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 …

This review summarizes recent advances in our understanding of anomalous transport in
spin chains, viewed through the lens of integrability. Numerical advances, based on tensor …

At the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and
John Mather launched a revolution in the venerable field of optimal transport founded by G …

This review article aims to stress and reunite some of the analytic formalism of the
anomalous diffusive processes that have succeeded in their description. Also, it has the …

The Heat Equation is one of the three classical linear partial differential equations of second
order that form the basis of any elementary introduction to the area of PDEs, and only …

Degenerate and singular parabolic equations have been the subject of extensive research
for the last 25 years. Despite important achievements, the issue of the Harnack inequality for …

Transport properties are among the defining characteristics of many important phases in
condensed-matter physics. In the presence of strong correlations they are difficult to predict …

We describe the mathematical theory of diffusion and heat transport with a view to including
some of the main directions of recent research. The linear heat equation is the basic …

We develop a theory of existence, uniqueness and regularity for the following porous
medium equation with fractional diffusion, with m> m⁎=(N− 1)/N, N⩾ 1 and f∈ L1 (RN). An …

We describe two models of flow in porous media including nonlocal (long-range) diffusion
effects. The first model is based on Darcy's law and the pressure is related to the density by …