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A Priori Regularity of Parabolic Partial Differential Equations

In this thesis, we consider parabolic partial differential equations such as the heat equation, the Fokker-Planck equation, and the porous media equation. Our aim is to develop methods that provide a priori estimates for solutions with singular initial data. These estimates are obtained by understanding the time decay of norms of solutions.
First, we derive regularity results for the heat equation by estimating the decay of Lebesgue norms. Then, we apply similar methods to the Fokker-Planck equation with suitable assumptions on the advection and diffusion. Finally, we conclude by extending our techniques to the porous media equation. The sharpness of our results is confirmed by examining known solutions of these equations.
The main contribution of this thesis is the use of functional inequalities to express decay of norms as differential inequalities. These are then combined with ODE methods to deduce estimates for the norms of solutions and their derivatives.

Identiferoai:union.ndltd.org:kaust.edu.sa/oai:repository.kaust.edu.sa:10754/627833
Date13 May 2018
CreatorsBerkemeier, Francisco
ContributorsGomes, Diogo A., Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, Tzavaras, Athanasios, Santamarina, Carlos
Source SetsKing Abdullah University of Science and Technology
LanguageEnglish
Detected LanguageEnglish
TypeThesis

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