Inverse problems for time-dependent nonlinear transport equations

TL;DR

This paper addresses the inverse problem of recovering a time-dependent coefficient in nonlinear transport equations on Riemannian manifolds and Euclidean space, establishing uniqueness and well-posedness using geometrical optics and light ray transforms.

Contribution

It introduces a novel approach combining linearization and geometrical optics solutions to uniquely determine the time-dependent coefficient from boundary and initial/final data.

Findings

Unique recovery of the time-dependent coefficient from boundary measurements.

Well-posedness of the inverse problem for small initial and incoming data.

Extension of methodology to inverse source problems in scattering-free media.

Abstract

In this work, we investigate inverse problems of recovering the time-dependent coefficient in the nonlinear transport equation in both cases: two-dimensional Riemannian manifolds and Euclidean space $\mathbb{R}^n$, $n\geq 2$. Specifically, it is shown that its initial boundary value problem is well-posed for small initial and incoming data. Moreover, the time-dependent coefficient appearing in the nonlinear term can be uniquely determined from boundary measurements as well as initial and final data. To achieve this, the central techniques we utilize include the linearization technique and the construction of special geometrical optics solutions for the linear transport equation. This allows us to reduce the inverse coefficient problem to the inversion of certain weighted light ray transforms. Based on the developed methodology, the inverse source problem for the nonlinear transport equation in the scattering-free media is also studied.