Stopping a viscous fluid by a feedback dissipative field. I: The stationary Stokes problem
The paper studies the stationary flow of an incompressible viscous fluid with velocity \({\mathbf u}= (u(x, y), v(x, y))\) in \(\Omega= (0,\infty)\times L\). Here, \({\mathbf u}\) satisfies the Stokes system: \[ -\nu\Delta{\mathbf u}={\mathbf f}-\nabla p;\quad \text{div\,}{\mathbf u}= 0;\quad{\mathbf u}(0,y)={\mathbf u}_*(y);\quad{\mathbf u}(x,y)|_{y= 0,L}=\mathbf{0}; \] and \(p(x, y)\) denotes the hydrostatic pressure function. The localized body forces \({\mathbf f}(x,y,{\mathbf u})\) (times \((-1)\)) is assumed in a sense coercive in \({\mathbf u}\) such that \({\mathbf f}(x,y,{\mathbf u})= 0\) for \(x\geq\exists\;x_{{\mathbf f}}(> 0)\). The purpose is to find an \(x_{{\mathbf u}}> 0\) such that \({\mathbf u}(x, y)= \mathbf{0}\) for \(x\geq x_{{\mathbf u}}\). This property is called the localization effect. Since \({\mathbf f}\) contains the term \({\mathbf u}\), it is a feedback dissipative field. However, the form of ``feedback'' looks different from the regular one in control theory. The localization effect is shown by applying an energy method to the associated stream function \(\psi\), i.e., \({\mathbf u}= (\psi_y, -\psi_x)\).
scientific article; zbMATH DE number 2161222