Asymptotic behaviour of a porous medium equation with fractional diffusion

By Luis Caffarelli, Juan Luis V'azquez. 2010

112 citations recorded by the source corpus.

We consider a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. In a previous paper we have found mass-preserving, nonnegative weak solutions of the equation satisfying energy estimates. The equation is posed in the whole space R^n. Here we establish the large-time behaviour. We first find selfsimilar nonnegative solutions by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density, which we call obstacle Barenblatt solutions. The theory for elliptic fractional problems with obstacles has been recently established. We then use entropy methods to show that the asymptotic behavior of general finite-mass solutions is described after renormalization by these special solutions.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/bc0598233bf6f0fbd14188e70771a4602fccc49b

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Citations in the corpus

Nonlinear Porous Medium Flow with Fractional Potential Pressure (2010, methodology): “The details of the construction, as well as the convergence of a typical solution to such asymptotic profiles after suitable scaling, are established in the next paper [ 13 ].”
Barenblatt profiles for a nonlocal porous medium equation (2010, unknown): “Then, of course (− ) 2−α 2 p = |u|m−1, see [3,4] for that notation.”
Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators (2014, background): “Uniqueness needs to go over to the potential equation as follows: we take the convolution of u(x, t) with the Riesz kernel and define: U(x, t) = Cn,s ∫ u(y, t) |x− y|N−2s dy .”
Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials (2016, background): “Here we would like to point out some interesting models involving the fractional Laplacian, such as, the fractional Schrödinger equation (see [14, 15, 22, 23, 24]), the fractional Kirchhoff equation (see [16, 32, 33, 46, 47]), the fractional porous medium equation (see [9, 45]), the fractional Yamab”
Getting Acquainted with the Fractional Laplacian (2017, unknown): “[33,34,49].”
Quantitative local and global a priori estimates for fractional nonlinear diffusion equations (2012, background): “This has been studied by Caffarelli and Vazquez in [14, 13] and Biler, Karch and Monneau in [7], where the equation is derived in the framework of the theory of dislocations.”
The Fisher-KPP problem with doubly nonlinear diffusion (2016, background): “The study of this model has been pursued [5, 6].”
A General Fractional Porous Medium Equation (2011, unknown): “Thus, the papers [14], [15] consider the following fractional diffusion PME ∂tu−∇· (u∇(−∆)−s/2u) = 0.”
Regularity of solutions of the fractional porous medium flow with exponent 1/2 (2014, methodology): “The long-time behaviour of such solutions was described in a second paper [7] using an entropy method that leads to a fractional obstacle problem for the asymptotic profile.”
Infinite Speed of Propagation and Regularity of Solutions to the Fractional Porous Medium Equation in General Domains (2015, unknown): “The question of finite versus infinite speed of propagation on R has also been investigated in [19, 20, 60, 62] for a different model of fractional PME introduced by Caffarelli and Vazquez [19,20]; an equivalence among self-similar solutions between the two main models of nonlocal porous medium equa”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, methodology): “• The particular value s = 1/2 of the fractional exponent turned out to be extremely delicate for the regularity analysis of [70] and needed a further article with new geometrical ideas, [72].”
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].”