Calculus of Variations and Geometric Measure Theory

S. Fornaro - S. Lisini - G. Savaré - G. Toscani

Measure valued solutions of sub-linear diffusion equations with a drift term

created by lisini on 22 Sep 2010
modified on 25 Jan 2012


Published Paper

Inserted: 22 sep 2010
Last Updated: 25 jan 2012

Journal: Discrete Contin. Dyn. Syst.-Series A
Volume: 32
Pages: 1675-1707
Year: 2012


In this paper we study nonnegative, measure valued solutions of the initial value problem for one-dimensional drift-diffusion equations when the nonlinear diffusion is governed by an increasing $C^1$ function $\beta$ with $\lim_{r\to +\infty} \beta(r)<+\infty$. By using tools of optimal transport, we will show that this kind of problems is well posed in the class of nonnegative Borel measures with finite mass $m$ and finite quadratic momentum and it is the gradient flow of a suitable entropy functional with respect to the so called $L^2$-Wasserstein distance.

Due to the degeneracy of diffusion for large densities, concentration of masses can occur, whose support is transported by the drift. We shall show that the large-time behavior of solutions depends on a critical mass ${m}_{\rm c}$, which can be explicitely characterized in terms of $\beta$ and of the drift term. If the initial mass is less then ${m}_{\rm c}$, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure.

Conversely, when the total mass $m$ of the solutions is greater than the critical one, the steady state has a singular part in which the exceeding mass ${m} - {m}_{\rm c}$ is accumulated.

Keywords: Wasserstein distance, Gradient flows, sublinear diffusion, concentration phenomena, propagation of singularities, nonlinear diffusion equations, measure valued solutions