Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. II: General case
This paper is the second part of the authors' work (for part I, see [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 122, 125--168 (2015; Zbl 1326.35248)]). The initial-boundary value problem to the Navier-Stokes system is considered in a specific domain \(\Omega\subset \mathbb{R}^m\), \(m=2,3\). \(\Omega\) is ``a connected finite union of thin finite cylinders (in the 2D case respectively thin rectangles)''. The small positive parameter \(\varepsilon\) characterizes the thickness of cylinders. \[ \begin{aligned} &\frac{\partial v}{\partial t}-\nu\Delta v+(v\cdot\nabla)v+\nabla p=0,\quad \mathrm{div}\,v=0,\quad x\in\Omega,\quad t>0,\\ & v=g\quad \text{on }\partial\Omega,\quad v(x,0)=0. \end{aligned} \] It is assumed that \[ g(x,0)=0,\quad \int\limits_{\partial\Omega} g\cdot n\,dS=0. \] The authors construct a family of asymptotic approximation \(u^{(j)}\) to the solution of the problem. The norms of the difference \(v-u^{(j)}\) are estimated via powers of the thickness parameter \(\varepsilon\). In the appendix, the authors present results concerning the existence, uniqueness and asymptotic behavior of the solution to the non-steady Stokes problem in unbounded domains with cylindrical outlets to infinity.
scientific article; zbMATH DE number 6479797
scientific article