Inviscid, zero Froude number limit of the viscous shallow water system

Jianwei Yang,Huiyun Hao,Mengyu Liu

doi:10.1515/math-2021-0043

Abstract

Abstract In this paper, we study the inviscid and zero Froude number limits of the viscous shallow water system. We prove that the limit system is represented by the incompressible Euler equations on the whole space. Furthermore, the rate of convergence is also obtained.

Highlights

We know that scale analysis provides a valuable insight into the behaviour of complex fluid systems in the regime, where some of the characteristic dimensionless parameters become small or infinitely large
We study the following scaled two-dimensional viscous shallow water equations that are called the Saint-Venant equations among the French scientific community:
In [2], Wu performed the mathematical derivation of the rotating lake equations from the classical solution of the rotating shallow water and Euler equations via the low Froude number limit

Summary

Introduction

We know that scale analysis provides a valuable insight into the behaviour of complex fluid systems in the regime, where some of the characteristic dimensionless parameters become small or infinitely large. In [1], Cheng proved the low Froude number limit and convergence rates of compressible Euler and rotating shallow water equations, ill-prepared initial data towards their incompressible counterparts. In [2], Wu performed the mathematical derivation of the rotating lake equations from the classical solution of the rotating shallow water and Euler equations via the low Froude number limit. In this present paper, we consider the viscous version. The objective of this paper is to prove the low Froude number limit of the shallow water system (1.1)–(1.2).

Main results

Proof of main results