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arXiv · 1009.4305

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

S. Fornaro·S. Lisini·G. Savare'·G. Toscani
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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 an increasing $C^1$ function $β$ with $\lim_{r\to +\infty} β(r)<+\infty$. 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 $β$ 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.

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S. Fornaro, S. Lisini, G. Savare', G. Toscani. 2010-09-22. Measure valued solutions of sub-linear diffusion equations with a drift term. https://doi.org/10.3934/dcds.2012.32.1675

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