Nonlinear Porous Medium Flow with Fractional Potential Pressure

By Luis Caffarelli, Juan Luis V'azquez. 2010

246 citations recorded by the source corpus.

AbstractWe study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator: $$\partial_t u-\nabla\cdot(u\nabla p)=0, \quad p=(-\Delta)^{-s}u,\quad 0 < s < 1.$$We pose the problem for $${x\in {\mathbb{R}^n}}$$ and t > 0 with bounded and compactly supported initial data, and prove the existence of weak and bounded solutions that propagate with finite speed, a property that is not shared by other fractional diffusion models.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/ccce9b2ed3ba5a432ae5440c909e718884f5bac2

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

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].”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, background): “• In paper [71] Luis Caffarelli and the author established the existence of weak energy solutions, the basic properties of the solutions, like conservation of mass”
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 Fisher-KPP problem with doubly nonlinear diffusion (2016, methodology): “In doing this we follow the outline of ([4]).”
Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators (2014, methodology): “This has been studied by Caffarelli and the author in [39, 40] and Biler, Karch and Monneau [21, 19], where the equation is derived in the framework of the theory of dislocations.”
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.”
Regularity of solutions of the fractional porous medium flow with exponent 1/2 (2014, methodology): “As was already indicated in our first paper [6], the functional treatment of the equation and its kernels is very similar for s < 1/2, but for s = 1/2 we find that L(x−y) = c log(x−y) which is unbounded at infinity.”