MULTIPLE VORTICES FOR THE SHALLOW WATER EQUATION IN TWO DIMENSIONS

• 作者: Daomin Cao Zhongyuan Liu
• 来源期刊: 应用数学年刊:英文版
• 学科: 数学
• 关键词: SHALLOW water EQUATION free BOUNDARY STREAM function VORTEX solution
• 年，卷（期）: 2019,（3）
• 页码范围: 221-249
• 页数: 29页
• 分类号: O17

摘要

In this paper,we construct stationary classical solutions of the shallow water equation with vanishing Froude number Fr in the so-called lake model.To this end we need to study solutions to the following semilinear elliptic problem -ε^2div(△u/b)=b(u-qlog1/ε)^p+,in Ω,u=0,on aΩfor small ε>0,where p>1,div(△q/b)=0 and ΩcR^2 is a smooth bounded domain.We show that if q^2/b has m strictly local minimum(maximum)points Zi,i=1,...,m,then there is a stationary classical solution approximating stationary m points vortex solution of shallow water equations with vorticity m∑i=1π2q(zi)/b(zi).Moreover,strictly local minimum points of q^2/b on the boundary can also give vortex solutions for the shallow water equation.