The asymptotic properties of the solution to the Stokes problem in domains that are layer-like at infinity (Q1303800)

The present paper is the continuation of the authors' article [J. Math. Fluid Mech. 1, 78-116 (1999)]. Using the results of this paper the authors study the boundary-value problem to the Stokes equations \[ -\nu\Delta v+\nabla p=f,\quad \text{div} v=g \quad \text{in} \Omega, \qquad v=h \quad \text{on} \partial\Omega \] where \(\Omega\subset \mathbb{R}^3\) is a domain with the smooth boundary \(\partial\Omega\), coinciding outside the ball \(B_R=\{x\in \mathbb{R}^3:\;= |x|<R\}\) with the infinite layer \[ \Pi=\{x=(y,z):\;y=(y_1,y_2)\in \mathbb{R}^2;\;z\in (0,1)\} \] The authors study the properties of the solution \((v,p)\) assuming only that \((v,p)\in L^2_\beta(\Omega)\times L^2_\beta(\Omega)\) with some \(\beta\). \(L^2_\beta(\Omega)\) is the weighted space with the norm \[ \|v\|_{L^2_\beta(\Omega)}=\biggl(\int\limits_\Omega (1+r^2)^\beta|v|^2 dx\biggr)^{\frac 12}. \] In terms of specific weighted spaces the authors prove regularity results and a coercive estimate for the solution \((v,p)\in L^2_\beta(\Omega)\times L^2_\beta(\Omega)\). The asymptotic representation at infinity for this solution is constructed too.