Measure valued solutions of sub-linear diffusion equations with a drift term
| Autor: | Fornaro, S., Lisini, S., Savare', G., Toscani, G. |
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| Rok vydání: | 2010 |
| Předmět: | |
| Zdroj: | Discrete Contin. Dyn. Syst. 32 (2012), 1675-1707 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.3934/dcds.2012.32.1675 |
| Popis: | 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. Comment: 30 pages |
| Databáze: | arXiv |
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