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Asymptotic Analysis of Partial Differential Equations Arising in Biological Processes of Anomalous Diffusion

Álvaro Mateos González 1, 2
2 BEAGLE - Artificial Evolution and Computational Biology
LBBE - Laboratoire de Biométrie et Biologie Evolutive - UMR 5558, Inria Grenoble - Rhône-Alpes, LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
Abstract : This thesis is devoted to 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, we study 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 L1 decay rate towards the arc-sine distribution and the introduction of a specific pivot function in a relative entropy method.The second part treats the hyperbolic limit of an age-structured space-jump renewal equation. We prove 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 work presents three 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 time-logarithmic correction termsin 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 inuence 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. Throughout this thesis, we have performed numerical simulations of different types: Monte Carlo, finite volume schemes, Lax-Friedrichs schemes and Weighted Essentially Non Oscillating schemes.
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Álvaro Mateos González. Asymptotic Analysis of Partial Differential Equations Arising in Biological Processes of Anomalous Diffusion. Analysis of PDEs [math.AP]. Université de Lyon, 2017. English. ⟨NNT : 2017LYSEN069⟩. ⟨tel-01701022⟩

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