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Continuous Data Assimilation Algorithm for the Two Dimensional Cahn–Hilliard–Navier–Stokes System

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Abstract

Based on the fact that dissipative dynamical systems possess finite degrees of freedom, a new continuous data assimilation algorithm for the two dimensional Cahn–Hilliard–Navier–Stokes system is introduced. In this paper, we provide some suitable conditions on the nudging parameters and the size of the spatial coarse mesh observables, which are sufficient to show that the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system under the assumption that the observed data are free of error. Thus, we can make the future predictions of the exact solution by the approximation solution of the continuous data assimilation algorithm if the initial data is missing, which usually appears in the fields of geophysical and biological sciences.

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References

  1. Daley, R.: Atmos. Data Anal. Cambridge University Press, New York (1991)

    Google Scholar 

  2. Azouani, A., Titi, E.S.: Feedback control of nonlinear dissipative systems by finite determining parameters-a reaction-diffusion paradigm. Evol. Equ. Control Theory 3(4), 579–594 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Lunasin, E., Titi, E.S.: Finite determining parameters feedback control for distributed nonlinear dissipative systems: A computational study. Evol. Equ. Control Theory 6(4), 535–557 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Azouani, A., Olson, E., Titi, E.S.: Continuous data assimilation using general interpolant observables. J. Nonlinear Sci. 24(2), 277–304 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cockburn, B., Jones, D.A., Titi, E.S.: Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems. Math. Comput. 66, 1073–1087 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Foias, C., Manley, O., Rosa, R., Temam, R.: Navier–Stokes Equations and Turbulence. Cambridge University Press, New York (2001)

    Book  MATH  Google Scholar 

  7. Foias, C., Prodi, G.: Sur le comportement global des solutions non-stationnaires des equations de Navier–Stokes en dimension 2. Rend. Semin. Mat. Univ. Padova 39, 1–34 (1967)

    MATH  Google Scholar 

  8. Foias, C., Temam, R.: Sur la détermination dun écoulement fluide par des observations discrétes. C. R. l’Acad. Sci. I 295(3), 239–241 (1982)

    MATH  Google Scholar 

  9. Foias, C., Temam, R.: Determination of the solutions of the Navier–Stokes equations by a set of nodal values. Math. Comput. 43, 117–133 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  10. Foias, C., Titi, E.S.: Determining nodes, finite difference schemes and inertial manifolds. Nonlinearity 4(1), 135–153 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Holst, M.J., Titi, E.S.: Determining projections and functionals for weak solutions of the Navier–Stokes equations. Contemp. Math. 204, 125–138 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jones, D.A., Titi, E.S.: Determining finite volume elements for the 2D Navier–Stokes equations. Physica D 60, 165–174 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jones, D.A., Titi, E.S.: Upper bounds on the number of determining modes, nodes and volume elements for the Navier–Stokes equations. Indiana Univ. Math. J. 42(3), 875–887 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Olson, E., Titi, E.S.: Determining modes for continuous data assimilation in 2D turbulence. J. Stat. Phys. 113, 799–840 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Olson, E., Titi, E.S.: Determining modes and Grashof number in 2D turbulence. Theor. Comput. Fluid Dyn. 22(5), 327–339 (2008)

    Article  MATH  Google Scholar 

  16. Foias, C., Jolly, M.S., Kravchenko, R., Titi, E.S.: A unified approach to determining forms for the 2D Navier–Stokes equations-the general interpolants case. Russ. Math. Surv. 69(2), 177–200 (2014)

    Article  MATH  Google Scholar 

  17. Foias, C., Mondaini, C.F., Titi, E.S.: A discrete data assimilation scheme for the solutions of the two-dimensional Navier–Stokes equations and their statistics. SIAM J. Appl. Dyn. Syst. 15(4), 2109–2142 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hoke, J., Anthes, R.: The initialization of numerical models by a dynamic relaxation technique. Monthly Weather Rev. 104, 1551–1556 (1976)

    Article  Google Scholar 

  19. Bessaih, H., Olson, E., Titi, E.S.: Continuous data assimilation with stochastically noisy data. Nonlinearity 28, 729–753 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Farhat, A., Lunasin, E., Titi, E.S.: Data assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements. J. Math. Anal. Appl. 438, 492–506 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Farhat, A., Jolly, M.S., Titi, E.S.: Continuous data assimilation for the 2D Bénard convection through velocity measurements alone. Physica D 303, 59–66 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Farhat, A., Lunasin, E., Titi, E.S.: Continuous data assimilation for a 2D Bénard convection system through horizontal velocity measurements alone. J. Nonlinear Sci. 27, 1065–1087 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Biswas, A., Foias, C., Mondaini, C.F., Titi, E.S.: Downscaling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations. Ann. l’Inst. Henri Poincaré C 36, 295–326 (2019)

    MathSciNet  MATH  Google Scholar 

  24. Carlson, E., Hudson, J., Larios, A.: Parameter recovery for the 2 dimensional Navier–Stokes equations via continuous data assimilation. SIAM J. Sci. Comput. 42(1), A250–A270 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. Carlson, E., Larios, A.: Sensitivity analysis for the 2D Navier–Stokes equations with applications to continuous data assimilation. J. Nonlinear Sci. 31(5), 84 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Farhat, A., Lunasin, E., Titi, E.S.: Abridged continuous data assimilation for the 2D Navier–Stokes equations utilizing measurements of only one component of the velocity field. J. Math. Fluid Mech. 18, 1–23 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Garcia-Archilla, B., Novo, J., Titi, E.S.: Uniform in time error estimates for a finite element method applied to a downscaling data assimilation algorithm for the Navier–Stokes equations. SIAM J. Numer. Anal. 58(1), 410–429 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gardner, M., Larios, A., Rebholz, L.G., Vargun, D., Zerfas, C.: Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier–Stokes equations. Electron. Res. Arch. 29(3), 2223–2247 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mondaini, C.F., Titi, E.S.: Uniform-in-time error estimates for the postprocessing Galerkin method applied to a data assimilation algorithm. SIAM J. Numer. Anal. 56(1), 78–110 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Albanez, D.A.F., Benvenutti, M.J.: Continuous data assimilation algorithm for simplified Bardina model. Evol. Equ. Control Theory 7(1), 33–52 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Markowich, P.A., Titi, E.S., Trabelsi, S.: Data assimilation for the three-dimensional Brinkman–Forchheimer-extended Darcy model. Nonlinearity 29, 1292–1328 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Albanez, D.A.F., Lopes, H.J.N., Titi, E.S.: Continuous data assimilation for the three-dimensional Navier–Stokes-\(\alpha \) model. Asymp. Anal. 97, 139–164 (2016)

    MathSciNet  MATH  Google Scholar 

  33. Jolly, M.S., Martinez, V.R., Olson, E.J., Titi, E.S.: Continuous data assimilation with blurred-in-time measurements of the surface quasi-geostrophic equation. Chin. Ann. Math. B 40(5), 721–764 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. Chen, N., Li, Y.C., Lunasin, E.: An efficient continuous data assimilation algorithm for the sabra shell model of turbulence. Chao 31(10), 103123 (2021)

    Article  Google Scholar 

  35. Chow, Y. T., Leung, W. T., Pakzad, A.: Continuous data assimilation for two-phase flow: analysis and simulations. http://arxiv.org/abs/2110.13491v1 (2021)

  36. Diegel, A. E., Rebholz, L. G.: Continuous data assimilation and long-time accuracy in a \(\cal{C}^0\) interior penalty method for the Cahn–Hilliard equation. http://arxiv.org/abs/2106.14744v1 (2021)

  37. Rebholz, L.G., Zerfas, C.: Simple and efficient continuous data assimilation of evolution equations via algebraic nudging. Numer. Methods Partial Diff. Equ. 37(3), 2588–2612 (2021)

    Article  MathSciNet  Google Scholar 

  38. Dimet, F.X.L., Talagrand, O.: Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects. Tellus A 38, 97–110 (1986)

    Article  Google Scholar 

  39. Fehrenbach, J., Masmoudi, M., Souchon, R., Trompette, P.: Detection of small inclusions using elastography. Inverse Problem 22, 1055–1069 (2006)

    Article  MATH  Google Scholar 

  40. Fehrenbach, J., Oudry, J., Sandrin, L.: Variational data assimilation to estimate the velocity in the wave equation. Inverse Problem 26, 115005 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Korn, P.: Strong solvability of a variational data assimilation problem for the primitive equations of large-scale atmosphere and ocean dynamics. J. Nonlinear Sci. 31(3), 56 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  42. Cherfils, L., Petcu, M.: On the viscous Cahn–Hilliard–Naveri–Stokes equations with dynamic boundary conditions. Commun. Pure Appl. Anal. 15(4), 1419–1449 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. Anderson, D.M., McFadden, G.B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  44. Hohenberg, P.C., Halperin, B.I.: Theory of dynamical critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977)

    Article  Google Scholar 

  45. Jasnow, D., Vinals, J.: Coarse-grained description of thermo-capillary flow. Phys. Fluids 8, 660–669 (1996)

    Article  MATH  Google Scholar 

  46. Feng, X.B.: Fully discrete element approximations of the Navier–Stokes–Cahn–Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44(3), 1049–1072 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kay, D., Styles, V., Welford, R.: Finite element approximation of a Cahn–Hilliard–Navier–Stokes system. Interfaces Free Bound 10, 15–43 (2008)

    Article  MathSciNet  Google Scholar 

  48. Kay, D., Welford, R.: Efficient numerical solution of Cahn–Hilliard–Navier–Stokes fluids in 2D. SIAM J. Sci. Comput. 29, 2241–2257 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  49. Cao, C.S., Gal, C.G.: Global solutions for the 2D Navier–Stokes–Cahn–Hilliard model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility. Nonlinearity 25, 3211–3234 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  50. Gal, C.G., Grasselli, M.: Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D. Ann. l’Inst. Henri Poincaré C 27, 401–436 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  51. Gal, C.G., Grasselli, M.: Instability of two-phase flows: a lower bound on the dimension of the global attractor of the Cahn–Hilliard–Navier–Stokes system. Physica D 240, 629–635 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  52. Starovoitov, V.N.: The dynamics of a two-component fluid in the presence of capillary forces. Math. Notes 62, 244–254 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  53. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1997)

    Book  MATH  Google Scholar 

  54. Li, Y., Choi, J., Kim, J.: Multi-component Cahn–Hilliard system with different boundary conditions in complex domains. J. Comput. Phys. 323, 1–16 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Science Foundation of China Grant (11401459,11801427, 11871389), the Natural Science Foundation of Shaanxi Province (2018JQ1009, 2018JM1012) and the Fundamental Research Funds for the Central Universities (xjj2018088).

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  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Bo You & Qing Xia

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You, B., Xia, Q. Continuous Data Assimilation Algorithm for the Two Dimensional Cahn–Hilliard–Navier–Stokes System. Appl Math Optim 85, 5 (2022). https://doi.org/10.1007/s00245-022-09863-2

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