中文摘要：

本文主要研究光滑超音速流从三维单位球面向无穷远处扩张的整体存在性和稳定性.我们用三维稳态位势流方程来描述超音速气体的运动,该方程在沿气流方向上是拟线性双曲型的（但是当半径趋于无限长时会发生退化）.它的线性部分满足形式δt^2-1/（1＋t）^（2（γ-1））（δ1^2＋δ2^2）＋2（（γ-1））/1＋tδt其中1〈γ〈2.从物理学的角度来看,由于随着半径的增大球的表面积也逐渐变大,以及气体质量的守恒性质,球面上的气流会逐渐地变稀薄,并且在无限远处将趋向于真空.这就隐含了当气体受到小的扰动后在无穷远处仍能保持稳定的状态.我们将通过严格的数学推理证明这种现象并且进一步说明在任何有限位置都不会出现真空.

英文摘要：

In this paper, we are concerned with the global existence and stability of a smooth supersonic flow which expands from a 3D unit ball to infinity. The flow is described by a 3-D steady potential equation, which is multi-dimensional quasilinear hyperbolic （but degenerate at infinity） with respect to the supersonic direction, and whose linearized part admits the form δt^2-1/（1＋t）^（2（γ-1））（δ1^2＋δ2^2）＋2（（γ-1））/1＋tδt for 1 〈 γ〈 2. From the physical point of view, due to the expansive geometric property of volume of expanded ball and the mass conservation of gas, the moving gas will gradually become rarefactive and tends to a vacuum state at infinity, which implies that such a smooth supersonic flow should be globally stable for small perturbations since there are no obvious resulting compressions in the motion of the flow. We will confirm such a global stability phenomena by rigorous mathematical proofs and further show that there do not exist vacuum domains in any finite part under the small perturbations.