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

ESAIM: Control, Optimisation and Calculus of Variations (2009)

- Volume: 15, Issue: 3, page 712-740
- ISSN: 1292-8119

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topLisini, Stefano. "Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces." ESAIM: Control, Optimisation and Calculus of Variations 15.3 (2009): 712-740. <http://eudml.org/doc/244645>.

@article{Lisini2009,

abstract = {We study existence and approximation of non-negative solutions of partial differential equations of the type\[\partial \_t u - \operatorname\{div\}(A(\nabla (f(u))+u\nabla V )) = 0 \qquad \mbox\{in \} (0,+\infty )\times \mathbb \{R\}^n,\qquad \qquad (0.1)\]where $A$ is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, $f:[0,+\infty ) \rightarrow [0,+\infty )$ is a suitable non decreasing function, $V:\mathbb \{R\}^n \rightarrow \mathbb \{R\}$ is a convex function. Introducing the energy functional $\phi (u)=\int _\{\mathbb \{R\}^n\} F(u(x))\,\{\rm d\}x+\int _\{\mathbb \{R\}^n\}V(x)u(x)\,\{\rm d\}x$, where $F$ is a convex function linked to $f$ by $f(u) = uF^\{\prime \}(u)-F(u)$, we show that $u$ is the “gradient flow” of $\phi $ with respect to the 2-Wasserstein distance between probability measures on the space $\mathbb \{R\}^n$, 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.},

author = {Lisini, Stefano},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {nonlinear diffusion equations; parabolic equations; variable coefficient parabolic equations; gradient flows; Wasserstein distance; asymptotic behaviour; asymptotic behavior},

language = {eng},

number = {3},

pages = {712-740},

publisher = {EDP-Sciences},

title = {Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces},

url = {http://eudml.org/doc/244645},

volume = {15},

year = {2009},

}

TY - JOUR

AU - Lisini, Stefano

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2009

PB - EDP-Sciences

VL - 15

IS - 3

SP - 712

EP - 740

AB - We study existence and approximation of non-negative solutions of partial differential equations of the type\[\partial _t u - \operatorname{div}(A(\nabla (f(u))+u\nabla V )) = 0 \qquad \mbox{in } (0,+\infty )\times \mathbb {R}^n,\qquad \qquad (0.1)\]where $A$ is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, $f:[0,+\infty ) \rightarrow [0,+\infty )$ is a suitable non decreasing function, $V:\mathbb {R}^n \rightarrow \mathbb {R}$ is a convex function. Introducing the energy functional $\phi (u)=\int _{\mathbb {R}^n} F(u(x))\,{\rm d}x+\int _{\mathbb {R}^n}V(x)u(x)\,{\rm d}x$, where $F$ is a convex function linked to $f$ by $f(u) = uF^{\prime }(u)-F(u)$, we show that $u$ is the “gradient flow” of $\phi $ with respect to the 2-Wasserstein distance between probability measures on the space $\mathbb {R}^n$, 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.

LA - eng

KW - nonlinear diffusion equations; parabolic equations; variable coefficient parabolic equations; gradient flows; Wasserstein distance; asymptotic behaviour; asymptotic behavior

UR - http://eudml.org/doc/244645

ER -

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