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Title: Stability for nonlinear diffusive PDEs
Author: Alasio, Luca Cesare Biagio
ISNI:       0000 0004 7654 0898
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2018
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This thesis focuses on well-posedness and stability estimates (i.e. continuous dependence with respect to the nonlinearities) for a family of cross-diffusion systems that includes several models used to describe cell diffusion in Mathematical Biology for one or more species. We discuss rigorous quantitative results for a class of (possibly degenerate) parabolic PDEs, including scalar equations as well as systems. The approach is typical of Analysis of PDEs, hence the concepts of weak solutions in Sobolev spaces, a priori estimates and well-posedness are crucial. Considering scalar, nonlinear diffusion, we extend the existing solvability, uniqueness, boundedness and stability results. Regarding non-degenerate models for multiple species, we work in the framework of systems with an entropy structure and a limited set of other assumptions that allow us to prove stability estimates, improving some of the existing results. Finally, we have been working on the multi-species, degenerate case, providing a continuous dependence estimate under suitable (but rather restrictive) assumptions. Two different types of numerical simulations are presented in order to complement the abstract results.
Supervisor: Capdeboscq, Yves ; Bruna, Maria Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Analysis of PDEs