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We develop a new approach and employ it to establish the global existence andnonlinear structural stability of attached weak transonic shocks in steadypotential flow past three-dimensional wedges; in particular, the restrictionthat the perturbation is away from the wedge edge in the previous results isremoved. One of the key ingredients is to identify a "good" direction of theboundary operator of a boundary condition of the shock along the wedge edge,based on the non-obliqueness of the boundary condition for the weak shock onthe edge. With the identification of this direction, an additional boundarycondition on the wedge edge can be assigned to make sure that the shock isattached on the edge and linearly stable under small perturbation. Based on thelinear stability, we introduce an iteration scheme and prove that there existsa unique fixed point of the iteration scheme, which leads to the globalexistence and nonlinear structural stability of the attached weak transonicshock. This approach is based on neither the hodograph transformation nor thespectrum analysis, and should be useful for other problems with similardifficulties.

Stability of Attached Transonic Shocks in Steady Potential Flow past Three-Dimensional Wedges