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The nonlocal porous medium equation: Barenblatt profiles and other weak solutions. (English) Zbl 1308.35197

Summary: A degenerate nonlinear nonlocal evolution equation is considered; it can be understood as a porous medium equation whose pressure law is nonlinear and nonlocal. We show the existence of sign-changing weak solutions to the corresponding Cauchy problem. Moreover, we construct explicit compactly supported self-similar solutions which generalize Barenblatt profiles – the well-known solutions of the classical porous medium equation.

MSC:

35Q35 PDEs in connection with fluid mechanics
76S05 Flows in porous media; filtration; seepage
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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[1] Alikakos N.D.: An application of the invariance principle to reaction-diffusion equations. J. Differ. Equ. 33, 201-225 (1979) · Zbl 0386.34046 · doi:10.1016/0022-0396(79)90088-3
[2] Barles G., Chasseigne E., Imbert C.: On the Dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ. Math. J. 57, 213-246 (2008) · Zbl 1139.47057 · doi:10.1512/iumj.2008.57.3315
[3] Biler, P., Imbert, C., Karch, G.: Finite speed of propagation for a non-local porous medium equation (preprint) · Zbl 1221.35209
[4] Biler P., Imbert C., Karch G.: Barenblatt profiles for a nonlocal porous medium equation. C. R., Math. Acad. Sci. Paris 349, 641-645 (2011) · Zbl 1221.35209 · doi:10.1016/j.crma.2011.06.003
[5] Biler P., Karch G., Monneau R.: Nonlinear diffusion of dislocation density and self-similar solutions.. Commun. Math. Phys. 294, 145-168 (2010) · Zbl 1207.82049 · doi:10.1007/s00220-009-0855-8
[6] Caffarelli L., Vázquez J.L.: Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202, 537-565 (2011) · Zbl 1264.76105 · doi:10.1007/s00205-011-0420-4
[7] Caffarelli L.A., Soria F., Vázquez J.L.: Regularity of solutions of the fractional porous medium flow. J. Eur. Math. Soc. (JEMS) 15, 1701-1746 (2013) · Zbl 1292.35312 · doi:10.4171/JEMS/401
[8] Caffarelli L.A., Vázquez J.L.: Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete Contin. Dyn. Syst. 29, 1393-1404 (2011) · Zbl 1211.35043
[9] Carrillo J.A., Jüngel A., Markowich P.A., Toscani G., Unterreiter A.: Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math. 133, 1-82 (2001) · Zbl 0984.35027 · doi:10.1007/s006050170032
[10] Córdoba A., Córdoba D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys., 249, 511-528 (2004) · Zbl 1309.76026 · doi:10.1007/s00220-004-1055-1
[11] de Pablo, A., Quirós, F., Rodríguez, A., Vázquez, J.L.: A fractional porous medium equation. Adv. Math. 226, 1378-1409 (2011) · Zbl 1208.26016
[12] de Pablo, A., Quirós, F., Rodríguez, A., Vázquez, J.L.: A general fractional porous medium equation. Commun. Pure Appl. Math. 65, 1242-1284 (2012) · Zbl 1248.35220
[13] Droniou J., Imbert C.: Fractal first order partial differential equations. Arch. Ration. Mech. Anal. 182, 299-331 (2006) · Zbl 1111.35144 · doi:10.1007/s00205-006-0429-2
[14] Dyda B.: Fractional calculus for power functions and eigenvalues of the fractional laplacian. Fract. Calc. Appl. Anal. 15, 535-555 (2012) · Zbl 1312.35176 · doi:10.2478/s13540-012-0038-8
[15] Getoor R.K.: First passage times for symmetric stable processes in space. Trans. Am. Math. Soc. 101, 75-90 (1961) · Zbl 0104.11203 · doi:10.1090/S0002-9947-1961-0137148-5
[16] Huang, Y.: Explicit Barenblatt profiles for fractional porous medium equations. arXiv:1312.0469 [math.AP]. (2013) · Zbl 1207.82049
[17] Imbert C., Mellet A.: Existence of solutions for a higher order non-local equation appearing in crack dynamics.. Nonlinearity 24, 3487-3514 (2011) · Zbl 1230.35053 · doi:10.1088/0951-7715/24/12/008
[18] Karch G., Miao C., Xu X.: On convergence of solutions of fractal Burgers equation toward rarefaction waves. SIAM J. Math. Anal. 39, 1536-1549 (2008) · Zbl 1154.35080 · doi:10.1137/070681776
[19] Ladyzhenskaya, O., Solonnikov, V., Ural’tseva, N.: Linear and Quasi-Linear Equations of Parabolic Type. Translated from the Russian by S. Smith, p. 648. Translations of Mathematical Monographs. 23. American Mathematical Society (AMS), Providence, RI. XI, 1968 · Zbl 0174.15403
[20] Liskevich, V.A., Semenov, Y.A.: Some problems on Markov semigroups. In: Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, Vol. 11 of Math. Top., pp. 163-217. Akademie Verlag, Berlin, 1996 · Zbl 0854.47027
[21] Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics, Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966 · Zbl 0143.08502
[22] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences. Springer, New York, 1983 · Zbl 0516.47023
[23] Rakotoson J.M., Temam R.: An optimal compactness theorem and application to elliptic-parabolic systems. Appl. Math. Lett. 14, 303-306 (2001) · Zbl 1001.46049 · doi:10.1016/S0893-9659(00)00153-1
[24] Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, vol. 3 of de Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin, 1996 · Zbl 0873.35001
[25] Stan, D., del Teso, F., Vázquez, J.L.: Finite and infinite speed of propagation for porous medium equations with fractional pressure. arXiv:1311.7007 [math.AP]. (2013) · Zbl 1379.35253
[26] Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, 1970 · Zbl 0207.13501
[27] Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean spaces. Princeton University Press, Princeton Mathematical Series, No. 32, Princeton, 1971 · Zbl 0232.42007
[28] Taylor, M.E.: Tools for PDE, vol. 81 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, 2000 (pseudodifferential operators, paradifferential operators, and layer potentials) · Zbl 0963.35211
[29] Taylor, M.E.: Partial Differential Equations. III: Nonlinear Equations, 2nd edn, xxii, p. 715. Applied Mathematical Sciences 117. Springer, New York, 2011 · Zbl 1206.35004
[30] Vázquez, J.L.: Smoothing and Decay Estimates for Nonlinear Diffusion Equations, vol. 33 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2006 (equations of porous medium type) · Zbl 1113.35004
[31] Vázquez, J.L.: The Porous Medium Equation. Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, Oxford, 2007 (mathematical theory)
[32] Vázquez, J.L.: Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type. Preprint arXiv:1205.6332v1. (2012) · Zbl 0386.34046
[33] Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995 (reprint of the second (1944) edition) · Zbl 0063.08184
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