On the primitive equations and the hydrostatic Stokes operator in L²

On the primitive equations and the hydrostatic Stokes operator in L²

Muguruza Lasa, Garazi (TU Delft Electrical Engineering, Mathematics and Computer Science)

Veraar, M.C. (mentor)
Sauer, J. (graduation committee)
Schuttelaars, H.M. (graduation committee)

Delft University of Technology

Applied Mathematics

2021-02-18

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.

hydrostatic stokes operator
h infinity calculus
primitive equations

http://resolver.tudelft.nl/uuid:bf74862f-5b7f-460f-8d09-712d32d29c91

Student theses

master thesis

© 2021 Garazi Muguruza Lasa