International Conference on Complex Dynamics: Modeling and Simulation

In this talk we will revisit the classical problem on the Hele—Shaw or incompressible limit for nonlinear degenerate diffusion equations. We will demonstrate that the theory of optimal transport via gradient flows can bring new perspectives, when it comes to considering confining potentials or nonlocal drift terms within the problem. In particular, we provide quantitative convergence rates in the 2-Wasserstein distance for the singular limit, which are global in time thanks to the contractive property arising from the external potentials. The talk will be based on a joint work with Noemi David and Filippo Santambrogio.

Many phenomena in the life sciences, ranging from the microscopic to macroscopic level, exhibit surprisingly similar structures. Behaviour at the microscopic level, including ion channel transport, chemotaxis, and angiogenesis, and behaviour at the macroscopic level, including herding of animal populations, motion of human crowds, and bacteria orientation, are both largely driven by long-range attractive forces, due to electrical, chemical or social interactions, and short-range repulsion, due to dissipation or finite size effects. Various modelling approaches at the agent-based level, from cellular automata to Brownian particles, have been used to describe these phenomena. An alternative way to pass from microscopic models to continuum descriptions requires the analysis of the mean-field limit, as the number of agents becomes large. All these approaches lead to a continuum kinematic equation for the evolution of the density of individuals known as the aggregation-diffusion equation. This equation models the evolution of the density of individuals of a population, that move driven by the balances of forces: on one hand, the diffusive term models diffusion of the population, where individuals escape high concentration of individuals, and on the other hand, the aggregation forces due to the drifts modelling attraction/repulsion at a distance. The aggregation-diffusion equation can also be understood as the steepest-descent curve (gradient flow) of free energies coming from statistical physics. Significant effort has been devoted to the subtle mechanism of balance between aggregation and diffusion. In some extreme cases, the minimisation of the free energy leads to partial concentration of the mass. Aggregation-diffusion equations are present in a wealth of applications across science and engineering. Of particular relevance is mathematical biology, with an emphasis on cell population models. The aggregation terms, either in scalar or in system form, is often used to model the motion of cells as they concentrate or separate from a target or interact through chemical cues. The diffusion effects described above are consistent with population pressure effects, whereby groups of cells naturally spread away from areas of high concentration. This talk will give an overview of the state of the art in the understanding of aggregation-diffusion equations, and their applications in mathematical biology.

We consider a KGS system as an infinite-dimensional dynamical system in the energy space $E=H^1\times H^1\times L^2$. Despite the fact that there is no smoothing effect in finite time for the flow of solutions, we prove that the global attractor $\mathcal{A}$ for this system is a compact finite-dimensional subset of $H^2\times H^2\times H^2$.

We are introducing a new class of cross-diffusion systems to model congestion phenomena in pedestrian dynamics. This approach is based on a steepest descent algorithm that incorporates a minimal flow process to measure the proximal work. The goal is to minimize the system's internal energy. We will present and discuss a few comparisons with existing cross-diffusion systems that use the Wasserstein distance.

Models issued from ecology, chemical reactions and several other application fields lead to semi-linear parabolic equations with super-linear growth. Many of these systems naturally satisfy the following two properties : (P) : the positivity of the solutions is preserved as long as they exist ; (M) : the total mass of the components is controlled (or even preserved). It turns out that the structure (P)+(M) does not keep the solution from blowing up in L∞- norm, even in finite time. Thus, some growth restrictions and extra structure on the reactive terms are needed for global existence of classical solutions. This issue has been intensively studied in the semi-linear case and the initial data are bounded. However, the situation is quite more complicated for nonlinear diffusion, more general nonlinearities and initial data of low regularity. So, it is necessary to deal with weak solutions if one expects global existence. In this talk, we will give a general overview on systems verifying (P) and (M). In particular, we will present some results obtained in collaboration with B. Perthame [1], M. Pierre [2, 4, 3] and M. Daoud and A. Baalal [5].

La croissance des tissus vivants, tels que les tumeurs solides par exemple, peut être décrite à différentes échelles, de la cellule à l'organne. Pour un grand nombre de cellules, une approche de mécanique des fluides a été proposée par les biomécaniciens. Nous ferons le point sur plusieurs approches. Ensuite, nous considèrons le cas où le tissu est vu comme un milieu poreux. Nous considèrerons la limite incompressible, i.e. l'asymptotique Hele-Shaw passant d'une densité de cellules à une frontière libre dans la limite de lois de pression raide. Cette limite ouvre la question de l'apparition de régimes instables.

In this talk we present a new general approach for the numerical analysis of stable equilibria to second order mean field games systems in cases where the uniqueness of solutions may fail. For the sake of simplicity, we focus on a simple stationary case. We propose an abstract framework to study these solutions by reformulating the mean field game system as an abstract equation in a Banach space. In this context, stable equilibria turn out to be regular solutions to this equation, meaning that the linearized system is well-posed. We provide three applications of this property, we study the sensitivity analysis of stable solutions, establish error estimates for their finite element approximations, and prove the local converge of Newton’s method in infinite dimensions.