Mathematical models for cellular aggregation: the chemotactic instability and clustering formation

In this thesis we present a mathematical formulation of the interaction between microorganisms such as bacteria or amoebae and chemicals, often produced by the organisms themselves. This interaction is called chemotaxis and leads to cellular aggregation. We derive some models to describe chemotaxis. The first is the pioneristic Keller-Segel parabolic-parabolic model and it is derived by two different frameworks: a macroscopic perspective and a microscopic perspective, in which we start with a stochastic differential equation and we perform a mean-field approximation. This parabolic model may be generalized by the introduction of a degenerate diffusion parameter, which depends on the density itself via a power law. Then we derive a model for chemotaxis based on Cattaneo's law of heat propagation with finite speed, which is a hyperbolic model. The last model proposed here is a hydrodynamic model, which takes into account the inertia of the system by a friction force. In the limit of strong friction, the model reduces to the parabolic model, whereas in the limit of weak friction, we recover a hyperbolic model. Finally, we analyze the instability condition, which is the condition that leads to aggregation, and we describe the different kinds of aggregates we may obtain: the parabolic models lead to clusters or peaks whereas the hyperbolic models lead to the formation of network patterns or filaments. Moreover, we discuss the analogy between bacterial colonies and self gravitating systems by comparing the chemotactic collapse and the gravitational collapse (Jeans instability).

Ultracold Bose-Fermi mixtures with pairing: quantum phase transition, mechanical stability, and trap confinement

Ultracold dilute gases occupy an important role in modern physics and they are employed to verify fundamental quantum theories in most branches of theoretical physics. The scope of this thesis work is the study of Bose-Fermi (BF) mixtures at zero temperature with a tunable pairing between bosons and fermions. The mixtures are treated with diagrammatic quantum many-body methods based on the so-called T-matrix formalism. Starting from the Fermi-polaron limit, I will explore various values of relative concentrations up to mixtures with a majority of bosons, a case barely considered in previous works. An unexpected quantum phase transition is found to occur in a certain range of BF coupling for mixture with a slight majority of bosons. The mechanical stability of mixtures has been analysed, when the boson-fermion interaction is changed from weak to strong values, in the light of experimental results recently obtained for a double-degenerate Bose-Fermi mixture of 23 Na - 40 K. A possible improvement in the description of the boson-boson repulsion based on Popov's theory is proposed. Finally, the effects of a harmonic trapping potential are described, with a comparison with the experimental data for the condensate fraction recently obtained for a trapped 23 Na - 40 K mixture.

Lipschitz regularity for weak solutions of parabolic p-Laplacian type equations in certain subriemannian structures

The main aim of the thesis is to prove the local Lipschitz regularity of the weak solutions to a class of parabolic PDEs modeled on the parabolic p-Laplacian. This result is well known in the Euclidean case and recently has been extended in the Heisenberg group, while higher regularity results are not known in subriemannian parabolic setting. In this thesis we will consider vector fields more general than those in the Heisenberg setting, introducing some technical difficulties. To obtain our main result we will use a Moser-like iteration. Due to the non linearity of the equation, we replace the usual parabolic cylinders with new ones, whose dimension also depends on the L^p norm of the solution. In addition, we deeply simplify the iterative procedure, using the standard Sobolev inequality, instead of the parabolic one.