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

Abstract

In this article, we present an inverse problem for the nonlinear 1D Kuramoto–Sivashinsky (KS) 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 also at some positive time in the whole space domain. The Lipschitz stability for this inverse problem is our main result and it relies on the Bukhgeĭm–Klibanov method. The proof is indeed based on a global Carleman estimate for the linearized KS equation.

Keywords:

• inverse problem
• Kuramoto–Sivashinsky equation
• Carleman estimate

AMS Subject Classifications::

• 35R30
• 35K55

Acknowledgements:

This work began while L. Baudouin and E. Crépeau were visiting the Universidad Técnica Federico Santa María on the framework of the MathAmsud project CIP-PDE. This study was partially supported by Fondecyt #11080130, Fondecyt #11090161, ANR C-QUID and CISIFS, and CMM-Basal grants.