On the structure of weak solutions to the Riemann problem for degenerate nonlinear diffusion equation

TL;DR

This paper derives explicit weak solutions for a Riemann problem in a degenerate nonlinear diffusion equation, revealing phase transition lines as minima of convex functions and extending to a variational formulation for infinite phases.

Contribution

It provides a novel explicit form of weak solutions for the Riemann problem in degenerate nonlinear diffusion equations, including a variational approach for infinitely many phases.

Findings

Phase transition lines are minima of convex functions.

Explicit weak solutions are derived for finite phases.

A variational formulation is obtained for infinite phases.

Abstract

We find an explicit form of weak solutions to a Riemann problem for a degenerate semilinear parabolic equation with piecewise constant diffusion coefficient. It is demonstrated that the phase transition lines (free boundaries) correspond to the minimum point of some strictly convex function of a finite number of variables. In the limit as number of phases tend to infinity we obtain a variational formulation of self-similar solution with an arbitrary nonnegative diffusion function.