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