On the essential spectrum of the linearized Navier-Stokes operator
Let \(\Omega \) be a bounded domain in \(\mathbb{R}^n\), \(n \geq 2\), whose boundary is of class \(C^2\). Let \(H^p(\Omega)\) denote a Sobolev space. Let \(\mathbf L\) be the closure of the Stokes operator \[ \mathbf L_0 = \left( \begin{matrix} -\Delta I &-\text{grad}\cr \text{div} & 0 \end{matrix} \right), \] where \(I\) is the \(n \times n\) unit matrix, \(\mathbf L_0\) is defined on \(L^2(\Omega) ^n \times L^2(\Omega)\) with the image domain \((H^2(\Omega) \cap H^1_0(\Omega)) ^n \times H^1(\Omega)\). Then the essential spectrum of \(\mathbf L\) is \(( -1, -1/2) \).
scientific article; zbMATH DE number 1542579