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

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 in $BV$ 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.Comment: 50 pages; 7 figue

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

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.Comment: 41 pages, 5 figures. arXiv admin note: text overlap with arXiv:2008.0240

Non-existence of strong regular reflections in self-similar potential flow

We consider shock reflection which has a well-known local non-uniqueness: the reflected shock can be either of two choices, called weak and strong. We consider cases where existence of a global solution with weak reflected shock has been proven, for compressible potential flow. If there was a global strong-shock solution as well, then potential flow would be ill-posed. However, we prove non-existence of strong-shock analogues in a natural class of candidates

Oblique shock reflection from an axis of symmetry: shock dynamics and relation to the Guderley singularity

Oblique shock reflection from an axis of symmetry is studied using Whitham's theory of geometrical shock dynamics, and the results are compared with previous numerical simulations of the phenomenon by Hornung (2000). The shock shapes (for strong and weak shocks), and the location of the shock-shock (for strong shocks), are in good agreement with the numerical results, though the detail of the shock reflection structure is, of course, not resolved by shock dynamics. A guess at a mathematical form of the shock shape based on an analogy with the Guderley singularity in cylindrical shock implosion, in the form of a generalized hyperbola, fits the shock shape very well. The smooth variation of the exponent in this equation with initial shock angle from the Guderley value at zero to 0.5 at 90° supports the analogy. Finally, steady-flow shock reflection from a symmetry axis is related to the self-similar flow

Multidimensional Conservation Laws: Overview, Problems, and Perspective

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.Comment: 43 pages, 3 figure