[Paper Review] Carleman estimate for linear viscoelasticity equations and an inverse source problem

Oleg Imanuvilov, Masahiro Yamamoto|arXiv (Cornell University)|Nov 25, 2017

Numerical methods in inverse problems15 references3 citations

TL;DR

This paper establishes a Carleman estimate for the linear system of viscoelasticity without assuming compact support for solutions, using pseudodifferential operators to handle decoupled divergence and curl components. It applies the estimate to prove Lipschitz stability in an inverse source problem, uniquely determining a spatially varying source factor from Neumann data on a lateral subboundary over a long time interval.

ABSTRACT

We consider the linear system of viscoelasticity with the homogeneous Dirichlet boundary condition. First we prove a Carleman estimate with boundary values of solutions of viscoelasticity system. Since a solution $u$ under consideration is not assumed to have compact support, in the decoupling of the Lamé operator by introducing div $u$ and rot $u$, we have no boundary condition for them, so that we have to carry out arguments by a pseudodifferential operator. Second we apply the Carleman estimate to an inverse source problem of determining a spatially varying factor of the external source in the linear viscoelastitiy by extra Neumann data on the lateral subboundary over a sufficiently long time interval and establish the stability estimate.

Motivation & Objective

• To develop a Carleman estimate for the linear viscoelasticity system when solutions do not have compact support.
• To overcome the challenge of strong coupling in the Lamé operator and integral memory terms in the viscoelastic model.
• To establish stability in an inverse source problem for identifying a spatially varying factor of the external force using Neumann boundary measurements.
• To extend Carleman estimates to systems with non-compact support, enabling inverse problem analysis under realistic observation conditions.

Proposed method

• Derive a Carleman estimate for the viscoelastic system using pseudodifferential operators to manage boundary conditions for decoupled div u and rot u components.
• Decouple the Lamé operator via introduction of divergence and curl of the displacement vector, treating them as independent variables.
• Use a weighted $ L^2 $-norm estimate with large parameter to control solutions, incorporating the full structure of the viscoelastic operator.
• Apply the Carleman estimate to an inverse source problem by utilizing Neumann data on a lateral subboundary over a sufficiently long time interval.
• Establish a stability estimate via G 5arding-type inequality and energy estimates in Sobolev spaces with weights.
• Employ cutoff functions and symbol estimates to control error terms arising from non-compact support and boundary layer effects.

Experimental results

Research questions

• RQ1 Can a Carleman estimate be established for the linear viscoelasticity system when solutions are not compactly supported in space-time?
• RQ2 How can the strong coupling of the Lamé operator be handled in the absence of compact support?
• RQ3 Is it possible to uniquely determine the spatially varying factor of an external source using only Neumann data on a lateral subboundary over a long time interval?
• RQ4 What stability estimate can be derived for the inverse source problem in viscoelasticity under such partial boundary observations?
• RQ5 Can pseudodifferential operator techniques effectively manage the lack of boundary conditions for divergence and curl components in the decoupled system?

Key findings

• A Carleman estimate is established for the linear viscoelasticity system without assuming compact support for the solution, using pseudodifferential operators to handle the decoupled divergence and curl components.
• The Carleman estimate holds uniformly in a large parameter and controls the solution in a weighted $ L^2 $-norm, enabling observability and uniqueness results.
• Lipschitz stability is proven for the inverse source problem: the spatially varying factor of the external force is uniquely determined from Neumann data on a lateral subboundary over a sufficiently long time interval.
• The stability estimate is quantified in terms of the distance between observation subboundary and the support of the source, with explicit dependence on the geometry and regularity of coefficients.
• The method relies on a G 5arding-type inequality for pseudodifferential operators with symbols in $ C^2_{cl}S^{2,s} $, controlling error terms via cutoff functions and Sobolev norms.
• The analysis accounts for the integral memory term in the viscoelastic equation by treating the kernel functions $ \widetilde{\lambda}, \widetilde{\mu} $ as smooth functions in space and time.