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Multiple Vortices for the Shallow Water Equation in Two Dimensions

Multiple Vortices for the Shallow Water Equation in Two Dimensions

Year:    2019

Author:    Daomin Cao, Zhongyuan Liu

Annals of Applied Mathematics, Vol. 35 (2019), Iss. 3 : pp. 221–249

Abstract

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

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for small ε > 0, where p > 1, div(\frac{∇q}{b}) = 0 and \mathbb{R}is a smooth bounded domain.
We show that if \frac{q^2}{b} has m strictly local minimum (maximum) points \widetilde{z}_i, i = 1, · · · , m, then there is a stationary classical solution approximating stationary m points vortex solution of shallow water equations with vorticity \sum\limits_{i=1}^m \frac{2πq(\widetilde{z}_i)}{b(\widetilde{z}_i)}. Moreover, strictly local minimum points of \frac{q^2}{b} on the boundary can also give vortex solutions for the shallow water equation.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/2019-AAM-18081

Annals of Applied Mathematics, Vol. 35 (2019), Iss. 3 : pp. 221–249

Published online:    2019-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    29

Keywords:    shallow water equation free boundary stream function vortex solution.

Author Details

Daomin Cao

Zhongyuan Liu