Skip to main content
SpringerLink
Log in

Existence of Weak Solutions to a Class of Fourth Order Partial Differential Equations with Wasserstein Gradient Structure

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

We prove the global-in-time existence of nonnegative weak solutions to a class of fourth order partial differential equations on a convex bounded domain in arbitrary spatial dimensions. Our proof relies on the formal gradient flow structure of the equation with respect to the L 2-Wasserstein distance on the space of probability measures. We construct a weak solution by approximation via the time-discrete minimizing movement scheme; necessary compactness estimates are derived by entropy-dissipation methods. Our theory essentially comprises the thin film and Derrida-Lebowitz-Speer-Spohn equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrosio, L., Gigli, N., Savaré, G. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, 2nd edn. Birkhäuser Verlag, Basel (2008)

  2. Bernis, F., Friedman, A.: Higher order nonlinear degenerate parabolic equations. J. Differ. Equat. 83(1), 179–206 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bertozzi, A.L., Pugh, M.: The lubrication approximation for thin viscous films: regularity and long-time behavior of weak solutions. Comm. Pure Appl. Math. 49(2), 85–123 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bertsch, M., Dal Passo, R., Garcke, H., Grün, G.: The thin viscous flow equation in higher space dimensions. Adv. Differ. Equat. 3(3), 417–440 (1998)

    MathSciNet  MATH  Google Scholar 

  5. Blanchet, A., Carlen, E.A., Carrillo, J.A.: Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model. J. Funct. Anal. 262 (5), 2142–2230 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blanchet, A., Carrillo, J.A., Kinderlehrer, D., Kowalczyk, M., Laurençot, P., Lisini, S.: A hybrid variational principle for the Keller-Segel system in \(\mathbb {R}^2\). ESAIM Math. Model. Numer Anal. 49(6), 1553–1576 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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(7), 923–942 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carlen, E.A., Ulusoy, S.: Asymptotic equipartition and long time behavior of solutions of a thin-film equation. J. Differ. Equat. 241(2), 279–292 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carrillo, J.A., Lisini, S., Savaré, G., Slepčev, D.: Nonlinear mobility continuity equations and generalized displacement convexity. J. Funct. Anal. 258(4), 1273–1309 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Carrillo, J.A., Slepčev, D.: Example of a displacement convex functional of first order. Calc. Var. Partial Differ. Equat. 36(4), 547–564 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Carrillo, J.A., Toscani, G.: Long-time asymptotics for strong solutions of the thin film equation. Comm. Math Phys. 225(3), 551–571 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dal Passo, R., Garcke, H., Grün, G.: On a fourth-order degenerate parabolic equation: global entropy estimates, existence, and qualitative behavior of solutions. SIAM J. Math. Anal. 29(2), 321–342. (electronic), 1998

  13. Daneri, S., Savaré, G.: Eulerian calculus for the displacement convexity in the Wasserstein distance. SIAM J. Math Anal. 40(3), 1104–1122 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. De Gennes, P.G.: Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827–863 (1985)

    Article  Google Scholar 

  15. De Giorgi, E.: New problems on minimizing movements. Notes Appl. Math. 2 (4), 81–89 (1983)

    MathSciNet  MATH  Google Scholar 

  16. Derrida, B., Lebowitz, J.L., Speer, E.R., Spohn, H.: Dynamics of an anchored Toom interface. J. Phys. A 24(20), 4805–4834 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Derrida, B., Lebowitz, J.L., Speer, E.R., Spohn, H.: Fluctuations of a stationary nonequilibrium interface. Phys. Rev. Lett. 67(2), 165–168 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dolbeault, J., Nazaret, B., Savaré, G.: A new class of transport distances between measures. Calc. Var. Partial Differ. Equat. 34(2), 193–231 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Giacomelli, L., Gnann, M.V., Knüpfer, H., Otto, F.: Well-posedness for the Navier-slip thin-film equation in the case of complete wetting. J Differ. Equat. 257 (1), 15–81 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Giacomelli, L., Gnann, M.V., Otto, F., Gnann, V., Otto, F.: Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3. Eur. J. Appl. Math. 24(5), 735–760 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Giacomelli, L., Knüpfer, H., Otto, F.: Smooth zero-contact-angle solutions to a thin-film equation around the steady state. J. Differ. Equat. 245(6), 1454–1506 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Giacomelli, L., Otto, F.: Variational formulation for the lubrication approximation of the Hele-Shaw flow. Calc. Var. Partial Differ. Equat. 13(3), 377–403 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Giacomelli, L., Otto, F.: Rigorous lubrication approximation. Interfaces Free Bound. 5(4), 483–529 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gianazza, U., Savaré, G., Toscani, G.: The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation. Arch. Ration. Mech. Anal. 194(1), 133–220 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gnann, M.V.: Well-posedness and self-similar asymptotics for a thin-film equation. SIAM J. Math Anal. 47(4), 2868–2902 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Grün, G.: Droplet spreading under weak slippage—existence for the Cauchy problem. Comm. Partial Differ. Equat. 29(11-12), 1697–1744 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jüngel, A., Matthes, D.: The Derrida-Lebowitz-Speer-Spohn equation: existence, non-uniqueness, and decay rates of the solutions. SIAM J. Math. Anal. 39(6), 1996–2015 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jüngel, A., Pinnau, R.: Global nonnegative solutions of a nonlinear fourth-order parabolic equation for quantum systems. SIAM J. Math. Anal. 32(4), 760–777. (electronic), 2000

  30. Knüpfer, H.: Well-posedness for a class of thin-film equations with general mobility in the regime of partial wetting. Arch. Ration. Mech. Anal. 218(2), 1083–1130 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Knüpfer, H., Masmoudi, N.: Darcy’s flow with prescribed contact angle: well-posedness and lubrication approximation. Arch. Ration. Mech. Anal. 218(2), 589–646 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Laurençot, P., Matioc, B.-V.: A gradient flow approach to a thin film approximation of the Muskat problem. Calc. Var. Partial Differ. Equat. 47(1-2), 319–341 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lisini, S., Marigonda, A.: On a class of modified Wasserstein distances induced by concave mobility functions defined on bounded intervals. Manuscripta Math. 133 (1-2), 197–224 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lisini, S., Matthes, D., Savaré, G.: Cahn-Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics. J. Differ. Equat. 253(2), 814–850 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Matthes, D., McCann, R.J., Savaré, G.: A family of nonlinear fourth order equations of gradient flow type. Comm. Partial Differ. Equat. 34(10-12), 1352–1397 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128(1), 153–179 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. Oron, A., Davis, S.H., Bankoff, S.G.: Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980 (1997)

    Article  Google Scholar 

  38. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differ. Equat. 26(1-2), 101–174 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  39. Renardy, M., Rogers, R.C. An introduction to partial differential equations, volume 13 of Texts in Applied Mathematics, 2nd edn. Springer-Verlag, New York (2004)

  40. Rossi, R., Savaré, G.: Tightness, Integral equicontinuity and compactness for evolution problems in Banach spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2(2), 395–431 (2003)

    MathSciNet  MATH  Google Scholar 

  41. Villani, C.: Topics in optimal transportation, volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2003)

    Google Scholar 

  42. Zinsl, J.: Existence of solutions for a nonlinear system of parabolic equations with gradient flow structure. Monatsh. Math. 174(4), 653–679 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Zentrum für Mathematik, Technische Universität München, 85747, Garching, Germany

    Daniel Loibl, Daniel Matthes & Jonathan Zinsl

Authors
  1. Daniel Loibl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Matthes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jonathan Zinsl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Matthes.

Additional information

This research was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Loibl, D., Matthes, D. & Zinsl, J. Existence of Weak Solutions to a Class of Fourth Order Partial Differential Equations with Wasserstein Gradient Structure. Potential Anal 45, 755–776 (2016). https://doi.org/10.1007/s11118-016-9565-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-016-9565-y

Keywords

Mathematics Subject Classification (2010)

Search

Navigation