Startseite Mathematik Carleman estimate for stochastic degenerate wave equation with drift and its application
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Carleman estimate for stochastic degenerate wave equation with drift and its application

  • Lin Yan EMAIL logo und Bin Wu
Veröffentlicht/Copyright: 1. Juli 2025

Abstract

This paper is devoted to establishing the Carleman estimate for the stochastic degenerate wave equation with drift in the weakly degenerate case. Due to the degeneracy, we first study an approximate nondegenerate system. In order to overcome the difficulty arising from the drift term, we introduce an auxiliary function by which the diffusion and the drift term are transformed into a complex whole. After complicated and detailed estimates, we then derive the Carleman estimate for the stochastic degenerate wave equation. As an application for our Carleman estimate, we solve an inverse problem of determining three unknowns simultaneously by some suitable observations for the stochastic degenerate wave equation with drift. More precisely, we obtain a global uniqueness result for our inverse problem.

MSC 2020: 35L05; 60H15; 35R30; 35A02

Award Identifier / Grant number: 12171248

Funding statement: This work is supported by NSFC (No. 12171248).

Acknowledgements

The authors gratefully thank the associate editor and the anonymous referees for their valuable comments and suggestions for improving this paper.

References

[1] G. Bao, S. N. Chow, P. Li and H. Zhou, Numerical solution of an inverse medium scattering problem with a stochastic source, Inverse Problems 26 (2010), Article ID 074014. 10.1088/0266-5611/26/7/074014Suche in Google Scholar

[2] G. Bao and X. Xu, An inverse random source problem in quantifying the elastic modulus of nanomaterials, Inverse Problems 29 (2013), no. 1, Article ID 015006. 10.1088/0266-5611/29/1/015006Suche in Google Scholar

[3] V. Barbu, A. Răşcanu and G. Tessitore, Carleman estimates and controllability of linear stochastic heat equations, Appl. Math. Optim. 47 (2003), no. 2, 97–120. 10.1007/s00245-002-0757-zSuche in Google Scholar

[4] A. L. Bukhgeĭm and M. V. Klibanov, Global uniqueness of a class of multidimensional inverse problems, Soviet Math. Dokl. 24 (1981), 244–247. Suche in Google Scholar

[5] L. Cavalier and A. Tsybakov, Sharp adaptation for inverse problems with random noise, Probab. Theory Related Fields 123 (2002), no. 3, 323–354. 10.1007/s004400100169Suche in Google Scholar

[6] O. Y. Èmanuilov, Controllability of parabolic equations, Sb. Math. 186 (1995), 879–900. 10.1070/SM1995v186n06ABEH000047Suche in Google Scholar

[7] P. Gao, M. Chen and Y. Li, Observability estimates and null controllability for forward and backward linear stochastic Kuramoto–Sivashinsky equations, SIAM J. Control Optim. 53 (2015), no. 1, 475–500. 10.1137/130943820Suche in Google Scholar

[8] Y. Hu and S. G. Peng, Adapted solution of a backward semilinear stochastic evolution equation, Stochastic Anal. Appl. 9 (1991), no. 4, 445–459. 10.1080/07362999108809250Suche in Google Scholar

[9] M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems 8 (1992), no. 4, 575–596. 10.1088/0266-5611/8/4/009Suche in Google Scholar

[10] M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse Ill-Posed Probl. 21 (2013), no. 4, 477–560. 10.1515/jip-2012-0072Suche in Google Scholar

[11] X. Liu and Y. Yu, Carleman estimates of some stochastic degenerate parabolic equations and application, SIAM J. Control Optim. 57 (2019), no. 5, 3527–3552. 10.1137/18M1221448Suche in Google Scholar

[12] Q. Lü, Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems, Inverse Problems 28 (2012), no. 4, Article ID 045008. 10.1088/0266-5611/28/4/045008Suche in Google Scholar

[13] Q. Lü, Observability estimate for stochastic Schrödinger equations and its applications, SIAM J. Control Optim. 51 (2013), no. 1, 121–144. 10.1137/110830964Suche in Google Scholar

[14] Q. Lü and X. Zhang, Global uniqueness for an inverse stochastic hyperbolic problem with three unknowns, Comm. Pure Appl. Math. 68 (2015), no. 6, 948–963. 10.1002/cpa.21503Suche in Google Scholar

[15] S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations, SIAM J. Control Optim. 48 (2009), no. 4, 2191–2216. 10.1137/050641508Suche in Google Scholar

[16] C. Wang, Y. Zhou, R. Du and Q. Liu, Carleman estimate for solutions to a degenerate convection-diffusion equation, Discrete Contin. Dyn. Syst. Ser. B 23 (2018), no. 10, 4207–4222. 10.3934/dcdsb.2018133Suche in Google Scholar

[17] B. Wu, Q. Chen and Z. Wang, Carleman estimates for a stochastic degenerate parabolic equation and applications to null controllability and an inverse random source problem, Inverse Problems 36 (2020), no. 7, Article ID 075014. 10.1088/1361-6420/ab89c3Suche in Google Scholar

[18] B. Wu, Y. Gao, Z. Wang and Q. Chen, Unique continuation for a reaction-diffusion system with cross diffusion, J. Inverse Ill-Posed Probl. 27 (2019), no. 4, 511–525. 10.1515/jiip-2017-0094Suche in Google Scholar

[19] B. Wu and J. Yu, Hölder stability of an inverse problem for a strongly coupled reaction-diffusion system, IMA J. Appl. Math. 82 (2017), no. 2, 424–444. 10.1093/imamat/hxw058Suche in Google Scholar

[20] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems 25 (2009), no. 12, Article ID 123013. 10.1088/0266-5611/25/12/123013Suche in Google Scholar

[21] L. Yan, Global uniqueness of an inverse problem for stochastic degenerate wave equation with three unknowns, Math. Methods Appl. Sci. 44 (2021), no. 17, 12545–12558. 10.1002/mma.7562Suche in Google Scholar

[22] Y. Yan, Carleman estimates for stochastic parabolic equations with Neumann boundary conditions and applications, J. Math. Anal. Appl. 457 (2018), no. 1, 248–272. 10.1016/j.jmaa.2017.08.003Suche in Google Scholar

[23] G. Yuan, Determination of two kinds of sources simultaneously for a stochastic wave equation, Inverse Problems 31 (2015), no. 8, Article ID 085003. 10.1088/0266-5611/31/8/085003Suche in Google Scholar

[24] G. Yuan, Determination of two unknowns simultaneously for stochastic Euler–Bernoulli beam equations, J. Math. Anal. Appl. 450 (2017), no. 1, 137–151. 10.1016/j.jmaa.2017.01.023Suche in Google Scholar

[25] X. Zhang, Carleman and observability estimates for stochastic wave equations, SIAM J. Math. Anal. 40 (2008), no. 2, 851–868. 10.1137/070685786Suche in Google Scholar

Received: 2024-08-22
Revised: 2025-05-21
Accepted: 2025-06-03
Published Online: 2025-07-01
Published in Print: 2025-08-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 10.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jiip-2024-0060/pdf
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