Article Contents

2012, Volume 32, Issue 5: 1675-1707. Doi: 10.3934/dcds.2012.32.1675

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

1.
Università degli Studi di Pavia, Dipartimento di Matematica “F. Casorati”, via Ferrata 1, 27100 Pavia

Received: November 2010

Revised: July 2011

Published: May 2012

Abstract / Introduction Related Papers Cited by

Abstract

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 $\lim_{r\to +\infty} \beta(r)<+\infty$. 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 explicitly 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 stationary solution has a singular part in which the exceeding mass $m- m_{\rm c}$ is accumulated.

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Mathematics Subject Classification: 35K15, 35A21, 35B40.

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