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

Lisini, Stefano

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

$${\partial }_{t}u-div\left(A(\nabla \left(f\left(u\right)\right)+u\nabla V)\right)=0\text{in}(0,+\infty )\times {ℝ}^n,(0.1)$$

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 $\phi \left(u\right)={\int }_{{ℝ}^n}F\left(u\left(x\right)\right)\mathrm{d}x+{\int }_{{ℝ}^n}V\left(x\right)u\left(x\right)\mathrm{d}x$, 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.

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