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An Inverse Problem for Multi-Dimensional Piston Models with Large Velocity Variations

Dian Hu, Qianfeng Li, Yongqian Zhang

Abstract

When a circular symmetric piston suddenly expands into a still gas, a leading shock wave is generated. This paper investigates an inverse problem of reconstructing the trajectory of the piston from the given leading shock front and the given initial flow conditions. We observe that in piston models, as the initial density goes to zero, the piston approaches the shock front; however, in the region between the piston and the shock front, the strict hyperbolicity of the system degenerates. By applying asymptotic analysis, we provide quantitative characterizations of the distance between the piston and the shock front, and the degeneration of strict hyperbolicity. Consequently, by designing appropriate a priori assumptions to balance the benefits and drawbacks arising as the initial density approaches zero, we employ the method of characteristics to prove the global-in-time existence of the piecewise smooth solution for this inverse problem. In particular, the resulting flow structure exhibits significant velocity variations.

An Inverse Problem for Multi-Dimensional Piston Models with Large Velocity Variations

Abstract

When a circular symmetric piston suddenly expands into a still gas, a leading shock wave is generated. This paper investigates an inverse problem of reconstructing the trajectory of the piston from the given leading shock front and the given initial flow conditions. We observe that in piston models, as the initial density goes to zero, the piston approaches the shock front; however, in the region between the piston and the shock front, the strict hyperbolicity of the system degenerates. By applying asymptotic analysis, we provide quantitative characterizations of the distance between the piston and the shock front, and the degeneration of strict hyperbolicity. Consequently, by designing appropriate a priori assumptions to balance the benefits and drawbacks arising as the initial density approaches zero, we employ the method of characteristics to prove the global-in-time existence of the piecewise smooth solution for this inverse problem. In particular, the resulting flow structure exhibits significant velocity variations.
Paper Structure (4 sections, 10 theorems, 126 equations, 2 figures)

This paper contains 4 sections, 10 theorems, 126 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Shock wave moving with constant speed
  3. Asymptotic expansion of the solution near the shock
  4. proof of Theorem \ref{['thm1']}

Key Result

Theorem 1.1

For any given positive constants $\kappa_1,\kappa_2,\kappa_3,\kappa_{4}, \varpi_0$ there exists $\epsilon>0$ such that if the given initial data $(\rho_{\infty},0)$ and the given shock trajectory $\mathsf{S} =\{(r,t):r=s(t), t>0\}$ jointly satisfy $0<\rho_{\infty}<\epsilon$ and then problem EE1-EE4 globally admits $b(t)\in C^2(\mathbb{R}^{+})$ and $(\rho,v)\in C^1(\Omega).$

Figures (2)

  • Figure 1.1: The schematic diagram for spherical piston inverse problem
  • Figure 4.1: $\Omega_{\mathsf{T}}$ and characteristic curves therein

Theorems & Definitions (22)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.1
  • Definition 2.1
  • Theorem 2.3: Zeroth-order estimates for the self-similar flow field
  • proof
  • Theorem 2.4: First-order estimates for the self-similar flow field
  • ...and 12 more