Existence and asymptotic representation of weak solutions to the flowing problem under the condition of regular slippage on solid walls
type: Item
instance of: scholarly article
title: Existence and asymptotic representation of weak solutions to the flowing problem under the condition of regular slippage on solid walls
zbMATH Open document ID: 0856.35099
DOI: 10.1007/BF02104770
author: S. N. Alekseenko
published in: Siberian Mathematical Journal
publication date: 1997-03-02T00:00:00Z
review text: We study the motion of a viscous incompressible fluid in a three-dimensional bounded domain \(\Omega\), which is described by the Navier-Stokes system \[ v_t- \nu \Delta v+ \sum^3_{k= 1} v_k v_{x_k}= \text{grad } p+ f,\quad\text{div } v= 0,\quad x\in \Omega,\quad t\in [0, T],\quad T=\text{const.} \] We consider the case in which the fluid flows through the domain \(\Omega\) and at the same time does not adhere to solid walls but slips over them; moreover, the instantaneous rotation axis of the fluid coincides with the normal vector of the boundary at each of its points (regular slippage). The principal result of the present article consists in proving an asymptotic formula of the form \[ v= u+ \Pi+ \sqrt{\nu\eta} \] for a weak solution \(v\) to the considered problem, where \(u\) is a smooth solution to the following problem for the system of Euler equations \[ u_t+ \sum^3_{k= 1} u_k u_{x_k}= \text{grad } q+ f,\quad \text{div } u= 0, \] \(\Pi\) is a boundary layer function near \(S_3\), and \(\eta(x, t)\) is a remainder bounded in \(L_2(\Omega)\).
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 916352
zbMATH Keywords: initial condition
zbMATH Keywords: regular slippage
zbMATH Keywords: Navier-Stokes system
zbMATH Keywords: asymptotic formula
zbMATH Keywords: Euler equations
MaRDI profile type: Publication
cites work: Q5618254
cites work: Q3685389
description: scientific article; zbMATH DE number 916352