Recovery of piecewise smooth parameters in an acoustic-gravitational system of equations from exterior Cauchy data

TL;DR

This paper proves that the wave speed and density within a bounded domain can be uniquely reconstructed from exterior measurements in an acoustic-gravitational system, despite the nonlocal effects of gravity and discontinuities.

Contribution

It introduces a method to uniquely determine piecewise smooth parameters in an acoustic-gravitational PDE system using exterior data and high-frequency wave analysis.

Findings

Unique determination of wave speed and density from exterior data

Handling of nonlocal gravitational effects in inverse problems

Reconstruction in the presence of discontinuities across interfaces

Abstract

In this paper, we study an inverse problem for an acoustic-gravitational system whose principal symbol is identical to that of an acoustic wave operator. The displacement vector of a gas or liquid between the unperturbed and perturbed flow is denoted by $u(t,x)$. It satisfies a partial differential equation (PDE) system with a principal symbol corresponding to an acoustic wave operator, but with additional terms to account for a global gravitational field and self-gravitation. These factors make the operator nonlocal, as it depends on the wave speed and density of mass. We assume that all parameters are piecewise smooth in $\mathbb{R}^3$ (i.e., smooth everywhere except for jump discontinuities across closed hypersurfaces called interfaces) but unknown inside a bounded domain $\Omega$. We are given the solution operator for this acoustic-gravitational system, but only outside $\Omega$ and only for initial data supported outside $\Omega$. Using high-frequency waves, we prove that the piecewise smooth wave speed and density are uniquely determined by this map under certain geometric conditions.