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 consider conservative and gradient flows for N-particle Riesz energies with meanfield
scaling on the torus Td, for d 1, and with thermal noise of McKean–Vlasov type. We prove …

We consider a model of fractional diffusion involving a natural nonlocal version of the p-
Laplacian operator. We study the Dirichlet problem posed in a bounded domain Ω of RN …

We consider weak solutions of the fractional heat equation posed in the whole n-
dimensional space, and establish their asymptotic convergence to the fundamental solution …

We analyze the one-dimensional pressureless Euler–Poisson equations with linear
dam** and nonlocal interaction forces. These equations are relevant for modeling …

In this expository work we discuss the asymptotic behaviour of the solutions of the classical
heat equation posed in the whole Euclidean space. After an introductory review of the main …

A mean-field-type limit from stochastic moderately interacting many-particle systems with
singular Riesz potential is performed, leading to nonlocal porous-medium equations in the …

We study the general nonlinear diffusion equation u_t= ∇ ⋅ (u^ m-1 ∇ (-Δ)^-su) ut=∇·(um-
1∇(-Δ)-su) that describes a flow through a porous medium which is driven by a nonlocal …

We study a porous medium equation with fractional potential pressure:∂ tu=∇⋅(um− 1∇
p), p=(− Δ)− su, for m> 1, 0< s< 1 and u (x, t)≥ 0. The problem is posed for x∈ RN, N≥ 1 …

We consider a family of porous media equations with fractional pressure, recently studied by
Caffarelli and Vázquez. We show the construction of a weak solution as the Wasserstein …