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

ESAIM: Control, Optimisation and Calculus of Variations (2008)

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

## Access Full Article

top## Abstract

top## How to cite

topLisini, Stefano. "Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces." ESAIM: Control, Optimisation and Calculus of Variations 15.3 (2008): 712-740. <http://eudml.org/doc/90933>.

@article{Lisini2008,

abstract = {
We study existence and approximation of non-negative solutions of partial differential equations of the type
$$\partial\_t u - \div (A(\nabla (f(u))+u\nabla V )) = 0 \qquad \mbox\{in \} (0,+\infty )\times \mathbb\{R\}^n,\qquad\qquad (0.1)$$
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 $f(u) = uF'(u)-F(u)$,
we show that u is the “gradient flow” of ϕ with respect to the
2-Wasserstein distance between probability measures on
the space $\mathbb\{R\}^n$, 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; nonlinear diffusion equations; Wasserstein distance; asymptotic behavior},

language = {eng},

month = {7},

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/90933},

volume = {15},

year = {2008},

}

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

DA - 2008/7//

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
$$\partial_t u - \div (A(\nabla (f(u))+u\nabla V )) = 0 \qquad \mbox{in } (0,+\infty )\times \mathbb{R}^n,\qquad\qquad (0.1)$$
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 $f(u) = uF'(u)-F(u)$,
we show that u is the “gradient flow” of ϕ with respect to the
2-Wasserstein distance between probability measures on
the space $\mathbb{R}^n$, 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; nonlinear diffusion equations; Wasserstein distance; asymptotic behavior

UR - http://eudml.org/doc/90933

ER -

## References

top- M. Agueh, Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. Adv. Differential Equations10 (2005) 309–360.
- L. Ambrosio, Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl.19 (1995) 191–246.
- L. Ambrosio, Transport equation and cauchy problem for non-smooth vector fields. Lecture Notes of the CIME Summer school (2005) available on line at . URIhttp://cvgmt.sns.it/people/ambrosio/
- L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000).
- L. Ambrosio, N. Gigli and G. Savarè, Gradient flows in metric spaces and in the Wasserstein spaces of probability measures. Birkhäuser (2005).
- A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Comm. Partial Diff. Eq.26 (2001) 43–100.
- J.D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math.84 (2000) 375–393.
- E.A. Carlen and W. Gangbo, Constrained steepest descent in the 2-Wasserstein metric. Ann. Math.157 (2003) 807–846.
- E.A. Carlen and W. Gangbo, Solution of a model Boltzmann equation via steepest descent in the 2-Wasserstein metric. Arch. Rational Mech. Anal.172 (2004) 21–64.
- J.A. Carrillo, A. Jüngel, P.A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math.133 (2001) 1–82.
- J.A. Carrillo, R.J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana19 (2003) 971–1018.
- J.A. Carrillo, R.J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Rational Mech. Anal.179 (2006) 217–263.
- J. Crank, The mathematics of diffusion. Clarendon Press, Oxford, second edition (1975).
- G. De Cecco and G. Palmieri, Intrinsic distance on a Lipschitz Riemannian manifold. Rend. Sem. Mat. Univ. Politec. Torino46 (1990) 157–170.
- G. De Cecco and G. Palmieri, Intrinsic distance on a LIP Finslerian manifold. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl.17 (1993) 129–151.
- G. De Cecco and G. Palmieri, LIP manifolds: from metric to Finslerian structure. Math. Z.218 (1995) 223–237.
- E. De Giorgi, New problems on minimizing movements, in Boundary value problems for partial differential equations and applications, RMA Res. Notes Appl. Math.29, Masson, Paris (1993) 81–98.
- E. De Giorgi, A. Marino and M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.68 (1980) 180–187.
- M. Degiovanni, A. Marino and M. Tosques, Evolution equations with lack of convexity. Nonlinear Anal.9 (1985) 1401–1443.
- C. Dellacherie and P.A. Meyer, Probabilities and potential. North-Holland Publishing Co., Amsterdam (1978).
- L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992).
- R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal.29 (1998) 1–17 (electronic).
- D. Kinderlehrer and A. Tudorascu, Transport via mass transportation. Discrete Contin. Dyn. Syst. Ser. B6 (2006) 311–338.
- S. Lisini, Characterization of absolutely continuous curves in Wasserstein spaces. Calc. Var. Partial Differential Equations28 (2007) 85–120.
- R.J. McCann, A convexity principle for interacting gases. Adv. Math.128 (1997) 153–179.
- F. Otto, Doubly degenerate diffusion equations as steepest descent. Manuscript (1996) available on line at . URIhttp://www-mathphys.iam.uni-bonn.de/web/forschung/publikationen/main-en.htm
- F. Otto, Evolution of microstructure in unstable porous media flow: a relaxational approach. Comm. Pure Appl. Math.52 (1999) 873–915.
- F. Otto, The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Diff. Eq.26 (2001) 101–174.
- L. Petrelli and A. Tudorascu, Variational principle for general diffusion problems. Appl. Math. Optim.50 (2004) 229–257.
- K.-T. Sturm, Convex functionals of probability measures and nonlinear diffusions on manifolds. J. Math. Pures Appl.84 (2005) 149–168.
- J.L. Vázquez, The porous medium equation, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford (2007).
- C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics58. American Mathematical Society, Providence, RI (2003).
- M.-K. von Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy and Ricci curvature. Comm. Pure Appl. Math.58 (2005) 923–940.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.