# American Institute of Mathematical Sciences

April  2018, 38(4): 2141-2169. doi: 10.3934/dcds.2018088

## Pullback $\mathbb{V}-$attractor of a three dimensional globally modified two-phase flow model

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

Received  May 2017 Revised  October 2017 Published  January 2018

The existence and final fractal dimension of a pullback attractor in the space $\mathbb{V}$ for a three dimensional system of a non-autonomous globally modified two phase flow on a bounded domain is established under appropriate properties on the time depending forcing term. The model consists of the globally modified Navier-Stokes equations proposed in [6] for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter. The existence of the pullback attractors is obtained using the flattening property. Furthermore, we prove that the fractal dimension in $\mathbb{V}$ of the pullback attractor is finite.

Citation: Theodore Tachim Medjo. Pullback $\mathbb{V}-$attractor of a three dimensional globally modified two-phase flow model. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2141-2169. doi: 10.3934/dcds.2018088
##### References:
 [1] H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506. doi: 10.1007/s00205-008-0160-2. Google Scholar [2] T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Physica D (Applied Physics), 32 (1999), 1119-1123. doi: 10.1088/0022-3727/32/10/307. Google Scholar [3] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. Google Scholar [4] T. Caraballo and P. E. Kloeden, The three-dimensional globally modified Navier-Stokes equations: Recent developments, In A. Johann, H. P. Kruse and F. Rupp, editors, Recent Trends in Dynamical Systems: Proceedings of a Conference in Honor of Jürgen Scheurle, Springer Proceedings in Mathematics and Statistics, 35 (2013), 473-492. Google Scholar [5] T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761-781. doi: 10.3934/dcdsb.2008.10.761. Google Scholar [6] T. Caraballo, J. Real and P.E. Kloeden, Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 6 (2006), 411-436. Google Scholar [7] T. Caraballo, J. Real and A.M. Márquez, Three-dimensional system of globally modified Navier-Stokes equations with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869-2883. doi: 10.1142/S0218127410027428. Google Scholar [8] S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flows, Phys. Rev. Lett., 81 (1998), 5338-5341. doi: 10.1103/PhysRevLett.81.5338. Google Scholar [9] S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Physica D, 133 (1999), 49-65. doi: 10.1016/S0167-2789(99)00098-6. Google Scholar [10] S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids, 11 (1999), 2343-2353. doi: 10.1063/1.870096. Google Scholar [11] S. Chen, D.D. Holm, L.G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Physica D, 133 (1999), 66-83. doi: 10.1016/S0167-2789(99)00099-8. Google Scholar [12] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988. Google Scholar [13] G. Deugoue and J.K. Djoko, On the time discretization for the globally modified three dimensional Navier-Stokes equations, J. Comput. Appl. Math, 235 (2011), 2015-2029. doi: 10.1016/j.cam.2010.10.003. Google Scholar [14] E. Feireisl, H. Petzeltová, E. 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. Google Scholar [15] C. 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. Google Scholar [16] 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. Google Scholar [17] 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. Google Scholar [18] P.C. Hohenberg and B.I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479. Google Scholar [19] D.D. Holm, J.E. Marsden and T.S. Ratiu, The Euler-Poincaré equations and semi-direct products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721. Google Scholar [20] D.D. Holm, J.E. Marsden and T.S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 349 (1998), 4173-4177. doi: 10.1103/PhysRevLett.80.4173. Google Scholar [21] P.E. Kloeden and J.A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753. Google Scholar [22] P.E. Kloeden, J.A. Langa and J. Real, Pullback ${V-}$attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 6 (2007), 937-955. doi: 10.3934/cpaa.2007.6.937. Google Scholar [23] P.E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 8 (2009), 785-802. doi: 10.3934/cpaa.2009.8.785. Google Scholar [24] P.E. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 1491-1508. doi: 10.1098/rspa.2007.1831. Google Scholar [25] G. Lukaszewicz, On pullback attractors in ${H}^1_0$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg, 20 (2010), 2637-2644. doi: 10.1142/S0218127410027258. Google Scholar [26] Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. Google Scholar [27] P. Marín-Rubio, J. Real and A.M. Márquez-Durán, On the convergence of solutions of globally modified Navier-Stokes equations with delays to solutions of Navier-Stokes equations with delays, Adv. Nonlinear Stud., 11 (2011), 917-927. Google Scholar [28] A.M. Márquez, Existence and uniqueness of solutions, and pullback attractor for a system of globally modified 3D-Navier-Stokes equations with finite delay, Bol. Soc. Esp. Mat. Apl. S$\overrightarrow{e}$MA, 51 (2010), 117-124. Google Scholar [29] J.E. Marsden and S. Shkoller, Global well-posedness for the {L}agrangian averaged Navier-Stokes (LANS-$α$) equations on bounded domains, Phil. Trans. R. Soc. Lond. A, 359 (2001), 1449-1468. doi: 10.1098/rsta.2001.0852. Google Scholar [30] A. Onuki, Phase transition of fluids in shear flow, J. Phys. Condens. Matter, 9 (1997), 6119-6157. Google Scholar [31] M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427. Google Scholar [32] H. Song, Pullback attractors of non-autonomous reaction-diffusion equations in ${H}^1_0$, J. Differential Equations, 249 (2010), 2357-2376. doi: 10.1016/j.jde.2010.07.034. Google Scholar [33] X.L. Song and Y. Hou, Pullback $\underset{\scriptscriptstyle\centerdot}{-}$attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain, Discrete Contin. Dyn. Syst., 32 (2012), 991-1009. Google Scholar [34] T. Tachim Medjo, Unique strong and ${V-}$attractor of a three dimensional globally modified Allen-Cahn-Navier-Stokes model, Appl. Anal., 96 (2017), 2695-2716. doi: 10.1080/00036811.2016.1236924. Google Scholar [35] T. Tachim Medjo, Unique strong and ${V}-$attractor of a three dimensional globally modified Cahn-Hilliard-Navier-Stokes model, Appl. Anal., http://dx.doi.org/10.1080/00036811.2016.1236924, 2016.Google Scholar [36] T. Tachim Medjo, Pullback ${V}-$attractor of a three dimensional globally modified Cahn-Hilliard-Navier-Stokes model, Appl. Anal., http://dx.doi.org/10.1080/00036811.2017.1296952, 2017.Google Scholar [37] R. Temam, Infinite Dynamical Systems in Mechanics and Physics, volume 68, Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997. Google Scholar [38] M.I. Vishik, A.I. Komech and A.V. Fursikov, Some mathematical problems of statistical hydromechanics, Uspekhi Mat. Nauk, 34 (1979), 135-210. Google Scholar [39] B. You and F. Li, The existence of a pullback attractor for the three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation, Nonlinear Anal., 112 (2015), 118-128. doi: 10.1016/j.na.2014.08.018. Google Scholar [40] C. Zhao, S. Zhou and Y. Li, Uniform attractor for a two-dimensiona nonautonomous incompressible non-Newtonian fluid, Appl. Math. Comput., 201 (2008), 688-700. doi: 10.1016/j.amc.2008.01.005. Google Scholar

show all references

##### References:
 [1] H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506. doi: 10.1007/s00205-008-0160-2. Google Scholar [2] T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Physica D (Applied Physics), 32 (1999), 1119-1123. doi: 10.1088/0022-3727/32/10/307. Google Scholar [3] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. Google Scholar [4] T. Caraballo and P. E. Kloeden, The three-dimensional globally modified Navier-Stokes equations: Recent developments, In A. Johann, H. P. Kruse and F. Rupp, editors, Recent Trends in Dynamical Systems: Proceedings of a Conference in Honor of Jürgen Scheurle, Springer Proceedings in Mathematics and Statistics, 35 (2013), 473-492. Google Scholar [5] T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761-781. doi: 10.3934/dcdsb.2008.10.761. Google Scholar [6] T. Caraballo, J. Real and P.E. Kloeden, Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 6 (2006), 411-436. Google Scholar [7] T. Caraballo, J. Real and A.M. Márquez, Three-dimensional system of globally modified Navier-Stokes equations with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869-2883. doi: 10.1142/S0218127410027428. Google Scholar [8] S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flows, Phys. Rev. Lett., 81 (1998), 5338-5341. doi: 10.1103/PhysRevLett.81.5338. Google Scholar [9] S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Physica D, 133 (1999), 49-65. doi: 10.1016/S0167-2789(99)00098-6. Google Scholar [10] S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids, 11 (1999), 2343-2353. doi: 10.1063/1.870096. Google Scholar [11] S. Chen, D.D. Holm, L.G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Physica D, 133 (1999), 66-83. doi: 10.1016/S0167-2789(99)00099-8. Google Scholar [12] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988. Google Scholar [13] G. Deugoue and J.K. Djoko, On the time discretization for the globally modified three dimensional Navier-Stokes equations, J. Comput. Appl. Math, 235 (2011), 2015-2029. doi: 10.1016/j.cam.2010.10.003. Google Scholar [14] E. Feireisl, H. Petzeltová, E. 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. Google Scholar [15] C. 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. Google Scholar [16] 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. Google Scholar [17] 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. Google Scholar [18] P.C. Hohenberg and B.I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479. Google Scholar [19] D.D. Holm, J.E. Marsden and T.S. Ratiu, The Euler-Poincaré equations and semi-direct products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721. Google Scholar [20] D.D. Holm, J.E. Marsden and T.S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 349 (1998), 4173-4177. doi: 10.1103/PhysRevLett.80.4173. Google Scholar [21] P.E. Kloeden and J.A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753. Google Scholar [22] P.E. Kloeden, J.A. Langa and J. Real, Pullback ${V-}$attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 6 (2007), 937-955. doi: 10.3934/cpaa.2007.6.937. Google Scholar [23] P.E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 8 (2009), 785-802. doi: 10.3934/cpaa.2009.8.785. Google Scholar [24] P.E. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 1491-1508. doi: 10.1098/rspa.2007.1831. Google Scholar [25] G. Lukaszewicz, On pullback attractors in ${H}^1_0$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg, 20 (2010), 2637-2644. doi: 10.1142/S0218127410027258. Google Scholar [26] Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. Google Scholar [27] P. Marín-Rubio, J. Real and A.M. Márquez-Durán, On the convergence of solutions of globally modified Navier-Stokes equations with delays to solutions of Navier-Stokes equations with delays, Adv. Nonlinear Stud., 11 (2011), 917-927. Google Scholar [28] A.M. Márquez, Existence and uniqueness of solutions, and pullback attractor for a system of globally modified 3D-Navier-Stokes equations with finite delay, Bol. Soc. Esp. Mat. Apl. S$\overrightarrow{e}$MA, 51 (2010), 117-124. Google Scholar [29] J.E. Marsden and S. Shkoller, Global well-posedness for the {L}agrangian averaged Navier-Stokes (LANS-$α$) equations on bounded domains, Phil. Trans. R. Soc. Lond. A, 359 (2001), 1449-1468. doi: 10.1098/rsta.2001.0852. Google Scholar [30] A. Onuki, Phase transition of fluids in shear flow, J. Phys. Condens. Matter, 9 (1997), 6119-6157. Google Scholar [31] M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427. Google Scholar [32] H. Song, Pullback attractors of non-autonomous reaction-diffusion equations in ${H}^1_0$, J. Differential Equations, 249 (2010), 2357-2376. doi: 10.1016/j.jde.2010.07.034. Google Scholar [33] X.L. Song and Y. Hou, Pullback $\underset{\scriptscriptstyle\centerdot}{-}$attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain, Discrete Contin. Dyn. Syst., 32 (2012), 991-1009. Google Scholar [34] T. Tachim Medjo, Unique strong and ${V-}$attractor of a three dimensional globally modified Allen-Cahn-Navier-Stokes model, Appl. Anal., 96 (2017), 2695-2716. doi: 10.1080/00036811.2016.1236924. Google Scholar [35] T. Tachim Medjo, Unique strong and ${V}-$attractor of a three dimensional globally modified Cahn-Hilliard-Navier-Stokes model, Appl. Anal., http://dx.doi.org/10.1080/00036811.2016.1236924, 2016.Google Scholar [36] T. Tachim Medjo, Pullback ${V}-$attractor of a three dimensional globally modified Cahn-Hilliard-Navier-Stokes model, Appl. Anal., http://dx.doi.org/10.1080/00036811.2017.1296952, 2017.Google Scholar [37] R. Temam, Infinite Dynamical Systems in Mechanics and Physics, volume 68, Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997. Google Scholar [38] M.I. Vishik, A.I. Komech and A.V. Fursikov, Some mathematical problems of statistical hydromechanics, Uspekhi Mat. Nauk, 34 (1979), 135-210. Google Scholar [39] B. You and F. Li, The existence of a pullback attractor for the three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation, Nonlinear Anal., 112 (2015), 118-128. doi: 10.1016/j.na.2014.08.018. Google Scholar [40] C. Zhao, S. Zhou and Y. Li, Uniform attractor for a two-dimensiona nonautonomous incompressible non-Newtonian fluid, Appl. Math. Comput., 201 (2008), 688-700. doi: 10.1016/j.amc.2008.01.005. Google Scholar
 [1] 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 [2] Tomás Caraballo, Peter E. Kloeden, José Real. Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 761-781. doi: 10.3934/dcdsb.2008.10.761 [3] 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 [4] P.E. Kloeden, Pedro Marín-Rubio, José Real. Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 785-802. doi: 10.3934/cpaa.2009.8.785 [5] Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Three dimensional system of globally modified Navier-Stokes equations with infinite delays. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 655-673. doi: 10.3934/dcdsb.2010.14.655 [6] G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123 [7] 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 [8] Bo You. Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2283-2298. doi: 10.3934/cpaa.2019103 [9] Jochen Merker. Strong solutions of doubly nonlinear Navier-Stokes equations. Conference Publications, 2011, 2011 (Special) : 1052-1060. doi: 10.3934/proc.2011.2011.1052 [10] 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 [11] 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 [12] Susan Friedlander, Nataša Pavlović. Remarks concerning modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 269-288. doi: 10.3934/dcds.2004.10.269 [13] 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 [14] Vladimir V. Chepyzhov, E. S. Titi, Mark I. Vishik. On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 481-500. doi: 10.3934/dcds.2007.17.481 [15] 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 [16] Hirokazu Ninomiya. Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2001-2019. doi: 10.3934/dcds.2019084 [17] 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 [18] 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 [19] 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 [20] 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

2018 Impact Factor: 1.143

## Metrics

• PDF downloads (66)
• HTML views (235)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]