Abstract

The theory for gravity driven avalanche flows is qualitatively similar to that of compressible fluid dynamics. I will present one of the models describing flow of granular avalanches – the Savage-Hutter model. The derivation of considered continuum flow models essentially bases on the fact that the characteristic length in the flowing direction is in general much larger than the thickness of an avalanche. Such an approach resulted in depth-averaged equation governed by generalized system of shallow water equations (Saint-Venant equations). The evolution of granular avalanches along an inclined slope is described by the mass conservation law and momentum balance law. Originally the model was derived in one-dimensional setting. Our interest is mostly directed to two-dimensional extension. As the solutions of the Savage-Hutter system develop shock waves and other singularities characteristic for hyperbolic system of conservation laws. Accordingly, any mathematical theory based on the classical concept of smooth solutions fails as soon as we are interested in global-in-time solutions to the system. I will start with presenting the concept of measure-valued solutions (generalization by DiPerna and Majda). Then I will show how the method of convex integration, recently adapted to the incompressible Euler system by De Lellis and Szekelyhidi, can be applied to show that the Savage-Hutter system is always solvable but not well posed in the class of weak solutions. The talk is based on the following results:

[1] P. Gwiazda On measure-valued solutions to a two-dimensional gravity-driven avalanche flow model. Math. Methods Appl. Sci. 28 (2005), no. 18, 2201-2223.

[2] E. Feireisl, P. Gwiazda, and A. Swierczewska Gwiazda. On weak solutions to the 2d Savage-Hutter model of the motion of a gravity driven avalanche flow, arXiv:1502.06223.

[3] P. Gwiazda, A. Swierczewska-Gwiazda, and E. Wiedemann. Weak-strong uniqueness for measure-valued solutions of the Savage-Hutter equations, arXiv:1503.05246