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

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This study solves inverse problems for supersonic potential flows past cones, establishing the existence of global entropy solutions. It confirms the stability of conical shock waves and determines cone shapes from flow properties.

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Area of Science:

  • Fluid Dynamics
  • Aerodynamics
  • Partial Differential Equations

Background:

  • Investigates supersonic potential flows governed by steady isentropic Euler equations for axisymmetric flows.
  • Addresses inverse problems for infinite axisymmetric Lipschitz cones, featuring a singular geometric source term.

Purpose of the Study:

  • To study the inverse problem for the stability of an oblique conical shock.
  • To establish the existence and asymptotic behavior of global entropy solutions for supersonic potential flows past cones.
  • To determine the generating curves of the cone surface from flow characteristics.

Main Methods:

  • Employs a modified Glimm-type scheme using self-similar solutions to handle the geometric source term.
  • Develops a Glimm-type functional incorporating interaction estimates between waves, shocks, and self-similar solutions.
  • Utilizes asymptotic analysis of reflection coefficients for large incoming flow Mach numbers.

Main Results:

  • Establishes the existence and bounded BV norm of global entropy solutions under specific conditions (large Mach number, small pressure variation).
  • Proves the decrease of the Glimm-type functional in the flow direction through appropriate weighting.
  • Determines the generating curves of the cone surface and confirms the existence of global entropy solutions with a leading conical shock.

Conclusions:

  • The study successfully determines cone shapes from supersonic flow data, confirming the existence of stable conical shock solutions.
  • Entropy solutions asymptotically approach self-similar solutions determined by incoming flow and asymptotic pressure.
  • Provides a robust mathematical framework for analyzing supersonic flows past cones via inverse problems.