On the semigroup of the Stokes operator for exterior domains in L q- spaces
For an exterior domain \(\Omega\) of \({\mathbb{R}}^ n,\) \(n\geq 3\), and \(q\in (1,\infty)\), consider the Stokes operator \(A_ q:=-P_ q\Delta_ q\) where \(\Delta_ q\) denotes the Laplacian in L q(\(\Omega)\) n, \(P_ q\) the linear projection of L q(\(\Omega)\) n onto the subspace \(H_ q(\Omega)\) of divergence-free vector fields parallel to the boundary of \(\Omega\). It is shown that \(A_ q\) is a sectorial operator in \(H_ q(\Omega)\) in the sense that, for arbitrary \(\omega\in (0,\frac{\pi}{2})\), there is a constant c such that \(\| (\lambda Id+A_ q)^{-1}\|_{H_ q}\leq c/| \lambda |\) holds for all \(\lambda\in {\mathbb{C}}\setminus \{0\}\) with \(| \arg (\lambda)| \leq \pi /2+\omega.\) This spectral property (well-known for \(q=2\), but new for the general case) yields important information on the (analytic) semigroup generated by \(-A_ q\), in particular uniform boundedness which in turn is crucial for the stability analysis of Stokes' boundary value problem. The proof relies on a localization procedure by means of which the problem is split into one for a bounded domain and another one for the entire space \({\mathbb{R}}^ n \)(both cases are well-known) under the restriction \(q<n/2\); the general case is dealt with by duality and interpolation arguments. Important tools are earlier estimates of \textit{Y. Giga} [Math. Z. 178, 297-329 (1981; Zbl 0473.35064)], \textit{V. A. Solonnikov} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklov 38, 153-231 (1973; Zbl 0346.35083); English translation in: J. Sov. Math. 8, 467-529 (1977)] and \textit{M. McCracken} [SIAM J. Math. Anal. 12, 201-228 (1981; Zbl 0475.35073)], and \textit{M. E. Bogovskij}'s theory on the equation of continuity in incompressible media [Dokl. Akad Nauk SSSR 248, 1037-1040 (1979; Zbl 0499.35022)].
scientific article; zbMATH DE number 4037431