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This paper concerns smooth supersonic flows with Lipschitz continuous speedin two-dimensional infinite expanding nozzles, which are governed by aquasilinear hyperbolic equation being singular at the sonic and vacuum state.The flow satisfies the slip condition on the walls and the flow velocity isprescribed at the inlet. First, it is proved that if the incoming flow is awayfrom the sonic and vacuum state and its streamlines are rarefactive at theinlet, then a flow in a straight nozzle never approaches the sonic and vacuumstate in any bounded region. Furthermore, a sufficient and necessary conditionof the incoming flow at the inlet is derived for the existence of a globalsmooth supersonic flow in a straight nozzle. Then, it is shown that for eachincoming flow satisfying this condition, there exists uniquely a global smoothsupersonic flow in a symmetric nozzle with convex upper wall. It is noted thatsuch a flow may contain a vacuum. If there is a vacuum for a global smoothtransonic flow in a symmetric nozzle with convex upper wall, it is proved thatfor the symmetric upper part of the flow, the first vacuum point along thesymmetric axis must be located at the upper wall and the set of vacuum pointsis the closed domain bounded by the tangent half-line of the upper wall at thispoint to downstream and the upper wall after this point. Moreover, the flowspeed is globally Lipschitz continuous in the nozzle, and on the boundarybetween the gas and the vacuum, the flow velocity is along this boundary andthe normal derivatives of the flow speed and the square of the sound speed bothare zero. As an immediate consequence, the local smooth transonic flow obtainedin [10] can be extended into a global smooth transonic flow in a symmetricnozzle whose upper wall after the local flow is convex.

Global Smooth Supersonic Flows in Infinite Expanding Nozzles