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

An explicit form of weak solutions to the Riemann problem for a degenerate nonlinear parabolic equation with a piecewise constant diffusion coe cient is found. It is shown that the lines of phase transitions (free boundaries) correspond to the minimum point of some strictly convex and coercive function of a nite number of variables. A similar result is true for Stefan’s problem. In the limit, when the number of phases tends to in nity, there arises a variational formulation of self-similar solutions to the equation with an arbitrary nonnegative diffusion function.