Entropy solution theory for fractional degenerate convection–diffusion equations

Cifani, Simone; Jakobsen, Espen R.

doi:10.1016/j.anihpc.2011.02.006

Cited by 79 publications

(117 citation statements)

References 34 publications

(62 reference statements)

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“…To overcome these and other difficulties, we will use the fact that our solution u is the trace of the solution of a local problem obtained by extending u m to a half-space whose boundary is our original space. See also the paper by Cifani and Jakobsen [31] for an alternative L 1 theory dealing with a more general class of nonlocal porous medium equations, including strong degeneration and convection.…”

Section: Preliminary Notionsmentioning

confidence: 99%

Nonlinear Diffusion with Fractional Laplacian Operators

2012

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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: Preliminary Notionsmentioning

confidence: 99%

Nonlinear Diffusion with Fractional Laplacian Operators

2012

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“…Some of the previous results also need the further restriction that solutions belong to L 1 ∩ L ∞ , see [28,14]. In particular, prior to this paper, there were no well-posedness results for merely bounded solutions of the non-local variant of (1.1) when ϕ is non-linear.…”

Section: Introductionmentioning

confidence: 99%

“…By [14,15], there exist entropy solutions u m , u n of (1.1) with initial data u 0,m , u 0,n respectively. By Corollary 3.1 a) and the triangle inequality,…”

Section: Consequencesmentioning

confidence: 99%

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$L^1$ Contraction for Bounded (Nonintegrable) Solutions of Degenerate Parabolic Equations

Endal¹,

Jakobsen²

2014

SIAM J. Math. Anal.

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Abstract. We obtain new L 1 contraction results for bounded entropy solutions of Cauchy problems for degenerate parabolic equations. The equations we consider have possibly strongly degenerate local or non-local diffusion terms. As opposed to previous results, our results apply without any integrability assumption on the solutions. They take the form of partial Duhamel formulas and can be seen as quantitative extensions of finite speed of propagation local L 1 contraction results for scalar conservation laws. A key ingredient in the proofs is a new and non-trivial construction of a subsolution of a fully non-linear (dual) equation. Consequences of our results are maximum and comparison principles, new a priori estimates, and in the non-local case, new existence and uniqueness results.

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“…Fractional calculus plays a significant part in studies of entropy [33][34][35][36][37][38]. It should be noted that entropy is also used in analysis of anomalous diffusion processes and fractional diffusion equation [39][40][41][42][43][44][45].…”

Section: Introductionmentioning

confidence: 99%

Fractional Heat Conduction in an Infinite Medium with a Spherical Inclusion

Povstenko

2013

Entropy

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Entropy solution theory for fractional degenerate convection–diffusion equations

Cited by 79 publications

References 34 publications

Nonlinear Diffusion with Fractional Laplacian Operators

Nonlinear Diffusion with Fractional Laplacian Operators

$L^1$ Contraction for Bounded (Nonintegrable) Solutions of Degenerate Parabolic Equations

Fractional Heat Conduction in an Infinite Medium with a Spherical Inclusion

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