A fractional porous medium equation

Abstract

We develop a theory of existence, uniqueness and regularity for a porous medium equation with fractional diffusion, $\frac{\partial u}{\partial t}+(-\Delta {)}^{1/2}(|u{|}^{m-1}u)=0$ in ${\mathbb{R}}^N$, with $m>m_{\ast }=(N-1)/N$, $N\ge 1$ and $f\in L^1({\mathbb{R}}^N)$. An $L^1$-contraction semigroup is constructed and the continuous dependence on data and exponent is established. Nonnegative solutions are proved to be continuous and strictly positive for all $x\in {\mathbb{R}}^N$, $t>0$.

References in corpus (3)

• Normal and Anomalous Diffusion: A Tutorial
• Hydrodynamic limit of particle systems with long jumps
• Positive solutions of nonlinear problems involving the square root of the Laplacian

Cited by in corpus (2)

• On a space discretization scheme for the Fractional Stochastic Heat Equations
• Weak Nonmild Solution of Stochastic Fractional Porous Medium Equation