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 \({m < 2}\).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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)
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)
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
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)
P. Biler, G. Karch, R. Monneau, Nonlinear diffusion of dislocation density and self-similar solutions. Commun. Math. Phys. 294, 145–168 (2010)
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)
M. Bonforte, Y. Sire, J.L. Vázquez, Optimal existence and uniqueness theory for the fractional heat equation. Nonlinear Anal. 153, 142–168 (2017)
M. Bonforte, J. Vázquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations. Adv. Math. 250, 242–284 (2014)
L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)
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)
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
L. Caffarelli, J.L. Vazquez, Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202, 537–565 (2011)
L.A. Caffarelli, J.L. Vázquez, Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete Contin. Dyn. Syst. 29, 1393–1404 (2011)
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)
A. de Pablo, F. Quirós, A. Rodríguez, J. Vázquez, A fractional porous medium equation. Adv. Math. 226, 1378–1409 (2011)
A. de Pablo, F. Quirós, A. Rodríguez, J. Vázquez, A general fractional porous medium equation. Commun. Pure Appl. Math. 65, 1242–1284 (2012)
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
F. del Teso, Finite difference method for a fractional porous medium equation. Calcolo 51, 615–638 (2014)
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)
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
del Teso, F., Jakobsen, E.R.: A convergent numerical method for the porous medium equation with fractional pressure, In preparation
del Teso, F., Vázquez, J.L.: Finite difference method for a general fractional porous medium equation 2013, arXiv:1307.2474
E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
J. Dolbeault, A. Zhang, Flows and functional inequalities for fractional operators Appl. Anal. 96, 1547–1560 (2018)
M. Duerinckx, Mean-field limits for some Riesz interaction gradient flows. SIAM J. Math. Anal. 48(3), 2269–2300 (2016)
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)
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
Grafakos, L.: Classical Fourier Analysis, 2nd edn. Graduate Texts in Mathematics, 249. Springer, New York 2008
Ignat, L.I., Rossi, J.D.: Decay estimates for nonlocal problems via energy methods. J. Math. Pures Appl. (9) 92(2), 163–187 2009
C. Imbert, Finite speed of propagation for a non-local porous medium equation. Colloq. Math. 143(2), 149–157 (2016)
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
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)
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)
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)
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
J.M. Rakotoson, R. Temam, An optimal compactness theorem and application to elliptic-parabolic systems. Appl. Math. Lett. 14(3), 303–306 (2001)
Rossi, J.D.: Approximations of local evolution problems by nonlocal ones. Bol. Soc. Esp. Mat. Apl. S \({{\rm e}}\) MA 42, 49–65 2008
S. Serfaty, Mean-Field Limits of the Gross-Pitaevskii and Parabolic Ginzburg-Landau Equations. J. Am. Math. Soc. 30(3), 713–768 (2017)
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)
J. Simon, Compact sets in the space \({L}^p(0,{T};{B})\). Ann. Mat. Pura Appl. 146, 65–96 (1987)
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)
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
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)
D. Stan, F. del Teso, J.L. Vázquez, Porous medium equation with nonlocal pressure. Curr. Res. Nonlinear Anal. 135, 277–308 (2018)
E. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, 1970)
Stroock, D.W.: An Introduction to the Theory of Large Deviations, p. vii+196. Universitext. Springer, New York 1984
J.L. Vázquez, The Porous Medium Equation (Oxford University Press, Oxford, Mathematical Theory. Oxford Mathematical Monographs, 2007)
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)
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)
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
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
X. Zhou, W. Xiao, J. Chen, Fractional porous medium and mean field equations in Besov spaces. Electron. J. Differ. Equ. 199, 1–14 (2014)
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
Corresponding author
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
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-019-01361-0