Existence and uniqueness of a positive connection for the scalar viscous shallow water systemin a bounded interval

Marta Strani

doi:10.3934/cpaa.2014.13.1653

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2014, Volume 13, Issue 4: 1653-1667. Doi: 10.3934/cpaa.2014.13.1653

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Existence and uniqueness of a positive connection for the scalar viscous shallow water system in a bounded interval

Marta Strani^1,

1.
Universitat Wuerzburg, Campus Hubland Nord, Emil-Fischer-Strasse 30, 97074 Wuerzburg

Received: September 2013

Revised: January 2014

Published: July 2015

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Abstract

We study the existence and the uniqueness of a {\bf positive connection}, that is a stationary solution connecting the boundary data, for the initial-boundary value problem for the viscous shallow water system \begin{eqnarray} \partial_t u + \partial_x v=0, \partial_t v+\partial_x( \frac{v^2}{u}+ P(u))= \varepsilon\partial_x ( u \partial_x(\frac{v}{u})) \end{eqnarray} in a bounded interval $(-l,l)$ of the real line. We firstly consider the general case where the term of pressure $P(u)$ satisfies \begin{eqnarray} P(0)=0, P(+\infty)=+\infty, P'(u) \quad and \quad P''(u)>0 \ \forall u >0, \end{eqnarray} and then we show properties of the steady state in the relevant case $P(u)=\kappa u^{\gamma}$, $\gamma>1$. The viscous Saint-Venant system, corresponding to $\gamma=2$, fits in the general framework.

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Mathematics Subject Classification: Primary: 35A01, 35L50; Secondary: 34A05.

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