American Institute of Mathematical Sciences

Advanced
March  2010, 28(1): 1-39. doi: 10.3934/dcds.2010.28.1

Longtime behavior for a model of homogeneous incompressible two-phase flows

Ciprian G. Gal 1, and  Maurizio Grasselli 2,

1. 

Department of Mathematics, University of Missouri, Columbia, MO 65211, United States
2. 

Dipartimento di Matematica, Politecnico di Milano, 20133 Milano

Received  August 2009 Revised  January 2010 Published  April 2010

We consider a diffuse interface model for the evolution of an iso-thermal incompressible two-phase flow in a two-dimensional bounded domain. The model consists of the Navier-Stokes equation for the fluid velocity u coupled with a convective Allen-Cahn equation for the order (phase) parameter $\phi$, both endowed with suitable boundary conditions. We analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. We first prove that the initial and boundary value problem generates a strongly continuous semigroup on a suitable phase space which possesses the global attractor $ \mathcal{A}$. Then we establish the existence of an exponential attractor $ \mathcal{E}$ which entails that $\mathcal{A}$ has finite fractal dimension. This dimension is then estimated in terms of some model parameters. Moreover, assuming the potential to be real analytic, we demonstrate that, in absence of external forces, each trajectory converges to a single equilibrium by means of a Łojasiewicz-Simon inequality. We also obtain a convergence rate estimate. Finally, we discuss the case where $\phi $ is forced to take values in a bounded interval, e.g., by a so-called singular potential.
Keywords: incompressible fluids, exponential attractors, Navier-Stokes equations, convergence to equilibria., Allen-Cahn equations, nematic liquid crystals, fractal dimension, global attractors.
Mathematics Subject Classification: Primary: 35B40, 35B41, 35K55, 35Q35, 37L30; Secondary: 76D05, 76T9.
Citation: Ciprian G. Gal, Maurizio Grasselli. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 1-39. doi: 10.3934/dcds.2010.28.1
[1]

Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239
[2]

Yinghua Li, Shijin Ding, Mingxia Huang. Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1507-1523. doi: 10.3934/dcdsb.2016009
[3]

Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113
[4]

V. V. Chepyzhov, A. A. Ilyin. On the fractal dimension of invariant sets: Applications to Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 117-135. doi: 10.3934/dcds.2004.10.117
[5]

Fang Li, Bo You. Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 55-80. doi: 10.3934/dcdsb.2019172
[6]

Stefano Scrobogna. Global existence and convergence of nondimensionalized incompressible Navier-Stokes equations in low Froude number regime. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5471-5511. doi: 10.3934/dcds.2020235
[7]

Roberta Bianchini, Roberto Natalini. Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations. Kinetic & Related Models, 2019, 12 (1) : 133-158. doi: 10.3934/krm.2019006
[8]

Daniel Pardo, José Valero, Ángel Giménez. Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3569-3590. doi: 10.3934/dcdsb.2018279
[9]

Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Pullback attractors for globally modified Navier-Stokes equations with infinite delays. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 779-796. doi: 10.3934/dcds.2011.31.779
[10]

Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067
[11]

Hirokazu Ninomiya, Masaharu Taniguchi. Global stability of traveling curved fronts in the Allen-Cahn equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 819-832. doi: 10.3934/dcds.2006.15.819
[12]

Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
[13]

Julia García-Luengo, Pedro Marín-Rubio, José Real. Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1603-1621. doi: 10.3934/cpaa.2015.14.1603
[14]

Songsong Lu, Hongqing Wu, Chengkui Zhong. Attractors for nonautonomous 2d Navier-Stokes equations with normal external forces. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 701-719. doi: 10.3934/dcds.2005.13.701
[15]

P.E. Kloeden, José A. Langa, José Real. Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 937-955. doi: 10.3934/cpaa.2007.6.937
[16]

Grzegorz Łukaszewicz. Pullback attractors and statistical solutions for 2-D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 643-659. doi: 10.3934/dcdsb.2008.9.643
[17]

J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647
[18]

Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209
[19]

Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127
[20]

Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (83)
  • HTML views (0)
  • Cited by (39)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences