Lipschitz stability in an inverse problem for the Kuramoto-Sivashinsky equation

TL;DR

This paper investigates the inverse problem of determining the anti-diffusion coefficient in the 1D Kuramoto-Sivashinsky equation using boundary measurements, establishing uniqueness and Lipschitz stability through Carleman estimates.

Contribution

It introduces a novel approach to prove Lipschitz stability for the inverse problem in the nonlinear Kuramoto-Sivashinsky equation using Carleman estimates.

Findings

Proved uniqueness of the inverse problem.

Established Lipschitz stability for the coefficient reconstruction.

Applied Carleman estimates to a nonlinear PDE context.

Abstract

This paper presents an inverse problem for the nonlinear 1-d Kuramoto-Sivashinsky (K-S) equation. More precisely, we study the nonlinear inverse problem of retrieving the anti-diffusion coefficient from the measurements of the solution on a part of the boundary and at some positive time everywhere. Uniqueness and Lipschitz stability for this inverse problem are proven with the Bukhgeim-Klibanov method. The proof is based on a global Carleman estimate for the linearized K-S equation.