Analysis of some diffusive and kinetic models in mathematical biology and physics

• Autores: Jesús Rosado Linares
• Directores de la Tesis: José Antonio Carrillo (dir. tes.)
• Lectura: En la Universitat Autònoma de Barcelona ( España ) en 2010
• Idioma: inglés
• Tribunal Calificador de la Tesis: Ángel Calsina Ballesta (presid.) , Ansgar Jüngel (secret.) , Miguel Escobedo Martínez (voc.)
• Enlaces
  • Tesis en acceso abierto en: TDX
• Resumen

  • In this work, on one hand we study several models based upon non-linear Fokker-Planck equations which apply to physics and biology. More precisely, we focus in the Fermi-Dirac model $$f_t = \mathrm{div}\{\nabla f+vf(1+kf)\},$$ with $k=-1$, which is a simplified version of the Boltzmann equation for fermions. We have proven global existence and uniqueness of solution in any dimension and, using entropy methods, the exponential convergence of these solutions to the stationary solution given by the Fermi-Dirac statistic of the same mass as the initial condition. The same arguments provide local existence for the Bose-Einstein model ($k=1$) in any dimension but analogous results about asymptotic behavior are obtained only in dimension 1.

    We also study the Keller-Segel model for chemotaxis $$ \rho_t = \mathrm{div}\{\varepsilon\nabla(\rho)+\rho\chi(\rho,S)\nabla S\}\qquad S_t = \Delta S + r(\rho,S) $$ where $\rho$ stands for the concentration of cells moving due to the influence of a chemical substance, $S$, $\chi$ is the sensibility of the cells to this substance and $r$ the secretion/destruction rate of $S$. In the case that the chemosensibility presents a saturation of the form $\chi=(C-\rho)$, it has been proven the existence and uniqueness of solution in any dimension and in the study of the asymptotic behavior we get a quadratic decay to zero of the infinite norm, both of $\rho$ and $S$ and polynomial convergence to self-similar solutions for the concentration of cells when $\varepsilon>1/4$. The convergence for the concentration of the chemical and the asymptotic behavior of the solutions when $\varepsilon<1/4$ \'es is still an open problem.

    On the other hand, we develop a fairly complete theory for the well-posedness of the aggregation equation $$\frac{\partial u}{\partial t} + div(uv) = 0\qquad\text{with}\qquad v=-\nabla K\ast u$$ in $L^p$. The $L^p$ framework we adopt allow us to make to significant advances in the understanding of the aggregation equation. First, we are able to consider potentials $K$ which are less singular than the Newtonian potential whilst up to now it was often requires to the ones which have been considered to be in the worst case Lipschitz singular at the origin. Second, we identify the critical regularity needed on the initial data in order to guarantee that the solution will remain absolutely continuous with respect to the Lebesgue measure, at least for short time. Also, we deal with the uniqueness question in this and some other models which involve an aggregation term and competition between aggregation and diffusion. Albeit in some of cases the result was already known, we are able to provide a simpler proof thanks to new point of view and tools coming from the optimal transport theory.

    Finally, we study different models of collective animal behavior. We first approach them from the ODE viewpoint, through the study of the system of $N$ particles, to focus afterwards in the kinetic equations obtained from the mean field limit, when we increase the number of particles. All the models we study can be seen as particular cases of $$ \label{eq:swarming-E} \partial_t f + v \nabla_x f + \nabla_v \cdot \left[\xi(f) (x,v,t) f(x,v,t)\right] + \mathrm{div}_v(\gamma(v) f) = 0, $$ where $\xi$ is an interaction potential and $\gamma$ gather the friction and self-propulsion terms. Here we are interested in the study of the stability of solutions. To deal with this problem we also use optimal transport techniques, providing new applications for these already known tools.