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    A Priori Regularity of Parabolic Partial Differential Equations

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    Type
    Thesis
    Authors
    Berkemeier, Francisco cc
    Advisors
    Gomes, Diogo A. cc
    Committee members
    Tzavaras, Athanasios cc
    Santamarina, Carlos cc
    Program
    Applied Mathematics and Computational Science
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2018-05-13
    Permanent link to this record
    http://hdl.handle.net/10754/627833
    
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    Abstract
    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.
    Citation
    Berkemeier, F. (2018). A Priori Regularity of Parabolic Partial Differential Equations. KAUST Research Repository. https://doi.org/10.25781/KAUST-TI27W
    DOI
    10.25781/KAUST-TI27W
    ae974a485f413a2113503eed53cd6c53
    10.25781/KAUST-TI27W
    Scopus Count
    Collections
    Applied Mathematics and Computational Science Program; Theses; Theses; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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