Metalenz is a spin­off company of Harvard University that works on finding newer and better designs for lenses. To prevent manufacturing designs that might not work, they first want to run computer simulations on their designs. This research focuses on a possible method for these simulations using a test case as design. It turns out that the simulations work, but that they are slow. Therefore, a technique to make the simulations faster, Multi-trace boundary integral equations, is researched. This technique manages to change simulations that take a week to finish into simulations that take a couple of hours to finish. Before Metalenz can start using the simulation techniques, they should be tested for more complicated cases, to make sure they still
work when more factors are at play

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.