Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-23T20:22:32.056Z Has data issue: false hasContentIssue false
  • English
  • Français

CARLEMAN ESTIMATE AND UNIQUE CONTINUATION PROPERTY FOR THE LINEAR STOCHASTIC KORTEWEG–DE VRIES EQUATION

Part of: Equations of mathematical physics and other areas of application Stochastic analysis

Published online by Cambridge University Press:  05 June 2014

PENG GAO
PENG GAO*
Affiliation:
Institute of Mathematics, Jilin University, Changchun 130012, PR China email gaopengjilindaxue@126.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we obtain the well posedness of the linear stochastic Korteweg–de Vries equation by the Galerkin method, and then establish the Carleman estimate, leading to the unique continuation property (UCP) for the linear stochastic Korteweg–de Vries equation. This UCP cannot be obtained from the classical Holmgren uniqueness theorem.

Keywords

linear stochastic Korteweg–de Vries equationCarleman estimateunique continuation property

MSC classification

Secondary: 35Q53: KdV-like equations (Korteweg-de Vries) 60H15: Stochastic partial differential equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 2 , October 2014 , pp. 283 - 294
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Barbu, V., Răscanu, A. and Tessitore, G., ‘Carleman estimate and controllability of linear stochastic heat equations’, Appl. Math. Optim. 47 (2003), 97–120.Google Scholar
Chen, M. and Gao, P., ‘A new unique continuation property for the Korteweg–de Vries equation’, Bull. Aust. Math. Soc., to appear. Published online 10 January 2014.Google Scholar
Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions (Cambridge University Press, Cambridge, 1992).Google Scholar
Korteweg, D. J. and de Vries, G., ‘On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves’, Philos. Mag. 39 (1895), 422–443.CrossRefGoogle Scholar
Larkin, N. A., ‘Modified KdV equation with a source term in a bounded domain’, Math. Methods Appl. Sci. 29 (2006), 751–765.Google Scholar
Lü, Q., ‘Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems’, Inverse Problems 28 045008 (2012), 18 pp.Google Scholar
Lü, Q., ‘Observability estimate for stochastic Schrödinger equations and its applications’, SIAM J. Control Optim. 51 (2013), 121–144.Google Scholar
Meléndez, P. G., ‘Lipschitz stability in an inverse problem for the main coefficient of a Kuramoto–Sivashinsky type equation’, J. Math. Anal. Appl. 408 (2013), 275–290.Google Scholar
Tang, S. and Zhang, X., ‘Null controllability for forward and backward stochastic parabolic equations’, SIAM J. Control Optim. 48 (2009), 2191–2216.CrossRefGoogle Scholar
Zhang, X., ‘Carleman and observability estimates for stochastic wave equations’, SIAM J. Math. Anal. 40 (2008), 851–868.Google Scholar