Artificial boundary conditions for the Stokes and Navier-Stokes equations in domains that are layer-like at infinity
The subject of the paper is approximations to solutions of Stokes and Navier-Stokes equations in domains that are layer-like at infinity. Such a domain is \(\Omega\subset \mathbb{R}^3\) with a smooth boundary \(\partial\Omega\), and \(\Omega\) coincides with the layer \[ \Lambda=\{x=(y,z):\;y=(y_1,y_2)\in \mathbb{R}^2,\;| z| <\frac{1}{2}\} \] outside the ball \(B_{R_0}=\{x\in \mathbb{R}^3:\;| x| <R_0\}\). \((v^\infty,p^\infty)\) is the solution to the Stokes problem \[ -\nu \Delta v^\infty\,+\nabla p^\infty=f, \quad\text{div}\,v^\infty=0\quad \text{in}\;\Omega,\quad v^\infty=0 \quad \text{on}\;\partial\Omega. \] \((v^R,p^R)\) is the approximation to \((v^\infty,p^\infty)\). \((v^R,p^R)\) is the solution to the problem in a bounded domain \(\Omega_R=\{x=(r,\varphi,z)\in\Omega:\;r<R\}\) with boundary \(\partial\Omega_R=\Sigma_R\cup\Gamma_R\), \(\Sigma_R=\{x\in\partial\Omega:\;r<R\}\), \(\Gamma_R=\{x\in\Omega:\;r=R,| z| <\frac{1}{2}\}\) \[ \begin{aligned} -\nu \Delta v^R\,+\nabla p^R=f, &\quad\text{div}\,v^R=0\quad \text{in}\;\Omega_R,\quad v^R=0 \quad \text{on}\;\Sigma_R\\ &M_R(v^R,p^R)=0 \quad \text{on}\;\Gamma_R \end{aligned} \] Here \(M_R\) is a suitable artificial operator which involves the Steklov-Poincare operator. The norms \(\| v^\infty-v^R\|_{H^1(\Omega_R)}\) and \(\| p^\infty-p^R\|_{L^2(\Omega_R)}\) are estimated on the base of asymptotics of \((v^\infty,p^\infty)\). These norms vanish of order \(O(R^{-\alpha})\), where \(\alpha\) can be arbitrary large. Similar result is obtained to the Navier-Stokes equations.
scientific article; zbMATH DE number 5296722