## Perturbation of the Navier-Stokes flow in an annular domain with the non-vanishing outflow condition

J. Math. Sci. Univ. Tokyo
Vol. 3 (1996), No. 1, Page 73--82.

Morimoto, Hiroko ; Ukai, Seiji
Perturbation of the Navier-Stokes flow in an annular domain with the non-vanishing outflow condition
The boundary value problem of the Navier-Stokes equations has been studied so far only under the vanishing outflow condition due to Leray. We consider this problem in an annular domain $D = \{ {\Vec x} \in {\bf R}^2 ; R_1 < |{\Vec x}| < R_2 \},$ under the boundary condition with non-vanishing outflow. In a previous paper of the first author, an exact solution is obtained for a simple boundary condition of non-vanishing outflow type: ${\Vec u} = \displaystyle{Î¼ \over R_i} {\Vec e}_r + b_i{\Vec e}_Î¸ \ \mbox{ on } Î_i, \ i=1, 2,$ where $Î¼,b_1,b_2$ are arbitrary constants. In this paper, we show the existence of solutions satisfying the boundary condition: ${\Vec u} = \{ \displaystyle{Î¼ \over {R_i}}+ \varphi_i(Î¸)\}{\Vec e}_r + \{b_i + Ï_i(Î¸)\} {\Vec e}_{Î¸} \ \mbox{ on } \ Î_i,\ i=1, 2,$ where $\varphi_i(Î¸),Ï_i(Î¸)$ are $2 Ï$-periodic smooth function of $Î¸$, under some additional condition.