Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators

By Juan Luis V'azquez. 2014

230 citations recorded by the source corpus.

We report on recent progress in the study of nonlinear diffusion equations involving nonlocal, long-range diffusion effects. Our main concern is the so-called fractional porous medium equation, $\partial_t u +(-\Delta)^{s}(u^m)=0$, and some of its generalizations. Contrary to usual porous medium flows, the fractional version has infinite speed of propagation for all exponents $0 0$; on the other hand, it also generates an $L^1$-contraction semigroup which depends continuously on the exponent of fractional differentiation and the exponent of the nonlinearity. After establishing the general existence and uniqueness theory, the main properties are described: positivity, regularity, continuous dependence, a priori estimates, Schwarz symmetrization, among others. Self-similar solutions are constructed (fractional Barenblatt solutions) and they explain the asymptotic behaviour of a large class of solutions. In the fast diffusion range we study extinction in finite time and we find suitable special solutions. We discuss KPP type propagation. We also examine some related equations that extend the model and briefly comment on current work.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/72598bd9f3cada45db2e2beff71ce2cb2b16c289

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In Applied Mathematics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Endpoint Sobolev Theory for the Muskat Equation (2020, unknown): “Let us mention papers by Vazquez [55], for the fractional porous media equation, as well as Granero-Belinchón and Scrobogna [36] and Scrobogna [47] for quadratic approximation of the Hele-Shaw equation (with or without surface tension).”
Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity (2014, background): “1) relies not only on mathematical purposes but also on their significance in real models, as explained by Caffarelli in [9]–[11], and by Vázquez in [30]–[31]; see also [14].”
Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains (2014, unknown): “…de Cantoblanco, 28049 Madrid, Spain (b) e-mail address: matteo.bonforte@uam.es (c) Université Aix-Marseille, I2M, Centre de Mathématique et Informatique, Technopôle de ChateauGiombert, Marseille, France (d) e-mail address: sire@cmi.univ-mrs.fr (e) e-mail address: juanluis.vazquez@uam.es Contents”
What is the fractional Laplacian? A comparative review with new results (2018, unknown): “It also referred to in the literature as the integral fractional Laplacian [15] or as the restricted (Dirichlet) fractional Laplacian [63,64]; for consistency, we do not use these terms in this article.”
Optimal Existence and Uniqueness Theory for the Fractional Heat Equation (2016, background): “For some recent work on fractional nonlinear diffusion equations see [41].”
The Dirichlet Problem for the fractional p-Laplacian evolution equation (2015, background): “Evolution problems of nonlocal diffusion type originated from material science and statistical mechanics, and led to much work on fractional porous medium models, see for instance [7, 22, 23, 20, 43, 44, 6, 10, 52, 60, 25], the work was summarized in [56, 58].”
Regularity of solutions of the fractional porous medium flow with exponent 1/2 (2014, unknown): “[21].”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, methodology): “Early results on the PMFP have been reported in Proceedings from the Abel Symposium [245], and then in [246], so we will concentrate on general facts and only develop in more detail some of the new material.”