| Summary: | 博士 === 國立交通大學 === 應用數學系所 === 106 === This dissertation investigates asymptotic behavior of stationary Navier-Stokes flows in the exterior domains of either two or three-dimensional hyperbolic spaces. We rst mention some classical results about stationary Navier-Stokes flows passing an obstacle in the 2D-Euclidean setting as a motivation for this thesis. Then we develop the far-range behavior of a solution to the Navier-Stokes equation on a negatively curved manifold, the background space with which we work is exactly the hyperbolic
space H^{n}(-a^{2}) with negative sectional curvature -a^{2} for the spatial dimension n = 2 and 3. Hence,
the stationary Navier-Stokes flows which we studied take place in an exterior domain in H^{n}(-a^{2}) with a circular obstacle as represented by a geodesic ball in H^{n}(-a^{2}).
In this thesis, we study the asymptotic behavior of the velocity, the associated vorticity and the pressure of a stationary Navier-Stokes ow in an exterior domain in the setting of two-dimensional hyperbolic spaces. More precisely, we can show that the velocity prole with nite Dirichlet norm in the hyperbolic setting decays to zero by means of new approach which is quite different from
Euclidean setting. We also address the exponential decay of the associated vorticity and the behavior of the pressure at far-range.
Furthermore, the other main result is about the behavior at far range of the velocity prole with
analogical nite Dirichlet integral to the stationary Navier-Stokes equation in the three dimensional hyperbolic setting.
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