Skip to main content
SpringerLink
Log in

Existence of Weak Solutions for a General Porous Medium Equation with Nonlocal Pressure

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We study the general nonlinear diffusion equation \({u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)}\) that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters \({m > 1}\) and \({0 < s < 1}\), we assume that the solutions are non-negative and that the problem is posed in the whole space. In this paper we prove the existence of weak solutions for all integrable initial data \({u_0 \ge 0}\) and for all exponents \({m > 1}\) by developing a new approximation method that allows one to treat the range \({m\geqq 3}\), which could not be covered by previous works. We also extend the class of initial data to include any non-negative measure \({\mu}\) with finite mass. In passing from bounded initial data to measure data we make strong use of an L1-\({L^\infty}\) smoothing effect and other functional estimates. Finite speed of propagation is established for all \({m \geqq 2}\), and this property implies the existence of free boundaries. The authors had already proved that finite propagation does not hold for \({m < 2}\).

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

    MATH  Google Scholar 

  2. F. Andreu-Vaillo, J.M. Mazon, J.D. Rossi, J.J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, vol. 65 (American Mathematical Society, Providence, RI, 2010)

    Book  MATH  Google Scholar 

  3. Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), Vol. 348. Springer, Cham. xx+552 pp 2014

  4. P. Biler, C. Imbert, G. Karch, The nonlocal porous medium equation: Barenblatt profiles and other weak solutions. Arch. Ration. Mech. Anal. 215, 497–529 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Biler, G. Karch, R. Monneau, Nonlinear diffusion of dislocation density and self-similar solutions. Commun. Math. Phys. 294, 145–168 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. M. Bonforte, A. Figalli, J.L. Vázquez, Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains. Anal. PDEs 11(4), 945–982 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Bonforte, Y. Sire, J.L. Vázquez, Optimal existence and uniqueness theory for the fractional heat equation. Nonlinear Anal. 153, 142–168 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Bonforte, J. Vázquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations. Adv. Math. 250, 242–284 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Caffarelli, F. Soria, J.L. Vázquez, Regularity of solutions of the fractional porous medium flow. J. Eur. Math. Soc. (JEMS) 15, 1701–1746 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Caffarelli, L., Vázquez, J.: Regularity of solutions of the fractional porous medium flow with exponent 1/2, Algebrai Analiz [St. Petersb. Math. J.], 27(3), 125–156 2015; translation in St. Petersburg Math. J. 27(3) (2016), 437–460

  12. L. Caffarelli, J.L. Vazquez, Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202, 537–565 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. L.A. Caffarelli, J.L. Vázquez, Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete Contin. Dyn. Syst. 29, 1393–1404 (2011)

    MathSciNet  MATH  Google Scholar 

  14. J.A. Carrillo, Y. Huang, M.C. Santos, J.L. Vázquez, Exponential convergence towards stationary states for the 1D porous medium equation with fractional pressure. J. Differ. Equ. 258, 736–763 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. A. de Pablo, F. Quirós, A. Rodríguez, J. Vázquez, A fractional porous medium equation. Adv. Math. 226, 1378–1409 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. A. de Pablo, F. Quirós, A. Rodríguez, J. Vázquez, A general fractional porous medium equation. Commun. Pure Appl. Math. 65, 1242–1284 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. de Pablo, A., Quirós, F., Rodríguez, A., Vázquez, J.: Classical solutions for a logarithmic fractional diffusion equation. J. Math. Pures Appl. (9) 101(6), 901–924 2014

  18. F. del Teso, Finite difference method for a fractional porous medium equation. Calcolo 51, 615–638 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. F. del Teso, J. Endal, E.R. Jakobsen, Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type. Adv. Math. 305, 78–143 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. del Teso, F., Endal, J., Jakobsen, E.R.: On the well-posedness of solutions with finite energy for nonlocal equations of porous medium type. EMS Series of Congress Reports: Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, pp. 129-167 2018

  21. del Teso, F., Jakobsen, E.R.: A convergent numerical method for the porous medium equation with fractional pressure, In preparation

  22. del Teso, F., Vázquez, J.L.: Finite difference method for a general fractional porous medium equation 2013, arXiv:1307.2474

  23. E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Dolbeault, A. Zhang, Flows and functional inequalities for fractional operators Appl. Anal. 96, 1547–1560 (2018)

    MATH  Google Scholar 

  25. M. Duerinckx, Mean-field limits for some Riesz interaction gradient flows. SIAM J. Math. Anal. 48(3), 2269–2300 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Giacomin, J.L. Lebowitz, Phase segregation dynamics in particle systems with long range interaction I. Macroscopic limits. J. Stat. Phys. 87(1–2), 37–61 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Giacomin, G., Lebowitz, J.L.: Phase segregation dynamics in particle systems with long range interactions. II. Interface motion. SIAM J. Appl. Math. 58(6), 1707–1729 1998

  28. Grafakos, L.: Classical Fourier Analysis, 2nd edn. Graduate Texts in Mathematics, 249. Springer, New York 2008

  29. Ignat, L.I., Rossi, J.D.: Decay estimates for nonlocal problems via energy methods. J. Math. Pures Appl. (9) 92(2), 163–187 2009

  30. C. Imbert, Finite speed of propagation for a non-local porous medium equation. Colloq. Math. 143(2), 149–157 (2016)

    MathSciNet  MATH  Google Scholar 

  31. Ladyženskaja, O.A., Solonnikov, V.A., Ural'ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I., xi+648 pp 1968

  32. V.A. Liskevich, Semenov, YuA: Some inequalities for sub-Markovian generators and their applications to the perturbation theory. Proc. Am. Math. Soc. 119(4), 1171–1177 (1993)

    Google Scholar 

  33. S. Lisini, E. Mainini, A. Segatti, A gradient flow approach to the porous medium equation with fractional pressure. Arch. Ration. Mech. Anal. 227(2), 567–606 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nguyen, Q.-H., Vázquez, J.: Porous medium equation with nonlocal pressure in a bounded domain. Commun. PDEs. https://doi.org/10.1080/03605302.2018.1475492)

  35. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, Vol. 44. Springer, New York, 1983. viii+279 pp. ISBN: 0-387-90845-5

  36. J.M. Rakotoson, R. Temam, An optimal compactness theorem and application to elliptic-parabolic systems. Appl. Math. Lett. 14(3), 303–306 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Rossi, J.D.: Approximations of local evolution problems by nonlocal ones. Bol. Soc. Esp. Mat. Apl. S \({{\rm e}}\) MA 42, 49–65 2008

  38. S. Serfaty, Mean-Field Limits of the Gross-Pitaevskii and Parabolic Ginzburg-Landau Equations. J. Am. Math. Soc. 30(3), 713–768 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  39. S. Serfaty, J.L. Vázquez, A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators. Calc. Var. Partial Differ. Equ. 49(3–4), 1091–1120 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. J. Simon, Compact sets in the space \({L}^p(0,{T};{B})\). Ann. Mat. Pura Appl. 146, 65–96 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  41. D. Stan, F. del Teso, J.L. Vázquez, Finite and infinite speed of propagation for porous medium equations with fractional pressure. C. R. Math. Acad. Sci. Paris 352, 123–128 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. Stan, D., Teso, F del., Vázquez, J.L.: Transformations of self-similar solutions for porous medium equations of fractional type. Nonlinear Anal. 119, 62–73 2015

  43. D. Stan, F. del Teso, J.L. Vázquez, Finite and infinite speed of propagation for porous medium equations with nonlocal pressure. J. Differ. Equ. 260(2), 1154–1199 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. D. Stan, F. del Teso, J.L. Vázquez, Porous medium equation with nonlocal pressure. Curr. Res. Nonlinear Anal. 135, 277–308 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  45. E. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, 1970)

    MATH  Google Scholar 

  46. Stroock, D.W.: An Introduction to the Theory of Large Deviations, p. vii+196. Universitext. Springer, New York 1984

  47. J.L. Vázquez, The Porous Medium Equation (Oxford University Press, Oxford, Mathematical Theory. Oxford Mathematical Monographs, 2007)

    MATH  Google Scholar 

  48. J.L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete Contin. Dyn. Syst. Ser. S 7(4), 857–885 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  49. J.L. Vázquez, Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type. J. Eur. Math. Soc. (JEMS) 16, 769–803 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  50. Vázquez, J.L.: The mathematical theories of diffusion: nonlinear and fractional diffusion, In: ``Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions, volume 2186 of Lecture Notes in Math., pp. 205–278. Springer, Cham 2017

  51. Xiao, W., Zhou, X.: Well-Posedness of a porous medium flow with fractional pressure in Sobolev spaces. Electron. J. Differ. Equ. 2017(238), 1–7 2017

  52. X. Zhou, W. Xiao, J. Chen, Fractional porous medium and mean field equations in Besov spaces. Electron. J. Differ. Equ. 199, 1–14 (2014)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are partially supported by the Spanish ProjectMTM2014-52240-P.Diana Stan and Félix del Teso are partially supported by theMEC-Juan de la Cierva postdoctoral fellowships number FJCI-2015-25797 and FJCI-2016-30148 respectively, and by the BCAM Severo Ochoa accreditation SEV-2017-0718. Félix del Teso is partially supported by the Toppforsk (research excellence) project Waves and Nonlinear Phenomena (WaNP), Grant 250070 from the Research Council of Norway. Juan Luis Vázquez has been a Visiting Professor at Univ. Complutense de Madrid during the academic year 2017–2018. The authors want to thank the anonymous referee for accurate suggestions that allowed them to improve the original text.

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Av. de los Castros, 39005, Santander, Cantabria, Spain

    Diana Stan

  2. Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009, Bilbao, Basque-Country, Spain

    Félix del Teso

  3. Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049, Madrid, Spain

    Juan Luis Vázquez

Authors
  1. Diana Stan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Félix del Teso
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Juan Luis Vázquez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan Luis Vázquez.

Additional information

Communicated by A. Figalli

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Stan, D., del Teso, F. & Vázquez, J.L. Existence of Weak Solutions for a General Porous Medium Equation with Nonlocal Pressure. Arch Rational Mech Anal 233, 451–496 (2019). https://doi.org/10.1007/s00205-019-01361-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-019-01361-0

Search

Navigation