Archive for Rational Mechanics and Analysis

, Volume 189, Issue 1, pp 159-188

First online:

Vanishing Viscosity Method for Transonic Flow

  • Gui-Qiang ChenAffiliated withDepartment of Mathematics, Northwestern University Email author 
  • , Marshall SlemrodAffiliated withDepartment of Mathematics, University of Wisconsin
  • , Dehua WangAffiliated withDepartment of Mathematics, University of Pittsburgh

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A vanishing viscosity method is formulated for two-dimensional transonic steady irrotational compressible fluid flows with adiabatic constant \(\gamma \in [1,3)\) . This formulation allows a family of invariant regions in the phase plane for the corresponding viscous problem, which implies an upper bound uniformly away from cavitation for the viscous approximate velocity fields. Mathematical entropy pairs are constructed through the Loewner–Morawetz relation via entropy generators governed by a generalized Tricomi equation of mixed elliptic–hyperbolic type, and the corresponding entropy dissipation measures are analyzed so that the viscous approximate solutions satisfy the compensated compactness framework. Then the method of compensated compactness is applied to show that a sequence of solutions to the artificial viscous problem, staying uniformly away from stagnation with uniformly bounded velocity angles, converges to an entropy solution of the inviscid transonic flow problem.