Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces

Lisini, Stefano

doi:10.1051/cocv:2008044

Lisini, Stefano

ESAIM: Control, Optimisation and Calculus of Variations, Volume 15 (2009) no. 3, p. 712-740

Abstract

We study existence and approximation of non-negative solutions of partial differential equations of the type $\partial_{t} u - div (A (\nabla (f (u)) + u \nabla V)) = 0 in (0, + \infty) \times ℝ^{n}, (0.1)$ where $A$ is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, $f : [0, + \infty) \to [0, + \infty)$ is a suitable non decreasing function, $V : ℝ^{n} \to ℝ$ is a convex function. Introducing the energy functional $φ (u) = \int_{ℝ^{n}} F (u (x)) d x + \int_{ℝ^{n}} V (x) u (x) d x$ , where $F$ is a convex function linked to $f$ by $f (u) = u F^{'} (u) - F (u)$ , we show that $u$ is the “gradient flow” of $φ$ 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.

MR 2542579 | Zbl 1178.35201 | 1 citation in Numdam

DOI : https://doi.org/10.1051/cocv:2008044
Classification: 35K55, 35K15, 35B40
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {3},
     year = {2009},
     pages = {712-740},
     doi = {10.1051/cocv:2008044},
     zbl = {1178.35201},
     mrnumber = {2542579},
     language = {en},
     url = {http://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, Volume 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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