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, Volume 121, Issue 1, pp 1–50 | Cite as

Stability of flows associated to gradient vector fields and convergence of iterated transport maps

  • Luigi AmbrosioEmail author
  • Stefano Lisini
  • Giuseppe Savaré


In this paper we address the problem of stability of flows associated to a sequence of vector fields under minimal regularity requirements on the limit vector field, that is supposed to be a gradient. We apply this stability result to show the convergence of iterated compositions of optimal transport maps arising in the implicit time discretization (with respect to the Wasserstein distance) of nonlinear evolution equations of a diffusion type. Finally, we use these convergence results to study the gradient flow of a particular class of polyconvex functionals recently considered by Gangbo, Evans and Savin. We solve some open problems raised in their paper and obtain existence and uniqueness of solutions under weaker regularity requirements and with no upper bound on the jacobian determinant of the initial datum.


Tangent Velocity Gradient Vector Internal Energy Homogeneous Neumann Boundary Condition Admissible Velocity 
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  1. 1.
    Agueh M. (2005) Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. Adv Differential Equations 10, 309–360MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ambrosio, L. Lecture notes on transport equation and cauchy problem for bv vector fields and applications. To appear in the proceedings of the School on Geometric Measure Theory, Luminy, October 2003, available at (2004)Google Scholar
  3. 3.
    Ambrosio L. (2004) Transport equation and cauchy problem for bv vector fields. Inventiones Mathematicae 158, 227–260CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Ambrosio L., Crippa G., Maniglia S. (2005) Traces and fine properties of a BD class of vector fields and applications. Ann Fac Sci Toulouse Math. 14(6): 527–561MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ambrosio L., Fusco N., Pallara D. (2000) Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. Clarendon Press, OxfordGoogle Scholar
  6. 6.
    Ambrosio L., Gigli N., Savaré G. (2005) Gradient flows – in metric spaces and in the space of probability measures. Birkhäuser, BaselzbMATHGoogle Scholar
  7. 7.
    Aronson D.G., Bénilan P. (1979) Régularité des solutions de l’équation des milieux poreux dans R N. CR Acad Sci Paris Sér A-B 288, A103–A105Google Scholar
  8. 8.
    Bouchut F., James F. (1998) One-dimensional transport equations with discontinuous coefficients. Nonlinear Anal 32, 891–933CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Bouchut F., James F., Mancini S. (2005) Uniqueness and weak stability for multi-dimensional transport equations with one-sided lipschitz coefficient. Annali SNS 4(5): 1–25MathSciNetzbMATHGoogle Scholar
  10. 10.
    Brenier Y. (1991) Polar factorization and monotone rearrangement of vector-valued. Comm Pure Appl Math 44, 375–417CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Brézis H., Crandall M.G. (1979) Uniqueness of solutions of the initial-value problem for \(u_{t}-\Delta \varphi (u)=0\). J Math Pures Appl 58(9): 153–163MathSciNetzbMATHGoogle Scholar
  12. 12.
    Brézis H., Strauss W.A. (1973) Semi-linear second-order elliptic equations in L 1. J Math Soc Jpn, 25, 565–590zbMATHGoogle Scholar
  13. 13.
    Caffarelli L.A. (1991) Some regularity properties of solutions of Monge Ampère equation. Comm Pure Appl Math 44, 965–969CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Caffarelli L.A. (1992) Boundary regularity of maps with convex potentials. Comm Pure Appl Math 45, 1141–1151CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Caffarelli L.A. (1992) The regularity of mappings with a convex potential. J Am Math Soc 5, 99–104CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Caffarelli L.A. (1996) Boundary regularity of maps with convex potentials. II. Ann Math 144(2): 453–496MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Caffarelli L.A., Friedman A. (1979) Continuity of the density of a gas flow in a porous medium. Trans Am Math Soc 252, 99–113CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Crandall M.G., Liggett T.M. (1971) Generation of semi-groups of nonlinear transformations on general Banach spaces. Am J Math 93, 265–298CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    DiBenedetto E. (1983) Continuity of weak solutions to a general porous medium equation. Indiana Univ Math J 32, 83–118CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    DiBenedetto, E. Degenerate parabolic equations. Universitext, Springer Berlin Heidelberg, New York 1993Google Scholar
  21. 21.
    DiPerna R.J., Lions P.-L. (1989) Ordinary differential equations, transport theory and Sobolev spaces. Invent Math 98, 511–547CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Evans, L.C. Partial differential equations and Monge-Kantorovich mass transfer. In: Cambridge MA (Ed.) Current developments in mathematics, 1997. Int Press Boston, 65–126 1999Google Scholar
  23. 23.
    Evans L.C., Savin O., Gangbo W. (2005) Diffeomorphisms and nonlinear heat flows. SIAM J Math Anal 37, 737–751 (electronic)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Friedman, A. Variational principles and free-boundary problems. Pure and Applied Mathematics. Wiley, New York. A Wiley-Interscience Publication 1982Google Scholar
  25. 25.
    Gangbo W., McCann R.J. (1996) The geometry of optimal transportation. Acta Math 177, 113–161CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Giaquinta, M., Modica, G., Souček, J. Cartesian currents in the calculus of variations. I. vol. 38 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A series of modern surveys in mathematics [Results in mathematics and related areas. 3rd series. A series of modern surveys in mathematics], Springer, Berlin Heidelberg New York. Variational integrals 1998Google Scholar
  27. 27.
    Hauray M. (2003) On two-dimensional Hamiltonian transport equations with \(L^P_{\rm loc}\) coefficients. Ann Inst H Poincaré Anal Non Linéaire 20, 625–644CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Hauray M. (2004) On Liouville transport equation with force field in BV loc. Comm Partial Differential Equations 29, 207–217CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Jordan R., Kinderlehrer D., Otto F. (1998) The variational formulation of the Fokker-Planck equation. SIAM J Math Anal 29, 1–17 (electronic)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Ladyženskaja, O.A., Solonnikov, V.A., Ural’cevaceva, N.N. Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, R.I. 1967.Google Scholar
  31. 31.
    Le Bris C., Lions P.-L. (2004) Renormalized solutions of some transport equations with partially W 1,1 velocities and applications. Ann Mat Pura Appl 183(4): 97–130CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Lerner N. (2004) Transport equations with partially BV velocities. Ann Sci Norm Super Pisa Cl Sci 3(5): 681–703MathSciNetzbMATHGoogle Scholar
  33. 33.
    Lieberman G.M. (1986) Intermediate Schauder theory for second order parabolic equations. I. Existence, uniqueness, and regularity. J Differential Equations 63, 32–57CrossRefMathSciNetGoogle Scholar
  34. 34.
    McCann R.J. (1997) A convexity principle for interacting gases. Adv Math 128, 153–179CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Otto, F. Doubly degenerate diffus ion equations as steepest descent. Manuscript (1996)Google Scholar
  36. 36.
    Otto F. (1999) Evolution of microstructure in unstable porous media flow: a relaxational approach. Comm Pure Appl Math 52, 873–915CrossRefMathSciNetGoogle Scholar
  37. 37.
    Otto F. (2001) The geometry of dissipative evolution equations: the porous medium equation. Comm Partial Differential Equations 26, 101–174CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    Porzio M.M., Vespri V. (1993) Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J Differential Equations 103, 146–178CrossRefMathSciNetzbMATHGoogle Scholar
  39. 39.
    Poupaud F., Rascle M. (1997) Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients. Comm Partial Differential Equations 22, 337–358CrossRefMathSciNetzbMATHGoogle Scholar
  40. 40.
    Urbas J.I.E. (1988) Global Hölder estimates for equations of Monge-Ampère type. Invent Math 91, 1–29CrossRefMathSciNetzbMATHGoogle Scholar
  41. 41.
    Urbas J.I.E. (1988) Regularity of generalized solutions of Monge-Ampère equations. Math Z 197, 365–393CrossRefMathSciNetzbMATHGoogle Scholar
  42. 42.
    Villani C. (2003) Topics in optimal transportation. vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, ProvidencezbMATHGoogle Scholar
  43. 43.
    Ziemer W.P. (1982) Interior and boundary continuity of weak solutions of degenerate parabolic equations. Trans Am Math Soc 271, 733–748CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Luigi Ambrosio
    • 1
    Email author
  • Stefano Lisini
    • 2
  • Giuseppe Savaré
    • 2
  1. 1.Scuola Normale Superiore PisaPisaItaly
  2. 2.Dipartimento di MatematicaUniversità di PaviaPaviaItaly

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