Multiple Vortices for the Shallow Water Equation

TL;DR

This paper constructs stationary vortex solutions for the shallow water equations using solutions to a semilinear elliptic problem, revealing how local extrema of certain functions influence vortex formation.

Contribution

It introduces a new method to construct multiple vortex solutions for the shallow water equation based on elliptic problem solutions and local extrema analysis.

Findings

Existence of stationary vortex solutions near local minima/maxima of q^2/b.

Vortex solutions can be approximated by solutions to a specific elliptic problem.

Construction of vortex pair solutions as an extension.

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 \[{cases} -\varepsilon^2\text{div}(\frac{\nabla u}{b})=b(u-q\log\frac{1}{\varepsilon})_+^{p},& \text{in}\; \Omega, u=0, &\text{on}\;\partial \Omega, {cases} \] for small $\varepsilon>0$, 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.