On a resolvent estimate for the Stokes system with Neumann boundary condition.
The authors consider the Stokes resolvent problem with Neumann boundary condition in the domain \(\Omega\subset \mathbb{R}^n\), \(n\geq 2\). \[ \begin{aligned} &\lambda v-\mu\Delta v+\nabla p=f,\quad \nabla\cdot u=0\quad \text{in \;}\Omega\\ &\nu\cdot T(v,p)|_{\partial\Omega}=h \quad \text{on\;}\partial\Omega\end{aligned} \tag{1} \] where \(T(v,p)\) is the stress tensor, \(\nu\) is the unit outward normal to \(\partial\Omega\), \[ \lambda\in\{z\in C: z\neq 0,\;| \text{arg}\, z| <\pi-\varepsilon\},\;0<\varepsilon<\frac{\pi}2, \] \(\Omega\) is a bounded or exterior domain in \(\mathbb{R}^n\) with smooth boundary. The existence, uniqueness of solution to the problem (1) and its \(L_p\)-estimates are proved.The proof is based on using the space \(W^{-1}_p\). The problem (1) was already treated by \textit{G. Grubb} and \textit{V. Solonnikov} in [Math. Scand. 69, 217--290 (1991; Zbl 0766.35034)], \textit{G. Grubb} [Differ. Integral Equ. 8, No. 5, 1013--1046 (1995; Zbl 0831.35126)] by using a theory of pseudo-differential operators.
scientific article; zbMATH DE number 2004905