×

Multiple vortices for the shallow water equation in two dimensions. (English) Zbl 1449.35222

Summary: 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 \[\begin{cases} -{\varepsilon^2}{\mathrm{div}}\left (\frac{\nabla u}{b}\right) = b\left (u-q\log\frac{1}{\varepsilon}\right)_+^p, & {\mathrm{in}}\;\Omega,\\ u=0, & {\mathrm{on}}\; \partial\Omega,\end{cases}\] for small \(\varepsilon > 0\), where \(p > 1\), \({\mathrm{div}}\left (\frac{\nabla q}{b}\right) = 0\) and \(\Omega \subset \mathbb{R}^2\) is a smooth bounded domain. We show that if \(\frac{q^2}{b}\) has \(m\) strictly local minimum (maximum) points \({{\bar z}_i}, i = 1,\cdots, m\), then there is a stationary classical solution approximating stationary \(m\) points vortex solution of shallow water equations with vorticity \(\sum\limits_{i=1}^m \frac{2\pi q ({\bar z}_i)}{b ({\bar z}_i)}\). Moreover, strictly local minimum points of \(\frac{q^2}{b}\) on the boundary can also give vortex solutions for the shallow water equation.

MSC:

35J61 Semilinear elliptic equations
35A09 Classical solutions to PDEs
35R35 Free boundary problems for PDEs
PDFBibTeX XMLCite