Carleman estimate and an inverse source problem for the Kelvin-Voigt model for viscoelasticity

TL;DR

This paper establishes a Carleman estimate for the Kelvin-Voigt viscoelasticity model and uses it to prove Lipschitz stability in inverse source problems, also extending the approach to a compressible fluid system.

Contribution

It introduces a Carleman estimate for the Kelvin-Voigt model without compact support and applies it to inverse source problems, including a related fluid system.

Findings

Proved a Carleman estimate for the Kelvin-Voigt model.

Established Lipschitz stability for inverse source problems.

Extended the method to a compressible fluid system.

Abstract

We consider the Kelvin-Voigt model for the viscoelasticity, and prove a Carleman estimate for functions without compact supports. Then we apply the Carleman estimate to prove the Lipschitz stability in determining a spatial varying function in an external source term of Kelvin-Voigt model by a single measurement. Finally as a related system, we consider an isothermal compressible fluid system and apply the Carleman estimate to establish the Lipschitz stability for an inverse source problem for the compressible fluid system.