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

TL;DR

This paper addresses an inverse problem in multi-dimensional piston models with large velocity variations, reconstructing piston trajectories from shock front data and initial conditions, using asymptotic analysis and characteristics methods.

Contribution

It introduces a novel approach to reconstruct piston motion in complex flow regimes with degenerating hyperbolicity, providing global existence results for the inverse problem.

Findings

Quantitative characterization of piston-shock front distance

Analysis of hyperbolicity degeneration in piston models

Proof of global-in-time existence of solutions

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.