Analyticity of the Stokes semigroup in \textit{BMO}-type spaces
The unsteady homogeneous Stokes equations are studied for a large class of uniformly \(C^{3}\) flow-domains \(\Omega \subset R^n, \,\, n \geq 2\), which contains bounded domains, the half-space and exterior domains. In [\textit{K. Abe} and \textit{Y. Giga}, Acta Math. 211, No. 1, 1--46 (2013; Zbl 1288.35383)] was proved that the Stokes operator generates an analytic semigroup in the closure of smooth solenoidal vector fields with compact support in \(L^p(\Omega)\) for a specific class of flow-domains. In the present paper, this result is generalized for a specific space of bounded mean oscillation functions -- BMO. A norm measuring the mean oscillation of functions in \(\Omega\) and the mean value of functions near the boundary \(\partial \Omega\) is introduced. The \(L^{\infty}\)-norm of the derivatives of the Stokes solution are estimated in terms of a specific BMO-norm of the initial data, as in [the first two authors, Math. Z. 284, No. 3--4, 1163--1183 (2016; Zbl 1354.35065)]. An important step is a boundary estimate of the pressure. For this, the solvability of the Neumann problem is analyzed, by using the Sobolev space on \(\partial \Omega\). The main tools are the John-Niremberg, Hölder, Poincaré and trace inequalities, the interpolation between \(L^2(\partial \Omega)\) and \(H^1(\partial \Omega)\) and the embedding of \(L^{\infty}(\Omega)\) in a specific BMO\((\Omega)\). In Sections 3 and 4, some very interesting results are given, concerning the boundedness in BMO-type spaces and the equivalences of specific BMO-norms.
scientific article; zbMATH DE number 6859848
scientific article