Regularity theory for parabolic nonlinear integral operators

Caffarelli, Luis A.; Chan, Chi Hin; Vasseur, Alexis

doi:10.1090/s0894-0347-2011-00698-x

Cited by 159 publications

(237 citation statements)

References 21 publications

(19 reference statements)

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“…Since σ − 1 − α ≥ σ 0 − 1 − α > 0 the right hand sides becomes smaller as k increases. Because I (1) and I (2) are close to I at every scale we get that,…”

Section: α Estimatesmentioning

confidence: 88%

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Regularity for solutions of nonlocal parabolic equations II

Chang-Lara

Dávila

2014

Journal of Differential Equations

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Abstract. We prove boundary regularity and a compactness result for parabolic nonlocal equations of the form ut − Iu = f , where the operator I is not necessarily translation invariant. As a consequence of this and the regularity results for translation invariant case, we obtain C 1,α interior estimates in space for non translation invariant operators under some hypothesis on the time regularity of the boundary data.

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“…Since σ − 1 − α ≥ σ 0 − 1 − α > 0 the right hand sides becomes smaller as k increases. Because I (1) and I (2) are close to I at every scale we get that,…”

Section: α Estimatesmentioning

confidence: 88%

“…The variational problem was studied by L. Caffarelli, C. Chan and A. Vasseur in [2] by using De Giorgi's technique. Also in a recent work from M. Felsinger and M. Kassmann [10] they prove a Harnack inequality in the divergence case, using Moser's technique.…”

Section: Introductionmentioning

confidence: 99%

Regularity for solutions of nonlocal parabolic equations II

Chang-Lara

Dávila

2014

Journal of Differential Equations

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“…Indeed, they are C α continuous with a uniform modulus. The proof done in [26] is lengthy and uses many techniques of the local regularity theory for elliptic and parabolic PDEs developed by Caffarelli and collaborators, and in particular some of the new ideas contained in Caffarelli-Chan-Vasseur [23]. The crucial point is to get a local version of the energy inequalities that can be iterated.…”

Section: Instantaneous Boundedness and Regularitymentioning

confidence: 99%

Nonlinear Diffusion with Fractional Laplacian Operators

Vázquez

2012

Nonlinear Partial Differential Equations

146

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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“…Let N ∈ Z + and p ∈ (1, N ) to be given, and let u be a a bounded viscosity solution on (0, T ) × R N to (1), with a Hamiltonian H(t, x, P ) satisfying coercivity property (2). Then, it follows that, for each δ ∈ (0, T ), we have u ∈ C α ([δ, T )×R N ), where α ∈ (0, 1), and u C α ([δ,T )×R N ) depend only on N , δ, Λ, p and u L ∞ ((0,T )×R N ) .…”

Section: Theoremmentioning

confidence: 99%

“…However the method of proof, based on the De Giorgi method [9] to study the regularity of elliptic equations with rough coefficients, is pretty unusual for the study of viscosity solutions, and should lead to new results in this area, as the study of regularity of solutions to nonlocal "Hamilton-Jacobi like" equations ( [7]). Our proof is inspired by previous applications of the De Giorgi method to integral-differential parabolic equations [2,3].…”

Section: Theoremmentioning

confidence: 99%