arXiv subjects

Clemens Kienzler

Publications and source records attributed to Clemens Kienzler.

2 recordsLinked to original sources

arXiv2016

Flatness implies smoothness for solutions of the porous medium equation

One of the major problems in the theory of the porous medium equation is the regularity of the solutions and the free boundaries. Here we assume flatness of the solution in space time cylinder and derive smoothness of the interface after a small time, as well as smoothness of the solution in the positivity set and up to the free boundary for some time interval. We use these facts to prove the following eventual regularity result: solutions with compactly supported initial data are smooth after a finite time T that depends on mass and the size of the initial support. This result eliminates the condition of non-degeneracy on the initial data that has been carried on for decades in the literature.

math.AP↗

arXiv2015

Flat Fronts and Stability for the Porous Medium Equation

This work is concerned with the equation $ \partial_t ρ= Δ_x ρ^m $, $ m > 1 $, known as the porous medium equation. It shows stability of the pressure of solutions close to flat travelling wave fronts in the homogeneous Lipschitz sense that is in a way optimal for the treatment of the equation. This is the first result of this type and implies global regularity estimates for any number of derivatives of the pressure. Consequences include smoothness, analyticity in temporal and tangential directions, and analyticity of the interface between empty and occupied regions. In the course of the argument a Gaussian estimate in an intrinsically arising space of homogeneous type is crucial to obtain linear estimates by means of the non-Euclidean Calderón-Zygmund singular integral theory.

math.AP↗