Research article

Identification of the source term in Navier-Stokes system with incomplete data

  • Received: 19 February 2019 Accepted: 12 May 2019 Published: 27 May 2019
  • MSC : 93B05, 49J20, 76D05

  • The aim of this work is to get instantaneous information at fixed instant T on pollution term in Navier-Stokes system in which the initial condition is incomplete. The best method which can solve this problem is the sentinel method; It allows estimating the pollution term at which we look for information independently of the missing term that we do not want to identify. So, we prove the existence of such instantaneous sentinel by solving a problem of controllability with constraint on the control.

    Citation: Berhail Amel, Rezzoug Imad. Identification of the source term in Navier-Stokes system with incomplete data[J]. AIMS Mathematics, 2019, 4(3): 516-526. doi: 10.3934/math.2019.3.516

    Related Papers:

  • The aim of this work is to get instantaneous information at fixed instant T on pollution term in Navier-Stokes system in which the initial condition is incomplete. The best method which can solve this problem is the sentinel method; It allows estimating the pollution term at which we look for information independently of the missing term that we do not want to identify. So, we prove the existence of such instantaneous sentinel by solving a problem of controllability with constraint on the control.


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    [1] I. Rezzoug and A. Ayadi, Controllability for the parabolic equations, PJM, 6 (2017), 638-648.
    [2] I. Rezzoug, Étude théorique et numérique des problèmes d'identification des systèmes gouvernés par des équations aux dérivées partielles Thèse de doctorat, Université de Oum El Bouaghi, Algérie, 2014.
    [3] J. P. Kernevez, The Sentinel Method and Its Applications to Environmental Pollution Problems, Boca Raton: CRC Press, 1997.
    [4] J. L. Lions, Sentinelle pour les systèmes distribués à données incomplètes. Paris: Masson, 1992, Vol 21.
    [5] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, 1984.
    [6] A. Berhail and A. Ayadi, Estimation of pollution term in Petrowsky system with incomplete data, Int. J. Open Problems Comp. Math., 3 (2010), 1-10.
    [7] I. Pranoto, Partial internal control recovery on 1-D Klein-Girdon systems, ITB J. Sci., 42A (2010), 11-22.
    [8] G. Massengo and J. P. Puel, Boundary sentinels with given sensitivity, Rev. Mat. Complut., 22 (2009), 165-185.
    [9] Y. Miloudi, O. Nakoulima and A. Omrane, On the instantaneous sentinels in pollution problems of incomplete data, Inverse Probl. Sci. Eng., (2008), 1-9.
    [10] O. Nakoulima, A revision of J.L.Lions notion of sentinels, Portugal. Math. (N.S)., 65 (2008), 1-22.
    [11] G. Massengo and O. Nakoulima, Sentinels with given sensitivity, Euro. J. Appl. Math., 19 (2008), 21-40. doi: 10.1017/S0956792507007267
    [12] J. Velin, Discriminating distributed sentinel involving a Navier-stokes problem and parameter identification, ESAIM Proc., 17 (2007), 143-166. doi: 10.1051/proc:071709
    [13] O. Nakoulima, Optimal control for distributed systems subject to null-controllability. Application to discriminating sentinels, ESAIM Contr. Optim. Ca., 13 (2007), 623-638. doi: 10.1051/cocv:2007038
    [14] A. Ayadi and M. Djebarni, Pollution terms estimations in parabolic system with incomplete data, Far East J. Math. Sci., 17 (2005), 317-328.
    [15] O. Nakoulima, Contrȏlabilité à zéro avec contraintes sur le contrȏle, C. R. Math., 339 (2004), 405-410. doi: 10.1016/j.crma.2004.07.005
    [16] O. A. Ladyzhenskaya, Sixth problem of the millennium: Navier-Stokes equations, existence and smoothness, Russ. Math. Surv., 58 (2003), 251-286. doi: 10.1070/RM2003v058n02ABEH000610
    [17] G. Chavent, Generalized sentinels defined via least squares, Rapport de Recherche de l'INRIA, N 1932, (1993), 1-32.
    [18] I. Rezzoug, Identification d'une partie de la frontière inconnue d'une membrane, Thèse de magister, Université de Oum El Bouaghi, Algérie, 2009.
    [19] M. Clerc, P. L. Tallec, M. Mallet, Contrȏle optimale de Navier-Stokes Parabolisé, Rapport de recherche, N 2653, (1995), 1-48.
    [20] A. V. Fursikov, Problèmes de contrȏle et résultats sur la résolution unique de problèmes mixtes pour les équations de Navier-Stokes et Euler tridimensionnelles, Mat. Sbornik, 115 (1981), 281-306.
    [21] J. L. Lions, G. Prodi, Un théorème d'existence et unicité dans les équations de Navier-Stokes en dimension 2, C. R. Acad. Sci. Paris, 248 (1959), 3519-3521.
    [22] O. A. Ladyzenskaya, Global solution of the boundary value problem for Navier-Stokes equations in 2 space dimensions, Doklady Akad. Nauk. SSSR, 123 (1958), 427-429.
    [23] Z. Mizohata, Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques, Mem. Coll. Sci. Univ. Kyoto A, 31 (1958), 219-239.
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