Stokes Operator on Lipschitz Domains

Working Groups: Lehrstuhl Angewandte Analysis

Collaborators (MAT): Fabian Gabel, M. Sc.

Collaborators (External): Patrick Tolksdorf

Description

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. Journal of Differential Equations 340, 2022, pp. 227–272. doi:10.1016/j.jde.2022.09.001