Volume 35, Issue 3
Multiple Vortices for the Shallow Water Equation in Two Dimensions

Daomin Cao & Zhongyuan Liu

Ann. Appl. Math., 35 (2019), pp. 221-249.

Published online: 2020-08

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  • 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

image.png

for small $ε$ > 0, where $p$ > 1, div($\frac{∇q}{b}$) = 0 and $Ω$ ⊂ $\mathbb{R}$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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@Article{AAM-35-221, author = {Cao , Daomin and Liu , Zhongyuan}, title = {Multiple Vortices for the Shallow Water Equation in Two Dimensions}, journal = {Annals of Applied Mathematics}, year = {2020}, volume = {35}, number = {3}, pages = {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

image.png

for small $ε$ > 0, where $p$ > 1, div($\frac{∇q}{b}$) = 0 and $Ω$ ⊂ $\mathbb{R}$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.

}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/aam/18081.html} }
TY - JOUR T1 - Multiple Vortices for the Shallow Water Equation in Two Dimensions AU - Cao , Daomin AU - Liu , Zhongyuan JO - Annals of Applied Mathematics VL - 3 SP - 221 EP - 249 PY - 2020 DA - 2020/08 SN - 35 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/aam/18081.html KW - shallow water equation, free boundary, stream function, vortex solution. AB -

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

image.png

for small $ε$ > 0, where $p$ > 1, div($\frac{∇q}{b}$) = 0 and $Ω$ ⊂ $\mathbb{R}$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.

Cao , Daomin and Liu , Zhongyuan. (2020). Multiple Vortices for the Shallow Water Equation in Two Dimensions. Annals of Applied Mathematics. 35 (3). 221-249. doi:
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