Hamburg University of Technology / Institute of Mathematics / Research Topics / Stokes Operator on Lipschitz Domains Stokes Operator on Lipschitz Domains

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  and builds mainly on resolvent estimates for the Stokes operator $$A$$ and the analyticity property of the Stokes semigroup.

 Fujita, H. and Kato, T. On the Navier-Stokes initial value problem I. Archive for Rational Mechanics and Analysis 16(1964), 269?315.

 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/.

 Gabel, F. On Resolvent Estimates in Lp for the Stokes Operator in Lipschitz Domains. Master thesis, Technische Universität Darmstadt, 2018.

 Gabel, F. and Tolksdorf, P. The Stokes operator in two-dimensional bounded Lipschitz domains. In preparation.