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We are concerned with the structural stability of conical shocks in thethree-dimensional steady supersonic flows past Lipschitz perturbed cones whosevertex angles are less than the critical angle. The flows under considerationare governed by the steady isothermal Euler equations for potential flow withaxisymmetry so that the equations contain a singular geometric source term. Wefirst formulate the shock stability problem as an initial-boundary valueproblem with the leading conical shock-front as a free boundary, and thenestablish the existence and asymptotic behavior of global entropy solutions ofbounded variation (BV) of the problem. To achieve this, we first develop amodified Glimm scheme to construct approximate solutions via self-similarsolutions as building blocks in order to incorporate with the geometric sourceterm. Then we introduce the Glimm-type functional, based on the localinteraction estimates between weak waves, the strong leading conical shock, andself-similar solutions, as well as the estimates of the center changes of theself-similar solutions. To make sure the decreasing of the Glimm-typefunctional, we choose appropriate weights by careful asymptotic analysis of thereflection coefficients in the interaction estimates, when the Mach number ofthe incoming flow is sufficiently large. Finally, we establish the existence ofglobal entropy solutions involving a strong leading conical shock-front,besides weak waves, under the conditions that the Mach number of the incomingflow is sufficiently large and the weighted total variation of the slopes ofthe generating curve of the Lipschitz perturbed cone is sufficiently small.Furthermore, the entropy solution is shown to approach asymptotically theself-similar solution that is determined by the incoming flow and theasymptotic tangent of the cone boundary at infinity.

Stability of Conical Shocks in the Three-Dimensional Steady Supersonic Isothermal Flows past Lipschitz Perturbed Cones