Stokes Operator and Navier-Stokes Equations on Lipschitz Domains

For a motivational intro, you can also watch the following video starting at 29:39.

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 Link to heading

[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 $L^p$-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 $L^p$ 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”, Journal of Differential Equations, vol. 340, 2022, pp. 227–72. [doi] [arXiv]