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

Zhigang Wang

doi:10.3934/dcds.2018015

Article Contents

2018, Volume 38, Issue 1: 310-327. Doi: 10.3934/dcds.2018015

This issue Previous Article Limit periodic upper and lower solutions in a generic sense Next Article Symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system

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

Zhigang Wang^,

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

* Corresponding author: Zhigang Wang
* Corresponding author: Zhigang Wang

Received: April 2017

Revised: July 2017

Published: October 2017

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

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Abstract

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 $\epsilon^{1-\frac{s'}{3}}$.

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Mathematics Subject Classification: Primary:35B40, 35A05;Secondary:35B35, 85A05.

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