doi: 10.3934/dcdsb.2021251
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Trajectory statistical solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

Received  March 2021 Revised  July 2021 Early access October 2021

The objective of this paper is to consider the long-time behavior of solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines. As we know, it is very difficult to obtain the uniqueness of an energy solution for this system even in two dimensions caused by the presence of the strong coupling at the boundary. Thus, we first prove the existence of a trajectory attractor for such system, which is a minimal compact trajectory attracting set for the natural translation semigroup defined on the trajectory space. Furthermore, based on the abstract results (trajectory attractor approach) developed in [38], we construct trajectory statistical solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines.

Citation: Bo You. Trajectory statistical solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021251
References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis : A Hitchhiker's Guide, Springer-Verlag, Berlin, 2006.  Google Scholar

[2]

D. M. AndersonG. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annual Review of Fluid Mechanics, 30 (1998), 139-165.  doi: 10.1146/annurev.fluid.30.1.139.  Google Scholar

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K. BaoY. ShiS. Sun and X. P. Wang, A finite element method for the numerical solution of the coupled Cahn-Hilliard and Navier-Stokes system for moving contact line problems, J. Comput. Phys., 231 (2012), 8083-8099.  doi: 10.1016/j.jcp.2012.07.027.  Google Scholar

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A. C. BronziC. F. Mondaini and R. M. S. Rosa, Trajectory statistical solutions for three-dimensional Navier-Stokes-like systems, SIAM J. Math. Anal., 46) (2014), 1893-1921.  doi: 10.1137/130931631.  Google Scholar

[5]

A. C. BronziC. F. Mondaini and R. M. S. Rosa, Abstract framework for the theory of statistical solutions, J. Differential Equations, 260 (2016), 8428-8484.  doi: 10.1016/j.jde.2016.02.027.  Google Scholar

[6]

M. D. Chekroun and N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications, Comm. Math. Phys., 316 (2012), 723-761.  doi: 10.1007/s00220-012-1515-y.  Google Scholar

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A. Cheskidov, Global attractors of evolutionary systems, J. Dynam. Differential Equations, 21 (2009), 249-268.  doi: 10.1007/s10884-009-9133-x.  Google Scholar

[8]

E. B. Dussan, The moving contact line: The slip boundary condition, Journal of Fluid Mechanics, 77 (1976), 665-684.   Google Scholar

[9]

E. B. DussanV and S. H. Davis, On the motion of a fluid-fluid interface along a solid surface, Journal of Fluid Mechanics, 65 (1974), 71-95.   Google Scholar

[10]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Cambridge, 2010. doi: 10.1090/gsm/019.  Google Scholar

[11] C. FoiasO. ManleyR. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754.  Google Scholar
[12]

C. Foiaș and G. Prodi, Sur les solutions statistiques des équations de Navier-Stokes, Ann. Mat. Pura Appl., 111 (1976), 307-330.  doi: 10.1007/BF02411822.  Google Scholar

[13]

C. FoiasR. M. S. Rosa and R. Temam, Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations, Ann. Inst. Fourier (Grenoble), 63 (2013), 2515-2573.  doi: 10.5802/aif.2836.  Google Scholar

[14]

C. G. Gal, M. Grasselli and A. Miranville, Cahn-Hilliard-Navier-Stokes system with moving contact lines, Calc. Var. Partial Differential Equations, 55 (2016), Art. 50, 47 pp. doi: 10.1007/s00526-016-0992-9.  Google Scholar

[15]

M. Gao and X.-P. Wang, A gradient stable scheme for a phase field model for the moving contact line problem, J. Comput. Phys., 231 (2012), 1372-1386.  doi: 10.1016/j.jcp.2011.10.015.  Google Scholar

[16]

M. E. GurtinD. Polignone and J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831.  doi: 10.1142/S0218202596000341.  Google Scholar

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M. Heida, On the derivation of thermodynamically consistent boundary conditions for the Cahn-Hilliard-Navier-Stokes system, Internat. J. Engrg. Sci., 62 (2013), 126-156.  doi: 10.1016/j.ijengsci.2012.09.005.  Google Scholar

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H. Jiang and C. Zhao, Trajectory statistical solutions and Liouville type theorem for nonlinear wave equations with polynomial growth, Adv. Differential Equations, 26 (2021), 107-132.   Google Scholar

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P. D. Lax, Functional Analysis, Wiley, New York, 2002.  Google Scholar

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G. ŁukaszewiczJ. Real and J. C. Robinson, Invariant measures for dissipative dynamical systems and generalised Banach limits, J. Dynam. Differential Equations, 23 (2011), 225-250.  doi: 10.1007/s10884-011-9213-6.  Google Scholar

[21]

G. Łukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 4211-4222.  doi: 10.3934/dcds.2014.34.4211.  Google Scholar

[22]

J. C. Maxwell, On stresses in rarified gases arising from inequalities of temperature, Philosophical Transactions of the Royal Society of London, 170 (1879), 704-712.  doi: 10.1017/CBO9780511710377.068.  Google Scholar

[23]

H. K. Moffatt, Viscous and resistive eddies near a sharp corner, Journal of Fluid Mechanics, 18 (1964), 1-18.   Google Scholar

[24]

T. QianX.-P. Wang and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Physical Review E, 68 (2003), 016306.  doi: 10.1103/PhysRevE.68.016306.  Google Scholar

[25]

T. QianX.-P. Wang and P. Sheng, A variational approach to moving contact line hydrodynamics, J. Fluid Mech., 564 (2006), 333-360.  doi: 10.1017/S0022112006001935.  Google Scholar

[26]

W. Rudin, Real and Complex Analysis, New York, McGraw-Hill Education, 1974.  Google Scholar

[27]

G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33.  doi: 10.1007/BF02218613.  Google Scholar

[28]

J. ShenX. Yang and H. Yu, Efficient energy stable numerical schemes for a phase field moving contact line model, J. Comput. Phys., 284 (2015), 617-630.  doi: 10.1016/j.jcp.2014.12.046.  Google Scholar

[29]

Y. ShiK. Bao and X.-P. Wang, 3D adaptive finite element method for a phase field model for the moving contact line problems, Inverse Probl. Imaging, 7 (2013), 947-959.  doi: 10.3934/ipi.2013.7.947.  Google Scholar

[30]

R. Temam, Infinite-Dimensional Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[31]

M. I. Višhik and A. V. Fursikov, Translationally homogeneous statistical solutions and individual solutions with infinite energy of a system of Navier-Stokes equations, Sibirsk. Mat. Zh., 19 (1978), 1005–1031, 1213.  Google Scholar

[32]

X. Wang, Upper-semicontinuity of stationary statistical properties of dissipative systems, Discrete Contin. Dyn. Syst., 23 (2009), 521-540.  doi: 10.3934/dcds.2009.23.521.  Google Scholar

[33]

X.-P. Wang and Y.-G. Wang, The sharp interface limit of a phase field model for moving contact line problem, Methods Appl. Anal., 14 (2007), 287-294.  doi: 10.4310/MAA.2007.v14.n3.a6.  Google Scholar

[34]

B. You, Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines, Commun. Pure Appl. Anal., 18 (2019), 2283-2298.  doi: 10.3934/cpaa.2019103.  Google Scholar

[35]

P. YueC. Zhou and J. J. Feng, Sharp-interface limit of the Cahn-Hilliard model for moving contact lines, J. Fluid Mech., 645 (2010), 279-294.  doi: 10.1017/S0022112009992679.  Google Scholar

[36]

C. Zhao and T. Caraballo, Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier-Stokes equations, J. Differential Equations, 266 (2019), 7205-7229.  doi: 10.1016/j.jde.2018.11.032.  Google Scholar

[37]

C. ZhaoT. Caraballo and G. Łukaszewicz, Statistical solution and Liouville type theorem for the Klein-Gordon-Schrödinger equations, J. Differential Equations, 281 (2021), 1-32.  doi: 10.1016/j.jde.2021.01.039.  Google Scholar

[38]

C. ZhaoY. Li and T. Caraballo, Trajectory statistical solutions and Liouville type equations for evolution equations: Abstract results and applications, J. Differential Equations, 269 (2020), 467-494.  doi: 10.1016/j.jde.2019.12.011.  Google Scholar

[39]

C. Zhao, Y. Li and Z. Song, Trajectory statistical solutions for the 3D Navier-Stokes equations: The trajectory attractor approach, Nonlinear Analysis: Real World Applications, 53 (2020), 103077, 10 pp. doi: 10.1016/j.nonrwa.2019.103077.  Google Scholar

[40]

C. Zhao, Z. Song and T. Caraballo, Strong trajectory statistical solutions and Liouville type equation for dissipative Euler equations, Appl. Math. Lett., 99 (2020), 105981, 6 pp. doi: 10.1016/j.aml.2019.07.012.  Google Scholar

show all references

References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis : A Hitchhiker's Guide, Springer-Verlag, Berlin, 2006.  Google Scholar

[2]

D. M. AndersonG. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annual Review of Fluid Mechanics, 30 (1998), 139-165.  doi: 10.1146/annurev.fluid.30.1.139.  Google Scholar

[3]

K. BaoY. ShiS. Sun and X. P. Wang, A finite element method for the numerical solution of the coupled Cahn-Hilliard and Navier-Stokes system for moving contact line problems, J. Comput. Phys., 231 (2012), 8083-8099.  doi: 10.1016/j.jcp.2012.07.027.  Google Scholar

[4]

A. C. BronziC. F. Mondaini and R. M. S. Rosa, Trajectory statistical solutions for three-dimensional Navier-Stokes-like systems, SIAM J. Math. Anal., 46) (2014), 1893-1921.  doi: 10.1137/130931631.  Google Scholar

[5]

A. C. BronziC. F. Mondaini and R. M. S. Rosa, Abstract framework for the theory of statistical solutions, J. Differential Equations, 260 (2016), 8428-8484.  doi: 10.1016/j.jde.2016.02.027.  Google Scholar

[6]

M. D. Chekroun and N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications, Comm. Math. Phys., 316 (2012), 723-761.  doi: 10.1007/s00220-012-1515-y.  Google Scholar

[7]

A. Cheskidov, Global attractors of evolutionary systems, J. Dynam. Differential Equations, 21 (2009), 249-268.  doi: 10.1007/s10884-009-9133-x.  Google Scholar

[8]

E. B. Dussan, The moving contact line: The slip boundary condition, Journal of Fluid Mechanics, 77 (1976), 665-684.   Google Scholar

[9]

E. B. DussanV and S. H. Davis, On the motion of a fluid-fluid interface along a solid surface, Journal of Fluid Mechanics, 65 (1974), 71-95.   Google Scholar

[10]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Cambridge, 2010. doi: 10.1090/gsm/019.  Google Scholar

[11] C. FoiasO. ManleyR. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754.  Google Scholar
[12]

C. Foiaș and G. Prodi, Sur les solutions statistiques des équations de Navier-Stokes, Ann. Mat. Pura Appl., 111 (1976), 307-330.  doi: 10.1007/BF02411822.  Google Scholar

[13]

C. FoiasR. M. S. Rosa and R. Temam, Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations, Ann. Inst. Fourier (Grenoble), 63 (2013), 2515-2573.  doi: 10.5802/aif.2836.  Google Scholar

[14]

C. G. Gal, M. Grasselli and A. Miranville, Cahn-Hilliard-Navier-Stokes system with moving contact lines, Calc. Var. Partial Differential Equations, 55 (2016), Art. 50, 47 pp. doi: 10.1007/s00526-016-0992-9.  Google Scholar

[15]

M. Gao and X.-P. Wang, A gradient stable scheme for a phase field model for the moving contact line problem, J. Comput. Phys., 231 (2012), 1372-1386.  doi: 10.1016/j.jcp.2011.10.015.  Google Scholar

[16]

M. E. GurtinD. Polignone and J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831.  doi: 10.1142/S0218202596000341.  Google Scholar

[17]

M. Heida, On the derivation of thermodynamically consistent boundary conditions for the Cahn-Hilliard-Navier-Stokes system, Internat. J. Engrg. Sci., 62 (2013), 126-156.  doi: 10.1016/j.ijengsci.2012.09.005.  Google Scholar

[18]

H. Jiang and C. Zhao, Trajectory statistical solutions and Liouville type theorem for nonlinear wave equations with polynomial growth, Adv. Differential Equations, 26 (2021), 107-132.   Google Scholar

[19]

P. D. Lax, Functional Analysis, Wiley, New York, 2002.  Google Scholar

[20]

G. ŁukaszewiczJ. Real and J. C. Robinson, Invariant measures for dissipative dynamical systems and generalised Banach limits, J. Dynam. Differential Equations, 23 (2011), 225-250.  doi: 10.1007/s10884-011-9213-6.  Google Scholar

[21]

G. Łukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 4211-4222.  doi: 10.3934/dcds.2014.34.4211.  Google Scholar

[22]

J. C. Maxwell, On stresses in rarified gases arising from inequalities of temperature, Philosophical Transactions of the Royal Society of London, 170 (1879), 704-712.  doi: 10.1017/CBO9780511710377.068.  Google Scholar

[23]

H. K. Moffatt, Viscous and resistive eddies near a sharp corner, Journal of Fluid Mechanics, 18 (1964), 1-18.   Google Scholar

[24]

T. QianX.-P. Wang and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Physical Review E, 68 (2003), 016306.  doi: 10.1103/PhysRevE.68.016306.  Google Scholar

[25]

T. QianX.-P. Wang and P. Sheng, A variational approach to moving contact line hydrodynamics, J. Fluid Mech., 564 (2006), 333-360.  doi: 10.1017/S0022112006001935.  Google Scholar

[26]

W. Rudin, Real and Complex Analysis, New York, McGraw-Hill Education, 1974.  Google Scholar

[27]

G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33.  doi: 10.1007/BF02218613.  Google Scholar

[28]

J. ShenX. Yang and H. Yu, Efficient energy stable numerical schemes for a phase field moving contact line model, J. Comput. Phys., 284 (2015), 617-630.  doi: 10.1016/j.jcp.2014.12.046.  Google Scholar

[29]

Y. ShiK. Bao and X.-P. Wang, 3D adaptive finite element method for a phase field model for the moving contact line problems, Inverse Probl. Imaging, 7 (2013), 947-959.  doi: 10.3934/ipi.2013.7.947.  Google Scholar

[30]

R. Temam, Infinite-Dimensional Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[31]

M. I. Višhik and A. V. Fursikov, Translationally homogeneous statistical solutions and individual solutions with infinite energy of a system of Navier-Stokes equations, Sibirsk. Mat. Zh., 19 (1978), 1005–1031, 1213.  Google Scholar

[32]

X. Wang, Upper-semicontinuity of stationary statistical properties of dissipative systems, Discrete Contin. Dyn. Syst., 23 (2009), 521-540.  doi: 10.3934/dcds.2009.23.521.  Google Scholar

[33]

X.-P. Wang and Y.-G. Wang, The sharp interface limit of a phase field model for moving contact line problem, Methods Appl. Anal., 14 (2007), 287-294.  doi: 10.4310/MAA.2007.v14.n3.a6.  Google Scholar

[34]

B. You, Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines, Commun. Pure Appl. Anal., 18 (2019), 2283-2298.  doi: 10.3934/cpaa.2019103.  Google Scholar

[35]

P. YueC. Zhou and J. J. Feng, Sharp-interface limit of the Cahn-Hilliard model for moving contact lines, J. Fluid Mech., 645 (2010), 279-294.  doi: 10.1017/S0022112009992679.  Google Scholar

[36]

C. Zhao and T. Caraballo, Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier-Stokes equations, J. Differential Equations, 266 (2019), 7205-7229.  doi: 10.1016/j.jde.2018.11.032.  Google Scholar

[37]

C. ZhaoT. Caraballo and G. Łukaszewicz, Statistical solution and Liouville type theorem for the Klein-Gordon-Schrödinger equations, J. Differential Equations, 281 (2021), 1-32.  doi: 10.1016/j.jde.2021.01.039.  Google Scholar

[38]

C. ZhaoY. Li and T. Caraballo, Trajectory statistical solutions and Liouville type equations for evolution equations: Abstract results and applications, J. Differential Equations, 269 (2020), 467-494.  doi: 10.1016/j.jde.2019.12.011.  Google Scholar

[39]

C. Zhao, Y. Li and Z. Song, Trajectory statistical solutions for the 3D Navier-Stokes equations: The trajectory attractor approach, Nonlinear Analysis: Real World Applications, 53 (2020), 103077, 10 pp. doi: 10.1016/j.nonrwa.2019.103077.  Google Scholar

[40]

C. Zhao, Z. Song and T. Caraballo, Strong trajectory statistical solutions and Liouville type equation for dissipative Euler equations, Appl. Math. Lett., 99 (2020), 105981, 6 pp. doi: 10.1016/j.aml.2019.07.012.  Google Scholar

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