On weak solutions to the 2D Savage-Hutter model of the motion of a gravity driven avalanche flow

TL;DR

This paper demonstrates that the 2D Savage-Hutter model for avalanche flow admits infinitely many weak solutions via convex integration, yet maintains weak-strong uniqueness when energy constraints are applied.

Contribution

It introduces the application of convex integration to the Savage-Hutter system, revealing the existence of multiple weak solutions and establishing conditions for uniqueness.

Findings

Existence of infinitely many weak solutions for the model.

Weak-strong uniqueness holds under the energy inequality.

Application of convex integration to shallow water equations.

Abstract

We consider the Savage-Hutter system consisting of two-dimensional depth-integrated shallow water equations for the incompressible fluid with the Coulomb-type friction term. Using the method of convex integration we show that the associated initial-value problem possesses infinitely many weak solutions for any finite-energy initial data. On the other hand, the problem enjoys the weak-strong uniqueness property provided the system of equations is supplemented with the energy inequality.