Fractional porous media equations: existence and uniqueness of weak solutions with measure data
type: Item
instance of: scholarly article
title: Fractional porous media equations: existence and uniqueness of weak solutions with measure data
zbMATH Open document ID: 1333.35322
author: Gabriele Grillo
author: Matteo Muratori
author: Fabio Punzo
published in: Calculus of Variations and Partial Differential Equations
publication date: 2015-11-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1312.6076
review text: The authors study solutions properties for the boundary nonlinear value problem \[ (*):\;\rho u_t(\cdot,\cdot)+(-\Delta)^s[u(\cdot,\cdot)]^m=0\text{ in } \mathbb R^d\times\mathbb R_+\quad \text{and}\quad \rho u(\cdot,\cdot)=\mu\text{ on }\mathbb R^d\times\{0\}, \] where \((s,d,m)\in (0,1)\times (2s,\infty) \times (1,\infty)\), \(\mu\) represents a positive Radon measure on \(\mathbb R^d\) and \(\rho\) is a frame measurable weight function on \(\mathbb R^d\) such that its frame bounds are given, up to multiplicative constants, in terms of the power of \(|x|^{-\gamma}\) for \(\gamma\in [0,2s)\) whenever \(x\) belongs to the complement of the open unit ball in \( \mathbb R^d\) (resp. \(|x|^{-\gamma_0}\) for \(\gamma_0\in [0,\gamma]\) and \(|x|<1\)). Then, the authors state their main result that \((*)\) has a weak solution \(u(\cdot,\cdot)\in L_p(\mathbb R^d,\rho(x)dx)\), the set of \(p\)-integrable functions with respect to the measure \(\rho dx\) for \(p\in [1,\infty]\) and the \(L_\infty\)-norm of \(u(\cdot,t)\) is bounded by \(K_{\gamma,d,s,m}t^{-\alpha}\|u(\cdot,t)\|^{\beta}_{L_1(\mathbb R^d,\rho(x)dx)}\) such that \(\alpha,\beta, K_{\gamma,d,m,s}\) are constants depending on \(\gamma\), \(d\), \(s\), \(m\), where the two first constants are given explicitly. Furthermore, the weak solution satisfies some energy estimates (Theorem 3.2). About the proof of the main result, the authors use an approximation process and substituting \(\mu\) by \(\rho u(\cdot,0)\), where \(u(\cdot,0)\) is a bounded function belonging to \(L_1(\mathbb R^d,\rho(x)dx)\) (Lemma 4.2). The uniqueness of the weak solution for \((*)\) is stated in Theorem 3.4 and due to the toughness for proving this theorem, the authors provide the fundamental ideas, see the fifth section. The authors conclude their work by stating that the operator \(\rho^{-1}(-\Delta)^{s}\) is self-adjoint, positive and generates a Markov semigroup on \(L_2(\mathbb R^d, \rho dx)\) (Theorem 3.7), and the proof is pinpointed i…
reviewed by: Mohammed El Aïdi
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35A02
Mathematics Subject Classification ID: 35R06
zbMATH DE Number: 6505927
zbMATH Keywords: fractional Laplacian
zbMATH Keywords: fractional Sobolev spaces
zbMATH Keywords: Weighted \(L^p\) spaces
zbMATH Keywords: Stroock-Varopoulos and Caffarelli-Kohn-Nirenberg inequalities
MaRDI profile type: Publication
OpenAlex ID: W2167937913
arXiv ID: 1312.6076
cites work: Q4954179
cites work: The Initial Trace of a Solution of the Porous Medium Equation
cites work: A Widder's type theorem for the heat equation with nonlocal diffusion
cites work: Barenblatt profiles for a nonlocal porous medium equation
cites work: Quantitative local and global a priori estimates for fractional nonlinear diffusion equations
cites work: Strong solutions in \(L^ 1\) of degenerate parabolic equations
cites work: Nonlinear parabolic equations involving measures as initial conditions
cites work: Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities
cites work: Extended solutions for general fast diffusion equations with optimal measure data
cites work: Fractional heat equations with subcritical absorption having a measure as initial data
cites work: Non-negative solutions of theo porous medium eqation
cites work: Stein-Weiss and Caffarelli-Kohn-Nirenberg inequalities with angular integrability
cites work: Q4041031
cites work: Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon
cites work: \(L^q\)-functional inequalities and weighted porous media equations
cites work: On the Bakry-Emery criterion for linear diffusions and weighted porous media equations
cites work: A fractional porous medium equation
cites work: A general fractional porous medium equation
cites work: Hitchhiker's guide to the fractional Sobolev spaces
cites work: The Cauchy problem for the nonlinear filtration equation in an inhomogeneous medium
cites work: The Filtration Equation in a Class of Functions Decreasing at Infinity
cites work: Sharp short and long time \(L^{\infty }\) bounds for solutions to porous media equations with homogeneous Neumann boundary conditions
cites work: Porous media equations with two weights: smoothing and decay properties of energy solutions via Poincaré inequalities
cites work: On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density
cites work: Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density
cites work: Propagation of thermal waves in an inhomogeneous medium
cites work: Non-linear diffusion in a finite mass medium
cites work: Q5666837
cites work: Capacitary estimates of solutions of semilinear parabolic equations
cites work: Nonlinear measure data problems
cites work: The fractional Laplacian in power-weighted \(L^{p}\) spaces: integration-by-parts formulas and self-adjointness
cites work: L 1 Properties of Two Classes of Singular Second Order Elliptic Operators
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Uniqueness of the solutions of ut−Δϕ(u) = 0 with initial datum a measure
cites work: Radon measure-valued solutions for a class of quasilinear parabolic equations
cites work: On the Cauchy problem for nonlinear parabolic equations with variable density
cites work: Well-posedness for the Cauchy problem for a fractional porous medium equation with variable density in one space dimension.
cites work: On the Cauchy problem for a general fractional porous medium equation with variable density
cites work: The Cauchy problem for the inhomogeneous porous medium equation
cites work: The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions
cites work: Long time behavior for the inhomogeneous PME in a medium with slowly decaying density
cites work: Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type
cites work: Q3409342
DOI: 10.1007/S00526-015-0904-4
description: scientific article