ABSTRACTS
Susanna Terracini (University of Turin)Spiralling and other solutions in limiting profiles of competition-diffusion systems
Reaction-diffusion systems with strong interaction terms appear in many multi-species physical problems as well as in population dynamics. The qualitative properties of the solutions and their limiting profiles in different regimes have been at the center of the community's attention in recent years. A prototypical example is the system of equations \[\left\{\begin{array}{l} -\Delta u+a_1u = b_1|u|^{p+q-2}u+cp|u|^{p-2}|v|^qu,\\ -\Delta v+a_2v = b_2|v|^{p+q-2}v+cq|u|^{p}|v|^{q-2}v \end{array} \right. \] in a domain $\Omega\subset \mathbb{R}^N$ which appears, for example, when looking for solitary wave solutions for Bose-Einstein condensates of two different hyperfine states which overlap in space. The sign of $b_i$ reflects the interaction of the particles within each single state. If $b_i$ is positive, the self interaction is attractive (focusing problems). The sign of $c$, on the other hand, reflects the interaction of particles in different states. This interaction is attractive if $c>0$ and repulsive if $c<0$. If the condensates repel, they eventually separate spatially giving rise to a free boundary. Similar phenomena occurs for many species systems. As a model problem, we consider the system of stationary equations: \[ \begin{cases} -\Delta u_i=f_i(u_i)-\beta u_i\sum_{j\neq i}g_{ij}(u_j)\;\\ u_i>0\;. \end{cases} \] The cases $g_{ij}(s)=\beta_{ij}s$ (Lotka-Volterra competitive interactions) and $g_{ij}(s)=\beta_{ij}s^2$ (gradient system for Gross-Pitaevskii energies) are of particular interest in the applications to population dynamics and theoretical physics respectively. Phase separation and has been described in the recent literature, both physical and mathematical. Relevant connections have been established with optimal partition problems involving spectral functionals. The classification of entire solutions and the geometric aspects of phase separation are of fundamental importance as well. We intend to focus on the most recent developments of the theory in connection with problems featuring anomalous diffusions, non-local and non symmetric interactions.
Alessia Nota (University of Bonn)
Homoenergetic Solutions for the Boltzmann Equation
We consider a particular class of solutions of the Boltzmann equation, known as homoenergetic solutions, which are useful to describe the dynamics of Boltzmann gases under shear, expansion or compression in nonequilibrium situations. While their well posedness theory has many similarities with the theory of homogeneous solutions of the Boltzmann equation, their long time asymptotics differs completely, due to the fact that these solutions describe far-from-equilibrium phenomena. Indeed, the long time asymptotics cannot always be described by Maxwellian distributions. For several collision kernels the asymptotics of homoenergetic solutions is given by particle distributions which do not satisfy the detailed balance condition. In this talk I will describe different possible long time asymptotics of homoenergetic solutions of the Boltzmann equation, as well as some open problems in this direction. (This is a joint work with R.D.James and J.J.L.Velázquez).
Ariane Trescases (Institut de Mathématiques de Toulouse)
Cross-diffusion, repulsion and attraction
In Population dynamics, reaction-cross diffusion systems model the evolution of populations of multiple species with (typically) repulsion between individuals of different species. For these strongly coupled nonlinear systems, a question as basic as the existence of solutions appears to be extremely complex. In this talk, we consider two-species cases where cross diffusion terms appear only in one of the two equations (triangular case). We will address questions related to the existence of solutions, a derivation from a microscopic model, and an application to chemotaxis.
Carola-Bibiane Schönlieb (University of Cambridge)
Wasserstein for learning image regularisers
In this talk I will present the results of our recent NeurIPS paper on adversarial regularisers for inverse problems which are trained with an objective function that approximates the 1-Wasserstein distance. I will discuss what we understand about this training procedure theoretically and showcase the performance of the learned regulariser for computed tomography imaging. This is joint work with Sebastian Lunz and Ozan Öktem.
Cinzia Soresina (Technical University of Munich)
On the bifurcation structure of the SKT model for competitive species
The SKT model was introduced by Shigesada-Kawasaki-Teramoto (J.Theor.Biol., 1979) to reproduce the spatial segregation of competitive species. Indeed, due to the presence of cross-diffusion terms, the system admits stable non-homogeneous steady states. For this reason, cross-diffusion terms has been considered as the key ingredient in pattern formation. Thanks to the Turing instability analysis, we can show that the presence of cross-diffusion terms not always enhances the formation of spatial patterns. Furthermore, using the continuation software pde2path, the effects of these terms on the global bifurcation structure of the model can be investigated for 1D and 2D domains, with a focus on coexisting stable steady states, as well as the convergence of the bifurcation structure of the corresponding fast-reaction system proposed by Iida, Mimura and Ninomiya (JMB 2006) to the one of the cross-diffusion model.
This is a joint work with M. Breden and C. Kuehn.
Ingrid Lacroix-Violet (University of Lille)
Energy preserving numerical methods for NLS
In this talk we will focus on the numerical time integration of the nonlinear Schrödinger equation using methods preserving the energy or a discrete analog of it. In particular we will present a linearly implicit method, called the relaxation method, for the cubic case and explain how to generalize it in order to be allow to deal with general power law nonlinearities. We will see the efficiency of the methods through numerical simulations for different physical models.
Li Chen (University of Mannheim)
Analysis on Keller-Segel Models in Chemotaxis
In this talk I will review some of our contributions in the analysis of parabolic elliptic Keller-Segel system, a typical model in chemotaxis. For the case of linear diffusion, after introducing the critical mass in two dimensions, I will show our result for blow-up conditions for higher dimension. The second part of the talk is concentrated in the critical exponent for Keller-Segel system with porous media type diffusion. In the end, motivated from the result on nonlocal Fisher-KPP equation, we show that the nonlocal reaction will also help in preventing the blow-up of the solutions.
Carina Geldhauser (Technical University of Dresden)
Space-discretizations of nonlinear PDEs and their stochastic perturbations
In this talk we will discuss two applications of a space-discrete reaction-diffusion equation: a model for interacting particles in a bistable potential, and a model for wave propagation in nerve fibres. We will present results on the continuum limit of these discrete models and the influence of stochastic perturbations.
Ewelina Zatorska (University College London)
On the Existence of Solutions to the Two-Fluids Systems
In this talk I will present the recent developments in the topic of existence of solutions to the two-fluid systems. I will discuss the application of approach developed by P.-L. Lions and E. Feireisl and explain the limitations of this technique in the context of multi-component flow models. A particular example of such a model is two-fluids Stokes system with single velocity field and two densities, and with an algebraic pressure law closure. The existence result that uses the compactness criterion introduced for the Navier-Stokes system by D. Bresch and P.-E. Jabin will be presented. I will also mention an innovative construction of solutions relying on the G. Crippa and C. DeLellis stability estimates for the transport equation.