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\title{Nonlinear Diffusions and Entropy Dissipation: From Geometry to Biology}

\author{BIRS workshop (10w5054) -- Program (update: \today)}

\date{May 9 - May 14, 2010}

\maketitle



\nom{AGUEH, MARTIAL}

\titre{Finsler structure in the p-Wasserstein space and applications to PDEs}

\resume{It is known from the work of F. Otto [{\it Comm. Partial Diff. Eq. 2001}], that the space of probability measures equipped with the quadratic Wasserstein distance, i.e., the 2-Wasserstein space, can be viewed as a "Riemannian manifold". Here we show that when the quadratic cost function is in general replaced by a homogeneous cost of degree $p>1$, then the corresponding space of probability measures, i.e., the $p$-Wasserstein space, can be endowed with a Finsler metric whose induced distance function is the $p$-Wasserstein distance. Using this Finsler structure of the $p$-Wasserstein space, we give definitions of the differential and gradient of functionals defined on this space, and then of gradient flows in this space. In particular we show in this framework that the parabolic  q-Laplacian equation is a gradient flow in the $p$-Wasserstein space, where $p=q/(q-1)$. When $p=2$, we recover the Riemannian structure  introduced by  Otto, which confirms that the 2-Wasserstein space is a Riemann-Finsler manifold. Our approach is confined to a smooth situation where probability measures are absolutely continuous with respect to the Lebesgue measure on $R^n$, and they have smooth and strictly positive densities.}



\nom{ANGENENT, SIGURD}

\titre{Contrasting Stochastic and non Stochastic Models for Cell Polarization}

\resume{We present the simplest possible model for cell polarization and
show that in this model polarization of the cell membrane is
cannot be explained by the continuum description of the model,
and that persistent polarization, when it occurs, must be
considered a stochastic effect.  (joint work with Altschuler, Wang, and Wu).}



\nom{BAKRY, DOMINIQUE}

\titre{Weighted Nash Inequalities}

\resume{In this talk, we shall present some families of weighted Nash type inequalities, which lead to non uniform upper bound to heat kernels. This may produce estimates on the Hilbert-Schmidt norm for Markov semigroups. We shall show on some 1-dimensional model  examples that those inequalities are quite easy to obtain, even in the case where no ultracontractive or even hypercontractive bounds may be expected.}



\nom{BILER, PIOTR}

\titre{Keller--Segel systems at $M=8\,\pi$}

\resume{We consider models for chemotaxis 
$$n_t=\Delta n-\nabla\cdot(n\,\nabla c)\,,$$
$$\tau\,c_t=\Delta c+n\,,$$

for $x\in{\mathbb R}^2$, $t>0$, and for each $\tau\ge 0$. 

As it is well known for the parabolic-elliptic system ($\tau=0$), 
there is infinite number of locally stable steady states of critical mass 
$M=8\pi$. On the other hand, initially not dispersed data lead to solutions 
that concentrate (or, in other words, blow up) in the infinite time: 
$n(\cdot\,, t)\rightharpoonup 8\,\pi\,\delta$ as $t\to\infty$. 

The picture is much more complicated for the doubly parabolic Keller--Segel model 
($\tau>0$) since for each $\tau>\tau_\ast$ (with some $\tau_\ast>0$), 
besides those steady states, there is a self-similar solution $(n,c)$ of the form 
\[\left(\frac{1}{t}u\left(\frac{x}{\sqrt{t}}\right),\, 
v\left(\frac{x}{\sqrt{t}}\right)\right)\]also of mass $M=8\,\pi$, 
which decays exponentially for $|x|\to\infty$. 
For this self-similar solution whole mass diffuses to infinity as $t\to\infty$. }



\nom{BLANCHET, ADRIEN}

\titre{Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model}

\resume{We investigate the long time behaviour of the critical mass
Patlak-Keller-Segel equation. This equation has a one parameter
family of steady-state solutions $\rho_{\infty,\lambda}$,
$\lambda>0$, with thick tails whose second moment is not bounded.
We show that these steady state solutions are stable, and find
basins of attraction for them using an entropy functional $\mathcal H$ coming from the critical fast diffusion equation in $\mathbb R^2$. We construct solutions of Patlak-Keller-Segel equation satisfying an entropy-entropy dissipation inequality for $\mathcal H$. While the entropy dissipation for $\mathcal H$ is strictly positive, it turns out to be a difference of two terms, neither of which need to be small when the dissipation is small. We introduce a strategy of {\em controlled concentration} to deal with this issue, and then use the regularity obtained from the entropy-entropy dissipation inequality to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards $\rho_{\infty,\lambda}$. In the present talk, we do not
provide any estimate of the rate of convergence, but we discuss
how this would result from a stability result for a certain sharp
Gagliardo-Nirenberg-Sobolev inequality.}



\nom{BONFORTE MATTEO}

\titre{Asymptotics of the Fast Diffusion Equation via Entropy methods}

\resume{We consider non-negative solutions of the fast diffusion equation
$u_t=\Delta u^m$ with $m \in (0,1)$, in the Euclidean space $\mathbb R^d$, $d\ge 3$, and study the asymptotic behavior of a natural
class of solutions, in the limit corresponding to $t\to\infty$ for $m\ge m_c=(d-2)/d$, or as $t$ approaches the extinction time when $m<m_c$. For a class of initial data we prove that the solution converges with a polynomial rate to a self-similar solution, for $t$ large enough  if $m\ge m_c$, or close enough to the extinction time  if $m< m_c$. Such results are new in the range $m\le m_c$ where previous approaches fail. In the range $m_c<m<1$ we improve on known results.

The precise value for the exponent $m=(d-4)/(d-2)$, $d\ge 3$, requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the linearized generator.}



\nom{CALVEZ, VINCENT}

\titre{Gradient flow interpretation for the one-dimensional Keller-Segel equation.}

\resume{{\em Joint work with J.A. Carrillo (ICREA, UAB, Barcelona, Spain).}\medskip

We wish to present a brief overview of recent developments in the analysis of the generalized Keller-Segel equation in dimension one:
\[\partial_t \rho(t,x) = \partial_{xx}\rho^m(t,x) + \chi \partial_x  \left(\rho(t,x) \partial_x\left(\dfrac{|x|^{k}}{k}*\rho(t,x)\right)\right)\, , \quad t>0\, , \quad x\in \mathbb{R}\, ,\]
for the following set of exponents: $m\geq 1$ and $k\in (-1,1)$. 
Here $\rho(t,x)$ denotes a density of cells which diffuse and attract each other through a mean field potential.
This model has raised a lot of interest in the field of mathematical biology since it generally exhibits a dichotomy between global existence (dispersion of the cells) and finite time blow-up (aggregation of the cells).

This equation possesses the structure of a gradient flow for the Wasserstein metric on the space of probability densities. Interestingly enough the energy functional $\mathcal{F}$ is the sum of two opposite contributions, being respectively displacement-convex and concave, and homogeneous. In particular we obtain some blow-up criterion as a direct consequence of the homogeneity property in the case $m+k<1$ ({\em attraction-dominating competition}):
\begin{equation*}
\mathcal{F}\left[\rho_0\right] <  C \dfrac{\left(m-1 + k \right)}{k} \left(\int_{\mathbb{R}}|x|^2\rho_0(x)\, dx\right)^{(1-m)/2}\, .
\end{equation*}

On the other hand we investigate the long-time asymptotics in the case $m + k=1$ ({\em fair competition}). We are able to prove the following inequalities along the gradient flow in appropriately rescaled variables:
\begin{equation*}
\dfrac12 \dfrac d{dt}W\left(\rho(t),\mu \right)^2 + W(\rho(t),\mu )^2\leq 0\,, \quad \mathbf{(m = 1)}\,  , 
\end{equation*} 
\begin{equation*}
\dfrac12 \dfrac d{dt}W\left(\rho(t),\mu \right)^2 + \dfrac{m+1}2 W(\rho(t),\mu )^2 \leq (m-1) \left(\mathcal F [\rho(t)] - \mathcal F [\mu  ]\right), \quad \mathbf{(m > 1)} \, ,
\end{equation*}
where $\mu$ denotes the minimizer of the energy functional (supplemented with a quadratic confinement potential due to rescaling). We deduce from the former a uniform rate of convergence towards the minimizer in the case $(m=1,k=0)$. These inequalities are strongly related to a convex-like property of the energy functional. }



\nom{CAñIZO, JOSé ALFREDO}

\titre{Entropy-entropy dissipation inequalities for fragmentation-drift equations}

\resume{The growth-fragmentation equation models a group of cells which grow (or age) at a certain rate, and divide into two or more pieces. After a suitable rescaling, the shape of the distribution function of the population converges to a universal profile. By means of entropy-entropy dissipation inequalities for some unbounded fragmentation coefficients, we show that the speed of this convergence is exponential. This kind of inequalities may be used for other linear equations for which the General Relative Entropy principle is valid.}



\nom{CHUGUNOVA, MARINA}

\titre{Entropy-energy analysis of coating flows}

\resume{The equation $$u_t + [u^n (u_{xxx} + \gamma^2 u_x - \sin(x))]_x = 0$$ with periodic boundary conditions is a model of the evolution of a thin liquid film on the outer surface of a horizontal cylinder in the presence of gravity field. We use entropy-energy methods to study different properties of generalized weak solutions of this equation. For example: finite speed of the compact support propagation for $n \in (1,3)$ is proved by application of local $\alpha$-entropy estimates.

Joint work with
A. Burchard, M. Pugh, B. Stephens, and R. Taranets}



\nom{CORRIAS, LUCILLA}

\titre{The role of the free energy and its dissipation in the mathematical analysis of the parabolic-parabolic Keller-Segel system.}

\resume{It is well known that the classical Keller-Segel system modeling chemotaxis is endowed with a free energy functional decreasing along the trajectories of the solutions. In this  talk, we shall show how to handle that energy with the help of several sharp functional inequalities, in order to obtain both global existence results  and density concentration phenomena for the fully parabolic Keller-Segel system on the whole space. The case of the high space dimension is also considered. (Joint works with V. Calvez and A. Ebde).}



\nom{DASKALOPOULOS, PANAGIOTA}

\titre{Ancient solutions to the Ricci flow on Surfaces}

\resume{We provide a classification for  ancient solutions to the Ricci flow on $S^2$.
We show that they are either the contracting spheres or the King-Rosenau solutions.
We also prove a similar result for ancient convex embedded solutions to the
curve shortening flow. This is joint work with Richard Hamilton and Natasa Sesum. }



\nom{DESVILLETTES, LAURENT}

\titre{entropy methods and cross diffusions}

\resume{ Some models in population dynamics include so-called cross diffusions terms. Those terms model the effect of the presence of one type of individuals on the diffusion rate of other individuals.
 We shall present in this talk results about the modeling and the mathematical analysis of systems including cross diffusion: those results share a common feature: the use of Lyapounov-like functionals (entropies).}



\nom{DI FRANCESCO, MARCO}

\titre{On the Hughes' model for pedestrian flow: The one-dimensional case}

\resume{In this talk we investigate the mathematical theory of Hughes' model for the flow of pedestrians (cf. Hughes 2002), consisting of a nonlinear conservation law for the density of pedestrians coupled with an eikonal equation for a potential modelling the common sense of the task. We first consider an approximation of the original model in which the eikonal equation is replaced by an elliptic approximation.

For such an approximated system we prove existence and uniqueness of entropy solutions (in one space dimension) in the sense of Kruzkov, in which the boundary conditions are posed following the approach of Bardos et al.. We use BV estimates on the density and stability estimates on the potential in order to prove uniqueness.

Furthermore, we analyse the evolution of characteristics for the original Hughes' model in one space dimension and study the behaviour of simple solutions, in order to reproduce interesting phenomena related to the formation of shocks and rarefaction waves. The characteristic calculus is supported by numerical simulations.}



\nom{FELLNER, KLEMENS}

\titre{On non-local interaction equations: Stbility of Stationary States and Limiting Behaviour}

\resume{We study non-local evolution equations for a density of individuals, which interact through a given symmetric potential. Such models appear in many applications such as swarming and flocking, opinion formation, inelastic materials, .... In particular, we are interested in interaction potentials, which behave locally repulsive, but aggregating over large scales. A particular example for such potentials was recently given in models of the alignment of the directions of filaments in the cytoskeleton.

We present results on the structure and stability of steady states. We shall show that stable stationary states of regular interaction potentials generically consist of sums of Dirac masses. However the amount of Diracs depends delicately on the interplay between local repulsion and aggregation. In particular we shall see that singular repulsive interaction potentials introduce diffusive effects in the sense that stationary state are rendered continuously.}



\nom{FIGALLI, ALESSIO}

\titre{A new transportation distance between non-negative measures}

\resume{Given a bounded domain $\O$, it is by now well-known that that the gradient flow of the entropy functional $\int_\O [\rho \log(\rho) - \rho]\,dx$ with respect to the Wasserstein distance produces a solution to the heat equation with Neumann boundary conditions. Recently, in collaboration with Nicola Gigli we introduced a new transportation distance between non-negative measures inside a domain $\O$.
This distance enjoys many nice properties, for instance it makes the space of non-negative measures inside $\O$ a geodesic space, without any convexity assumption on $\O$.
Moreover, the gradient flow of the entropy functional
$\int_\O [\rho \log(\rho) - \rho]\,dx$ w.r.t. this distance coincides with the heat equation subject to the Dirichlet boundary condition equal to $1$.
The aim of this talk will be to briefly review the classical theory, and then to introduce this new distance and its main properties.}



\nom{GONZALEZ, MARIA DEL MAR}

\titre{A free-boundary model in price formation}

\resume{We study a model, due to J.M. Lasry and P.L. Lions, describing the evolution of a scalar price which is realized as a free boundary in a 1D diffusion equation with dynamically evolving, non-standard sources. We establish global existence and uniqueness. This is joint work with L. Chayes, M. Gualdani and I. Kim.}



\nom{KINDERLEHRER, DAVID}

\titre{Coarsening in cellular systems}

\resume{Mesoscale experiment and simulation permit harvesting information about both geometric features and texture in material microstructures.  The grain boundary character distribution (GBCD) is an empirical distribution of the relative length (in 2D) or area (in 3D) of interface with a given lattice misorientation and grain boundary normal. During the growth process, an initially random texture distribution reaches a steady state that is strongly correlated to the interfacial energy density.  In simulation, we found that if the given energy depends only on lattice misorientation, then the steady state GBCD and the energy are related by a Boltzmann distribution.   This is among the simplest non-random distributions, corresponding to independent trials with respect to the energy.  Why does such a simple distribution arise from such a complex system?

We outline an entropy based theory which suggests that the evolution of the GBCD  satisfies a Fokker-Planck Equation.  Coarsening in polycrystalline systems is a complicated process involving details of material structures, chemistry, arrangement of grains in the configuration, and environment.  In this context, we consider just two global features:  cell growth according to a local evolution law of curvature driven growth and space filling constraints.  Space filling requirements are managed by critical events, rearrangements of the network involving deletion of small contracting cells and facets.  The interaction between the evolution law and the constraints is governed primarily by the Herring Condition, the boundary condition associated with the equation of curvature driven growth. It determines a dissipation relation.  To assist in the derivation, a simpler system is introduced which is driven by the boundary conditions and reflects the dissipation relation of the grain growth system.  It resembles an ensemble of inertia-free spring-mass-dashpots.  
For this simpler coarsening network, we learn how entropic or diffusive behavior at the large scale emerges from a dissipation relation at the scale of local evolution. 

Joint work with K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, R. Sharp, and S. Ta'asan.
}



\nom{LAURENCOT, PHILIPPE}

\titre{Large time behaviour for quasilinear a degenerate diffusion equation with gradient source term}

\resume{The qualitative behaviour of nonnegative solutions to the Cauchy-Dirichlet problem for the equation $\partial_t u -\Delta_p u = |\nabla u|^q$ in a bounded domain depends on the relative strength of the diffusion and the source terms. According to the values of $p$ and $q$, multiple steady states may exist or convergence to a ``friendly giant'' may take place or finite time blowup may occur (joint works with G. Barles, C. Stinner, and M. Winkler). }



\nom{LAURENT THOMAS}

\titre{Mechanism of formation of Dirac masses in the solutions of the aggregation equation}

\resume{The aggregation equation is a continuum model for interacting particle systems with attractive/repulsive pairwise interaction potential.  It arises in a number of models for biological aggregation, but also in materials science and granular media. As a mathematics problem, the aggregation equation is a gradient flow of the interaction energy with respect to the Wasserstein distance. The main phenomenon of interest is that, even with smooth initial data, the solutions can concentrate mass in finite time (i.e. a delta Dirac appears in the solution in finite time). We study the mechanism of formation of these delta Dirac masses.}



\nom{LEE, PAUL WOON YIN}

\titre{Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian Manifolds}

\resume{Measure contraction property (MCP) is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. However, the definition of MCP is not computable in general. In this talk, I'll discuss computable sufficient conditions for a three dimensional contact subriemannian manifold to satisfy such property. This is a joint work with Andrei Agrachev.}



\nom{LISINI, STEFANO}

\titre{Measure solutions of sub-linear diffusion equations with drift}

\resume{\newcommand{\R}{\mathbb{R}}
\newcommand{\de}{\partial}
\renewcommand{\d}{{\mathrm d}}
We present some recent results,
obtained in collaboration with Giuseppe Savar\'e, Giuseppe Toscani and Simona Fornaro,
about non-negative measure valued solutions of the Cauchy problem for
a one-dimensional drift-diffusion equation of the form
\begin{equation}\label{eq:rho}
    \de_t\rho-\de_x\big(\de_x(\beta(\rho))+V'\rho\big)=0, \quad {\rm in }\ (0,+\infty)\times  \R;
    \quad
    \rho(0,\cdot)=\rho_0 \quad \text{in } \R.
\end{equation}
The diffusion function $\beta$ is increasing, smooth, and satisfies 
\begin{equation}\label{beta}
\lim_{r\to+\infty}\beta(r) <+\infty,
\end{equation}
whereas $V:\R\to\R$ is a $\lambda$-convex driving potential.
The initial datum $\rho_0$ is a non-negative Borel measure with finite total mass
$\mathfrak m_0:=\rho_0(\R)$ and finite quadratic momentum.

By the assumption \eqref{beta}, in the regions where $\rho$ is large
the diffusion term in \eqref{eq:rho} could be negligible 
and the drift term in \eqref{eq:rho} could be predominant.
Based on the previous observation,
when the initial measure $\rho_0$ has a not bounded density or has a singular part,
the presence of a singular part in the solution
$\rho$ of \eqref{eq:rho} has to be taken into account.

We give a suitable notion of measure valued solution of problem \eqref{eq:rho}
and we show existence and uniqueness.
Moreover we show a characterization of the evolution of the singular part of the solution.

Finally we show that the large time behavior of the solutions of \eqref{eq:rho}
depends on a critical mass $\mathfrak m_c$ ($\mathfrak m_c$ depending on $\beta$ and $V$).
If the initial mass $\mathfrak m_0 \leq \mathfrak m_c$, then there is a non singular
steady state, while in the case $\mathfrak m_0 > \mathfrak m_c$ the steady
state has a singular part.

The proof of existence and uniqueness is based on the
interpretation of the solutions of problem \eqref{eq:rho} as trajectories of the gradient flow
of the energy functional
\begin{equation*}
  \mathcal E(\rho):=\int_\R E(u(x))\,\d x +  \int_\R V(x)\,\d \rho(x)  \quad \text{where}\quad \rho=u \d x +\rho^\perp,
\end{equation*}
(here $\beta'(r)=rE''(r)$)
with respect to the Wasserstein distance between non negative measures with mass $\mathfrak m_0$.}



\nom{MATTHES, DANIEL}

\titre{Gradient flow of the Diriclet functional}

\resume{On a formal level, the Cahn-Hillard equation $\partial_t u = -{\rm div}\big( \mathbf{m}(u) \nabla\Delta u \big) + \Delta P(u)$ is readily seen to constitute a gradient flow of a perturbed Diriclet energy functional.
The properties of the associated metric --- which is a Wasserstein-like distance, but with the \emph{non-linear mobility} function $\mathbf{m}$ --- have been investigated recently by Dolbeault, Nazareth and Savaré.
In this talk, we show how their results can be used to make the formal variational structure of the Cahn-Hilliard equation rigorous.
Our basic assumption is that $\mathbf{m}$ is concave and satisfies some growth condition.

In application of the rigorous gradient flow structure, we derive sharp estimates on the intermediate asymptotics of weak solutions to the Hele-Shaw flow.
The latter corresponds to the Cahn-Hilliard equation with \emph{linear} mobility $\mathbf{m}$, when the associated metric is just the classical $L^2$-Wasserstein distance.
The key idea is to use a very particular connection between the Diriclet energy, an entropy and the Wasserstein distance.

This is joint work with Stefano Lisini and Giuseppe Savaré (Pavia).
}



\nom{MCCANN, ROBERT}

\titre{Higher order asymptotics of fast diffusion in Euclidean space}

\resume{With Denzler and Koch, we quantify the speed of convergence and higher-order asymptotics
of fast diffusion dynamics on ${\mathbf R}^n$ to the Barenblatt (self similar) solution.
Degeneracies in the parabolicity of this equation are cured by
re-expressing the dynamics on a manifold with a
cylindrical end, called the cigar.
The nonlinear evolution semigroup becomes differentiable
in H\"older spaces on the cigar.  The
linearization of the dynamics is
given by the Laplace-Beltrami operator
plus a transport term (which can be suppressed by introducing
appropriate weights into the function space norm), plus a
finite-depth potential well with a universal profile.
In the limiting case of the (linear) heat equation, the depth diverges,
the number of eigenstates increases without bound,  and the continuous
spectrum recedes to infinity.  We provide a detailed study of
the linear and nonlinear problems in H\"older spaces on the cigar, including
a sharp boundedness estimate for the semigroup, and use this as a tool to
obtain sharp convergence results toward the Barenblatt solution
(as Bonforte, Dolbeault, Grillo and Vazquez also did independently
 in a weaker metric), and higher order asymptotics. In finer
convergence results (after modding out symmetries of the problem), a subtle
interplay between convergence rates and tail behavior is revealed.
The difficulties involved in choosing the right functional
spaces in which to carry out the analysis can be interpreted as genuine
features of the equation rather than mere annoying technicalities.
}



\nom{NAZARET, BRUNO}

\titre{Distances on probability measures induced by concave mobilities : geodesics via PDE}

\resume{In the framework of distances on probability measures introduced by Dolbeault, Nazaret and Savaré, we present some cases where the geodesics are obtained as solutions to a system of PDE on the density and the velocity of the intermediate configurations (joint work with F. Santambrogio).}



\nom{RAOUL GAëL}

\titre{Selection dynamics for the evolution of traits in a population}

\resume{We consider a model describing the evolution of a large population of individuals with different biological "traits". The reproduction rate of each individual depends on its trait and on the competition for resources within the population. For a certain class of coefficients, an entropy exists, and can be used to study the large time behaviour of the population, and in particular the formation of Dirac masses, that corresponds to species formation.}



\nom{ROSADO, JESUS}

\titre{A refined result of flocking for the Cucker-Smale model}

\resume{I will present and analyse the asymptotic behaviour of solutions of the continuous kinetic version of flocking by Cucker and Smale [2], which describes the collective behaviour of an ensemble of organisms, animals or devices. This kinetic version introduced by Ha and Tadmor is obtained from a particle model. The large-time behaviour of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of Cucker and Smale will be shown to hold for the solutions on the kinetic model. More precisely, the solutions concentrate exponentially fast their velocity to their mean while in space they will converge towards a translational flocking solution. The presentation is based on [1].
}



\nom{SAVARé, GIUSEPPE}

\titre{Mobility functions, transport distances, and nonlinear diffusion}

\resume{ We present a new class of distances between nonnegative measures in
  the Euclidean spaces. They are modeled on the dynamical characterization of the
  Wasserstein distances proposed by
  Benamou-Brenier and provide a wide family interpolating between the
  Wasserstein and the homogeneous Sobolev distances.

  From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure.

  New existence, stability, and contraction results for solutions of nonlinear diffusion equations in a convex bounded domain will be discussed.
  (In collaboration with J.~Dolbeault, B.~Nazaret, J.~Carrillo, S.~Lisini, D.~Slepcev)}



\nom{SCHMEISER, CHRISTIAN}

\titre{Hypocoercivity of a kinetic model for fast diffusion}

\resume{Convergence to equilibrium will be discussed for BGK kinetic models, which have fast diffusion equations as their macroscopic limits.}



\nom{SESUM. NATASA}

\titre{Curvature flows}

\resume{We will discuss various curvature flows such as the harmonic
mean curvature flow (HMCF), the mean curvature flow (MCF) and the Ricci
flow (RF). We will discuss the long time behaviour of the HMCF in the
case of star shaped surfaces and also weakly convex hypersurfaces in
higher dimensions. We will also discuss the curvature conditions that
guarantee the smooth existence of the MCF and the RF.}



\nom{STA\'NCZY, ROBERT}

\titre{Entropy methods for self-gravitating particles systems}

\resume{Systems describing the interaction of gravitationally attracting particles that obey some statistics are studied. The role of the entropy for these system is thoroughly analyzed both in the isothermal case and in the case with the energy fixed. Systems  of a similar form and structure appear also in modelling the chemotaxis phenomena in biology.}



\nom{STURM, KARL-THEODOR}

\titre{Monotone Approximation for the Wasserstein Diffusion}

\resume{We present a system of interacting Brownian particles on the real line for which the empirical distributions -- in the scaling limit of large particle numbers -- converge to the Wasserstein diffusion. The latter is a reversible Markov process with continuous paths on the space of probability measures whose square field operator -- which governs the short time behavior -- is the squared norm of the Wasserstein gradient.
We indicate also some extensions to higher dimensional spaces.}



\nom{ULUSOY, SULEYMAN}

\titre{Long Time Behavior of  Weak Solutions in Hele-Shaw Flow Problem}

\resume{ We investigate the long-time behavior of weak solutions to the thin- film type equation $u_t = -(uu_{xxx})_x.$ We employ a semidiscrete variational scheme to generate weak solutions as a gradient flow with respect to so called Wasserstein distance and we show that these weak solutions converge to the unique self-similar source type solution exponentially fast. This paper complements our results in Carlen, E. A., Ulusoy, S. : Asymptotic equipartition and long time behavior of solutions of a thin-film equation, J. Differential Equations., 241, pp. 279-292, 2007. This is a joint work with Eric A. Carlen.
}



\nom{WESTDICKENBERG, MICHAEL}

\titre{Optimal Transport for the System of Isentropic Euler Equations}

\resume{The isentropic Euler equations form a system of conservation laws modeling compressible fluid flows with constant thermodynamical entropy. Due to the occurence of shock discontinuities, the total energy of the system is decreasing in time. We review the second order calculus on the Wasserstein space of probability measures and show how the isentropic Euler equations can be interpreted as a steepest descent equation in this framework. We introduce a variational time discretization based on a sequence of minimization problems, and show that this approximation converges to a suitably defined measure-valued solution of the conservation law. Finally, we present some preliminary results about the numerical implementation of our time discretization. }



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