On well/ill-posedness for the generalized surface quasi-geostrophic equation in Hölder spaces

IF 2.3 2区 数学 Q1 MATHEMATICS
Young-Pil Choi , Jinwook Jung , Junha Kim
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引用次数: 0

Abstract

We establish the well/ill-posedness theories for the inviscid α-surface quasi-geostrophic (α-SQG) equation in Hölder spaces, where α=0 and α=1 correspond to the two-dimensional Euler equation in the vorticity formulation and SQG equation of geophysical significance, respectively. We first prove the local-in-time well-posedness of α-SQG equation in L([0,T];C0,β(R2)) with β(α,1) for some T>0. We then analyze the strong ill-posedness in C0,α(R2) constructing smooth solutions to the α-SQG equation that exhibit C0,α–norm growth in a short time. In particular, we develop the nonexistence theory for α-SQG equation in C0,α(R2).
Hölder空间中广义曲面拟地转方程的适位性
建立了Hölder空间中无粘α-表面拟地转(α-SQG)方程的井/不适定性理论,其中α=0和α=1分别对应于涡度公式中的二维欧拉方程和具有地球物理意义的SQG方程。首先证明了α- sqg方程在L∞([0,T];C0,β(R2))上,当β∈(α,1)时的局域时适定性。然后,我们分析了在C0,α(R2)上的强病态性,构造了在短时间内表现出C0,α -范数增长的α- sqg方程的光滑解。特别地,我们发展了α- sqg方程在C0,α(R2)中的不存在性理论。
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来源期刊
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
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