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 - <p style="text-align: justify;">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<br/></p><p style="text-align:center"><img src="http://admin.global-sci.org/uploads/ueditor/image/20200824/1598255508654041.png" title="1598255508654041.png" alt="image.png"/></p><p style="text-align: justify;">for small $ε$ &gt; 0, where $p$ &gt; 1, div($\frac{∇q}{b}$)
= 0 and $Ω$ ⊂ $\mathbb{R}$<sup>2&nbsp;</sup>is a smooth bounded
domain.<br/>We show that if <span style="text-align: justify;">$\frac{q^2}{b}$&nbsp;</span>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 <span style="text-align: justify;">$\frac{q^2}{b}$&nbsp;</span>on the boundary can also give
vortex solutions for the shallow water equation.</p>