Local Boundedness of Weak Solutions to the Diffusive Wave Approximation of the Shallow Water Equations

TL;DR

This paper proves that weak solutions to a diffusive wave model of shallow water equations are locally bounded, ensuring mathematical stability and regularity of water height under various land and source conditions.

Contribution

It establishes local boundedness of weak solutions for a nonlinear PDE modeling shallow water flow, extending mathematical understanding of the model's regularity.

Findings

Weak solutions are locally bounded under broad conditions.

The result applies to equations with nonlinear diffusion terms.

Provides a foundation for further regularity and stability analysis.

Abstract

In this paper we prove that weak solutions to the Diffusive Wave Approximation of the Shallow Water equations $$ \partial_t u - \nabla\cdot ((u-z)^\alpha|\nabla u|^{\gamma-1}\nabla u) = f $$ are locally bounded. Here, $u$ describes the height of the water, $z$ is a given function that represents the land elevation and $f$ is a source term accounting for evaporation, infiltration or rainfall.