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In this paper, we are concerned with the global existence and stability of asmooth supersonic flow with vacuum state at infinity in a 3-D infinitely longdivergent nozzle. 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 theform $\p_t^2-\ds\f{1}{(1+t)^{2(\g-1)}}(\p_1^2+\p_2^2)+\ds\f{2(\g-1)}{1+t}\p_t$for $1<\g<2$. From the physical point of view, due to the expansive geometricproperty of the divergent nozzle and the mass conservation of gas, the movinggas in the nozzle will gradually become rarefactive and tends to a vacuum stateat infinity, which implies that such a smooth supersonic flow should beglobally stable for small perturbations since there are no strong resultingcompressions in the motion of the flow. We will confirm such a global stabilityphenomena by rigorous mathematical proofs and further show that there do notexist vacuum domains in any finite part of the nozzle.

On global multidimensional supersonic flows with vacuum states at infinity