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Fatma Mohamed Defense
On Some Parabolic Type Problems from Thin Film Theory and Chemical Reaction-Diffusion Networks
Abstract: In the first part of this dissertation we study the evolution of a thin film of fluid modeled by the lubrication approximation for thin viscous films. We prove existence of (dissipative) strong solutions for the Cauchy problem when the sub-diffusive exponent ranges between 3/8 and 2; then we show that these solutions tend to zero at rates matching the decay of the source-type self-similar solutions with zero contact angle. We introduce the weaker concept of dissipative mild solutions and we show that in this case the surface-tension energy dissipation is the mechanism responsible for the $H^1$--norm decay to zero of the thickness of the film at an explicit rate. Relaxed problems, with second-order nonlinear terms of porous media type are also successfully treated by the same means.
In the second part we are concerned with the convergence of a certain space-discretization scheme (the so-called method of lines) for mass-action reaction-diffusion systems. First, we start with a toy model, namely AB and prove convergence of the method of lines for this linear case (weak convergence in L^2 is enough in this case). Then we show that solutions of the chemical reaction-diffusion system associated to A+BC in one spatial dimension can be approximated in L^2 on any finite time interval by solutions of a space discretized ODE system which models the corresponding chemical reaction system replicated in the discretization subdomains where the concentrations are assumed spatially constant. Same-species reactions through the virtual boundaries of adjacent subdomains lead to diffusion in the vanishing limit. We show convergence of our numerical scheme by way of a consistency estimate, with features generalizable to reaction networks other than the one considered here, and to multiple space dimensions.
Place: 315 Armstrong Hall
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