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On Classical Solutions to 2D Shallow Water Equations with Degenerate Viscosities

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Abstract

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).

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Authors and Affiliations

  1. School of Mathematical Sciences, MOE-LSC, and SHL-MAC, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Yachun Li

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, 30332, USA

    Ronghua Pan & Shengguo Zhu

  3. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Shengguo Zhu

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  1. Yachun Li
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  2. Ronghua Pan
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Correspondence to Yachun Li.

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Communicated by E. Feireisl

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Li, Y., Pan, R. & Zhu, S. On Classical Solutions to 2D Shallow Water Equations with Degenerate Viscosities. J. Math. Fluid Mech. 19, 151–190 (2017). https://doi.org/10.1007/s00021-016-0276-3

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