On Classical Solutions to 2D Shallow Water Equations with Degenerate Viscosities
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- 作者：Yachun Li ; Ronghua Pan ; Shengguo Zhu
- 关键词：Mathematics Subject ClassificationPrimary 35B40 ; 35A05 ; 76Y05 ; Secondary 35B35 ; 35L65 ; 85A05
- 刊名：Journal of Mathematical Fluid Mechanics
- 出版年：2017
- 出版时间：March 2017
- 年：2017
- 卷：19
- 期：1
- 页码：151-190
- 全文大小：
- 刊物类别：Physics and Astronomy
- 刊物主题：Fluid- and Aerodynamics; Mathematical Methods in Physics; Classical and Continuum Physics;
- 出版者：Springer International Publishing
- ISSN：1422-6952
- 卷排序：19
文摘
2D shallow water equations have degenerate viscosities proportional to surface height, which vanishes in many physical considerations, say, when the initial total mass, or energy are finite. Such a degeneracy is a highly challenging obstacle for development of well-posedness theory, even local-in-time theory remains open for a long time. In this paper, we will address this open problem with some new perspectives, independent of the celebrated BD-entropy (Bresch et al in Commun Math Phys 238:211–223, 2003, Commun Part Differ Eqs 28:843–868, 2003, Analysis and Simulation of Fluid Dynamics, 2007). After exploring some interesting structures of most models of 2D shallow water equations, we introduced a proper notion of solution class, called regular solutions, and identified a class of initial data with finite total mass and energy, and established the local-in-time well-posedness of this class of smooth solutions. The theory is applicable to most relatively physical shallow water models, broader than those with BD-entropy structures. We remark that our theory is on the local strong solutions, while the BD entropy is an essential tool for the global weak solutions. Later, a Beale-Kato-Majda type blow-up criterion is also established. This paper is mainly based on our early preprint (Li et al. in 2D compressible Navier–Stokes equations with degenerate viscosities and far field vacuum, preprint. arXiv:1407.8471, 2014).