Invited Speakers: Titles and Abstracts
Anton Arnold
Technische Universität Wien (Austria)
Entropy methods for the relativistic Fokker-Planck equation
This talk is concerned with the (linear) relativistic Fokker-Planck equation in whole space with confining potential. In this setting the unique steady state is given explicitly as a Maxwell-Jüttner distribution. Using entropy methods we shall prove exponential convergence to equilibrium for initial data in both a weighted L^2 space and weighted H^1 space. The first part is based on hypocoercivity methods due to Dolbeault-Mouhot-Schmeiser, but the relativistic transport term leads to many difficulties compared to classical Fokker-Planck equations. Moreover, we establish the instantaneous regularization (with a rate) from a weighted L^2 space into a weighted H^1 space. This proof uses an energy method with a functional that depends on time.
References: A. Arnold, G. Toshpulatov: Trend to equilibrium and hypoelliptic regularity for the relativistic Fokker-Planck equation, SIAM J. Math. Analysis 57, No. 3 (2025).
Alessandro Biancalani
ESILV (France)
Numerical simulations of turbulence and energetic particles in tokamak plasmas
Microinstabilities grow in tokamak plasmas, due to gradients of density and temperature. Their nonlinear interaction forms turbulence. Zonal (i.e. axisymmetric) structures (ZS) have an important role in the turbulence saturation. Similarly to the horizontal bands of Jupiter, zonal structures in tokamak plasmas are nonlinearly generated by turbulence and therefore they absorb part of its energy but also feed-back on turbulence. Fusion reactions and heating mechanisms generate a population of energetic, i.e. suprathermal particles (EP). The interaction of EPs and turbulence is being considered as a key issue to investigate in order to achieve a comprehensive model of transport. Addressing such an intrinsically multiscale kinetic problem requires a global gyrokinetic code. In this work, we investigate the nonlinear interaction of turbulence and EPs, and the role of ZSs. Simulations with the global particle-in-cell code ORB5 are discussed. We find that ZS forced-driven by Alfvén instabilities substantially decrease the growth rate of microturbulence modes. Alfvén instabilities can also decrease the temperature gradients, thus affecting the turbulence intensity.
Emanuele Caglioti
Sapienza Università di Roma (Italy)
Random matching in densities with unbounded support: some recent results
I will consider the 2-dimensional random matching problem in two dimensional sets. In a challenging paper, Caracciolo et. al., on the basis of a subtle linearization of the Monge Ampere equation, conjectured that the expected value of the square of the Wasserstein distance, with exponent 2, between two samples of N uniformly distributed points in the unit square is logN / 2πN plus corrections.This and other related conjectures has been proved by Ambrosio et al. in a series of challenging papers. In the talk I will review the results cited above and then I will focus on the case of radial densities on unbounded domains, e.g. the Gaussian, in generic dimension and exponent p.
Martin Campos Pinto
Max-Planck-Institut für Plasmaphysik, Garching (Germany)
Variational particle schemes for Vlasov-Maxwell equations on singular domains
In this talk I will present a discrete action principle for the Vlasov-Maxwell equations that applies to particle approximations coupled with structure-preserving finite elements. One practical case of interest involves domains with a polar singularity, discretized using tensor-product splines.
Guido Cavallaro
Sapienza Università di Roma (Italy)
Time evolution of concentrated vortex rings
The motion of fluids can sometimes be effectively modeled by a finite-dimensional dynamical system. A paradigmatic example is the point vortex model, introduced by Helmholtz in a seminal paper, which describes a regime where the vorticity of an ideal, incompressible fluid is confined to a set of infinitely thin, straight, parallel vortex filaments. In this talk, I will present recent results concerning an incompressible, inviscid fluid with axial symmetry and no swirl, establishing a connection between the motion of a system of N concentrated vortex rings with large radii and a dynamical system closely related to the point vortex model. Based on joint work with Paolo Buttà and Carlo Marchioro.
Li Chen
Universität Mannheim (Germany)
Mean-Field Derivation of Vlasov type equation (from classical and quantum mechanics)
In this talk, I will present the derivation of Vlasov-type equations from both classical interacting particle systems and the fermionic many-particle Schrödinger equation. For the classical case, we establish convergence to the Vlasov dynamics with Coulomb interaction, obtaining strong L^1 convergence via a combination of convergence in probability and relative entropy methods. For the quantum case, we prove, for smooth interactions, the weak convergence of the Husimi measure under the joint semiclassical and mean-field limit in the fermionic scaling. This talk is based on joint work with J. Jung, P. Pickl, Z. Wang, J. Lee, Y. Li, and M. Liew.
Anaïs Crestetto
Université de Nantes (France)
Multiscale numerical schemes for collisional Vlasov equation
The numerical approximation of collisional kinetic equations can be challenging due to multiscale effects combined with high dimensionality. In this talk, I will present some strategies to obtain efficient Asymptotic-Preserving schemes for collisional Vlasov-type equations.
Yu Deng
University of Chicago (USA)
Long time derivation of Boltzmann equation from hard sphere dynamics
We present recent works with Zaher Hani and Xiao Ma, in which we derive the Boltzmann equation from the hard sphere dynamics in the Boltzmann-Grad limit, for the full time range in which the (strong) solution to the Boltzmann equation exists. This is done in the Euclidean setting in any dimension $d\geq 2$, and in the periodic setting in dimensions $d\in\{2,3\}$. As a corollary, we also derive the corresponding fluid equations from the the hard sphere dynamics. This executes the original program , proposed in Hilbert's Sixth Problem in 1900, pertaining to the derivation of hydrodynamic equations from colliding particle systems, via the Boltzmann equation as the intermediate step.
Bruno Desprès
Sorbonne Université (France)
On sprays, linear Landau damping and more
The mathematical well-posedness of thick sprays (neutral particles) coupled with a compressible fluid is not clear, neither for strong solutions nor for weak solutions. In the linear regime, thick sprays can exhibit a linear Landau damping effect, as for plasma models for charged particles. I will review some results obtained recently, and I will detail a extension for a general system of conservation coupled with a kinetic equation.
François Golse
Ecole Polytechnique (France)
Numerical Analysis of Quantum Dynamics in the Semiclassical Regime (joint work with F. Filbet)
We propose a new approach to discretize the quantum Liouville, or von Neumann equation, which is efficient in the semi-classical limit. This method is first based on using the variables that appear in Weyl’s quantization, which successfully address the stiffness associated with the equation. Numerical simulations illustrating the benefits of this approach use a truncated Hermite expansion of the density operator in one of the Weyl variables and a finite volume approximation in the other. This asymptotic preserving numerical approximation provides a useful tool for solving the von Neumann equation in all regimes, near classical or not.
Megan Griffin-Pickering
University of Zurich (Switzerland)
Particle Approximations for the Vlasov-Poisson system for ions
The ionic Vlasov-Poisson system is a kinetic model for the ions in a dilute plasma. Compared to the well-known electron Vlasov-Poisson system, the ionic model features an additional exponential nonlinearity in the equation for the electrostatic potential, which creates several additional mathematical difficulties. The equation arises formally as the mean field limit from an underlying microscopic system representing individual ions interacting with a thermalized electron distribution. However, it remains an open problem to prove rigorously that the mean field limit holds. I will discuss recent progress on the derivation of kinetic models for ionic plasma from ‘regularized’ microscopic systems, including a recent result in the style of the probabilistic approach developed by Lazarovici and Pickl for the electron Vlasov-Poisson system. The method developed allows interactions with nonlinear dependence on the particle density to be handled.
Maria Pia Gualdani
University of Texas at Austin (USA)
The Fuzzy Landau Equation
The Landau equation, introduced by Lev Landau in 1936, is a modification of the Boltzmann equation to specific applications in plasma physics, and describes the interactions and collisions among charged particles in a plasma. The mathematical investigation of the Landau equation has been active for several decades, with researchers exploring various aspects of its behavior and properties. While the homogeneous version of this equation has been fully understood, the inhomogeneous equation remains a very difficult problem. In this talk I will present the first global well-posedness theory for a fuzzy version of the inhomogeneous Landau equation. Fuzzy equations allow delocalized collision. We will see that this delocalization offers a wider range of analytical tools and provides additional structure that can be exploit to show smoothness and global existence. I will also show examples of new Lyapunov functions of Fisher type.
Daniel Han-Kwan
CNRS, Université de Nantes (France)
Semiclassical limit of the cubic nonlinear Schrödinger equation for mixed states
We study the semiclassical limit of cubic NLS for mixed states. We justify the limit to a singular Vlasov equation (namely Vlasov-Benney in the defocusing case), for data with finite Sobolev regularity that satisfy a Penrose stability condition. Joint work with F. Rousset.
Mikaela Iacobelli (cancelled)
ETH Zürich (Switzerland)
Kinetic Models, Stability, and Quasineutral Limits in Plasma Dynamics
I will begin with a short introduction to kinetic models for collisionless plasmas, focusing on Vlasov-type systems (Vlasov-Poisson and the variant with thermalized/massless electrons). I will then turn to the quasineutral regime. For Vlasov-Poisson and for the model with massless electrons, I will present almost-optimal stability of the quasineutral limit obtained using the kinetic Wasserstein distance, together with refined control of the exponential Poisson coupling. In the magnetized setting, I will describe a new result establishing the quasineutral limit from the relativistic Vlasov-Maxwell system to electron-MHD within an analytic-regularity framework; the analysis yields uniform-in-ε estimates and strong (filtered) convergence to the limiting dynamics.
Ansgar Jüngel
Technische Universität Wien (Austria)
Charged particle transport in memristor devices for brain-inspired neuromorphic computing
More than 50 years ago, Moore predicted that the number of transistors on a microchip doubles every two years. This exponential growth is approaching its physical limit, highlighting the need for alternative computing paradigms. One promising avenue is neuromorphic computing, which aims to emulate the structure and function of the human brain. A key enabling technology is the memristor, a nonlinear resistor with memory. Memristors are capable of mimicking the dynamic conductance behavior of biological synapses, making them well-suited for implementing energy-efficient neural networks.
This talk focuses on the mathematical analysis of three-species drift-diffusion equations for memristors. We investigate the existence and boundedness of global-in-time weak solutions. The mathematical difficulties originate from the three-species situation and the different types of boundary conditions. These issues are addressed by combining free energy estimates with local and global compactness arguments. Additionally, we analyze memristor models coupled with electrical networks. One-dimensional numerical simulations capture the characteristic hysteresis behavior in the current-voltage curves, which are a fingerprint for memristive devices.
Giovanni Manfredi
CNRS Strasbourg (France)
Density functional theory for classical collisionless plasmas – equivalence of fluid and kinetic approaches
Density functional theory (DFT) is a powerful theoretical tool widely used in such diverse fields as computational condensed-matter physics, atomic physics and quantum chemistry. DFT establishes that a system of N interacting electrons can be described uniquely by its single-particle density, instead of the N-body wave function, yielding an enormous gain in terms of computational speed and memory storage space. Here, we use time-dependent DFT to show that a classical collisionless plasma can always, in principle, be described by a set of fluid equations for the single-particle density and current. The results of DFT guarantee that an exact closure relation, fully reproducing the Vlasov dynamics, necessarily exists, although it may be complicated (non-local in space and time, for instance) and difficult to obtain in practice. This goes against the common wisdom in plasma physics that the Vlasov and fluid descriptions are mutually incompatible, with the latter inevitably missing some \"purely kinetic\" effects.
Reference: G. Manfredi, “Density functional theory for collisionless plasmas – equivalence of fluid and kinetic approaches,” Journal of Plasma Physics, vol. 86, no. 2, p. 825860201, 2020. doi:10.1017/S0022377820000240
Evelyne Miot
Université Grenoble Alpes (France)
On the dynamics of point vortices and vortex filaments in two fluid models
This talk will be based on joint works, in collaboration with Martin Donati, Lars Eric Hientzsch and Christophe Lacave. We study the asymptotic evolution of sharply concentrated vorticities in two models for incompressible inviscid fluids: the lake equation on the one hand, and the three- dimensional Euler equations in helical symmetry without swirl on the other hand. Under quite general concentration assumptions on the initial data, we prove that the vorticity remains concentrated to point vortices (in the lake equations) or to helical vortex filaments (for the Euler equation with helical symmetry) for all times. We derive the asymptotic motion law of vortices: the trajectories follow the level lines of the depth of the lake in the first case, while the filaments are translating and rotating helices in the second case. In both cases we show that the localization is weak in the direction of the vortex motion but strong in its normal direction, and that it holds on an arbitrary long time interval in the naturally rescaled time scale. It turns out that the second model admits a two-dimensional reduction, which enables to reproduce the methods developed for the lake equations. In particular we derive in both situations a new explicit formula for the singular part of the Biot-Savart kernel, which allows us to obtain an appropriate decomposition of the velocity field.
Omar Morandi
Università degli Studi di Firenze (Italy)
Localization limit and optimal control problem applied to quantum particles with spin
Systems characterized by strong spin-orbit coupling are ideal candidates for spintronics as information is encoded in the electron spin degrees of freedom, whose orientation can be manipulated electrically. Optimal control procedures play a prominent role in designing and engineering spintronic devices performing specific tasks such as, for example, generating continuous or modulated current of spin or modifying the spin orientation of localized or moving charges. We investigate the well-posedness of the mathematical formulation of an optimization problem where the evolution of a quantum gas with spin is controlled in the presence of Zeeman-Rashba interaction, with particular emphasis on the classical limit of such an optimal control problem. Our optimal control procedure aims to design a set of time-dependent parameters modulating the total electric potential applied to the system achieving two objectives: maximize the probability of finding the particle at the final time close to a given phase-space coordinate, and the spin direction aligned to a certain direction. At the same time, the control should be implemented by using the minimum possible energy. This is achieved by formulating the optimal control problem in terms of a Lagrangian constrained minimization problem. We discuss the study of the classical limit of the optimal control problem, in which the quantum dynamics degenerates to a deterministic classical transport described by a single trajectory in the phase-space and a spin vector rotating around to the total magnetic field given by the sum of the external and the Rashba effective field (localization limit). We set a limit procedure in which the initial state of the quantum gas is described by a coherent state with prescribed scaling with respect to the Planck constant so that the associated Wigner distribution admits limit as measure in the phase-space, which can be easily characterized by Dirac's delta.
Claudia Negulescu
Université de Toulouse (France)
Spectral scheme for an energetic Fokker-Planck equation with kappa-distribution steady states
The concern of the present talk is the presentation of two efficient numerical schemes for a specific Fokker-Planck equation describing the dynamics of energetic particles occurring in thermonuclear fusion plasmas (runaway electrons for example). In the long-time limit, the velocity distribution function of these particles tends towards a (thermal) non-equilibrium κ-distribution function which is a steady-state of the considered Fokker-Planck equation. These κ-distribution functions have the particularity of being only algebraically decaying for large velocities, thus describing very well suprathermal particle populations. The aim is hence to present two efficient spectral methods for the simulation of such energetic particle dynamics. The first method will be based on rational Chebychev basis functions, rather than on Hermite basis sets which are the basis of choice for Maxwellian steady states. The second method is based on a different polynomial basis set, constructed via the Gram-Schmidt orthogonalisation process. These two new spectral schemes, specifically adapted to the here considered physical context, shall permit to cope with the long-time asymptotics without too much numerical costs. This is a joint work with Hugo Parada.
Gianluca Panati
Sapienza Università di Roma (Italy)
Transport, topology, and localization in Chern insulators and Quantum (Spin) Hall systems
The understanding of transport properties of quantum systems out of equilibrium is a crucial challenge in Mathematical Physics. A long term goal is to explain the conductivity properties of solids starting from first principles, as e. g. from the Schrödinger equation governing the dynamics of electrons and ionic cores. While the general goal appears to be still far, we are in position to describe the transport properties of independent electrons in a periodic background, possibly including a periodic magnetic field, as e. g. in Chern insulators and in Quantum Hall systems. We proved the existence of a strict relation between non-vanishing Hall conductivity, non-trivial topology of the Bloch bundle corresponding to the Fermi projector, and delocalization of the electronic states, measured in terms of (de-)localization of Wannier bases. If time permits, I will describe as the previous picture generalizes to spin transport (Quantum Spin Hall insulators) and to some non-periodic systems. The talk is based on joint papers with D. Monaco, A. Pisante, S. Teufel for the first part, and with G. Marcelli and M. Moscolari, and V. Rossi for the second part.
Thierry Paul
CNRS LYSM Roma (Italy)
Mean field for multiple/infinite-wise interactions and Hewitt-Savage theorem
We consider multi-agent systems; i.e. systems of N non-indistinguishable particles, at microscopic, mesoscopic/kinetic and macroscopic/hydrodynamic scales. The agents are subject to multiple-wise interactions, i.e. when each particle interacts with m other ones at the same time. We derive the associated Vlasov equation and prove propagation of chaos. We then consider the hydrodynamic limit for monokinetic solution and derive the corresponding Euler equation. The precise estimates in N and m of the rate of convergence allows to consider the joint (conditional) limit of diverging N and m towards a new type of macroscopic equation involving a vector field derived out of the Hewit-Savage theorem and an unpublished result by Pierre-Louis Lions.
Vittorio Romano
Università di Catania (Italy)
Abstract
Fulvio Zonca
CNPS and ENEA, C.R. Frascati (Italy)
The Dyson-Schrödinger Model: a paradigm for predicting fusion reactor performance
(F. Zonca, L. Chen, M.V. Falessi and Z. Qiu) The role of energetic particles (EPs) in fusion plasmas is unique as they could act as mediators of cross-scale couplings. Energetic particle driven shear Alfvén waves (SAWs), on one hand, provide a nonlinear feedback onto the macro-scale system via the interplay of plasma equilibrium and fusion reactivity profiles. Meanwhile, EP-driven instabilities could also excite singular radial mode structures at SAW continuum resonances, which, by mode conversion, yield microscopic fluctuations that may propagate and be absorbed elsewhere, inducing nonlocal behaviors that require a global analysis. Energetic particle transport must be described in phase space because of the underlying kinetic nature of wave-particle interactions and fluctuation excitations. The proper structures to describe such transport processes are phase space zonal structures (PSZS), which self-consistently evolve in time and space according to a Dyson-like equation.
Energetic particles, furthermore, may linearly and nonlinearly excite zonal electromagnetic fields (ZFs), acting, thereby, as generators of nonlinear equilibria, or zonal states (ZS) that generally evolve on the same time scale of the underlying fluctuations. These issues are presented within a general theoretical framework. In particular, we present the nonlinear Schrödinger-like envelope equations that are needed to solve for the self-consistent evolution of the SAW fluctuation spectrum driven by EPs and the Dyson-like PSZS transport equations, which determine the renormalized response of EPs including fluctuation induced transport.
Young Researchers' Session Speakers: Titles and Abstracts
Stefan Egger
Technische Universität Wien (Austria)
Compatibility of hypocoercivity and hypocontractivity with the midpoint method
We consider linear, dissipative ODE systems with values in (possibly infinite-dimensional) Hilbert spaces. A natural question in this context is which properties of these systems are preserved when transforming them to linear recurrences using discretization methods. It is well-established that the midpoint method maintains exponential stability and transforms dissipative ODE systems into semi-contractive linear recurrences. We extend those results showing that the recently established notions of hypocoercivity and hypocontractivity index are also preserved. Furthermore, those quantities are directly related to the short-time decay of the corresponding solution maps and we expect a considerably better convergence rate when the step size tends to zero, in comparison with results from the standard theory of numerical analysis. This is joint work with Anton Arnold (TU Wien, Vienna, Austria), Volker Mehrmann (TU Berlin, Berlin, Germany) and Eduard Nigsch (TU Wien, Vienna, Austria).
Mariia Filipkovska
Technische Universität Wien (Austria)
Analysis of partial differential-algebraic equations of electrical networks containing memristors
A memristor is a promising candidate for applications in high-density computer memories and for building neurons and synapses in neuromorphic computing. The memristor can be modeled by drift–diffusion equations for the densities of electrons, holes and oxygen vacancies, coupled to the Poisson equation for an electric potential. This PDE system coupled with differential-algebraic equations (DAEs) describing an electrical network forms the system of partial differential-algebraic equations (PDAEs) which will be considered in this talk. Conditions for the global solvability of the PDAEs will be provided and the behavior of solutions with increasing time will be discussed.
In addition, similar results will be presented for abstract DAEs, to which the PDAEs describing electrical networks coupled to meristors, as well as other PDEs and PDAEs arising in practical problems, can be reduced.
Maria Heitzinger
Technische Universität Wien (Austria)
Weak-strong uniqueness for cross-diffusion systems with volume filling
We prove the weak–strong uniqueness property of solutions to a general class of cross-diffusion systems with volume filling constraints. The theoretical treatment of these systems crucially relies on the underlying entropy structure which we assume to be given by the Blotzmann-Shannon entropy. In addition to this entropy structure we assume certain continuity and non-degeneracy properties of the arising matrices. By exploiting the inherent boundedness provided by the volume filling constraints, we are able to show relative entropy estimates. These prove the weak–strong uniqueness. We showcase the applicability with selected examples.
Peter Hirvonen
Technische Universität Wien (Austria)
A Power-Nonlinearity Busenberg-Travis System
Cross-diffusion equations are used to model a wide range of naturally occurring phenomena, including population dynamics or gas mixtures. A well-known example is the Busenberg-Travis system, which has been studied in various forms. Due to the complexity of the interactions between species, the mathematical treatment of these equations is a challenging task. In this talk, the Busenberg-Travis system and already studied modifications are introduced. The main focus lies on a new variation, including a power-nonlinearity pressure. Existence results are shown and mathematical difficulties for different parameter regimes are discussed.
Noah Geltner
Technische Universität Wien (Austria)
Analysis of a Multiphase Local Sensing Chemotaxis Model with Volume-Filling Constraints
We analyze the existence, uniqueness and long-term behaviour of a system describing the movement of multiple cell species guided by chemical signals. These signals are produced by the cells themselves and obey diffusion equations. Due to the local sensing mechanism involving general nonlinear sensitivity coefficients, the cells are attracted or repulsed by these chemicals. The multiphase model includes volume filling effects, preventing blow-up. The proofs are based on the boundedness-by-entropy method. Furthermore, we prove weak-strong uniqueness using a relative entropy estimate. For small chemotactic effects, we can also show the convergence to a constant steady state.
Tuấn Tùng Nguyễn
Technische Universität Wien (Austria)
Existence analysis of a three-species memristor drift-diffusion system coupled to electric networks
The existence of global weak solutions to a partial-differential-algebraic system is proved. The system consists of the drift-diffusion equations for the electron, hole, and oxide vacancy densities in a memristor device, the Poisson equation for the electric potential, and the differential-algebraic equations for an electric network. The memristor device is modeled by a two-dimensional bounded domain, and mixed Dirichlet-Neumann boundary conditions for the electron and hole densities as well as the potential are imposed. The coupling is realized via the total current through the memristor terminal and the network node potentials at the terminals. The network equations are decomposed in a differential and an algebraic part. The existence proof is based on the Leray-Schauder fixed-point theorem, a priori estimates coming from the free energy inequality, and a logarithmic-type Gagliardo-Nirenberg inequality. It is shown, under suitable assumptions, that the solutions are bounded and strictly positive.
Flora Philipp
Technische Universität Wien (Austria)
Global Existence of Weak Solutions to Navier-Stokes-Korteweg systems
We investigate the global existence of weak solutions to Navier-Stokes-Korteweg systems with drag forces. In particular, we assume a density-dependent viscosity of the form ν ∇·(ρ ∇v) and Korteweg terms κ ρ ∇(ψ’(ρ) Δ ψ(ρ)) where ψ’(ρ) = ρ^(α/2) with α in [-1, 0]. This parameter range interpolates between the quantum Navier–Stokes case (α = -1) and the classical capillarity case (α = 0), thus establishing existence results for a broader class of capillarity-viscosity couplings. The analysis is based on the use of a free energy and the BD-entropy, and exploits additional regularity properties derived via the systematic integration-by-parts technique introduced by Jüngel and Matthes.
Annamaria Pollino
Technische Universität Wien (Austria)
An ES-BGK model for polyatomic gases
An ellipsoidal statistical BGK model for polyatomic gases is derived in a general framework, covering also the non polytropic case. The distribution function of the polyatomic gas depends on its velocity and a general internal energy function, which can be discrete or continuous. We solve the minimization problem of Boltzmann entropy under suitable moment constraints and determine the transport coefficients (Prandtl number and first and second viscosity) in the hydrodynamic limit.
Stefano Rossi
ETH Zürich (Switzerland)
Long-time Behavior of Solutions to the Screened Vlasov-Poisson Equation
I will present a recent result concerning the long-time behavior of small-data solutions around vacuum for the screened Vlasov equation in the d-dimensional Euclidean space, with a particular focus on the low-dimensional case d≤3. This result explores how the asymptotic properties of the solutions strongly depend on the spatial dimension d and the corresponding decay rate of the electric field. In particular, in two dimensions, I will present a result on the asymptotic behavior of solutions with finite regularity, while in one dimension, for data with analytic regularity, I will describe their long-time behavior on time scales that depend on the size of the initial data. This is based on a joint work with M. Iacobelli and K. Widmayer.
Artur Stephan
Technische Universität Wien (Austria)
Derivation of the fourth order DLSS equation with nonlinear mobility via chemical reactions
We provide a derivation of the fourth-order DLSS equation based on an interpretation as a chemical reaction network. We consider on the discretized circle the rate equation for the process where pairs of particles sitting on the same site jump simultaneously to the two
neighboring sites, and the reverse jump where a pair of particles sitting on a common site jump simultaneously to the site in the middle.
Depending on the reaction rates, in the vanishing mesh size limit we obtain either the classical DLSS equation or a variant with nonlinear mobility of power type. We identify the limiting gradient structure to be driven by entropy with respect to a generalization of diffusive
transport with nonlinear mobility via evolutionary convergence for gradient systems. Furthermore, the DLSS equation with nonlinear mobility of the power type shares qualitative similarities with the fast diffusion and porous medium equations, since we find traveling wave
solutions with algebraic tails and polynomial compact support, respectively.
The talk is based on joint work with Alexander Mielke (Berlin) and André
Schlichting (Ulm).
Sara Xhahysa
Technische Universität Wien (Austria)
Multiphase cross-diffusion models with entropy structure for tissue growth
We introduce a class of cross-diffusion systems for tissue growth, based on the framework proposed by G. Lemon et al. The model possesses a formal entropy structure if the matrix of intraphase pressure coefficients is symmetric positive definite; this yields the a priori estimates required by the boundedness-by-entropy method, thereby establishing the existence of weak solutions. The theoretical analysis is complemented by biologically motivated case studies and numerical simulations including the case where the intraphase pressure matrix is no longer symmetric positive definite.