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We are concerned with the stability of multidimensional (M-D) transonicshocks in steady supersonic flow past multidimensional wedges. One of ourmotivations is that the global stability issue for the M-D case is much moresensitive than that for the 2-D case, which requires more careful rigorousmathematical analysis. In this paper, we develop a nonlinear approach andemploy it to establish the stability of weak shock solutions containing atransonic shock-front for potential flow with respect to the M-D perturbationof the wedge boundary in appropriate function spaces. To achieve this, we firstformulate the stability problem as a free boundary problem for nonlinearelliptic equations. Then we introduce the partial hodograph transformation toreduce the free boundary problem into a fixed boundary value problem near abackground solution with fully nonlinear boundary conditions for second-ordernonlinear elliptic equations in an unbounded domain. To solve this reducedproblem, we linearize the nonlinear problem on the background shock solutionand then, after solving this linearized elliptic problem, develop a nonlineariteration scheme that is proved to be contractive.

Stability of Transonic Shocks in Steady Supersonic Flow past Multidimensional Wedges