期刊
JOURNAL OF DIFFERENTIAL EQUATIONS
卷 286, 期 -, 页码 557-577出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2021.03.037
关键词
Primitive equations; Rotation; Blow-up; Ill-posedness
类别
资金
- Einstein Stiftung/Foundation - Berlin, through the Einstein Visiting Fellow Program [EVF-2017-358(FU)]
- NSERC [371637-2019]
This study investigates the dynamics of inviscid primitive equations with rotation, proving their ill-posedness in Sobolev spaces and suggesting that a suitable space for well-posedness is Gevrey class of order s = 1.
Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid PEs without rotation is known to be ill-posed in Sobolev spaces, and its smooth solutions can form singularity in finite time. In this paper, we extend the above results in the presence of rotation. We construct finite-time blowup solutions to the inviscid PEs with rotation, and establish that the inviscid PEs with rotation is ill-posed in Sobolev spaces in the sense that its perturbation around a certain steady state background flow is both linearly and nonlinearly ill-posed in Sobolev spaces. Its linear instability is of the Kelvin-Helmholtz type similar to the one appears in the context of vortex sheets problem. This implies that the inviscid PEs is also linearly ill-posed in Gevrey class of order s > 1, and suggests that a suitable space for the well-posedness is Gevrey class of order s = 1, which is exactly the space of analytic functions. (C) 2021 Elsevier Inc. All rights reserved.
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