• Optimal Transportation
• Workshop: Gradient flows and entropy methods

• Schedule
• Participants

We present a method for the derivation of lower bounds on free boundary propagation for the thin-film equation, one of the most prominent examples of a higher-order degenerate parabolic equation. In particular, we obtain sufficient conditions for instantaneous forward motion of the free boundary, upper bounds on waiting times, as well as lower bounds on asymptotic propagation rates. Our estimates coincide (up to a constant factor) with the previously known reverse bounds and are therefore optimal. To the best of our knowledge, these results constitute the first lower bounds on free boundary propagation for any higher-order degenerate parabolic equation. Our technique is based on new monotonicity formulas for solutions to the thin-film equation which involve weighted entropies with singular weight functions. It turns out that our method is not restricted to the thin-film equation, but also applicable to other higher-order parabolic equations like quantum drift-diffusion equations.

In this talk I will show a construction of weak global solutions for a family of fractional porous medium equation. The proof, alternative to the one given by Caffarelli and Vazquez, is based on the gradient flow interpretation and works for a general class of initial data.
An energy dissipation inequality and the decay of the L^p norms along the solutions will be illustrated.
Finally I will show the convergence of the solutions of the s-fractional porous medium equation to the unique solution of the classical porous medium equation as the parameter s goes to 0.
Joint work with E. Mainini and A. Segatti.

In this talk, I will discuss existence and long time behavior of two gradient flows in the Wasserstein and a combined Wasserstein-L2-metric, respectively. The common feature of the two is that although their potentials are not geodesically convex, exponentially fast convergence to equilibrium can be proven by variational methods. The first example (joint with R.McCann and G.Savare) is a family of fourth order degenerate parabolic equations, which arise e.g. in models for lubrication. The second example (joint work with J.Zinsl) is a variant of the Keller-Segel system of two nonlinear diffusion equations modeling the aggregation of bacteria.

It is well known that gradient flows in linear or metric spaces can be constructed by studying the limit of the discrete solutions obtained by the so called Minimizing Movement scheme. The lectures will present an introduction to another variational method, consisting in a family of minimum problems for suitable integral functionals defined in the space of (absolutely) continuous paths. In Hilbert spaces the method corresponds to an “elliptic regularization” of the differential equation (Lions-Magenes) and it has been studied by many authors in the past (Ilmanen, De Giorgi, Mielke-Stefanelli) with various applications. After a short and heuristic introduction for finite dimensional flows, we will discuss the general metric setting, showing the relationships with optimal control problems and Hamilton-Jacobi equations, and proving some convergence results (obtained in collaboration with Rossi, Segatti and Stefanelli,
<http://dx.doi.org/10.1016/j.crma.2011.11.002>) under various assumptions on the energy functional. We will eventually compare the WED approach with the Minimizing Movement one.

The long-time behavior of solutions to the porous medium equation is governed by the self-similar Barenblatt solution. This convergence is established in various topologies and optimal rates are known. In this talk we go one step further and study the higher order asymptotics. In a first step, we linearize the equation formally around the Barenblatt solution by performing computations of Riemannian nature in Waserstein space ("Otto calculus"). The resulting operator can be diagonalized rigorously and all eigenvalues and eigenfunctions are computed explicitly. In a second part of the talk, we present a way of how to linearize the equation rigorously and we use an invariant manifold theorem to derive an asymptotic expansion of solutions using the previously found diagonalization.

This work introduces a notion of gradient and an infimum-convolution operator that extend properties of solutions of Hamilton Jacobi equations to more general spaces, in particular to graphs. As a main application, the hypercontractivity of this class of infimal-convolution operators is connected to some discrete version of the log-Sobolev inequality and to a discrete version of Talagrand's transport inequality.

Nonlocal-interaction equations serve as one of the basic models of biological aggregation. The interaction between individuals is typically attractive at large distances and repulsive at short distances. We will discuss several phenomena appearing in systems governed by such interactions: the variety of patterns that stable steady states exhibit, rolling traveling swarms in heterogeneous environments, and phase separation (flock / empty space) in systems with a local dispersal mechanism.
A new model of aggregation which takes into account that long-range interactions are often not additive will also be discussed. The gradient flow structure with respect to geometries related to Wasserstein metric plays a key in the results.
The talk is based on joint works with Wu, Simione and Topaloglu, and Eisenbeis and Pego.

Metastability is a phenomenon that occurs in the dynamics of a multi-stable non-linear system subject to noise. It is characterized by the existence of multiple, well separated time scales. On a short time scale the system reaches a local equilibrium within a small subset of the available state space, while on a long time scale transitions between different subsets occur.
In the context of reversible stochastic dynamics, the potential theoretic approach to metastability has proven to be a powerful tool to derive sharp estimates for quantities characterizing the metastable behaviour. In this talk, I will focus on the metastable behaviour of reversible Markov chains. In particular, I will discuss an approach to derive optimal constants in the Poincar\'e and logarithmic Sobolev inequality. The proof is based on a refined two-scale decomposition. A key ingredient is a variational representation for negative Sobolev norm and its relation to capacities.

This is joint work with André Schlichting (Univ. Bonn)

A McKean-Vlasov diffusion corresponds to a particle in a mean-field system of particles which dimension goes to infinity. Benachour, Roynette and Vallois have proved the convergence of this process. Cattiaux, Guillin and Malrieu have extended this result by adding the gradient of a convex potential. Carrillo, McCann and Villani prove a similar result in a non-convex case by assuming that the center of mass is fixed. By using the exact number of invariant probabilities and the free-energy functional, the long-time convergence will be proved by easily checked assumptions.