Oseen resolvent estimates with small resolvent parameter (Q1743810)

The Oseen resolvent system in 3D exterior domain is studied in the paper. \[ -\Delta u+\tau\frac{\partial u}{\partial x_1}+\lambda u+\nabla \pi=f,\quad \text{div}\,u=0,\quad x\in\Omega, \] \[ u=0\quad \text{on}\;\partial\Omega, \] where \(\Omega\) is an open set in \(\mathbb{R}^3\) with bounded complement, \(\tau\) is a positive constant corresponding to Reynolds number, \(f\) is a given vector function, vector \(u\) and scalar \(\pi\) are unknown functions, \(\lambda\in\mathbb{C}\) is a spectral parameter. The series of estimates for the function \(u\) are obtained for \[ \lambda\in\mathbb{C}\setminus\{0\},\quad \text{Re}\,\lambda\geq 0,\quad |\lambda|\leq\frac{\tau^2}{4}. \] It is proved in particular that \[ \| u \|_p\leq\frac{C}{\lambda^2}\| f \|_p \] for any \(p>1\). The application of results of the article are discussed.