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

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.

KEYWORDS:

• Avalanche flow
• convex integration
• Savage–Hutter system
• weak solution

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 35Q86
• 35L60
• Secondary: 76T25