Stefano Lisini

Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics

Abstract In this paper, we establish a novel approach to proving existence of non-negative weak solutions for degenerate parabolic equations of fourth order, like the Cahn–Hilliard and certain thin film equations. The considered evolution equations are in the form of a gradient flow for a perturbed Dirichlet energy with respect to a Wasserstein-like transport metric, and weak solutions are obtained as curves of maximal slope. Our main assumption is that the mobility of the particles is a concave function of their spatial density. A qualitative difference of our approach to previous ones is that essential properties of the solution – non-negativity, conservation of the total mass and dissipation of the energy – are automatically guaranteed by the construction from minimizing movements in the energy landscape.

A remark on the anisotropic outer Minkowski content

Abstract. We study an anisotropic version of the outer Minkowski content of a closed set in ℝ n

A gradient flow approach to the porous medium equation with fractional pressure

{\mathbb {R}}^n

Measure valued solutions of sub-linear diffusion equations with a drift term

. In particular, we show that it exists on the same class of sets for which the classical outer Minkowski content coincides with the Hausdorff measure, and we give its explicit form.

Characterization of absolutely continuous curves in Wasserstein spaces

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 gradient flow of a square fractional Sobolev norm. The energy dissipation inequality, regularizing effect and decay estimates for the Lp norms are established. Moreover, we show that a classical porous medium equation can be obtained as a limit case.

Nonlinear mobility continuity equations and generalized displacement convexity

In this paper we study nonnegative, measure valued solutions of the initial value problem for one-dimensional drift-diffusion equations when the nonlinear diffusion is governed by an increasing

Stability of flows associated to gradient vector fields and convergence of iterated transport maps

C^1

A HYBRID VARIATIONAL PRINCIPLE FOR THE KELLER-SEGEL SYSTEM IN R 2

function

UNIQUENESS FOR KELLER-SEGEL-TYPE CHEMOTAXIS MODELS

\beta

Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces

with