Weighted \(L^q\)-theory for the Stokes resolvent in exterior domains
The authors consider the generalized Stokes resolvent system \[ \lambda u-\Delta u+\nabla p= f,\quad \text{div }u= g,\quad\text{in }\Omega,\quad u= 0\quad\text{on }\partial\Omega. \] The domain \(\Omega\) is either the whole space \(\mathbb{R}^n\) or an exterior domain of \(\mathbb{R}^n\) \((n\geq 2)\) and \[ \lambda\in \{z\in\mathbb{C}\mid z\neq 0,\;|\text{arg }z|< \pi-\varepsilon\}\quad \Biggl(\varepsilon\in \Biggl(0,{\pi\over 2}\Biggr)\Biggr). \] The main results of the paper establish the unique solvability of the resolvent problem for the above domains in weighted Sobolev spaces. These spaces are introduced via weighted \(L^q\)-spaces of the form \[ L^q_w(\Omega):= \Biggl\{u\in L^1_{\text{loc}}(\overline\Omega)\mid\Biggl(\int_\Omega| u|^q wdx\Biggr)^{1/q}< \infty\Biggr\}, \] where the nonnegative weight function is supposed to belong to the Muckenhoupt class \({\mathcal A}_q\). This choice is motivated by the fact that the multiplier theorem of Hörmander and Michlin is still valid in \(L^q_w(\Omega)\) for \({\mathcal A}_q\)-weights. Furthermore, the authors obtain the Helmholtz decomposition of \(L^q_w(\Omega)\) which allows them to define the Stokes operator on this space. They also show that the Stokes operator generates a bounded analytic semigroup which may subsequently be used in order to construct solutions of the instationary Stokes and Navier-Stokes systems in weighted spaces.
scientific article; zbMATH DE number 1172193