  We are concerned with inverse problems for supersonic potential flows past
infinite axisymmetric Lipschitz cones. The supersonic flows under consideration
are governed by the steady isentropic Euler equations for axisymmetric
potential flows, which involve a singular geometric source term. We first study
the inverse problem for the stability of an oblique conical shock as an
initial-boundary value problem with both the generating curve of the cone
surface and the leading conical shock front as free boundaries. We then
establish the existence and asymptotic behavior of global entropy solutions
with bounded BV norm of this problem, when the Mach number of the incoming flow
is sufficiently large and the total variation of the pressure distribution on
the cone is sufficiently small. To this end, we first develop a modified
Glimm-type scheme to construct approximate solutions by self-similar solutions
as building blocks to balance the influence of the geometric source term. Then
we define a Glimm-type functional, based on the local interaction estimates
between weak waves, the strong leading conical shock, and self-similar
solutions, along with the construction of the approximate generating curves of
the cone surface. Next, when the Mach number of the incoming flow is
sufficiently large, by asymptotic analysis of the reflection coefficients in
those interaction estimates, we prove that appropriate weights can be chosen so
that the corresponding Glimm-type functional decreases in the flow direction.
Finally, we determine the generating curves of the cone surface and establish
the existence of global entropy solutions containing a strong leading conical
shock, besides weak waves. Moreover, the entropy solution is proved to approach
asymptotically the self-similar solution determined by the incoming flow and
the asymptotic pressure on the cone surface at infinity.
