In the solution theory for nonlinear partial differential equations, an integral part of the solution process is often to develop a semigroup theory for the linearization of the equation.
In the case of the famous *Navier-Stokes equations* which for a given domain $\Omega \subseteq \mathbb{R}^d$, $d \geq 2$, describe the behavior of a Newtonian fluid over time, the linearization is given by the *Stokes equations*
$$
\partial_t u - \Delta u + \nabla \pi = 0 \quad\text{in } \Omega\,, \;t > 0\,, \quad
\operatorname{div}(u) = 0 \quad\text{in } \Omega\,,\; t > 0\,,
$$
$$
u(0) = a \text{ in } \Omega\,,
u = 0 \text{ on } \partial\Omega\,,\; t > 0\,,
$$
where $u \colon \mathbb{R}^+ \times \Omega \to \mathbb{R}^d$ stands for the velocity field and $\pi \colon \mathbb{R}^+ \times \Omega \to \mathbb{R}$ represents the pressure of the fluid.
The so-called *Stokes semigroup* $(\mathrm{e}^{-tA})_{t \geq 0}$ describes the evolution of the velocity $u$ and the *Stokes operator* $A$ corresponds to the term ''$-\Delta u + \nabla \pi$'' in the Stokes equations.
Having a semigroup makes it possible to look for *mild solutions* to the Navier-Stokes equations using a variation of constants formula to construct an iteration method.
This approach was introduced by Fujita and Kato [1] and builds mainly on resolvent estimates for the Stokes operator $A$ and the analyticity property of the Stokes semigroup.
## References
[1] Fujita, H. and Kato, T. On the Navier-Stokes initial value problem I. Archive for Rational Mechanics and Analysis 16(1964), 269–315.
[2] Tolksdorf, P. On the Lp-theory of the Navier-Stokes equations on Lipschitz domains. PhD thesis, Technische Universität Darmstadt, 2017. Available at http://tuprints.ulb.tu-darmstadt.de/5960/.
[3] Gabel, F. On Resolvent Estimates in Lp for the Stokes Operator in Lipschitz Domains. Master thesis, Technische Universität Darmstadt, 2018.
[4] Gabel, F. and Tolksdorf, P. The Stokes operator in two-dimensional bounded Lipschitz domains. *In preparation.*
