用次数: 0

摘要

我们分析了 R2 中不粘性广义表面准地转方程（gSQG）的局部自相似型有限时间炸裂情形。在自相似剖面及其梯度的 Lr 增长假设下，我们确定了自相似参数的适当范围，在这些范围内，对于合适的 r，p，剖面要么为完全相同的零，因此不会发生炸裂，要么可以描述其 Lp 渐近行为。我们的结果扩展了 Xue [38] 关于 SQG 方程的研究，也部分恢复了 Cannone 和 Xue [3] 关于 gSQG 方程全局自相似解的结果。

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On the locally self-similar blowup for the generalized SQG equation

We analyze finite-time blowup scenarios of locally self-similar type for the inviscid generalized surface quasi-geostrophic equation (gSQG) in

R^{2}

. Under an

L^{r}

growth assumption on the self-similar profile and its gradient, we identify appropriate ranges of the self-similar parameter where the profile is either identically zero, and hence blowup cannot occur, or its

L^{p}

asymptotic behavior can be characterized, for suitable

r, p

. Our results extend the work by Xue [38] regarding the SQG equation, and also partially recover the results proved by Cannone and Xue [3] concerning globally self-similar solutions of the gSQG equation.

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来源期刊

Journal of Differential Equations 数学-数学

CiteScore

4.40

自引率

8.30%

发文量

543

审稿时长

9 months

期刊介绍： The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools. Research Areas Include: • Mathematical control theory • Ordinary differential equations • Partial differential equations • Stochastic differential equations • Topological dynamics • Related topics