Sharp Boundary Estimates and Harnack Inequalities for Fractional Porous Medium Type Equations

In this research article we focus on the study of existence of global solution for a three-dimensional fractional Porous medium equation. The main objectives of studying the fractional porous medium equation in the corresponding critical function spaces are to show the existence of unique global mild solution under the condition of small initial data. Applying Fourier transform methods gives an equivalent integral equation of the model equation. The linear and nonlinear terms are then estimated in the corresponding Lei and Lin spaces. Further, the analyticity of solution to the fractional Porous medium equation is also obtained.

We consider a nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual $$L^p$$ spaces or to larger (weighted) spaces determined either in terms of a ground state of $$\Delta _{\mathbb {H}^{N}}$$ , or of the (fractional) Green’s function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative $$L^1-L^\infty $$ estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples.

Minimizing movement for a fractional porous medium equation in a periodic setting

This is the first of a series of two papers which studies the fractional porous medium equation on a Riemannian manifold with isolated conical singularities. In this article, we show R-sectoriality for the fractional powers of possibly non-invertible R-sectorial operators. Applications concern existence, uniqueness and maximal \(L^{q}\)-regularity results for solutions of the fractional porous medium equation on manifolds with conical singularities. Space asymptotic behavior of the solutions close to the singularities is provided, and its relation to the local geometry is established. Our method extends the freezing-of-coefficients method to the case of non-local operators that are expressed as linear combinations of terms in the form of a product of a function and a fractional power of a local operator.

A fractional porous medium equation

Previous article Next article Regularity Properties of Flows Through Porous Media: A CounterexampleD. G. AronsonD. G. Aronsonhttps://doi.org/10.1137/0119027PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] D. G. Aronson, Regularity propeties of flows through porous media, SIAM J. Appl. Math., 17 (1969), 461–467 10.1137/0117045 MR0247303 (40:571) 0187.03401 LinkISIGoogle Scholar[2] A. M. Il'In, , A. S. Kalashnikov and , O. A. Oleinik, Second order linear equations of parabolic type, Russian Math. Surveys, 17 (1962), 1–143 10.1070/rm1962v017n03ABEH004115 CrossrefGoogle Scholar[3] A. S. Kalašnikov, Formation of singularities in solutions of the equation of nonstationary filtration, Z. Vyčisl. Mat. i Mat. Fiz., 7 (1967), 440–444 MR0211058 (35:1940) 0184.53201 Google Scholar[4] S. N. Kruzhkov, Results on the character of the regularity of solutions of parabolic equations and some of their applications, Math. Z., 6 (1969), 97–108 Google Scholar[5] O. A. Ladyženskaja, , V. A. Solonnikov and , N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967xi+648 MR0241822 (39:3159b) 0174.15403 Google Scholar[6] O. A. Oleinik, , A. S. Kalashnikov and , Yui-Lin' Chzou, The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR. Ser. Mat., 22 (1958), 667–704 MR0099834 (20:6271) Google Scholar[7] R. E. Pattle, Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. Math., 12 (1959), 407–409 MR0114505 (22:5326) 0119.30505 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Degenerate Diffusion ProblemsPhysical Mathematics and Nonlinear Partial Differential Equations | 11 Dec 2020 Cross Ref Infinite Speed of Propagation and Regularity of Solutions to the Fractional Porous Medium Equation in General DomainsCommunications on Pure and Applied Mathematics, Vol. 70, No. 8 | 19 October 2016 Cross Ref Regularity and geometric character of solution of a degenerate parabolic equationBulletin of Mathematical Sciences, Vol. 6, No. 3 | 17 May 2016 Cross Ref A rescaling algorithm for the numerical solution to the porous medium equation in a two-component domainCommunications in Nonlinear Science and Numerical Simulation, Vol. 39 | 1 Oct 2016 Cross Ref Linearly Implicit Approximations of Diffusive Relaxation SystemsActa Applicandae Mathematicae, Vol. 125, No. 1 | 25 September 2012 Cross Ref Finite speed of propagation in 1-D degenerate Keller-Segel systemMathematische Nachrichten, Vol. 285, No. 5-6 | 2 January 2012 Cross Ref Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion EquationsFausto Cavalli, Giovanni Naldi, and Ilaria PerugiaSIAM Journal on Scientific Computing, Vol. 34, No. 1 | 31 January 2012AbstractPDF (837 KB)A self-similar solution for the porous medium equation in a two-component domainNonlinear Analysis: Theory, Methods & Applications, Vol. 75, No. 2 | 1 Jan 2012 Cross Ref The porous medium equation in a two-component domainJournal of Differential Equations, Vol. 247, No. 9 | 1 Nov 2009 Cross Ref Self-Similar Solutions as Large Time Asymptotics for Some Nonlinear Parabolic EquationsLarge Time Asymptotics for Solutions of Nonlinear Partial Differential Equations | 5 October 2009 Cross Ref The continuous dependence on nonlinearities of solutions of the Neumann problem of a singular parabolic equationNonlinear Analysis: Theory, Methods & Applications, Vol. 67, No. 7 | 1 Oct 2007 Cross Ref Continuity of Weak Solutions for Quasilinear Parabolic Equations with Strong Degeneracy*Chinese Annals of Mathematics, Series B, Vol. 28, No. 4 | 9 July 2007 Cross Ref High-Order Relaxation Schemes for Nonlinear Degenerate Diffusion ProblemsFausto Cavalli, Giovanni Naldi, Gabriella Puppo, and Matteo SempliceSIAM Journal on Numerical Analysis, Vol. 45, No. 5 | 28 September 2007AbstractPDF (478 KB)A nonlinear singular diffusion equation with sourceJournal of Mathematical Analysis and Applications, Vol. 325, No. 1 | 1 Jan 2007 Cross Ref A new method for the analytical solution of a degenerate diffusion equationAdvances in Water Resources, Vol. 28, No. 10 | 1 Oct 2005 Cross Ref Numerical approach to the waiting time for the one-dimensional porous medium equationQuarterly of Applied Mathematics, Vol. 61, No. 4 | 1 January 2003 Cross Ref Formation of discontinuities in flux-saturated degenerate parabolic equationsNonlinearity, Vol. 16, No. 6 | 20 August 2003 Cross Ref Theory of creeping gravity currents of a non-Newtonian liquidPhysical Review E, Vol. 60, No. 6 | 1 December 1999 Cross Ref Regularity of the free boundary for the porous medium equationJournal of the American Mathematical Society, Vol. 11, No. 4 | 1 January 1998 Cross Ref Shape and size of the current head in creeping flowsPhysical Review E, Vol. 55, No. 4 | 1 April 1997 Cross Ref Existence and nonexistence of global positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundaryTransactions of the American Mathematical Society, Vol. 349, No. 3 | 1 January 1997 Cross Ref The diffusive limit of two velocity models: The porous medium equationTransport Theory and Statistical Physics, Vol. 26, No. 1-2 | 1 Jan 1997 Cross Ref On the interfaces in a nonlocal quasilinear degenerate equation arising in population dynamicsJapan Journal of Industrial and Applied Mathematics, Vol. 13, No. 3 | 1 Oct 1996 Cross Ref Waiting-time solutions of a nonlinear diffusion equation: Experimental study of a creeping flow near a waiting frontPhysical Review E, Vol. 54, No. 3 | 1 September 1996 Cross Ref Localization of age-dependent anti-crowding populationsQuarterly of Applied Mathematics, Vol. 53, No. 1 | 1 January 1995 Cross Ref An age-dependent population model with nonlinear diffusion in 𝑅ⁿQuarterly of Applied Mathematics, Vol. 52, No. 2 | 1 January 1994 Cross Ref Populations with Age and Diffusion: Effects of the Fertility FunctionRocky Mountain Journal of Mathematics, Vol. 24, No. 1 | 1 Mar 1993 Cross Ref Diffusivity versus absorption through the boundaryJournal of Differential Equations, Vol. 99, No. 2 | 1 Oct 1992 Cross Ref On degenerate diffusion with very strong absorptionMathematical Methods in the Applied Sciences, Vol. 15, No. 7 | 1 Sep 1992 Cross Ref Reaction-diffusion equations in one dimension: particular solutions and relaxationPhysica D: Nonlinear Phenomena, Vol. 57, No. 3-4 | 1 Aug 1992 Cross Ref Anticrowding population models in several space variablesQuarterly of Applied Mathematics, Vol. 49, No. 1 | 1 January 1991 Cross Ref Large systems of interacting particles and the porous medium equationJournal of Differential Equations, Vol. 88, No. 2 | 1 Dec 1990 Cross Ref C1,α regularity of the free boundary for the n-dimensional porous media equationCommunications on Pure and Applied Mathematics, Vol. 43, No. 7 | 1 Oct 1990 Cross Ref Similarity solutions to three-dimensional nonlinear diffusion equationsJournal of Physics A: Mathematical and General, Vol. 23, No. 3 | 1 January 1999 Cross Ref Regularity of solutions and interfaces of the porous medium equation via local estimatesProceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 112, No. 1-2 | 14 November 2011 Cross Ref On the Initial Growth of the Interfaces in Nonlinear Diffusion-Convection ProcessesNonlinear Diffusion Equations and Their Equilibrium States I | 1 Jan 1988 Cross Ref Regularity of Flows in Porous Media: A SurveyNonlinear Diffusion Equations and Their Equilibrium States I | 1 Jan 1988 Cross Ref Homogeneous diffusion in ? with power-like nonlinear diffusivityArchive for Rational Mechanics and Analysis, Vol. 103, No. 1 | 1 Jan 1988 Cross Ref Initial and boundary problems for the degenerate or singular system of the filtration typePartial Differential Equations | 28 September 2006 Cross Ref Free boundary problems for degenerate parabolic equationsPartial Differential Equations | 28 September 2006 Cross Ref Lagrangian non-oscillatory and FEM schemes for the porous media equationComputers & Mathematics with Applications, Vol. 15, No. 6-8 | 1 Jan 1988 Cross Ref RÉGULARITÉ DES SOLUTIONS DE L'EQUATION DES MILIEUX POREUX EN UNE DIMENSION D'ESPACEMathematical Analysis and its Applications | 1 Jan 1988 Cross Ref Dynamics of populations with age-difference and diffusion: localizationApplicable Analysis, Vol. 29, No. 1-2 | 2 May 2007 Cross Ref Stability of similarity solutions by two-equation models of turbulenceAIAA Journal, Vol. 25, No. 6 | 1 Jun 1987 Cross Ref Hyperbolic Aspects in the Theory of the Porous Medium EquationMetastability and Incompletely Posed Problems | 1 Jan 1987 Cross Ref Eventual C?-regularity and concavity for flows in one-dimensional porous mediaArchive for Rational Mechanics and Analysis, Vol. 99, No. 4 | 1 Jan 1987 Cross Ref C ?-Regularity for the porous medium equationMathematische Zeitschrift, Vol. 192, No. 2 | 1 Jun 1986 Cross Ref The porous medium equationNonlinear Diffusion Problems | 11 September 2006 Cross Ref A Discrete Model for Spatially Aggregating PhenomenaPatterns and Waves - Qualitative Analysis of Nonlinear Differential Equations | 1 Jan 1986 Cross Ref Existence of solutions in a population dynamics problemQuarterly of Applied Mathematics, Vol. 43, No. 4 | 1 Jan 1986 Cross Ref A Scheme for Computing Solutions and Interface Curves for a Doubly-Degenerate Parabolic EquationDavid HoffSIAM Journal on Numerical Analysis, Vol. 22, No. 4 | 14 July 2006AbstractPDF (1791 KB)Standing pulse-like solutions of a spatially aggregating population modelJapan Journal of Applied Mathematics, Vol. 2, No. 1 | 1 Jun 1985 Cross Ref A Nonlinear Problem in Age-Dependent Population DiffusionMichel LanglaisSIAM Journal on Mathematical Analysis, Vol. 16, No. 3 | 1 August 2006AbstractPDF (1597 KB)A linearly implicit finite-difference scheme for the one-dimensional porous medium equationMathematics of Computation, Vol. 45, No. 171 | 1 January 1985 Cross Ref Waiting and propagating fronts in nonlinear diffusionPhysica D: Nonlinear Phenomena, Vol. 12, No. 1-3 | 1 Jul 1984 Cross Ref Shocks generated in a confined gas due to rapid heat addition at the boundary. I. Weak shock wavesProceedings of the Royal Society of London. A. Mathematical and Physical Sciences, Vol. 393, No. 1805 | 1 January 1997 Cross Ref Some properties of the solution of a filtration problem in partially saturated porous mediaActa Mathematicae Applicatae Sinica, Vol. 1, No. 1 | 1 Mar 1984 Cross Ref The flow of two immiscible fluids through a porous mediumNonlinear Analysis: Theory, Methods & Applications, Vol. 8, No. 4 | 1 Jan 1984 Cross Ref A coordinate transformation for the porous media equation that renders the free boundary stationaryQuarterly of Applied Mathematics, Vol. 42, No. 3 | 1 January 1984 Cross Ref An interface tracking algorithm for the porous medium equationTransactions of the American Mathematical Society, Vol. 284, No. 2 | 1 January 1984 Cross Ref The interfaces of one-dimensional flows in porous mediaTransactions of the American Mathematical Society, Vol. 285, No. 2 | 1 January 1984 Cross Ref Nonlinear Heat Conduction with Absorption: Space Localization and Extinction in Finite TimeRobert KersnerSIAM Journal on Applied Mathematics, Vol. 43, No. 6 | 12 July 2006AbstractPDF (976 KB)How an Initially Stationary Interface Begins to Move in Porous Medium FlowD. G. Aronson, L. A. Caffarelli, and S. KaminSIAM Journal on Mathematical Analysis, Vol. 14, No. 4 | 17 July 2006AbstractPDF (1453 KB)A Problem in Nonlinear Age Dependent Population DiffusionPopulation Biology | 1 Jan 1983 Cross Ref Some nonlinear degenerate diffusion equations with a nonlocally convective term in ecologyHiroshima Mathematical Journal, Vol. 13, No. 1 | 1 Jan 1983 Cross Ref Numerical methods for flows through porous media. IMathematics of Computation, Vol. 40, No. 162 | 1 January 1983 Cross Ref Waiting-Time Behavior in a Nonlinear Diffusion EquationStudies in Applied Mathematics, Vol. 67, No. 2 | 28 September 2015 Cross Ref Nonstationary filtration in partially saturated porous mediaArchive for Rational Mechanics and Analysis, Vol. 78, No. 2 | 1 Jun 1982 Cross Ref Large time behaviour of solutions of the porous medium equation in bounded domainsJournal of Differential Equations, Vol. 39, No. 3 | 1 Mar 1981 Cross Ref Global classical solutions to ut − Δ(u2m + 1) + λu = ƒJournal of Differential Equations, Vol. 38, No. 2 | 1 Nov 1980 Cross Ref Degenerate parabolic equations with general nonlinearitiesNonlinear Analysis: Theory, Methods & Applications, Vol. 4, No. 6 | 1 Nov 1980 Cross Ref Evolution of a stable profile for a class of nonlinear diffusion equations. III. Slow diffusion on the lineJournal of Mathematical Physics, Vol. 21, No. 6 | 1 Jun 1980 Cross Ref Horizontal line analysis of the multidimensional porous medium equation: Existence, rate of convergence and maximum principlesNumerical Analysis | 14 October 2006 Cross Ref Density-Dependent Interaction–Diffusion SystemsDynamics and Modelling of Reactive Systems | 1 Jan 1980 Cross Ref Asymptotic expansions for the solutions of certain nonlinear parabolic problems IICommunications in Partial Differential Equations, Vol. 5, No. 12 | 8 May 2007 Cross Ref Time Dependent Free Boundary ProblemsAvner FriedmanSIAM Review, Vol. 21, No. 2 | 17 February 2012AbstractPDF (849 KB)The behavior of the support of solutions of the equation of nonlinear heat conduction with absorption in one dimensionTransactions of the American Mathematical Society, Vol. 249, No. 2 | 1 Feb 1979 Cross Ref Continuity of the density of a gas flow in a porous mediumTransactions of the American Mathematical Society, Vol. 252, No. 0 | 1 January 1979 Cross Ref Properties of solutions of an equation in the theory of infiltrationArchive for Rational Mechanics and Analysis, Vol. 65, No. 3 | 1 Sep 1977 Cross Ref ReferencesNavier–Stokes Equations - Theory and Numerical Analysis | 1 Jan 1977 Cross Ref The porous medium equation in one dimensionTransactions of the American Mathematical Society, Vol. 234, No. 2 | 1 January 1977 Cross Ref The cauchy problem for an equation in the theory of infiltrationArchive for Rational Mechanics and Analysis, Vol. 61, No. 2 | 1 Jun 1976 Cross Ref Some estimates for solution of the Cauchy problem for equations of a nonstationary filtrationJournal of Differential Equations, Vol. 20, No. 2 | 1 Mar 1976 Cross Ref On approximation by semigroups of nonlinear contractions. IIJournal of Approximation Theory, Vol. 15, No. 1 | 1 Sep 1975 Cross Ref Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in timeJournal of Differential Equations, Vol. 16, No. 2 | 1 Sep 1974 Cross Ref The asymptotic behaviour of the solution of the filtration equationIsrael Journal of Mathematics, Vol. 14, No. 1 | 1 Mar 1973 Cross Ref A remark on fluid flows through porous mediaProceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 49, No. 1 | 1 Jan 1973 Cross Ref A Finite Difference Approach to Some Degenerate Nonlinear Parabolic EquationsJ. L. Graveleau and P. JametSIAM Journal on Applied Mathematics, Vol. 20, No. 2 | 12 July 2006AbstractPDF (1764 KB)Semigroups of Nonlinear Transformations in Banach SpacesContributions to Nonlinear Functional Analysis | 1 Jan 1971 Cross Ref Regularity properties of flows through porous media: The interfaceArchive for Rational Mechanics and Analysis, Vol. 37, No. 1 | 1 Jan 1970 Cross Ref BibliographyBlow-Up in Quasilinear Parabolic Equations Cross Ref Volume 19, Issue 2| 1970SIAM Journal on Applied Mathematics266-485 History Submitted:17 July 1969Published online:17 February 2012 InformationCopyright © 1970 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0119027Article page range:pp. 299-307ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

We investigate the behaviour of the solutions u_m(x,t) of the fractional porous medium equation u_t+(-\Delta)^s (u^m)=0, \quad x\in \mathbb R^N, \ t>0 with initial data u(x,0)\ge 0 , x\in \mathbb R^N , in the limit m\to\infty with fixed s\in (0,1) . We first identify the limit F_\infty of the Barenblatt solutions U_m(x,t) \normalcolor as the solution of a stationary fractional obstacle problem, and we observe that, contrary to the case s=1 , the limit is not compactly supported but exhibits a typical fractional tail with power-like decay. In other words, we do not get a plain mesa in the limit, but a mesa with a tail. This is not the whole story since the limit of V_m(x,t)= mt\, U_m^m(x,t) exists and is compactly supported (in x ). We then study the limit m\to\infty for a wide class of solutions with nonnegative initial data, and show also in this setting the phenomenon of initial discontinuity, whereby the solution does not take on the prescribed initial data. Finally, we derive counterexamples to expected propagation and comparison properties based on symmetrization and pose a related open problem.

In this article we derive the fractional porous medium equation for any power of the fractional Laplacian as the hydrodynamic limit of a microscopic dynamics of random particles with long range interactions, but the jump rate highly depends on the occupancy near the sites where the interactions take place.

We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional p-Laplace equations which includes the fractional parabolic p-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case, and in the nonlocal elliptic case, to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting.

This is the second of a series of two papers that studies the fractional porous medium equation, with and , posed on a Riemannian manifold with isolated conical singularities. The first aim of the article is to derive some useful properties for the Mellin–Sobolev spaces including the Rellich–Kondrachov theorem and Sobolev–Poincaré, Nash and Super Poincaré type inequalities. The second part of the article is devoted to the study the Markovian extensions of the conical Laplacian operator and its fractional powers. Then based on the obtained results, we establish existence and uniqueness of a global strong solution for initial data and all . We further investigate a number of properties of the solutions, including comparison principle, contraction and conservation of mass. Our approach is quite general and thus is applicable to a variety of similar problems on manifolds with more general singularities.

Entropy solution theory for fractional degenerate convection–diffusion equations

We prove optimal regularity results in Lp-based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator P of order 2a (0<a<1) with nonsmooth x-dependent coefficients. This includes the prominent case of the fractional Laplacian (-Δ)a, as well as elliptic operators (-∇·A(x)∇+b(x))a. The proofs are based on general results on maximal Lp-regularity and its relation to R-boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation. The nonlinear results are new in the case of domains with boundary; the linear results are so when P is x-dependent nonsymmetric.

In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 α ≤ 2, we prove global well-posedness for initial data with 1 ≤ p q ≤ ∞, and analyticity of solutions with 1 p q ≤ ∞. In particular, we also proved that when α = 1, both u and belong to . We solve this equation through the contraction mapping method based on Littlewood-Paley theory and Fourier multiplier. Furthermore, we can get time decay estimates of global solutions in Besov spaces, which is as t → ∞.

Existence of weak solutions for fractional porous medium equations with nonlinear term

On the Cauchy problem for a general fractional porous medium equation with variable density