• Previous Article
    Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones
  • DCDS-B Home
  • This Issue
  • Next Article
    A prey-predator model with a free boundary and sign-changing coefficient in time-periodic environment
November  2017, 22(9): 3273-3294. doi: 10.3934/dcdsb.2017137

A two-phase flow model with delays

Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, USA

Received  September 2016 Revised  February 2017 Published  April 2017

In this article, we study a coupled Allen-Cahn-Navier-Stokes model with delays in a two-dimensional domain. The model consists of the Navier-Stokes equations for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter. We prove the existence and uniqueness of the weak and strong solution when the external force contains some delays. We also discuss the asymptotic behavior of the weak solutions and the stability of the stationary solutions.

Citation: Theodore Tachim Medjo. A two-phase flow model with delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3273-3294. doi: 10.3934/dcdsb.2017137
References:
[1]

T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Pysica D (Applied Physics), 32 (1999), 1119-1123. doi: 10.1088/0022-3727/32/10/307.

[2]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[3]

T. Caraballo and X. Han, A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 1079-1101. doi: 10.3934/dcdss.2015.8.1079.

[4]

T. CaraballoA. M. Márquez-Durán and J. Real, Pullback and forward attractors for a 3D LANS-α model with delay, Discrete Contin. Dyn. Syst., 15 (2006), 559-578. doi: 10.3934/dcds.2006.15.559.

[5]

T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453. doi: 10.1098/rspa.2001.0807.

[6]

T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194. doi: 10.1098/rspa.2003.1166.

[7]

T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. doi: 10.1016/j.jde.2004.04.012.

[8]

R. Datko, Representation of solutions and stability of linear differential-difference equations in a Banach space, J. Differential Equations, 29 (1978), 105-166. doi: 10.1016/0022-0396(78)90043-8.

[9]

E. FeireislH. PetzeltováE. Rocca and G. Schimperna, Analysis of a phase-field model for two-phase compressible fluids, Math. Models Methods Appl. Sci., 20 (2010), 1129-1160. doi: 10.1142/S0218202510004544.

[10]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436. doi: 10.1016/j.anihpc.2009.11.013.

[11]

C. G. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst., 28 (2010), 1-39. doi: 10.3934/dcds.2010.28.1.

[12]

C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B, 31 (2010), 655-678. doi: 10.1007/s11401-010-0603-6.

[13]

P.C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479.

[14]

T. Tachim Medjo, Pullback attractors for a non-autonomous homogeneous two-phase flow model, J. Diff. Equa., 253 (2012), 1779-1806. doi: 10.1016/j.jde.2012.06.004.

[15]

H. MeiG. Yin and F. Wu, Properties of stochastic integro-differential equations with infinite delay: regularity, ergodicity, weak sense Fokker-Planck equations, Stochastic Process. Appl., 126 (2016), 3102-3123. doi: 10.1016/j.spa.2016.04.003.

[16]

S. A. Messaoudi, A. Fareh and N. Doudi, Well posedness and exponential stability in a wave equation with a strong damping and a strong delay, J. Math. Phys. , 57 (2016), 111501, 13 pp. doi: 10.1063/1.4966551.

[17]

C. Niche and G. Planas, Existence and decay of solutions in full space to Navier-Stokes equations with delays, Nonlinear Anal., 74 (2011), 244-256. doi: 10.1016/j.na.2010.08.038.

[18]

A. Onuki, Phase transition of fluids in shear flow, J. Phys. Condens. Matter, 9 (1997), 6119-6157. doi: 10.1017/CBO9780511534874.012.

[19]

T. tachim Medjo, Attractors for a two-phase flow model with delays, Differential Integral Equations, 29 (2016), 1071-1092.

[20]

T. Taniguchi, The exponential behavior of Navier-Stokes equations with time delay external force, Discrete Contin. Dyn. Syst., 12 (2005), 997-1018. doi: 10.3934/dcds.2005.12.997.

[21]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68. Appl. Math. Sci. , Springer-Verlag, New York, second edition, 1997. doi: 10.1007/978-1-4612-0645-3.

[22]

X. S. Wang and J. Wu, Seasonal migration dynamics: periodicity, transition delay and finite-dimensional reduction, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468 (2012), 634-650. doi: 10.1098/rspa.2011.0236.

show all references

References:
[1]

T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Pysica D (Applied Physics), 32 (1999), 1119-1123. doi: 10.1088/0022-3727/32/10/307.

[2]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[3]

T. Caraballo and X. Han, A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 1079-1101. doi: 10.3934/dcdss.2015.8.1079.

[4]

T. CaraballoA. M. Márquez-Durán and J. Real, Pullback and forward attractors for a 3D LANS-α model with delay, Discrete Contin. Dyn. Syst., 15 (2006), 559-578. doi: 10.3934/dcds.2006.15.559.

[5]

T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453. doi: 10.1098/rspa.2001.0807.

[6]

T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194. doi: 10.1098/rspa.2003.1166.

[7]

T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. doi: 10.1016/j.jde.2004.04.012.

[8]

R. Datko, Representation of solutions and stability of linear differential-difference equations in a Banach space, J. Differential Equations, 29 (1978), 105-166. doi: 10.1016/0022-0396(78)90043-8.

[9]

E. FeireislH. PetzeltováE. Rocca and G. Schimperna, Analysis of a phase-field model for two-phase compressible fluids, Math. Models Methods Appl. Sci., 20 (2010), 1129-1160. doi: 10.1142/S0218202510004544.

[10]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436. doi: 10.1016/j.anihpc.2009.11.013.

[11]

C. G. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst., 28 (2010), 1-39. doi: 10.3934/dcds.2010.28.1.

[12]

C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B, 31 (2010), 655-678. doi: 10.1007/s11401-010-0603-6.

[13]

P.C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479.

[14]

T. Tachim Medjo, Pullback attractors for a non-autonomous homogeneous two-phase flow model, J. Diff. Equa., 253 (2012), 1779-1806. doi: 10.1016/j.jde.2012.06.004.

[15]

H. MeiG. Yin and F. Wu, Properties of stochastic integro-differential equations with infinite delay: regularity, ergodicity, weak sense Fokker-Planck equations, Stochastic Process. Appl., 126 (2016), 3102-3123. doi: 10.1016/j.spa.2016.04.003.

[16]

S. A. Messaoudi, A. Fareh and N. Doudi, Well posedness and exponential stability in a wave equation with a strong damping and a strong delay, J. Math. Phys. , 57 (2016), 111501, 13 pp. doi: 10.1063/1.4966551.

[17]

C. Niche and G. Planas, Existence and decay of solutions in full space to Navier-Stokes equations with delays, Nonlinear Anal., 74 (2011), 244-256. doi: 10.1016/j.na.2010.08.038.

[18]

A. Onuki, Phase transition of fluids in shear flow, J. Phys. Condens. Matter, 9 (1997), 6119-6157. doi: 10.1017/CBO9780511534874.012.

[19]

T. tachim Medjo, Attractors for a two-phase flow model with delays, Differential Integral Equations, 29 (2016), 1071-1092.

[20]

T. Taniguchi, The exponential behavior of Navier-Stokes equations with time delay external force, Discrete Contin. Dyn. Syst., 12 (2005), 997-1018. doi: 10.3934/dcds.2005.12.997.

[21]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68. Appl. Math. Sci. , Springer-Verlag, New York, second edition, 1997. doi: 10.1007/978-1-4612-0645-3.

[22]

X. S. Wang and J. Wu, Seasonal migration dynamics: periodicity, transition delay and finite-dimensional reduction, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468 (2012), 634-650. doi: 10.1098/rspa.2011.0236.

[1]

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

[2]

Jochen Merker. Strong solutions of doubly nonlinear Navier-Stokes equations. Conference Publications, 2011, 2011 (Special) : 1052-1060. doi: 10.3934/proc.2011.2011.1052

[3]

Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015

[4]

Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679

[5]

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

[6]

T. Tachim Medjo. A Cahn-Hilliard-Navier-Stokes model with delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2663-2685. doi: 10.3934/dcdsb.2016067

[7]

Yan Hu. Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 947-964. doi: 10.3934/cpaa.2016.15.947

[8]

Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577

[9]

Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319

[10]

Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024

[11]

Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353

[12]

Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027

[13]

Tomás Caraballo, Xiaoying Han. A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1079-1101. doi: 10.3934/dcdss.2015.8.1079

[14]

Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613

[15]

Ken Shirakawa. Stability analysis for two dimensional Allen-Cahn equations associated with crystalline type energies. Conference Publications, 2009, 2009 (Special) : 697-707. doi: 10.3934/proc.2009.2009.697

[16]

Grégory Faye. Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2473-2496. doi: 10.3934/dcds.2016.36.2473

[17]

Luyi Ma, Hong-Tao Niu, Zhi-Cheng Wang. Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111

[18]

Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703

[19]

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

[20]

Giorgio Fusco. Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1807-1841. doi: 10.3934/cpaa.2017088

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (10)
  • HTML views (21)
  • Cited by (0)

Other articles
by authors

[Back to Top]