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We are concerned with inverse problems for supersonic potential flows pastinfinite axisymmetric Lipschitz cones. The supersonic flows under considerationare governed by the steady isentropic Euler equations for axisymmetricpotential flows, which involve a singular geometric source term. We first studythe inverse problem for the stability of an oblique conical shock as aninitial-boundary value problem with both the generating curve of the conesurface and the leading conical shock front as free boundaries. We thenestablish the existence and asymptotic behavior of global entropy solutionswith bounded BV norm of this problem, when the Mach number of the incoming flowis sufficiently large and the total variation of the pressure distribution onthe cone is sufficiently small. To this end, we first develop a modifiedGlimm-type scheme to construct approximate solutions by self-similar solutionsas building blocks to balance the influence of the geometric source term. Thenwe define a Glimm-type functional, based on the local interaction estimatesbetween weak waves, the strong leading conical shock, and self-similarsolutions, along with the construction of the approximate generating curves ofthe cone surface. Next, when the Mach number of the incoming flow issufficiently large, by asymptotic analysis of the reflection coefficients inthose interaction estimates, we prove that appropriate weights can be chosen sothat the corresponding Glimm-type functional decreases in the flow direction.Finally, we determine the generating curves of the cone surface and establishthe existence of global entropy solutions containing a strong leading conicalshock, besides weak waves. Moreover, the entropy solution is proved to approachasymptotically the self-similar solution determined by the incoming flow andthe asymptotic pressure on the cone surface at infinity.

Stability of Inverse Problems for Steady Supersonic Flows Past Lipschitz Perturbed Cones