MEASURE VALUED SOLUTIONS OF SUB-LINEAR DIFFUSION EQUATIONS WITH A DRIFT TERM

作者：Fornaro Simona^*; Lisini Stefano; Savare Giuseppe; Toscani Giuseppe

来源：Discrete and Continuous Dynamical Systems, 2012, 32(5): 1675-1707.

DOI：10.3934/dcds.2012.32.1675

• Sublinear diffusion
• concentration phenomena
• propagation of singularities
• gradient flows
• nonlinear diffusion equations
• Wasserstein distance
• measure valued solutions

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 a strictly increasing C-1 function beta with lim(r -%26gt;+infinity)beta(r) %26lt; +infinity. 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. %26lt;br%26gt;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(c), which can be explicitly characterized in terms of beta and of the drift term. If the initial mass is less then mc, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure.

• 出版日期2012-5