A fractional porous medium equation

Pablo, Arturo de; Quirós, Fernando; Rodríguez, Ana Alonso; Vázquez, Juan Luís

doi:10.1016/j.aim.2010.07.017

Cited by 198 publications

(276 citation statements)

References 31 publications

(39 reference statements)

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“…Joint work with Arturo de Pablo, Fernando Quirós and Ana Rodriguez, Madrid. Two papers contain the progress done so far, [35,36]. On the other hand, I. Athanasopoulos and L. Caffarelli studied in [7] the continuity of the weak solutions in the framework of more general boundary heat control problems.…”

Section: Nonlinear Evolution Modelsmentioning

confidence: 99%

“…Of course, this simplification pays a prize, namely, introducing an extra space variable. The application of such an idea is not so simple when σ ̸ = 1; it involves a number of difficulties that we address in [36]. We have to use the characterization of the Laplacian of order σ , (−∆ ) σ /2 , 0 < σ < 2, recently described by Caffarelli and Silvestre [25], in terms of the so-called σ -harmonic extension, which is the solution of an elliptic problem with a degenerate or singular weight.…”

Section: Preliminary Notionsmentioning

confidence: 99%

“…To be specific, the theory of existence and uniqueness as well the main properties are studied by De Pablo, Quirós, Rodríguez, and Vázquez in [35,36] for the Cauchy problem…”

Section: Mathematical Problem and General Notionsmentioning

confidence: 99%

“…In the particular case σ = 1 studied in [35], the problem is reformulated by means of the well-known representation of the half-Laplacian in terms of the Dirichlet-Neumann operator. This allowed us to transform the nonlocal problem into a local one (i. e., involving only derivatives and not integral operators).…”

Section: Preliminary Notionsmentioning

confidence: 99%

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Nonlinear Diffusion with Fractional Laplacian Operators

Vázquez

2012

Nonlinear Partial Differential Equations

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103

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We describe two models of flow in porous media including nonlocal (longrange) diffusion effects. The first model is based on Darcy's law and the pressure is related to the density by an inverse fractional Laplacian operator. We prove existence of solutions that propagate with finite speed. The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density. We use entropy methods to show that they describe the asymptotic behaviour of a wide class of solutions. The second model is more in the spirit of fractional Laplacian flows, but nonlinear. Contrary to usual Porous Medium flows (PME in the sequel), it has infinite speed of propagation. Similarly to them, an L 1 -contraction semigroup is constructed and it depends continuously on the exponent of fractional derivation and the exponent of the nonlinearity. Nonlinear diffusion and fractional diffusionSince the work by Einstein [39] and Smoluchowski [62] at the beginning of the last century (cf. also Bachelier [9]), we possess an explantation of diffusion and Brownian motion in terms of the heat equation, and in particular of the Laplace operator. This explanation has had an enormous success both in Mathematics and Physics. In the decades that followed, the Laplace operator has been often replaced by more general types of so-called elliptic operators with variable coefficients, and later by nonlinear differential operators; a huge body of theory is now available, both for the evolution equations [49] and for the stationary states, described by elliptic equations of different kinds [50,42].

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Section: Nonlinear Evolution Modelsmentioning

confidence: 99%

Section: Preliminary Notionsmentioning

confidence: 99%

“…To be specific, the theory of existence and uniqueness as well the main properties are studied by De Pablo, Quirós, Rodríguez, and Vázquez in [35,36] for the Cauchy problem…”

Section: Mathematical Problem and General Notionsmentioning

confidence: 99%

Section: Preliminary Notionsmentioning

confidence: 99%

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Nonlinear Diffusion with Fractional Laplacian Operators

Vázquez

2012

Nonlinear Partial Differential Equations

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103

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“…One of the Łukasz Płociniczak lukasz.plociniczak@pwr.edu.pl 1 Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50370 Wrocław, Poland most successful is anomalous diffusion [34,36,37], which can be observed in variety of situations such as moisture percolation in porous media [39], protein random walks in cells [53], telomere motion [7,23] and diffusion of cosmic rays across the magnetic fields [11]. When considering self-similar solutions to a sub-diffusive evolution equation [19,47], the fractional derivative operator (either Riemann-Liouville or Caputo) becomes the so-called Erdélyi-Kober (E-K) fractional integral [15,27] which possesses many interesting mathematical and physical features [20,41,50].…”

Section: Introductionmentioning

confidence: 99%

Numerical schemes for integro-differential equations with Erdélyi-Kober fractional operator

Płociniczak

Sobieszek

2016

Numer Algor

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This work investigates several discretizations of the Erdélyi-Kober fractional operator and their use in integro-differential equations. We propose two methods of discretizing E-K operator and prove their errors asymptotic behaviour for several different variants of each discretization. We also determine the exact form of error constants. Next, we construct a finite-difference scheme based on a trapezoidal rule to solve a general first order integro-differential equation. As is known from the theory of Abel integral equations, the rate of convergence of any finite-different method depends on the severity of kernel's singularity. We confirm these results in the E-K case and illustrate our considerations with numerical examples.

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A General Fractional Porous Medium Equation

Pablo

Quirós

Rodríguez

et al. 2012

Comm Pure Appl Math

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We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion,We consider data f ∈ L 1 (R N ) and all exponents 0 < σ < 2 and m > 0. Existence and uniqueness of a weak solution is established for m > m * = (N − σ) + /N , giving rise to an L 1 -contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range 0 < m ≤ m * existence and uniqueness of solutions with good properties happen under some restrictions, and the properties are different from the case above m * . We also study the dependence of solutions on f, m and σ. Moreover, we consider the above questions for the problem posed in a bounded domain.2000 Mathematics Subject Classification. 26A33, 35A05, 35K55, 76S05

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A fractional porous medium equation

Cited by 198 publications

References 31 publications

Nonlinear Diffusion with Fractional Laplacian Operators

Nonlinear Diffusion with Fractional Laplacian Operators

Numerical schemes for integro-differential equations with Erdélyi-Kober fractional operator

A General Fractional Porous Medium Equation

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