Regularity of solutions of the fractional porous medium flow

Caffarelli, Luis A.; Soria, Fernando; Vázquez, Juan Luís

doi:10.4171/jems/401

Cited by 102 publications

(202 citation statements)

References 17 publications

(24 reference statements)

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“…Calculation of convergence rates. This is the question of speed of convergence in formulas (24)- (26). The study was initiated by Carrillo and Toscani in 2000, [30], and there many interesting contributions (by Carrillo, Del Pino, Dolbeault, Markowich, McCann, Vazquez, and many others).…”

Section: Nonlinear Central Limitmentioning

confidence: 99%

“…The work that is presented next is explained in whole detail in the following papers [27,26,28]. The first deals with existence and basic propagation properties, the second about boundedness and regularity in the spirit of De Giorgi [33], and the third deals with asymptotic behaviour through the associated obstacle problem and entropy dissipation methods.…”

Section: Mathematical Theory For the Model Of Fractional Porous Mediumentioning

confidence: 99%

“…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%

“…Or we can use energy estimates based on the properties of the quadratic and bilinear forms associated to fractional operator, as done in [19]. For the equation and generality at hand, this is done in our paper [26] by the De Giorgi method.…”

Section: Instantaneous Boundedness and Regularitymentioning

confidence: 99%

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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: Nonlinear Central Limitmentioning

confidence: 99%

Section: Mathematical Theory For the Model Of Fractional Porous Mediumentioning

confidence: 99%

Section: Instantaneous Boundedness and Regularitymentioning

confidence: 99%

Section: Instantaneous Boundedness and Regularitymentioning

confidence: 99%

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Nonlinear Diffusion with Fractional Laplacian Operators

2012

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“…As far as existence issues as well as basic properties of the solutions of (1.3) are concerned, we refer to [9,8] . Schochet's original paper [27] was actually only concerned with the mean-field limit for a particle approximation of the 2D Euler equation, but the same argument directly applies to the present setting.…”

Section: Introductionmentioning

confidence: 99%

Mean-Field Limits for Some Riesz Interaction Gradient Flows

Duerinckx¹

2016

SIAM J. Math. Anal.

104

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This paper is concerned with the mean-field limit for the gradient flow evolution of particle systems with pairwise Riesz interactions, as the number of particles tends to infinity. Based on a modulated energy method, using regularity and stability properties of the limiting equation, as inspired by the work of Serfaty [28] in the context of the Ginzburg-Landau vortices, we prove a mean-field limit result in dimensions 1 and 2 in cases for which this problem was still open. IntroductionWe consider the energy of a system of N particles in the Euclidean space R d (d ≥ 1) interacting via (repulsive) Riesz pairwise interactions:where the interaction kernel is given bywith c d,s > 0 some normalization constants. We note that the Coulomb case corresponds to the choice s = d − 2, d ≥ 2. Particle systems with more general Riesz interactions as considered here are extensively motivated in the physics literature (cf. for instance [2,20]), as well as in the context of approximation theory with the study of Fekete points (cf.[14] and the references therein). Recently, a detailed description of such systems beyond the mean-field limit in the static case was obtained in [21], and also in [17] for non-zero temperature. In the present contribution, we are rather interested in the dynamics of such systems, and more precisely in a rigorous justification of the mean-field limit of their gradient flow evolution as the number N of particles tends to infinity, which has indeed remained an open problem wheneverWe thus consider the trajectories x t i,N driven by the corresponding flow, i.e. the solutions to the following system of ODEs:where (xis a sequence of N distinct initial positions. Since energy can only decrease in time and since the interaction is repulsive, particles cannot collide, and moreover it is easily seen that a particle cannot escape to infinity in finite time; from these observations and from the Picard-Lindelöf theorem, we may conclude that the trajectories x t i,N are smooth and well-defined on the whole of R + := [0, ∞). As the number of particles gets large, we would naturally like to pass to a continuum description of the system, in terms of the particle density distribution. For that purpose, we define the empirical measure associated with the point-vortex dynamics:2) 1 and the question is then to understand the limit of µ t N as N ↑ ∞. More precisely, assuming convergence at initial time This equation in the weak sense just means the following:As far as existence issues as well as basic properties of the solutions of (1.3) are concerned, we refer to [9,8] . Schochet's original paper [27] was actually only concerned with the mean-field limit for a particle approximation of the 2D Euler equation, but the same argument directly applies to the present setting. However, due to a possible lack of uniqueness of L 1 weak solutions to equation (1.3), Schochet [27] could only prove that the empirical measure µ t N converges up to a subsequence to some solution of (1.3). The key idea, which only holds for logarithmic i...

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Critical local well‐posedness for the fully nonlinear Peskin problem

Cameron,

Strain

2023

Comm Pure Appl Math

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We study the problem where a one‐dimensional elastic string is immersed in a two‐dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non‐linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time wellposedness for arbitrary initial data in the scaling critical Besov space . We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancelation structure.

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Regularity of solutions of the fractional porous medium flow

Cited by 102 publications

References 17 publications

Nonlinear Diffusion with Fractional Laplacian Operators

Nonlinear Diffusion with Fractional Laplacian Operators

Mean-Field Limits for Some Riesz Interaction Gradient Flows

Critical local well‐posedness for the fully nonlinear Peskin problem

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