In this talk we present some PDE models which describe the activity of neural networks by means of the membrane potential. We focus on models based on nonlinear PDEs of Fokker-Planck type. We study the wide range of phenomena that appear in this kind of models: blow-up, asynchronous/synchronous solutions, instability/stability of the steady states ...

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

Fractional kinetics is derived from Gaussian processes when the medium where the diffusion takes place is characterized by a population of length-scales [1]. This approach is analogous to the generalized grey Brownian motion [2], and it can be used for modeling anomalous diffusion in complex media. In particular, the resulting stochastic process can show sub-diffusion with a behavior in qualitative agreement with single-particle tracking experiments in living cells, such as the ergodicity breaking, p variation, and aging. Moreover, for a proper distribution of the length-scales, a single parameter controls the ergodic-to-nonergodic transition and, remarkably, also drives the transition of the diffusion equation of the process from nonfractional to fractional, thus demonstrating that fractional kinetics emerges from ergodicity breaking [3]. References: [1] Pagnini G. and Paradisi P., A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation. Fract. Cacl. Appl. Anal. 19, 408–440 (2016) [2] Mura A. and Pagnini G., Characterizations and simulations of a class of stochastic processes to model anomalous diffusion. J. Phys. A: Math. Theor. 41, 285003 (2008) [3] Molina–García D., Pham T. Minh, Paradisi P., Manzo C. and Pagnini G., Fractional kinetics emerging from ergodicity breaking in random media. Phys. Rev. E. 94, 052147 (2016)

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

A class of parabolic cross-diffusion systems modeling the interaction of an arbitrary number of population species is analyzed in a bounded domain with no-flux boundary conditions. The equations are formally derived from a random-walk lattice model in the diffusion limit. Compared to previous results in the literature, the novelty is the combination of general degenerate diffusion and volume-filling effects. Conditions on the nonlinear diffusion coefficients are identified, which yield a formal gradient-flow or entropy structure. This structure allows for the proof of global-in-time existence of bounded weak solutions and the exponential convergence of the solutions to the constant steady state. The existence proof is based on an approximation argument, the entropy inequality, and new nonlinear Aubin-Lions compactness lemmas. The proof of the large-time behavior employs the entropy estimate and convex Sobolev inequalities. Moreover, under simplifying assumptions on the nonlinearities, the uniqueness of weak solutions is shown by using the H^{-1} method, the E-monotonicity technique of Gajewski, and the subadditivity of the Fisher information.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

In this talk, we will describe how to mix entropy structure and duality estimates in order to build global weak solutions to a class of cross-diffusion systems.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

We investigate the rigorous fast-reaction limit from a reaction-cross-diffusion system with known entropy to a new class of cross-diffusion systems using entropy and duality estimates. Performing the fast-reaction limit leads to a limiting entropy of the limiting cross-diffusion system. In this way, we are able to obtain new entropies for new classes of cross-diffusion systems. This is a joint work with L. Desvillettes and A. Juengel.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

We show global-in-time existence of bounded weak solutions to systems of cross-diffusion equations in a one dimensional moving domain. These equations stem from the modelization of the evolution of the concentration of chemical species composing a crystalline solid during a physical vapor deposition process. To this aim, we use the so called boundedness-by-entropy technique developed in [1], [2] and [3] based on the formal gradient flow structure of the system. Moreover, we are interested in controlling the fluxes of the different atomic species during the process in order to reach a certain desired final profile of concentrations. This problem is formulated as an optimal control problem to which the existence of a solution is proven. In addition, an investigation of the long time behavior is presented in the case of constant positive external fluxes. Finally, some numerical results and comparison with actual experiments are presented. The material of this talk is a joint work with Virginie Ehrlacher. References [1] M.Burger, M.Di Francesco, J-F. Pietschmann and B. Schalke. Non linear cross diffusion with size exclusion. SIAM J. Math Anal 42 (2010). [2] A. Jüngel and Nicola Zamponi boundedness of weak solutions to cross-diffusion systems from population dynamics. arxiv:1404.6054v1 (2014). [3] A. Jüngel. The boundedness-by-entropy method for cross-diffusion systems. To appear in Nonlinearity, http://www.asc.tuwien.ac.at/ juengel/ (2015).

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

We consider the one-dimensional multi-species Boltzmann system of equations [2] in the diffusive scaling. Suppose that the Mach and the Knudsen numbers are of the same order of magnitude epsilon > 0 small enough. For each species i of the mixture, we define the macroscopic quantity of matter and flux as the moments 0 and 1 in velocity of the distribution functions f_i, solutions of the Boltzmann system associated to the scaling parameter epsilon. Using the moment method [4], we introduce a proper ansatz for each distribution function f_i in order to recover a Maxwell-Stefan diffusion limit-type as in [1]. In this way we build a suitable numerical scheme for the evolution of these macroscopic quantities in different regimes of the parameter epsilon. We prove some a priori estimates (mass conservation and nonnegativity) and well-posedness of the discrete problem. We also present numerical examples where we observe that the scheme shows an asymptotic preserving property similar to the one presented in [3]. This is a joint work with L. Boudin and B. Grec. References [1] L. Boudin, B. Grec and V. Pavan, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures with general cross sections, Nonlinear Analysis: Theory, Methods and Applications, 2017. [2] L. Desvillettes, R. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Eur. J. Mech. B Fluids, 24(2005), 219-236. [3] S. Jin and Q. Li, A BGK-penalization-based asymptotic-preserving scheme for the multispecies Boltzmann equation, Numer. Methods Partial Differential Equations, 29(3), pp. 1056-1080, 2013. [4] C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83(5-6):1021-1065, 1996

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

In this talk, we consider some specific reaction-diffusion equations in order to understand the equivalence between asymptotic Turing instability of a steady state and backwardness of some parabolic equations or cross-diffusion equations in the formal limit of fat reaction terms. We will see that the structure of the studied equations involves some Lyapunov functions which leads to a priori estimates allowing to pass rigorously for the fast reaction terms in the case without Turing instabilities.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

The main feature of the two-dimensional Keller-Segel model is the blow-up behaviour of solutions for supercritical masses. We introduce a regularisation of the fully parabolic system by adding a cross-diffusion term to the equation for the chemical substance. This regularisation provides another helpful entropy dissipation term allowing to prove global existence of weak solutions for any initial mass. For the proof we first analyse an approximate problem obtained from a semi-discretisation and a carefully chosen regularisation by adding higher order derivatives. Compactness arguments are used to carry out the limit to the original system. A similar approach can be used to analyse a pedestrian dynamics model for two groups moving in opposite direction. The evolutionary equations are driven by cohesion and aversion and are formally derived from a 2d lattice based approach. Also numerical simulations illustrating lane formation will be presented. These methods are extended to a crossing pedestrian model, where we additionally analyse the stability of stationary states in the corresponding 1d model.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)

We study interacting particles behaving according to a reaction-diffusion equation with non-linear diffusion and non-local attractive interaction. This class of equations has a very nice gradient flow structure that allows us to make links to homogeneous functionals and variations of well-known functional inequalities (Hardy-Littlewood-Sobolev inequality, logarithmic Sobolev inequality). Depending on the non-linearity of the diffusion, the choice of interaction potential and the dimensionality, we obtain different regimes. Our goal is to understand better the asymptotic behaviour of solutions in each of these regimes, starting with the fair-competition regime where attractive and repulsive forces are in balance. This is joint work with José A. Carrillo and Vincent Calvez.

• Thematic program: Models in Biology and Medicine (2016/2017)
• Event: Workshop on "Cross diffusion systems and kinetic equations for biology" (2017)