On the primitive equations and the hydrostatic Stokes operator in L²

Master Thesis (2021)
Author(s)

G. Muguruza Lasa (TU Delft - Electrical Engineering, Mathematics and Computer Science)

Contributor(s)

Mark Veraar – Mentor (TU Delft - Analysis)

J. Sauer – Graduation committee member (TU Delft - Analysis)

H.M. Schuttelaars – Graduation committee member (TU Delft - Mathematical Physics)

Faculty
Electrical Engineering, Mathematics and Computer Science
Copyright
© 2021 Garazi Muguruza Lasa
More Info
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Publication Year
2021
Language
English
Copyright
© 2021 Garazi Muguruza Lasa
Graduation Date
18-02-2021
Awarding Institution
Delft University of Technology
Programme
['Applied Mathematics']
Faculty
Electrical Engineering, Mathematics and Computer Science
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Abstract

In 2016 Hieber and Kashiwabara showed that the three dimensional primitive equations admit a unique, global, strong solution for all initial data in a closed subspace of the Bessel space $H^{2/p,p}(\Omega)$ provided $p\geq6/5$, being this the first result in the general $L^p$-setting. Their approach consisted in studying the properties of the hydrostatic Stokes operator $A_p$ defined on the solenoidal subspace $L^p_{\overline{\sigma}}(\Omega)$ of $L^p(\Omega)$. In 2017 Giga et. al. further proved that the hydrostatic Stokes operator $A_p$ admits a bounded $H^\infty$-calculus, obtaining maximal $L^q-L^p$ regularity estimates for the linearized primitive equations in a much simpler way. In this work we will study Giga et. al.'s and Hieber and Kashiwabara’s works particularized for the $L^2$-case as well as all the necessary literature to replicate the proofs. The goal of the thesis is to present an extended version of Giga et. al.’s proof to make it more accessible. Although the $L^p$-case is not studied for lack of time, we differentiate between the Sobolev-Slobodeckij, Bessel potential and Besov spaces to accentuate how we could extend the proofs to the $L^p$-setting.

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