Séminaire de EDP - Physique Mathématique

TBA

In this talk, we will present the study of a model for particle suspensions in a non-Newtonian Ostwald-DeWaele fluid with a potentially degenerate viscosity coefficient.

The analysis of problems associated with such systems is a very active research topic, the source of many recent results, particularly in the case of particles that sediment in a Newtonian fluid (see, for example, the works of D. Cobb, R. Höfer, A. Mecherbet, R. Schubert, F. Sueur). We will consider the case where particles are suspended in a non-Newtonian fluid. From a mathematical point of view, this is characterized by a Stokes-Transport equation with the particularity that the Stokes equation is nonlinear, the viscosity term being expressed as a p-Laplacian for the symmetrized gradient. After briefly contextualizing the problem, we will present our result: the existence of global weak energy solutions.

In order to show that we do define an active scalar equation, i.e. one in which the relative density of the particles suspended in the fluid gives meaning to a solution of the system through an inverse mapping, it is then necessary to use monotonicity methods in conjunction with techniques derived from DiPerna-Lions theory to establish the existence of suitable weak solutions. We will therefore present the main ideas for establishing the existence of such ones.

This talk is concerned with asymptotic persistence, extinction and spreading properties for structured population models resulting in non-cooperative Fisher-KPP systems with space-time periodic coefficients, motivated by a wide class of models in population biology. Results are formulated in terms of a family of generalized principal eigenvalues associated with the linearized problem. When the maximal generalized principal eigenvalue is negative, all solutions to the Cauchy problem become locally uniformly positive in long-time, at least one space-time periodic uniformly positive entire solution exists, and solutions with compactly supported initial condition asymptotically spread in space at a speed given by a Freidlin-Gärtner-type formula. When another, possibly smaller, generalized principal eigenvalue is nonnegative, then on the contrary all solutions to the Cauchy problem vanish uniformly and the zero solution is the unique space-time periodic nonnegative entire solution. When the two generalized principal eigenvalues differ and zero is in between, the long-time behavior depends on the decay at infinity of the initial condition. The proofs rely upon double-sided controls by solutions of cooperative systems. The control from below is new for such systems and makes it possible to shorten the proofs and extend the generality of the system simultaneously.