# 28.11 MANNHEIM HALF DAY WORKSHOP ON ANALYSIS, 1:30PM-5:30PM

The main topic of this mini-workshop is on the analysis of diffusion type (reaction diffusion, cross diffusion, diffusion fluid coupled) systems. In the last decades, there are quite a lot of new diffusion type models proposed from biology, which cannot be handled by classical parabolic theory. These have attracted more and more applied mathematicians to generate new tools. The aim of this workshop is to show the new trend of this area and to encourage cooperation on common interested problems.** **

**Schedule**

**1:30-2:20 Ansgar Juengel (TU Wien)**

**2:20-3:10 Klemens Fellner (University Graz)**

**3:10-3:50 Coffee break**

**3:50-4:40 Laurent Desvillettes (CNRS, Paris) **

**4:40-5:30 Michael Winkler (University Paderborn)**

**Room: A5,6 012**

**Speaker: Ansgar Juengel**

Title:The boundedness-by-entropy principle for cross-diffusion systems from biology

Abstract: Many systems of collective behavior for multiple species can be described in the continuum limit by cross-diffusion systems, derived e.g. from lattice models. Examples are coming from population dynamics, cell biology, and gas dynamics. A common feature of these strongly coupled parabolic differential equations is that the diffusion matrix is often neither symmetric nor positive definite, which makes the mathematical analysis very challenging.In this talk,we explain that for certain cross-diffusion systems, these difficulties can be overcome by exploiting a formal gradient-flow structure. This means that there exists a transformation of variables (called entropy variables) such that the transformed diffusion matrix becomes positive definite, and there exists a Lyapunov functional (called entropy) which enables suitable a priori estimates. Although the maximum principle generally does not hold for systems, we show that the entropy concept helps to prove lower and upper bounds for the solutions to certain systems, without the use of a maximum principle. We refer to this technique as the boundedness-by-entropy principle. We detail the theory for several examples coming from tumor-growth modeling, population dynamics, and multicomponent gas dynamics. The existence of global weak solutions and their long-time behavior is investigated and some numerical examples are presented.

**Speaker: Klemens Fellner**

Title: On Systems of Reaction-Diffusion Equations: Global Existence and Large Time Analysis.

Abstract: Starting with a volume-surface reaction-diffusion modelling asymmetric protein localisation on stem cells, we shall discuss questions of global existence, regularity and convergence to equilibrium for systems of reaction-diffusion equations. In particular, we shall present recents results on entropy and duality methods and how they apply to the existence theory and the large time analysis of systems of reaction-diffusion equations. Moreover, we shall point out recent considerations concerning the availability of entropy functional in the case of so-called complex reaction kinetics, where no detailed balance condition holds.

**Speaker: Laurent Desvillettes **

Title: Some new results for cross diffusion equations

Abstract: We present recent progresses on the cross diffusion systems first introduced by Shigesada, Teramoto and Kawasaki in the late seventies, obtained in collaboration with Thomas Lepoutre, Ayman Moussa and Ariane Trescases. These improvements include a new analysis of the entropy structure of the system, and new ideas for the approximation of the system.

**Speaker: Michael Winkler**

Title: Mathematical challenges arising in the analysis of chemotaxis-fluid interaction

Abstract: We consider models for the spatio-temporal evolution of populations of microorganisms, moving in an incopressible fluid, which are able to partially orient their motion along gradients of a chemical signal. According to modeling approaches accounting for the mutual interaction of the swimming cells and the surrounding fluid, we study study parabolic chemotaxis systems coupled to the (Navier-)Stokes equations through transport and buoyancy-induced forces. The presentation discusses mathematical challenges encountered even in the context of basic issues such as questions concerning global existence and boundedness, and attempts to illustrate this by reviewing some recent developments. A particular focus will be on strategies toward achieving priori estimates which provide information sufficient not only for the construction of solutions, but also for some qualitative analysis.

# 27.11 Self-organized Hydrodynamics in an Annular Domain: Modal Analysis and Nonlinear Effects (Hui Yu)

Title: Self-organized Hydrodynamics in an Annular Domain: Modal Analysis and Nonlinear Effects

Speaker: Hui Yu (Imperial College)

Abstract: The Self-Organized Hydrodynamics model of collective behavior is studied on an annular domain. A modal analysis of the linearized model around a perfectly polarized steady-state is conducted. It shows that the model has only pure imaginary modes in countable number and is hence stable. Numerical computations of the low-order modes are provided. The fully non-linear model is numerically solved and nonlinear mode-coupling is then analyzed. Finally, the efficiency of the modal decomposition to analyze the complex features of the nonlinear model is demonstrated.

Time: 27.11 Do. 15:30

Place: B6 A301

# 20.10 Local existence of smooth solutions to an nonsymmetric hyperbolic system-continued (Jing Wang)

Title: Local existence of smooth solutions to an nonsymmetric hyperbolic system (Continued)

Speaker: Jing Wang

Abstract:

Time: 20.10.2014, Mo.** 11:00**

Place: A5 B139

# 16.10 Local existence of smooth solutions to an nonsymmetric hyperbolic system (Jing Wang)

16.10 Local existence of smooth solutions to an nonsymmetric hyperbolic system

Title: Local existence of smooth solutions to an nonsymmetric hyperbolic system

Speaker: Jing Wang

Abstract:

Time: 16.10.2014, **Do. 15:00**

Place: A5 B139

# 30.09, Existence and Blow-Up of Some Parabolic Problems/ Lotka-Volterra Dynamics, (Evangelos Latos)

Title: Existence and Blow-up of some parabolic problems/ Lotka-Volterra Dynamics

Speaker: Evangelos Latos

Abstract: The local existence and uniqueness of solutions $u=u(x,t;\lambda)$ to the semilinear filtration equation, with initial data $u_0\geq0$ and appropriate boundary conditions are examined. There is a critical value $\lambda^*$ of the parameter $\lambda$ such that for $\lambda>\lambda^*$ there is not any kind of stationary solution of the problem, while for $\lambda\leq\lambda^*$ there exist classical stationary solutions. It is proved that the solution $u$, for $\lambda>\lambda^*$, blows-up in finite time $t^*$ for any $u_0\geq0$.

A non-local Filtration equation is considered, with some boundary and initial data $u_0$. It is proved that solutions blow-up for large enough values of the parameter $\lambda>0$ and for any $u_0>0$, or for large enough values of $u_0>0$ and for any $\lambda>0$. We will prove the global grow-up of critical solutions $u^*=u(x,t;\lambda^*)$ ($u^*(x,t)\rightarrow\infty$, as $t\rightarrow\infty$ for all $x\in(-1,1)$).

A prey-predator system associated with the classical Lotka-Volterra nonlinearity is studied. It is shown that the dynamics of the system are controlled by the ODE part. First the solution becomes spatially homogeneous and is subject to the ODE part as $t\to\infty$. Next the shadow system approximates the original system as $D\to\infty$. The asymptotics of the shadow system are also controlled by those of the ODE. The transient dynamics of the original system approaches to the dynamics of its ODE part with the initial mean as $D\to\infty$.

Time: 30.09, **Di. 13:45**

Place: A5 **C012**

# 08.05, Lifschitz tails on the Bethe lattice (Francisco Hoecker-Escuti)

Title: Lifschitz tails on the Bethe lattice

Speaker: Francisco Hoecker-Escuti

Abstract: It is well known that the integrated density of states (the normalized eigenvalue counting function) of models of disordered media exhibits an exponential decay near the band edges. This phenomenon is known as Lifschitz tails and it is a hallmark of Anderson localization (the absence of diffusion of waves). In this talk we will discuss the decay of the integrated density of states of the Anderson model whose underlying physical space is a Bethe lattice (the Cayley graph of the free group).

Time: 08.05, 2pm

Place: A5 B139

# 08.04 Die Csisz{\'a}r--Kullback--Ungleichung und Anwendungen (Michael Dreher)

Titel: Die Csisz{\'a}r--Kullback--__Ungleichung und Anwendungen
__

Speaker: Michael Dreher (UK)

Abstrakt:

Die Csisz{\'a}r--Kullback--__Ungleichung entstammt aus der Wahrscheinlichkeitstheorie bzw. der Informationstheorie und erlaubt es, aus einem Entropiema"s weitere Aussagen "uber die Verteilungsfunktion zu gewinnen. Der Vortrag stellt diese Ungleichung vor und diskutiert Anwendungen auf station"are Probleme aus der Halbleiterphysik.
__

Time: 08.04, 2pm

Place: B6 A302.

# 02.04, Derivation of Mean-field Dynamics for Fermions (Soeren Petrat)

Title: Derivation of Mean- field Dynamics for Fermions

Speaker: Soeren Petrat (LMU)

Abstract: The talk is about the derivation of the time-dependent Hartree(-Fock) equations as an e ective dynamics for fermionic many-particle systems in quantum mechanics. That is, I start from the microscopic Schrodinger dynamics for N particles and show that the Hartree(-Fock) equations approximate this dynamics well for large N. I concentrate on the natural scenario where the total kinetic energy is bounded by a constant times N and the interaction has long range, such that there is interesting quantum mechanical mean- fi eld behavior. The main results hold for a large class of interactions, including singular interactions. For Coulomb interaction, the results hold under certain assumptions on the properties of the solutions. The results are obtained by using a new method to derive mean- field limits developed by Pickl, which focuses on the correlations that develop due to the interaction.

Time: 02.04, 3.30pm

Place: A5C115

# 26.03 A note on Aubin-Lions-Dubinskii lemmas (Xiuqing Chen, Peking)

Title: A note on Aubin-Lions-Dubinskii lemmas

Speaker: Xiuqing Chen (Peking)

Abstract: Strong compactness results for families of functions in seminormed nonnegative cones in the spirit of the Aubin-Lions-Dubinskii lemma are proven, refining some recent results in the literature. The first theorem sharpens slightly a result of Dubinski\u{\i} (1965) for seminormed cones. The second theorem applies to piecewise constant functions in time and sharpens slightly the results of Dreher and Juengel (2012) and Chen and Liu (2012). An application is given, which is useful in the study of porous-medium or fast-diffusion type equations. This is a joint work with Juengel (Wien) and Liu (Duke)

Time: 26.03, 2pm

Place: A5 C115

# 18.02, Mean-field evolution of fermionic systems (Niels Benedikter)

**Title:** Mean-field evolution of fermionic systems

**Speaker:** Niels Benedikter (Bonn)

**Abstract:** Physical systems typically consist of a large number of interacting particles, making it difficult to predict measurements from the time-dependent Schroedinger equation. Therefore one is interested in deriving effective evolution equations approximating the Schroedinger equation. We consider the mean-field regime (high density and weak interaction) for fermionic systems and derive the time-dependent Hartree-Fock equations. We point out a semiclassical structure in typical initial data which is crucial for controlling the approximation up to arbitrary times. (Based on joint work with Marcello Porta and Benjamin Schlein.)

**Time: **18.02, 2pm

**Place:** B6 A104