Abstract:
We study the asymptotic analysis of partial differential equations modelling subdiffusive random motion in cell biology. The biological motivation for this work is the numerous recent observations of cytoplasmic proteins whose random motion deviates from normal Fickian diffusion. In the first part of this talk, I will present the self-similar decay towards a steady state of the solution of a heavy-tailed renewal equation. The ideas therein are inspired from relative entropy methods. Our main contributions are the proof of an L 1 decay rate towards the arc-sine distribution and the introduction of a specific pivot function in a relative entropy method. The second part will treat the hyperbolic limit of an age-structured space-jump renewal equation. We have proved a “stability” result: the solutions of the rescaled problems at ε > 0 converge as ε → 0 towards the viscosity solution of the limiting Hamilton-Jacobi equation of the ε > 0 problems. The main mathematical tools used come from the theory of Hamilton-Jacobi equations. This second work relies on three main interesting ideas. The first is that of proving the convergence result on the boundary condition of the studied problem rather than using perturbed test functions. The second consists in the introduction of timelogarithmic correction terms in a priori estimates that do not follow directly from the maximum principle. That is due to the non-existence of a suitable equilibrium for the space-homogenous problem. The third is a precise estimate of the decay of the influence of the initial condition on the renewal term. This is tantamount to a refined estimate of a non-local version of the time derivative of the solution.
Biography:
Álvaro Mateos González was born in Spain in 1989, where he did his school and high school education at the French school of Madrid. During that period of time, he won certain national and international prizes in mathematical competitions, and also a prize in French poetry. He studied mathematics in France, starting at the Classe Préparatoire Pierre de Fermat in Toulouse. He then entered the École Normale Supérieure de Lyon in 2010, where he finished his studies and did a Master's degree in Analysis and Partial Differential Equations. He completed his Ph.D. at the École Normale Supérieure de Lyon and the Inria Rhône-Alpes under the supervision of Vincent Calvez, Thomas Lepoutre, and Hugues Berry. The subject matter of his talk is taken from his doctor's thesis on asymptotic analysis of PDE models of random motion processes stemming from Biology.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai