American Institute of Mathematical Sciences

January  2018, 38(1): 311-328. doi: 10.3934/dcds.2018015

Vanishing viscosity limit of the rotating shallow water equations with far field vacuum

 School of Mathematics and Statistics, Fuyang Normal College, Fuyang 236037, China

* Corresponding author: Zhigang Wang

Received  April 2017 Revised  July 2017 Published  January 2018

Fund Project: Zhigang Wang is supported by Chinese National Natural Science Foundation under grant 11401104 and China Postdoctoral Science Foundation under grant 2015M581579.

In this paper, we consider the Cauchy problem of the rotating shallow water equations, which has height-dependent viscosities, arbitrarily large initial data and far field vacuum. Firstly, we establish the existence of the unique local regular solution, whose life span is uniformly positive as the viscosity coefficients vanish. Secondly, we prove the well-posedness of the regular solution for the inviscid flow. Finally, we show the convergence rate of the regular solution from the viscous flow to the inviscid flow in $L^{\infty}([0, T]; H^{s'})$ for any $s'\in [2, 3)$ with a rate of $\epsilon^{1-\frac{s'}{3}}$.

Citation: Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015
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