Discrete data assimilation algorithm for the three-dimensional Leray-α model

• Autorzy: Anh C. T. , Bach B. H.

Identyfikatory

DOI

10.4064/ba8162-11-2018

• Języki publikacji: EN

Abstrakty

EN

We study the data assimilation algorithm for the three-dimensional Leray-α model when the measurements are obtained discretely in time and may be contaminated by systematic errors. Under suitable conditions on the relaxation (nudging) parameter, the spatial mesh resolution, and the time step between successive measurements, we obtain an asymptotic in time estimate of the difference between the approximating solution and the unknown reference solution corresponding to the measurements, in an appropriate norm, which shows exponential convergence up to a term which depends on the size of the errors.

Słowa kluczowe

EN

discrete data assimilation; 3D Leray-α model; nudging; low Fourier modes projector

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2018
• Tom: Vol. 66, no. 2
• Strony: 143--156
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Anh C. T.

• Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

autor

Bach B. H.

• Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

Bibliografia

• [ANT16] D. A. Albanez, H. J. Nussenzveig Lopes and E. S. Titi, Continuous data assimilation for the three-dimensional Navier-Stokes-α model, Asymptot. Anal. 97 (2016), 139-164.
• [AK16] H. Ali and P. Kaplický, Existence and regularity of solutions to the Leray-α model with Navier slip boundary conditions, Electron. J. Differential Equations 2016, paper 235, 13 pp.
• [AFS14] F. D. Araruna, E. Fernández-Cara and D. A. Souza, Uniform local null control of the Leray-α model, ESAIM Control Optim. Calc. Var. 20 (2014), 1181-1202.
• [AOT14] A. Azouani, E. Olson and E. S. Titi, Continuous data assimilation using general interpolant observables, J. Nonlinear Sci. 24 (2014), 277-304.
• [CT09] Y. Cao and E. S. Titi, On the rate of convergence of the two-dimensional α -models of turbulence to the Navier-Stokes equations, Numer. Funct. Anal. Optim. 30 (2009), 1231-1271.
• [CTV07] V. V. Chepyzhov, E. S. Titi and M. I. Vishik, On the convergence of solutions of the Leray-α model to the trajectory attractor of the 3D Navier-Stokes system, Discrete Contin. Dynam. Systems 17 (2007), 481-500.
• [CHOT05] A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-α model of turbulence, Proc. R. Soc. London Ser. A Math. Phys. Engrg. Sci. 461 (2005), 629-649.
• [D17] G. Deugoué, On the convergence of the uniform attractor for the 2D Leray-α model, Abstr. Appl. Anal. 2017, art. ID 1681857, 11 pp.
• [FLT15] A. Farhat, E. Lunasin and E. S. Titi, Continuous data assimilation for a 2D Bénard convection system through horizontal velocity measurements alone, Phys. D 303 (2015), 59-66.
• [FLT16a] A. Farhat, E. Lunasin and E. S. Titi, Data assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements, J. Math. Anal. Appl. 438 (2016), 492-506.
• [FLT16b] A. Farhat, E. Lunasin and E. S. Titi, Abridged continuous data assimilation for the 2D Navier-Stokes equations utilizing measurements of only one komponent of the velocity field, J. Math. Fluid Mech. 18 (2016), 1-23.
• [FLT17] A. Farhat, E. Lunasin and E. S. Titi, Data assimilation algorithm: The paradigm of the 3D Leray-α model of turbulence, arXiv:1702.01506v1 (2017).
• [FMT16] C. Foias, C. F. Mondaini and E. S. Titi, A discrete data assimilation scheme for the solutions of the two-dimensional Navier-Stokes equations and their statistics, SIAM J. Appl. Dynam. Systems 15 (2016), 2109-2142.
• [GH08] J. D. Gibbon and D. D. Holm, Estimates for the LANS-α, Leray-α and Bardina models in terms of a Navier-Stokes Reynolds number, Indiana Univ. Math. J. 57 (2008), 2761-2773.
• [HOT11] K. Hayden, E. Olson and E. S. Titi, Discrete data assimilation in the Lorenz and 2D Navier-Stokes equations, Phys. D 240 (2011), 1416-1425.
• [JMT17] M. S. Jolly, V. R. Martinez and E. S. Titi, A data assimilation algorithm for the subcritical surface quasi-geostrophic equation, Adv. Nonlinear Stud. 17 (2017), 167-192.
• [K09] P. Korn, Data assimilation for the Navier-Stokes-α equations, Phys. D 238 (2009), 1957-1974.
• [OT03] E. Olson and E. S. Titi, Determining modes for continuous data assimilation in 2D turbulence, J. Statist. Phys. 113 (2003), 799-840.
• [QDY17] H. Qiu, Y. Du and Z. Yao, Global Cauchy problem for a Leray-α model, Acta Math. Appl. Sin. Engl. Ser. 33 (2017), 207-220.
• [R01] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts Appl. Math., Cambridge Univ. Press, 2001.
• [T95] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd ed., CBMS-NSF Reg. Conf. Ser. Appl. Math. 66, SIAM, 1995.
• [Y12] K. Yamazaki, On the global regularity of generalized Leray-alpha type models, Nonlinear Anal. 75 (2012), 503-515.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).