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Fellner, Klemens  WPI Seminarroom C 714  Mon, 21. Nov 11, 14:15 
"Aggregation patterns in nonlocal equations: discrete stochastic and continuum modelling"  
Nonlocal evolution equations featuring interaction of individuals due to a repulsiveaggregative potential are observed to produce a rich dynamical behaviour, which leads to a multitude of stationary pattern. For interaction potential with suitable attractive singularities convergence to measure solutions is deduced from a gradient flow structure in Wasserstein metric. Alternatively propagate singular repulsive interaction potential regular solutions. However, the case of interaction potential with both singular and repulsive singularities remains an open problem, for which we present an interesting comparison of numerical results with a stochastic lattice model.  

Stelzer, Ines  WPI Seminarroom C 714  Mon, 21. Nov 11, 15:00 
"Entropy Structure of a CrossDiffusion TumourGrowth Model"  
We consider the mechanical tumourgrowth model of Jackson and Byrne describing tumour encapsulation influenced by a cellinduced pressure coefficient. It consists of nonlinear parabolic crossdiffusion equations in one space dimension for the volume fractions of the tumour cells and the extracellular matrix. Exploiting the existence of entropy variables and of an entropy functional, yielding gradient estimates, we can prove the globalintime existence of nonnegative and bounded weak solutions to the initialboundaryvalue problem when the cellinduced pressure coefficient is smaller than a certain explicit critical value. Moreover, when the production rates vanish, the volume fractions converge exponentially fast to the homogeneous steady state. Finally, we will present some numerical results.  

Di Francesco, Marco  WPI Seminarroom C 714  Mon, 21. Nov 11, 16:15 
"Stationary states of quadratic diffusion equations with nonlocal attraction"  

Lachowicz, Miroslav  WPI Seminarroom C 714  Tue, 22. Nov 11, 10:30 
Individuallybased Markov processes modeling nonlinear systems in mathematical biology  
The general approach that allows to construct the Markov processes describing various processes in mathematical biology (or in other applied sciences) is presented. The Markov processes are of a jump type and the starting point is the related linear equations. They describe at the micro–scale level the behavior of a large number N of interacting individuals (entities). The large individual limit ("N ! 1") is studied and the intermediate level (the meso– scale level) is given in terms of nonlinear kinetic–type equations. Finally the corresponding systems of nonlinear ODEs (or PDEs) at the macroscopic level (in terms of densities of the interacting subpopulations) are obtained. Mathematical relationships between these three possible descriptions are presented and explicit error estimates are given. The general framework is applied to propose the microscopic and mesoscopic models that correspond to well known systems of nonlinear equations in biomathematics. Reference: M. Lachowicz, Individuallybased Markov processes modeling nonlinear systems in mathematical biology, Nonlinear Analysis Real World  

Frouvelle, Amic  WPI Seminarroom C 714  Tue, 22. Nov 11, 11:15 
"Macroscopic limits of a system of selfpropelled particles with phase transition"  
The Vicsek model, describing alignment and selforganisation in large systems of selfpropelled particles, such as fish schools or flocks of birds, has attracted a lot of attention with respect to its simplicity and its ability to reproduce complex phenomena. We consider here a timecontinuous version of this model, in the spirit of the one proposed by P. Degond and S. Motsch, but where the rate of alignment is proportional to the mean speed of the neighboring particles. In the hydrodynamic limit, this model undergoes a phase transition phenomenon between a disordered and an ordered phase, when the local density crosses a threshold value. We present the two different macroscopic limits we can obtain under and over this threshold, namely a nonlinear diffusion equation for the density, and a firstorder nonconservative hydrodynamic system of evolution equations for the local density and orientation.  

Canizo, Jose A.  WPI, Seminarroom C 714  Tue, 22. Nov 11, 14:15 
"Asymptotic behavior for the aggregation equation with diffusion"  
We consider the equation @t = r ((rW )) + , a diffusive equation with a nonlinear and nonlocal term given by a selfinteraction through a potential W. We will give wellposedness results and study its asymptotic behavior. If W satisfies some suitable bounds, one can prove that the behavior is dominated by diffusion, this is, solutions behave for large times essentially like those of the heat equation. Precise estimates on rates of convergence to the fundamental solution to the heat equation can be given by using entropy methods.  

HIttmeir, Sabine  WPI Seminarroom C 714  Tue, 22. Nov 11, 15:00 
"Nonlinear diffusion and additional crossdiffusion in the KellerSegel model"  
The main feature of the twodimensional KellerSegel model is the blowup behaviour of solutions for supercritical masses. We introduce a regularisation of the fully parabolic system by adding a crossdiffusion term to the equation for the chemical substance. This regularisation provides another helpful entropy dissipation term allowing to prove global existence of solutions for any initial mass. In the parabolicelliptic case this model can be reformulated to the KellerSegel model with nonlinear cell diffusion. Therefore solutions are known to be globally bounded. In the second part of the talk we return to the fully parabolic model and replace the cell diffusion and the additional crossdiffusion by nonlinear versions. This generalisation allows for global existence results in up to three space dimensions. Numerical simulations will be presented.  

Desvillettes, Laurent  WPI, Seminarroom C 714  Wed, 23. Nov 11, 10:30 
"Selection/Competition/Mutations equations: an asymptotic study"  
This work is based on a collaboration with A. Calsina (Univ. Autonoma of Barcelona), S. Cuadrado (Univ. Autonoma of Barcelona), and G. Raoul (Univ. Cambridge). It is possible to model populations which are structured with respect to a quantitative trait by integrodifferential equations which take into account the effects of selection, competition,and mutations. The large time behavior of those equations (and the asymptotic behavior when the mutation rate tends to 0) is complex and strongly depends on the assumptions satisfied by the coefficients in the integrodifferential equations. In the simplest case (when the asymptotic state [when both time tends to infinity and mutation rate tends to 0] is one Dirac mass), we provide an expansion with respect to both small parameters for the solution of the integrodifferential equations.  

Haskovec, Jan  WPI Seminarroom C 714  Wed, 23. Nov 11, 11:15 
"On two models of biological diffusive aggregation"  
We introduce two models of biological aggregation, based on randomly moving particles with individual diffusivities depending on the perceived average population density in their neighbourhood. In the firstorder model the location of each individual is subject to a densitydependent random walk, while in the secondorder model the densitydependent random walk acts on the velocity variable, together with a densitydependent damping term. The main novelty of our models is that we do not assume any explicit aggregative force acting on the individuals; instead, aggregation is obtained exclusively by reducing the diffusivity in response to higher perceived density. We formally derive the corresponding meanfield limits, leading to nonlocal degenerate diffusions. Then, we carry out the mathematical analysis of the firstorder model, in particular, we prove the existence of weak solutions and show that it allows for measurevalued steady states. We also perform linear stability analysis and identify conditions for pattern formation. Moreover, we discuss the role of the nonlocality for wellposedness of the firstorder model. Finally, we present results of numerical simulations for both the first and secondorder model on the individualbased and continuum levels of description. This is a joint work with Martin Burger and MarieTherese Wolfram.  

Dolbeault, Jean  Seminarroom C 209  Wed, 23. Nov 11, 15:30 
“Free energies, nonlinear flows and functional inequalities”  
This lecture will be devoted to a review of results based on "entropy methods" in nonlinear diffusion equations. The basic example is the fast diffusion equation in the euclidean space and the study of the asymptotic behaviour of the solutions in selfsimilar variables. Recent results (in collaboration with G. Toscani) provide interesting refinements for the study of the asymptotic behaviour of the solutions, based on best matching asymptotic profiles rather than on selfsimilar rescalings. As a result, we obtain for instance improved Sobolev inequalities which solve an old open question raised by H. Brezis and E. Lieb. Nonlocal improvements of standard functional inequalities will also be introduced, based on duality and nonlinear flows approaches. They suggest deep links connecting mean field models like the KellerSegel system with purely local nonlinear diffusion.  

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