Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 712-740.

We study existence and approximation of non-negative solutions of partial differential equations of the type

 ${\partial }_{t}u-div\left(A\left(\nabla \left(f\left(u\right)\right)+u\nabla V\right)\right)=0\phantom{\rule{2em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\left(0,+\infty \right)×{ℝ}^{n},\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\left(0.1\right)$
where $A$ is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, $f:\left[0,+\infty \right)\to \left[0,+\infty \right)$ is a suitable non decreasing function, $V:{ℝ}^{n}\to ℝ$ is a convex function. Introducing the energy functional $\phi \left(u\right)={\int }_{{ℝ}^{n}}F\left(u\left(x\right)\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x+{\int }_{{ℝ}^{n}}V\left(x\right)u\left(x\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$, where $F$ is a convex function linked to $f$ by $f\left(u\right)=u{F}^{\text{'}}\left(u\right)-F\left(u\right)$, we show that $u$ is the “gradient flow” of $\phi$ with respect to the 2-Wasserstein distance between probability measures on the space ${ℝ}^{n}$, endowed with the riemannian distance induced by ${A}^{-1}.$ In the case of uniform convexity of $V$, long time asymptotic behaviour and decay rate to the stationary state for solutions of equation (0.1) are studied. A contraction property in Wasserstein distance for solutions of equation (0.1) is also studied in a particular case.

DOI : https://doi.org/10.1051/cocv:2008044
Classification : 35K55,  35K15,  35B40
Mots clés : nonlinear diffusion equations, parabolic equations, variable coefficient parabolic equations, gradient flows, Wasserstein distance, asymptotic behaviour
@article{COCV_2009__15_3_712_0,
author = {Lisini, Stefano},
title = {Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {712--740},
publisher = {EDP-Sciences},
volume = {15},
number = {3},
year = {2009},
doi = {10.1051/cocv:2008044},
zbl = {1178.35201},
mrnumber = {2542579},
language = {en},
url = {www.numdam.org/item/COCV_2009__15_3_712_0/}
}
Lisini, Stefano. Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 712-740. doi : 10.1051/cocv:2008044. http://www.numdam.org/item/COCV_2009__15_3_712_0/

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