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| Content Provider | Springer Nature Link |
|---|---|
| Author | Zhu, Shengguo Pan, Ronghua Li, Yachun |
| Copyright Year | 2016 |
| 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). |
| Ending Page | 190 |
| Page Count | 40 |
| Starting Page | 151 |
| File Format | |
| ISSN | 14226928 |
| e-ISSN | 14226952 |
| Journal | Journal of Mathematical Fluid Mechanics |
| Issue Number | 1 |
| Volume Number | 19 |
| Language | English |
| Publisher | Springer International Publishing |
| Publisher Date | 2016-07-21 |
| Publisher Place | Cham |
| Access Restriction | One Nation One Subscription (ONOS) |
| Subject Keyword | Shallow water equations compressible Navier–Stokes equations Stability Fluid- and Aerodynamics Asymptotic behavior of solutions Quantum hydrodynamics and relativistic hydrodynamics Conservation laws Mathematical Methods in Physics local well-posedness vacuum blow-up criterion Classical and Continuum Physics degenerate viscosity regular solutions Galactic and stellar dynamics |
| Content Type | Text |
| Resource Type | Article |
| Subject | Applied Mathematics Mathematical Physics Condensed Matter Physics Computational Mathematics |
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