Find out more about the Chair in Applied Analysis' research activities throughout the years.
Title: Rate of convergence towards Hartree dynamics
Speaker: Lee Jinyeop (KAIST)
Abstract: We consider interacting $N$bosons in three dimensions. It is known that the difference between the manybody Schrödinger evolution in the meanfield regime and the corresponding Hartree dynamics is of order $1/
To the investigation, we will review quantum mechanics briefly from very basic ideas to the concept of BoseEinstein condensation. Moreover, to introduce the main technique of the proof, we will also cover the basics of Fock space representation. Strichartz estimate and time decay estimate are main new analytic tools for the proof.
Date: 23 July 2019 (Tues)
Time: 10:00  11:30
Location: TBA
Title: The Global Dynamics on 1D compressible MHD
Speaker: Ronghua Pan (Geogia Institie of Technology)
Abstract: Global dynamics of classical Solutions of 1D Compressible MHD with large initial data has an interesting history and is challenging. We will report a recent Progress made by my Joint work with X. Qin.
Time: 16:0017:30, 11.07.2019
Place: B6 A303
Title : Title: Multispecies crossdiffusion population models: existence of solutions and derivation from underlying particle models
Speaker: Esther Daus (TU Wien)
Abstract : In the first part of this talk, we focus on the proof of the existence of globalintime weak solutions to reactioncrossdiffusion Systems for an arbitrary number of competing population species. In the case of linear transition rates, the model extends the twospecies population model of Shigesada, Kawasaki, and Teramoto. The existence proof is based on a refined entropy method and a new approximation scheme. Global existence follows under a detailed balance or weak crossdiffusion condition, where the detailed balance condition is related to the symmetry of the mobility matrix, which mirrors Onsager's principle in thermodynamics. The second part of the talk links at the formal level the entropy structure of the crossdiffusion system satisfying the detailed balance condition with the entropy structure of a reversible microscopic manyparticle Markov process on a discretised space. Moreover, we present a very recent proof of a rigorous meanfield limit from a stochastic particle model to a cross diffusion model. These results are based on a joint work with Xiuqing Chen and Ansgar Juengel, a Joint work with Helge Dietert and Laurent Desvillettes, and a joint work with Li Chen and Ansgar Juengel.
Time: 14:3016:00, 11.07. 2019
Place: B6 A303
Title : Random horizon principalagent problem
Speaker: Junjian Yang (TU Wien )
Abstract :We consider a general formulation of the principalagent problem with a continuous payment and a lumpsum payment on a random horizon. We ﬁrst ﬁnd the contract that is optimal among those for which the agents value process allows a dynamic programming representation, in which case the agents optimal eﬀort is straightforward to ﬁnd. We then show that the optimization over this restricted family of contracts represents no loss of generality. Using this approach, we reduced a nonzerosum stochastic diﬀerential game to a stochastic control problem which may be solved by standard methods of stochastic control theory. At the beginning of the talk, I will also give an overview about the interplay between PDE and
stochastics.
Time: 13:0014:30, 11.07. 2019
Place: B6 A303
Title : Logarithmic delays in non local Fisher KPP problems.
Speaker: Emeric Bouin (Université ParisDauphine)
Abstract : We consider the nonlocal FisherKPP equation modeling a population with individuals competing with each other for resources with a strength related to their distance, and obtain the asymptotics for the position of the invasion front starting from a localized population. Depending on the behavior of the competition kernel at infinity, the location of the front is either as in the local case, or polynomial. This is a joint work with Christopher Henderson (U. Chicago) and Lenya Ryzhik (Stanford).
Time: 10:1511:45, 13.12. 2018
Place: C116
Title: Well/
Speaker: Eduard Feireisl (Institute of Mathematics AS CR )
Abstract: We discuss the state of the art of well/
arising in dynamics of compressible viscous/
a solution to this problem based on stochastic approach selecting a
suitable Markovian semigroup.
Time: 11:0012:30
Place: B 6, A 303
Title: Effective equations for focusing bosonic systems
Speaker: Thanh Nam Phan (LMU)
Abstract: We study the effective descriptions of manybody Schroedinger dynamics of interacting bosons. To the leading order, the system exhibits the BoseEinstein condensation and the condensate is described by a onebody nonlinear Schroedinger equation. To the second order, the excitations around the condensate are effectively described by Bogoliubov theory. I will discuss some recent results on the rigorous derivation of the effective equations. A special attention will be paid on the focusing case, where the stability of the system is not obvious.
Time: 10:1511:45 27.02.2018
Place: C116, A5,6
Title: Infinite time blowup of many solutions to a general quasilinear parabolicelliptic KellerSegel system
Speaker: Johannes Lankeit (Paderborn)
Abstract: We consider a parabolicelliptic chemotaxis system generalizing
\[ \begin{cases}\begin{split}
& u_t=\nabla\cdot((u+1)^{m1}\
& 0 = \Delta v  v + u
\end{split}\end{cases} \]
in bounded smooth domains $\Omega\subset \mathbb{R}^N$, $N\ge 3$, and with homogeneous Neumann boundary conditions. We show that
*) solutions are global and bounded if ${\sigma}
*) close to given radially symmetric functions there are many initial data producing unbounded solutions if $\sigma >m\frac{N2}N$.
In particular, if ${\sigma}\le 0$ and $\sigma > m\frac{N2}N$, there are many initial data evolving into solutions that blow up after infinite time.
Time: 24.01.2018 14:0015:30
Place: A5 6, C116
Title: Global existence analysis of crossdiffusion population systems for multiple species
Speaker: Esther Daus (TU Wien)
Abstract: We prove the existence of globalintime weak solutions to reactioncrossdiffusion systems for an arbitrary number of competing population species. The equations can be derived from an onlattice randomwalk model with general transition rates. In the case of linear transition rates, it extends the twospecies population model of Shigesada, Kawasaki, and Teramoto. The existence proof is based on a refined entropy method and a new approximation scheme. Global existence follows under a detailed balance or weak crossdiffusion condition. The detailed balance condition is related to the symmetry of the mobility matrix, which mirrors Onsager's principle in thermodynamics. Under detailed balance (and without reaction), the entropy is nonincreasing in time, but counterexamples show that the entropy may increase initially if detailed balance does not hold.
This is a joint work with X. Chen and A. Juengel.
Time: 10.01.2018 9:0010:30
Place: A5 6 C116
Title: Stability of Compressible NavierStokes/
Speaker: Yuehong Feng (Beijing)
Abstract: In this talk, we give the long time decay rates and stabilities of solutions for Euler/
Time: 19.12.2017 12:3013:30
Room: A5 C116
15.12.201716.12.2017, A5 6, C012
15.12.2017 Afternoon 
Speaker 
Topic 
14:3015:20 
Merz 
On the Atomic Density on the Semiclassical Length Scale in Relativistic Quantum Mechanics 
15:2016:10 
Griesemer 
On the dynamics of polarons in the strongcoupling limit 
16:1016:40 
Coffee break 

16:4017:30 
Morozov 
FourierMellin theory of the relativistic massless Coulomb operator 
 
16.12.2017 Morning 


9:009:50 
König 
Classification of positive solutions to a 
9:5010:40 
Chen 
An inequality on the inverse of x projected to the positive spetral subspace of the free Dirac operator: with applications to the ionization problem of the BrownRavenhall operator 
10:4011:10 
Coffee break 

11:1012:00 
Liew 
HewittSavage Theorem and its application in Fermionic semiclassical measures on phase space. 
 
16.12.2017 Afternoon 


14:0014:50 
Wang 
Some PDEs with competition effects and functional inequalities 
14:5015:40 
Cuenin 
Embedded eigenvalues of generalized Schrödinger operators 
15:4016:10 
Coffee break 
Title: Understanding blood cancer dynamics  insights from mathematical modeling
Speaker: Thomas Stiehl (Institute of Applied Mathematics, Heidelberg University)
Abstract: Acute leukemias are cancerous diseases of the blood forming (hematopoietic) system. The leukemic cell bulk is derived from a small and heterogeneous population of leukemic stem cells. Upon expansion, the leukemic cells outcompete healthy blood production which results in severe clinical symptoms. To study the interaction of leukemic and healthy cells, we propose mathematical models of hierarchical cell populations. Cell competition and selection are mediated by various biologically inspired feedback mechanisms. The models relate disease dynamics to basic cell properties, such as proliferation rate (number of cell divisions per unit of time) and selfrenewal fraction (probability that a progeny of a stem cell is again a stem cell). Depending on the posed questions, we use different mathematical approaches, including nonlinear ordinary differential equations, integrodifferential equations and stochastic simulations.
A combination of mathematical analysis, computer simulations and patient data analysis provides insights in the following questions:
The talk is based on joint works with Anna MarciniakCzochra (Institute of Applied Mathematics, Heidelberg University), Anthony D. Ho, Natalia Baran and Christoph Lutz (Heidelberg University Hospital).
Time: 17.10.2017, 12:0013:30
Room: A5 C116
Title: Nonlinear Evolutionary Systems and Green's Function
Speaker: Weike Wang (Shanghai Jiao Tong University)
Abstract: In this talk, I will show how real analysis and Green’s function method are applied for pointwise estimates of nonlinear evolutionary systems. Especially, for compressible NavierStokes equations, the general Huygan’s principle is obtained. I will also combine the Green’s function method and energy method to solve the initialboundary value problem.
Time: 16.08.2017 14:3016:00
Room: B139, A5
Title：Meanfield limit for the KellerSegel system and the theory of propagation of Chaos
Speaker: Hui Huang (Tsinghua University)
Abstract：We study the propagation of chaos for the Nparticle chemotaxis system subject to the Brownian diffusion. More precisely, we present a probabilistic proof of the distance between the exact microscopic and the approximate meanfiled dynamics, which leads to a derivation of the KellerSegel equation from the microscopic Nparticle system.
Time: 17.07.2017, 14:0015:30
Room: A5 C116
Title: Fractional Laplacian and the KellerSegel system with the nonlocal diffusion.
Speaker: Hui Huang (Tsinghua University)
Abstract: In this talk, I will give a brief introduction of the fractional Laplacian and describe our work of the KellerSegel system subject to the Levy diffusion.
Time: 17.07.2017, 14:0015:30
Room: A5 C116
Title: Dissipative reaction diffusion systems with quadratic growth
Speaker: Takashi Suzuki (Osaka)
Abstract: We introduce a class of reaction diffusion systems of which weak solution exists globalintime with relatively compact orbit in L1. Reaction term in this class is quasipositive, dissipative, and up to with quadratic growth rate. If the space dimension is less than or equal to two, the solution is classical and uniformly bounded. Provided with the entropy structure, on the other hand, this weak solution is asymptotically spatially homogeneous. Joint work with Michel Pierre and Yoshio Yamada.
Time: 05.07.2017, 14:0015:30
Room: A5 C116
Title: On the regularity of the 3D NavierStokes equations
Speaker: Daoyuan Fang (School of Mathematics Sciences, Zhejiang University)
Abstract: In this talk, I will show some recent results on the 3D NavierStokes equations, which were obtained by our group during these years.
Time: 03.05.2017, 10:3012:00
Room: A5 C116
Title: Propagation of chaos for the VlasovPoisson system
Speaker: Phillip Grass (LMU)
Abstract: The Vlasov equation is used to describe the macroscopic time evolution of a system consisting of many particles interacting by newtonian dynamics. In case of singular interaction a rigorous proof justifying this approach can be very challenging and is still an open problem for the most interesting case which is coulomb interaction. The desired result is to show that for typical initial conditions the empirical density given by the positions of the particles in phase space converges to the solution of Vlasov equation. In this talk, I will introduce a method recently developed by Boers, Lazarovici and Pickl which allows to consider the coulomb case with some cutoff depending on the particle number $N$. Additionally, I will suggest some adaptions to this approach which can be helpful to shrink the size of the $N$dependent cutoff.
Time: 15.03.2017, 10:3011:30
Room: A5 C116
Title: Nonlinear aggregationdiffusions in the diffusiondominated and faircompetitions regimes
Speaker: Jose Antonio Carrillo (Imperial College, London)
Abstract: We analyse under which conditions equilibration between two competing effects, repulsion modelled by nonlinear diffusion and attraction modelled by nonlocal interaction, occurs. I will discuss several regimes that appear in aggregation diffusion problems with homogeneous kernels. I will first concentrate in the fair competition case distinguishing among porous medium like cases and fast diffusion like ones. I will discuss the main qualitative properties in terms of stationary states and minimizers of the free energies. In particular, all the porous medium cases are critical while the fast diffusion are not. In the second part, I will discuss the diffusion dominated case in which this balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrisation techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregationdiffusion equations towards this unique equilibrium profile up to translations as time tends to infinity. This talk is based on works in collaboration with S. Hittmeir, B. Volzone and Y. Yao and with V. Calvez and F. Hoffmann.
Time: 03.03.2017, 15:3016:30
Room: C116
Title: Quantitative Isoperimetric Type Inequalities and Applications
Speaker: Giovanni Pisante (University of Campania “Luigi Vanvitelli”)
The simplicity in the statement of the isoperimetric inequality, together with the subtle difficulties that its rigorous proof hid, had been a source of increasing interest for mathematicians. In the past decades several quantitative versions and many related applications have been presented, often with more than one proof. Aim of the lectures is to give an introduction to the classical isoperimetric inequality and to various stability results proved in recent years for this inequality and other related geometric and analytic inequalities.
Time: 22.02.2017, 10:1511:45
Room: A5 C116
Title: On the selection of solutions to a nonlinear pde system
Speaker: Giovanni Pisante (University of Campania “Luigi Vanvitelli”)
In the last decades a great effort has been devoted to the study of nonlinear systems of partial differential equations of implicit type. Different and quite general methods have been developed to prove the existence of almost everywhere Lipschitz regular solutions. The usual approaches are not constructive and usually, when they can be applied, provide the existence of infinitely many solutions. Thus the question of selecting a preferred solution among them raised. In the recent literature some selection criteria have been proposed to somehow minimize the irregularities of the solutions to scalar problem.
The aim of the talk is to propose a selection criterion for a particular 2dimensional vectorial problem, hoping to shade some light on the different difficulties that one can encounter when passing from the scalar to the vectorial case. We provide a variational method to select, among the infinitely many solutions, the ones that minimize an appro priate weighted measure of some set of singularities of the gradient. The talk is based on a recent joint work with G. Croce.
Time: 15.02.2017, 10:1511:45
Room: A5 C116
Title: Duality approach for some variational problems involving polyconvex integrands
Speaker: Giovanni Pisante (University of Campania “Luigi Vanvitelli”)
Duality methods have beed proved to be a very useful tool in the Theory of Calculus of Variations. The theory is however fully developed only in the convex setting. Several attempts have been made to try to generalize the method to nonconvex problems. Aim of the talk is to present an approach that has been recently proposed to identify dual problems for a class of polyconvex integral functionals and to discuss how this theory can be successfully applied to recover informations on the minimizers
Time: 08.02.2017, 10:1511:45
Room: A5 C116
Title: Infinite mass boundary conditions for Dirac operators and its relation to graphene
Speaker: Edgardo Stockmeyer (Institute of Physics, PUC, Chile)
Abstract: In this talk I will review some recent results on certain Dirac operators defined on a planar domain . In particular, we will present some basic spectral properties of these operators for different boundary conditions and will discuss its relation to graphene flakes. Moreover, for some particular boundary conditions, I will show that the corresponding operator is the limit, as M ! 1, of a Dirac operator defined on the whole plane, with a mass term of size M supported outside.
Time: 6.12.2016, 14:0015:00
Room: A5 C116
Title: On a class of cross diffusion problems arising in population dynamics
Speaker: Laurent Desvillettes (Paris)
Abstract: We are going to discuss about systems of reactioncross diffusion equations arising in population dynamics. The mechanism of cross diffusion has been introduced by Shigesada Kawasaki and Terramoto to model the trend of a species to avoid another one and thereby, possibly segregate. The equations model the evolution of individuals belonging to two species in competition, which increase their diffusion rate in order to avoid the individuals of the other (or the same) species. More precisely, we will discuss about new results on the uniqueness, wellposedness, existence and nonexistence of weak solutions to such parabolic problems and systems.
Time: 01.12.2016, 14:0015:00
Room: A5 B139
Title: Transport models of data flows in high performance computing
Speaker: Richard C. Barnard (Oak Ridge National Laboratory USA)
Abstract: Scientific computing in high performance computing environments is characterized by distributing over many processors a task which is then performed by iteratively performing subtasks on each processor/
Time: 14.11.2016, 10:0011:00
Room: A5 C116
Effective oneparticle equations for fermionic manyparticle Coulomb system: derivation and properties.
The focus of this meeting is to get people from three working groups (LMU and UniMannheim) together and to concentrate on the derivation and properties of the effective oneparticle equations for (fermionic) many particle systems. The emphasis will be on electrons that interact via Coulomb forces among each other and the nucleus. Both, the stationary and the dynamic case will be discussed. In the stationary case the emphasis will be on relativistic Coulomb systems. In the dynamical case, there are few results for Coulomb systems even in the nonrelativistic setting. Some of the known results will be presented and further development of mathematical methods which allow the treatment of Coulomb singularities will be discussed. Finally, the effective oneparticle equations both in the mesoscopic level (Vlasov type) and hydrodynamic level (EulerPoisson type) will be introduced and further discussion on the difficulties resulted from external attractive Coulomb case will be carried out.
Schedule
October 22
9:0010:30 Konstantin Merz (LMU) Die atomare Dichte auf der Skala
10:3012:00 Heinz Siedentop (LMU) Die starke Scottvermutung
14:0015:30 Peter Pickl (LMU) Mean field limits for Fermions
15:3016:15 Francis Nier (Paris 13) Mean field quantum problems as semiclassical limits: a review
16:1517:00 Sergey Morozov (LMU) Lower bound on the modulus of DiracCoulomb operators by fractional Laplacians
17:0017:30 Discussion session “On possible development of mathematical methods in dealing with additional attractive Coulomb potential”, led by Hongshuo Chen (LMU)
October 23
9:0010:30 Qitao Yin (Mannheim) On the VlasovPoisson system
10:3012:00 Li Chen (Mannheim) Solvability of the EulerPoisson system
Room: A5,6 C012
Title: Characterising pathindependence of Girsanov transform for stochastic differential equations
Speaker: Jianglun Wu (Swansea, UK)
Abstract: This talk will address a new link from stochastic differential equations (SDEs) to nonlinear parabolic PDEs. Starting from the necessary and sufficient condition of the pathindependence of the density of Girsanov transform for SDEs, we derive characterisation by nonlinear parabolic equations of BurgersKPZ type. Extensions to the case of SDEs on differential manifolds and the case od SDEs with jumps as well as to that of (infinite dimensional) SDEs on separable Hilbert spaces will be discussed. A perspective to stochastically deformed dynamical systems will be briefly considered.
Time: 27.09.2016, 14:0015:00
Room: A5 C116
Title: Pseudoparabolic equation with small perturbation
Speaker: Yang Cao (Dalian University of Technology)
Time: 20.07.2016, 9:3010:30
Room: A5 B143
Title: Acceleration in reactiondiffusion equations
Speaker: Christopher Henderson (ENS de Lyon)
Abstract: Widely used in mathematical biology, reactiondiffusion equations, and in particular the FisherKPP equation, are used to model the spreading of a population through a new environment. The earliest results showed that populations moved at a constant speed (i.e. linear in time). However, about five years ago, Hamel and Roques discovered acceleration, or superlinear in time propagation of the population, when the initial population is very spread out. Over the last few years, acceleration has been discovered in a number of other settings. In this talk, I will discuss several of these settings, with the aim of developing an intuition for what causes and what blocks acceleration. The work in this talk is joint with Emeric Bouin and Lenya Ryzhik.
Time: 03.05.2016, 14:0015:00
Room: A5 C116
Title: On existence results for finite energy weak solutions to a class of Quantum Hydrodynamic systems
Speaker: Paolo Antonelli (L'Aquila)
Abstract: Quantum Hydrodynamic models arise in the description of superfluid, BoseEinstein condensates, or in the modeling of semiconductor devices. I will discuss some existence results for finite energy weak solutions, globally in time. By using the underlying wave function dynamics and a polar factorisation technique, which circumvents the use of the WKB ansatz, we are able to set up a consistent theory in terms of mass and current densities, without the need to define the velocity field in the vacuum region. I will then present some recent progresses about models for quantum fluids with selfgenerated electromagnetic fields and I will try to discuss the open problems in this direction.
Time: 19.04.2016, 14:0015:00
Room: A5 C116
Title: Derivation and analysis of a system modeling the chemotactic movement of hematopoietic stem cells
Speaker: Maria NeussRadu (Erlangen)
Abstract: It has been shown that hematopoietic stem cells migrate in vitro and in vivo towards a gradient of a chemotactic factor produced by stroma cells. In this paper, a mathematical model for this process is presented. The model consists of chemotaxis equations coupled with an ordinary differential equation on the boundary of the domain and subjected to nonlinear boundary conditions. The existence and uniqueness of a local solution is proved and the model is simulated numerically. It turns out that for adequate parameter ranges, the qualitative behavior of the stem cells observed in the experiment is in good agreement with the numerical results.
Time: 05.04.2016, 14:0015:00
Room: A5 C116
Title: Global WellPosedness for a Model of Incompressible NavierStokes in Three Dimensions
Speaker: Shuguang Shao (Beijing)
Time: 02.03.2016, 12:3013:15
Room: A5 B139
Title: Axially Symmetric Incompressible Flow of Threedimensional Magnetohydrodynamics
Speaker: Jihui Wu (Beijing)
Time: 02.03.2016, 13:1514:00
Room: A5 B139
Title: The Solution of a Quasilinear Differential Equation with Closed Nonconvex Initial Discontinuity
Speaker: Haiping Niu (Beijing)
Time: 02.03.2016, 14:1515:00
Room: A5 B139
Title: SignChanging TwoPeak Solutions For An Elliptic Free Boundary Problem Related To Confined Plasmas
Speaker: Giovanni Pisante (Napoli)
Abstract: Motivated by the description of equilibrium states for plasmas in a tokamak, we consider a singular problem defined on a planar domain that generalizes the classical model for plasma confinement. This problem shares some structural similarities with the equation describing the desingularized solutions for the Euler equation of two point vortices with opposite signs and it is well known that in these type of problems there is a strong connection between the existence of roughly saying ``bubbling solutions'' and the critical points of the associated KirchoffRouth type functional. The aim of the talk is to show how to exploit this relation to prove, via a perturbative method, the existence of solutions with two oppositesigned sharp peaks and to establish some physically relevant qualitative properties for such solutions. These are results from a recent project in collaboration with Tonia Ricciardi (Univ. „Federico II“ in Naples).
Time: 23.02.2016, 14:0015:00
Room: A5 C116
Title: Effective one particle equations II
Speaker: Heinz Siedentop (LMU)
Time: 17.02.2016, 14:0015:00
Room: A5 C116
Title: In compressible limits
Speaker: Michael Dreher (HeriotWatt University Edinburgh)
Time: 09.12.2015, 15:0016:00
Room: B139
Title: The SelfOrganized Hydrodynamics models with density dependent velocity
Speaker: Hui Yu (Aachen)
Abstract: We propose a selforganized hydrodynamics model whose velocity depends on the local density. The analysis shows that the correlation between velocity and the local density is essential to the stability. The results are demonstrated by numerical simulations of the particle and the hydrodynamic models with several experimental functions that define the relation between density and velocity. The growth rate of the unstable modes are analysed among the particle, linearized and nonlinear hydrodynamic models.
Time: 26.11.2015, 14:0015:00
Room: B6 A302
Title of talk: Discrete Beckner Inequalities via a BochnerBakryEmery Method for Markov Chains
Speaker: Wen Yue (TU Wien)
Abstract: Beckner inequalities, which interpolate between the logarithmic Sobolev Inequalities and Poincare inequalities, are derived in the context of Markov chains. The proof is based on the BakryEmery method and the use of discrete Bochnertype inequalities. We apply our result to several Markov chains including BirthDeath process, ZeroRange process, BernoulliLaplace models and Random Transposition models and thus get the exponential convergence rates of the ''distributions'' of these Markov chains to their invariant measures.
Time: 12.11.2015, 14:0015:00
Room: B6 A302
Title: Derivation of the Vlasov Equation
Speaker: Peter Pickl (LMU)
Time: 02.11.2015, 9:3010:30
Room: TBA
Title: On the diffusion limit of the Boltzmann system for gaseous mixtures.
Speaker: Francesco Salvarani (University of Pavia, Italy and Université ParisDauphine, France)
Abstract: In this talk, we consider the nonreactive elastic Boltzmann equation for monatomic gaseous mixtures, and we analyse its behaviour under the standard di ffusive scaling. In particular, we point out the relationships between the cross sections of the model and the diffusion coefficients, emphasizing the differences with respect to the monospecies case.
Time: 29.10.2015, 14:0015:00
Room: B6 A302
Title: Finite time versus infinite time blowup for a fully parabolic KellerSegel system
Speaker: Christian Stinner (TU Kaiserslautern)
Abstract: Several variants of the KellerSegel model are used in mathematical biology to describe the evolution of cell populations due to both diffusion and chemotactic movement. In particular, the emergence of cell aggregation is related to blowup of the solution. Critical nonlinearities with respect to the occurrence of blowup had been identified for a quasilinear parabolicparabolic KellerSegel system, but it was not known whether the solution blows up in finite or infinite time. We show that indeed both blowup types appear and that the growth of the chemotactic sensitivity function is essential to distinguish between them. We provide conditions for the existence of each blowup type and discuss their optimality. An important ingredient of our proof is a detailed analysis of the Liapunov functional. This is a joint work with T. Cie´slak (Warsaw).
Time: 17.09.2015, 14:0015:00
Room: B6 A302
Title: Effective one particle equations I
Speaker: Heinz Siedentop (LMU)
Time: 14.09.2015, 9:3011:00
Room: A5 C116
Title: Hypocoercivity for a linearized multispecies Boltzmann system
Speaker: Esther Daus (TU Wien)
Abstract: A new coercivity estimate on the spectral gap of the linearized Boltzmann collision operator for multiple species is proved. The assumptions on the collision kernels include hard and Maxwellian potentials under Grad's angular cutoff condition. Two proofs are given: a nonconstructive one, based on the decomposition of the collision operator into a compact and a coercive part, and a constructive one, which exploits the ``crosseffects'' coming from collisions between different species and which yields explicit constants. Furthermore, the essential spectra of the linearized collision operator and the linearized Boltzmann operator are calculated. Based on the spectralgap estimate, the exponential convergence towards global equilibrium with explicit rate is shown for solutions to the linearized multispecies Boltzmann system onthe torus. The convergence is achieved by the interplay between the dissipative collision operator and the conservative transport operator and is proved by using the hypocoercivity method of Mouhot and Neumann.
Time: 21.07.2015, 14:0015:00
Room: C012
Title: Relaxation für parabolische Systeme gemischter Ordnung
Speaker: Michael Dreher (HeriotWatt University Edinburgh)
Abstract: Motiviert durch numerische Verfahren betrachten wir die Approximation von parabolischen Problemen gemischter Ordnung, wie sie zum Beispiel bei der Beschreibung von fluiddynamischen Systemen mit Kapillaritäts oder Quanteneffekten auftreten, durch relaxierte Systeme einheitlicher Ordnung.
Time: 09.06.2015, 14:0015:00
Room: B143
Title: Blowup mechanism of 2D SmoluchowskiPoisson equation: in infinite time quantization and finite time simplicity
Speaker: T. Suzuki (Osaka)
Abstract: The SmoluchowskiPoisson equation was formulated by Sire and Chavanis in the study of Brownian particles, where the variational structure matches the scaling invariance. We show that the blowup in infinite time occurs only when the initial total mass is so quantized as 8\pi times integer. Thus we meet the quantized blowup mechanism of this model in three phases; stationary, finite time, and infinite time. From the argument, simplicity of the collapse formed in finite time is proven. The KantorovichRubinstein metric takes a role in accordance with the improved TrudingerMoser inequality.
Time: 05.05.2015, 14:0015:00
Room: C116
Title: Lowdimensional structures of stochastic partial Adaptive methods for exploring differential equations
Speaker: Zhiwen Zhang (UCLA)
Time: 25.03.2015 14:0015:00
Room: B139
The main topic of this miniworkshop 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:
1:302:20 Ansgar Juengel (TU Wien)
2:203:10 Klemens Fellner (University Graz)
3:103:50 Coffee break
3:504:40 Laurent Desvillettes (CNRS, Paris)
4:405:30 Michael Winkler (University Paderborn)
Room: A5,6 012
Speaker: Ansgar Juengel
Title:The boundednessbyentropy principle for crossdiffusion systems from biology
Abstract: Many systems of collective behavior for multiple species can be described in the continuum limit by crossdiffusion 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 crossdiffusion systems, these difficulties can be overcome by exploiting a formal gradientflow 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 boundednessbyentropy principle. We detail the theory for several examples coming from tumorgrowth modeling, population dynamics, and multicomponent gas dynamics. The existence of global weak solutions and their longtime behavior is investigated and some numerical examples are presented.
Speaker: Klemens Fellner
Title: On Systems of ReactionDiffusion Equations: Global Existence and Large Time Analysis.
Abstract: Starting with a volumesurface reactiondiffusion modelling asymmetric protein localisation on stem cells, we shall discuss questions of global existence, regularity and convergence to equilibrium for systems of reactiondiffusion 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 reactiondiffusion equations. Moreover, we shall point out recent considerations concerning the availability of entropy functional in the case of socalled 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 chemotaxisfluid interaction
Abstract: We consider models for the spatiotemporal 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 buoyancyinduced 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.
Title: Selforganized Hydrodynamics in an Annular Domain: Modal Analysis and Nonlinear Effects
Speaker: Hui Yu (Imperial College)
Abstract: The SelfOrganized Hydrodynamics model of collective behavior is studied on an annular domain. A modal analysis of the linearized model around a perfectly polarized steadystate is conducted. It shows that the model has only pure imaginary modes in countable number and is hence stable. Numerical computations of the loworder modes are provided. The fully nonlinear model is numerically solved and nonlinear modecoupling 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
Title: Local existence of smooth solutions to an nonsymmetric hyperbolic system (Continued)
Speaker: Jing Wang
Time: 20.10.2014, Mo. 11:00
Place: A5 B139
Title: Local existence of smooth solutions to an nonsymmetric hyperbolic system
Speaker: Jing Wang
Time: 16.10.2014, Do. 15:00
Place: A5 B139
Title: Existence and Blowup of some parabolic problems/ LotkaVolterra 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 $\lambda>\lambda^*$, blowsup in finite time $t^*$ for any $u_0\geq0$.
A nonlocal Filtration equation is considered, with some boundary and initial data $u_0$. It is proved that solutions blowup 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 growup of critical solutions $u^*=u(x,t;\lambda^*)$ ($u^*(x,t)\rightarrow\infty$, as $t\rightarrow\infty$ for all $x\in(1,1)$).
A preypredator system associated with the classical LotkaVolterra 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
Title: Lifschitz tails on the Bethe lattice
Speaker: Francisco HoeckerEscuti
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
Titel: Die Csisz{\'a}rKullbackUngleichung und Anwendungen
Speaker: Michael Dreher (UK)
Abstrakt: Die Csisz{\'a}rKullbackUngleichung 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.
Title: Derivation of Mean field Dynamics for Fermions
Speaker: Soeren Petrat (LMU)
Abstract: The talk is about the derivation of the timedependent Hartree(Fock) equations as an e ective dynamics for fermionic manyparticle 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
Title: A note on AubinLionsDubinskii lemmas
Speaker: Xiuqing Chen (Peking)
Abstract: Strong compactness results for families of functions in seminormed nonnegative cones in the spirit of the AubinLionsDubinskii 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 porousmedium or fastdiffusion type equations. This is a joint work with Juengel (Wien) and Liu (Duke)
Time: 26.03, 2pm
Place: A5 C115
Title: Meanfield 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 timedependent Schroedinger equation. Therefore one is interested in deriving effective evolution equations approximating the Schroedinger equation. We consider the meanfield regime (high density and weak interaction) for fermionic systems and derive the timedependent HartreeFock 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