Carleman Estimate and Inverse Source Problem for Biot's Equations Describing Wave Propagation in Porous Media

TL;DR

This paper establishes a new Carleman estimate for Biot's equations describing wave propagation in porous media, leading to a stability result for an inverse source problem based on surface traction observations.

Contribution

It introduces a novel Carleman estimate for Biot's system and applies it to prove a Hölder stability estimate for an inverse source problem.

Findings

Hölder stability estimate for the inverse source problem

Development of a new Carleman estimate for Biot's equations

Application to determine body force from surface traction data

Abstract

According to Biot's paper in 1956, by using the Lagrangian equations in classical mechanics, we consider a problem of the filtration of a liquid in porous elastic-deformation media whose mechanical behavior is described by the Lam'e system coupled with a hyperbolic equation. Assuming the null surface displacement on the whole boundary, we discuss an inverse source problem of determining a body force only by observation of surface traction on a suitable subdomain along a sufficiently large time interval. Our main result is a H\"older stability estimate for the inverse source problem, which is proved by a new Carleman estimat for Biot's system.