Darcy's Law and the Theory of Shrinking Solutions of Fast Diffusion Equations

Abstract

The behavior of the solutions of the degenerate parabolic equation \[ v_{t}=v\,v_{xx}+{\kappa}|v_x|^{2}, \qquad {\kappa}\in {\mathbb{R}}, \] a basic model in the theory of flows in porous media, depends strongly on the parameter ${\kappa}$. We show here a striking example of that variability in the case of compactly supported solutions having freeboundaries. We consider the initial-value problem with continuous and compactly supported initial data $v(x,0)=v_0(x)\ge 0$. When ${\kappa}>0$ it is well known that this problem admits a unique weak solution which is compactly supported and the following free-boundary conditions are satisfied: \ $ v=0,$ \ $ v_t={\kappa}\,|v_x|^2.$ The latter relation is a form of Darcy's law and determines a unique solution, hence a unique choice of the interface. We prove here that for ${\kappa}\le 0$ there exist infinitely many solutions $v\ge 0$ with the same initial data and an interface on which Darcy's law holds. Actually, the interfaces can be chosen as arbitrary Lipschitz continuous curves as long as the support shrinks. Therefore, Darcy's law does not play a selecting role for this free-boundary problem.