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

Stefano Lisini

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

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

Abstract

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We study existence and approximation of non-negative solutions of partial differential equations of the type ∂ t u - div ( A ( ∇ ( f ( u ) ) + u ∇ V ) ) = 0 in ( 0 , + ∞ ) × ℝ n , ( 0 . 1 ) where A is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, f : [ 0 , + ∞ ) → [ 0 , + ∞ ) is a suitable non decreasing function, V : ℝ n → ℝ is a convex function. Introducing the energy functional φ ( u ) = ∫ ℝ n F ( u ( x ) ) d x + ∫ 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.

How to cite

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Lisini, 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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