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 \(L_2(\Omega)\).

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) |

##### Keywords:

initial condition; regular slippage; Navier-Stokes system; asymptotic formula; Euler equations
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\textit{S. N. Alekseenko}, Sib. Math. J. 35, No. 2, 209--230 (1994; Zbl 0856.35099); translation from Sib. Mat. Zh. 35, No. 2, 235--257 (1994)

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##### References:

[1] | S. N. Alekseenko, ”A weak solution to the Navier-Stokes system with the condition of regular slippage on part of a boundary,” in: Studies of Integro-Differential Equations [in Russian], Frunze,23, 1991, pp. 90–102. |

[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. |

[3] | O. A. Ladyzhenskaya, Mathematical Problems of the Dynamics of a Viscous Incompressible Fluid [in Russian], Nauka, Moscow (1970). · Zbl 0215.29004 |

[4] | S. N. Antontsev, A. V. Kazhikhov, and V. N. Monakhov, Boundary Problems in Mechanics of Nonhomogeneous Fluids [in Russian], Nauka, Novosibirsk (1983). · Zbl 0568.76001 |

[5] | È. B. Bykhovskiî and N. V. Smirnov, ”On orthogonal decompositions of the space of vector-functions square-summable in a given domain,” Trudy Mat. Inst. Steklov Akad. Nauk SSSR,59, 5–36 (1960). |

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