Steady solutions to viscous shallow water equations. The case of heavy water

TL;DR

This paper proves the existence of steady solutions to the viscous shallow water equations with density-dependent viscosity, for large forcing and mass, highlighting the mathematical construction of solutions and their relation to compressible Navier-Stokes systems.

Contribution

It demonstrates the existence of regular steady solutions under large forcing and mass for the shallow water equations with density-dependent viscosity, a novel mathematical result.

Findings

Existence of solutions for large forcing and mass

Construction method for solutions to the equations

Connection to singular limits of compressible Navier-Stokes

Abstract

In this note, we show the existence of regular solutions to the stationary version of the Navier-Stokes system for compressible fluids with a density dependent viscosity, known as the shallow water equations. For arbitrary large forcing we are able to construct a solution, provided the total mass is sufficiently large. The main mathematical part is located in the construction of solutions. Uniqueness is impossible to obtain, since the gradient of the velocity is of magnitude of the force. The investigation is connected to the corresponding singular limit as Mach number goes to zero and methods for weak solutions to the compressible Navier-Stokes system.