NONLINEAR DIFFUSION EQUATIONS WITH VARIABLE COEFFICIENTS AS GRADIENT FLOWS IN WASSERSTEIN SPACES

被引:32
作者
Lisini, Stefano
机构
[1] Dipartimento di Scienze e Tecnologie Avanzate, Università Degli Studi Del Piemonte Orientale
关键词
Nonlinear diffusion equations; parabolic equations; variable coefficient parabolic equations; gradient flows; Wasserstein distance; asymptotic behaviour; ENTROPY DISSIPATION; EVOLUTION-EQUATIONS; STEEPEST DESCENT; INEQUALITIES; TRANSPORT; PRINCIPLE; MEDIA;
D O I
10.1051/cocv:2008044
中图分类号
TP [自动化技术、计算机技术];
学科分类号
0812 ;
摘要
We study existence and approximation of non-negative solutions of partial differential equations of the type partial derivative(t)u - div(A(del(f(u)) + u del V)) = 0 in (0, +infinity) x R-n, (0.1) where A is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, f : [0, +infinity) -> [0, +infinity) is a suitable non decreasing function, V : R-n -> R is a convex function. Introducing the energy functional phi(u) = integral(Rn) F(u(x))dx + integral(Rn) V (x)u(x)dx, where F is a convex function linked to f by f(u) = uF'(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 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.
引用
收藏
页码:712 / 740
页数:29
相关论文
共 34 条
[1]  
Agueh M, 2005, ADV DIFFERENTIAL EQU, V10, P309
[2]  
Ambrosio L., 1995, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., V19, P191
[3]  
Ambrosio L., 2000, Oxford Mathematical Monographs
[4]  
Ambrosio L., 2005, LECT NOTES CIME SUMM
[5]  
[Anonymous], 2007, POROUS MEDIUM EQUATI
[6]   On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations [J].
Arnold, A ;
Markowich, P ;
Toscani, G ;
Unterreiter, A .
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 2001, 26 (1-2) :43-100
[7]  
Benamou JD, 2000, NUMER MATH, V84, P375, DOI 10.1007/s002119900117
[8]   Solution of a model Boltzmann equation via steepest descent in the 2-Wasserstein metric [J].
Carlen, EA ;
Gangbo, W .
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 2004, 172 (01) :21-64
[9]   Constrained steepest descent in the 2-Wasserstein metric [J].
Carlen, EA ;
Gangbo, W .
ANNALS OF MATHEMATICS, 2003, 157 (03) :807-846
[10]   Contractions in the 2-Wasserstein length space and thermalization of granular media [J].
Carrillo, JA ;
McCann, RJ ;
Villani, CD .
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 2006, 179 (02) :217-263