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Existence and asymptotic representation of weak solutions to the flowing problem under the condition of regular slippage on solid walls. (English. Russian original) Zbl 0856.35099
Sib. Math. J. 35, No. 2, 209-230 (1994); translation from Sib. Mat. Zh. 35, No. 2, 235-257 (1994).
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 \(L_2(\Omega)\).

MSC:
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35D05 Existence of generalized solutions of PDE (MSC2000)
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[2] S. N. Alekseenko, ”Asymptotics in viscosity of weak solutions to the flowing problem for the Navier-Stokes system in a cylindrical domain,” in: Studies of Integro-Differential Equations [in Russian], Frunze,23, 1991, pp. 83–90.
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