Orateur
Description
In this talk I will present recent analytical and numerical results on pressureless Euler systems which combine non-local interactions and chemotaxis. The system under investigation arises as the macroscopic counterpart of a class of hybrid discrete-continuum models for collective cell migration, in which cells interact mechanically through alignment and attraction-repulsion effects, and respond to the gradient of a chemical signal they themselves produce. The absence of a pressure term, replaced by non-local integral terms retaining information about the microscopic dynamics, represents the main issue relating to loss of regularity in finite time.
I will present analytical results concerning local well-posedness of the solution together with a continuation criterion showing that breakdown, if it occurs, can only be caused by the blow-up of the velocity gradient. A comparison with the chemotaxis-free case, already investigated in the literature of the field, shows that the subcritical or supercritical character of the dynamics is no longer determined by the initial configuration alone, but also by the cumulative, memory-type contribution of the chemotactic coupling.
Results are tested numerically in one space dimension, through a study of the transition between regular and singular regimes, of the robustness of the detected breakdown time with respect to discretization parameters, and of the role of the chemotactic strength in the onset of the singularity. In a two-dimensional setting, numerical simulations highlight the ability of the model to reproduce network-like patterns and immune-cancer interactions observed in cancer-on-chip experiments.