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 …

The aim of this survey is to serve as an introduction to the different techniques available in
the broad field of aggregation-diffusion equations. We aim to provide historical context, key …

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 consider the fractional porous medium flow introduced by Caffarelli and Vazquez (2011)
and obtain local in time existence, uniqueness, and blow-up criterion for smooth solutions …

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 prove existence of weak solutions of a fractional thin film type equation with linear
mobility in any space dimension and for any order of the equation. The proof is based on a …

Motivated by the possibility of noise to cure equations of finite-time blowup, the recent work
by the second and third named authors showed that with quantifiable high probability …

This paper deals with a nonlinear degenerate parabolic equation of order α between 2 and
4 which is a kind of fractional version of the Thin Film Equation. Actually, this one …

Aggregation equations, such as the parabolic-elliptic Patlak–Keller–Segel model, are known
to have an optimal threshold for global existence versus finite-time blow-up. In particular, if …