Multiple Vortices for the Shallow Water Equation

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 \[{cases} -\varepsilon^2\text{div}(\frac{\nabla u}{b})=b(u-q\log\frac{1}{\varepsilon})_+^{p},& \text{in}\; Ω, u=0, &\text{on}\;\partial Ω, {cases} \] for small $\epsilon >0$, where $p>1$, $\text{div}(\frac{\nabla q}{b})=0$ and $\Omega \subset {\mathbb{R}}^2$ is a smooth bounded domain,. We showed that if $\frac{q^2}{b}$ has $m$ strictly local minimum(maximum) points ${\bar{z}}_i,i=1,...,m$, then there is a stationary classical solution approximating stationary $m$ points vortex solution of shallow water equations with vorticity ${\sum }_{i=1}^m\frac{2\pi q({\bar{z}}_i)}{b({\bar{z}}_i)}$. Moreover, strictly local minimum points of $\frac{q^2}{b}$ on the boundary can also give vortex solutions for the shallow water equation. As a further study we construct vortex pair solutions as well. Existence and asymptotic behavior of single point non-vanishing vortex solutions were studied by S. De Valeriola and J. Van Schaftingen.

27 pages. arXiv admin note: text overlap with arXiv:1208.3002

References in corpus (1)

• Regularization of point vortices for the Euler equation in dimension two, part II