Advertisement

Archive for Rational Mechanics and Analysis

, Volume 194, Issue 1, pp 133–220 | Cite as

The Wasserstein Gradient Flow of the Fisher Information and the Quantum Drift-diffusion Equation

  • Ugo Gianazza
  • Giuseppe SavaréEmail author
  • Giuseppe Toscani
Article

Abstract

We prove the global existence of non-negative variational solutions to the “drift diffusion” evolution equation \({{\partial_t} u+ div \left(u{\mathrm{D}}\left(2 \frac{\Delta \sqrt u}{\sqrt u}-{f}\right)\right)=0}\) under variational boundary condition. Despite the lack of a maximum principle for fourth order equations, non-negative solutions can be obtained as a limit of a variational approximation scheme by exploiting the particular structure of this equation, which is the gradient flow of the (perturbed) Fisher information functional \({\fancyscript F^f(u):=\frac 12\int \left|{\mathrm{D}} \log u\right|^2 {u} dx+\int fu dx}\) with respect to the Kantorovich–Rubinstein–Wasserstein distance between probability measures. We also study long-time behavior of the solutions, proving their exponential decay to the equilibrium state g = eV characterized by \({-\Delta V+\frac12 \left|{\mathrm{D}} V\right|^2=f,\quad \int {\rm e}^{-V} dx=\int u_{0}dx,}\) when the potential V is uniformly convex: in this case the functional \({\fancyscript F^f}\) coincides with the relative Fisher information\({\fancyscript F^f(u)=\frac12\fancyscript I(u|g)= \int \left|{\mathrm{D}}\log(u/g)\right|^2u dx}\).

Keywords

Fisher Information Lower Semicontinuity Relative Entropy Planck Equation Logarithmic Sobolev Inequality 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambrosetti A., Prodi G.: A Primer of Nonlinear Analysis. Cambridge Studies in Advanced Mathematics, vol. 34. Cambridge University Press, Cambridge (1993)Google Scholar
  2. 2.
    Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. Clarendon Press, Oxford (2000)zbMATHGoogle Scholar
  3. 3.
    Ambrosio L., Gigli N., Savaré G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2005)zbMATHGoogle Scholar
  4. 4.
    Ambrosio L., Savaré G.: Gradient Flows of Probability Measures. Handbook of Evolution Equations (III). Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  5. 5.
    Ambrosio, L., Savaré, G., Zambotti, L.: Existence and Stability for Fokker–Planck Equations with log-concave Reference Measure, ArXiv Mathematics e-prints (2007)Google Scholar
  6. 6.
    Bakry D., Émery M.: Inégalités de Sobolev pour un semi-groupe symétrique. C. R. Acad. Sci. Paris Sér. I Math. 301, 411–413 (1985)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Benamou J.-D., Brenier Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84, 375–393 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Blachman N.M.: The convolution inequality for entropy powers. IEEE Trans. Inform. Theory IT-11, 267–271 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bleher P.M., Lebowitz J.L., Speer E.R.: Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations. Comm. Pure Appl. Math. 47, 923–942 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bogachev, V.I.: Gaussian Measures, Mathematical Surveys and Monographs, vol. 62. American Mathematical Society, Providence, 1998Google Scholar
  11. 11.
    Brenier Y.: Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44, 375–417 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Brezis H.: On a characterization of flow-invariant sets. Comm. Pure Appl. Math. 23, 261–263 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Brezis H.: Analyse fonctionnelle—Théorie et applications. Masson, Paris (1983)zbMATHGoogle Scholar
  14. 14.
    Buttazzo, G.: Semicontinuity, relaxation and integral representation in the calculus of variations. Pitman Research Notes in Mathematics Series, vol. 207. Longman Scientific & Technical, Harlow, 1989Google Scholar
  15. 15.
    Cáceres M.J., Carrillo J.A., Toscani G.: Long-time behavior for a nonlinear fourth-order parabolic equation. Trans. Amer. Math. Soc. 357, 1161–1175 (2005) (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Carlen E.A.: Superadditivity of Fisher’s information and logarithmic Sobolev inequalities. J. Funct. Anal. 101, 194–211 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Carlen, E.A., Loss M.: Logarithmic Sobolev inequalities and spectral gaps. Recent Advances in the Theory and Applications of Mass Transport. Contemp. Math., vol. 353. American Mathematical Society (ed.), Providence, pp. 53–60, 2004Google Scholar
  18. 18.
    Carrillo J.A., Toscani G.: Long-time asymptotics for strong solutions of the thin film equation. Comm. Math. Phys. 225, 551–571 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Da Prato G., Lunardi A.: Elliptic operators with unbounded drift coefficients and Neumann boundary condition. J. Differ. Equ. 198, 35–52 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Da Prato G., Lunardi A.: Elliptic operators with unbounded drift coefficients and Neumann boundary condition. J. Differ. Equ. 198, 35–52 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    De Giorgi, E.: New problems on minimizing movements. Boundary Value Problems for PDE and Applications (eds. Baiocchi C. and Lions J. L.) Masson, Paris, pp. 81–98, 1993Google Scholar
  22. 22.
    De Giorgi E., Marino A., Tosques M.: Problems of evolution in metric spaces and maximal decreasing curve. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 68(8), 180–187 (1980)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Derrida B., Lebowitz J.L., Speer E.R., Spohn H.: Fluctuations of a stationary nonequilibrium interface. Phys. Rev. Lett. 67, 165–168 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Dolbeault J., Gentil I., Jüngel A.: A logarithmic fourth-order parabolic equation and related logarithmic Sobolev inequalities. Commun. Math. Sci. 4, 275–290 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gangbo W., McCann R.J.: The geometry of optimal transportation. Acta Math. 177, 113–161 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Goffman C., Serrin J.: Sublinear functions of measures and variational integrals. Duke Math. J. 31, 159–178 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Grisvard P.: Elliptic Problems in Nonsmooth Domains. Pitman, London (1985)zbMATHGoogle Scholar
  28. 28.
    Gross L.: Logarithmic Sobolev inequalities. Am. J. Math. 97, 1061–1073 (1976)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Gualdani M.P., Jüngel A., Toscani G.: A nonlinear fourth-order parabolic equation with nonhomogeneous boundary conditions. SIAM J. Math. Anal. 37, 1761–1779 (2006) (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Jordan R., Kinderlehrer D., Otto F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29, 1–17 (1998) (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Jüngel A., Matthes D.: An algorithmic construction of entropies in higher-order nonlinear PDEs. Nonlinearity 19, 633–659 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Jüngel, A., Matthes, D.: The Derrida–Lebowitz–Speer–Spohn equation: existence, non-uniqueness, and decay rates of the solutions. SIAM J. Math. Anal. (2007, to appear)Google Scholar
  33. 33.
    Jüngel A., Pinnau R.: Global nonnegative solutions of a nonlinear fourth-order parabolic equation for quantum systems. SIAM J. Math. Anal. 32, 760–777 (2000) (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Jüngel A., Toscani G.: Exponential time decay of solutions to a nonlinear fourth-order parabolic equation. Z. Angew. Math. Phys. 54, 377–386 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lions P.-L., Toscani G.: A strengthened central limit theorem for smooth densities. J. Funct. Anal. 129, 148–167 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Markowich, P.A., Villani, C.: On the trend to equilibrium for the Fokker–Planck equation: an interplay between physics and functional analysis. Mat. Contemp. 19, 1–29 (2000). VI Workshop on Partial Differential Equations, Part II (Rio de Janeiro, 1999)MathSciNetzbMATHGoogle Scholar
  37. 37.
    McCann R.J.: A convexity principle for interacting gases. Adv. Math. 128, 153–179 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    McKean H.P. Jr.: Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas. Arch. Rational Mech. Anal. 21, 343–367 (1966)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Nagumo M.: Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen. Proc. Phys.-Math. Soc. Jpn. 24(3), 551–559 (1942)zbMATHGoogle Scholar
  40. 40.
    Otto F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differ. Equ. 26, 101–174 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Otto F., Villani C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361–400 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Rachev, S.T., Rüschendorf, L.: Mass transportation problems, Vol. I, Probability and its Applications, Theory. Springer, New York, 1998Google Scholar
  43. 43.
    Smith C.S., Knott M.: Note on the optimal transportation of distributions. J. Optim. Theory Appl. 52, 323–329 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Stam A.J.: Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inform. Control 2, 101–112 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Talagrand M.: Transportation cost for gaussian and other product measures. Geom. Funct. Anal. 6, 587–600 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Toscani G.: Sur l’inégalité logarithmique de Sobolev. C. R. Acad. Sci. Paris Sér. I Math. 324, 689–694 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Villani, C.: Topics in optimal transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence, 2003Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Ugo Gianazza
    • 1
  • Giuseppe Savaré
    • 1
    Email author
  • Giuseppe Toscani
    • 1
  1. 1.Dipartimento di Matematica “F. Casorati”Universita di PaviaPaviaItaly

Personalised recommendations