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July  2016, 21(5): 1507-1523. doi: 10.3934/dcdsb.2016009

Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities

Yinghua Li 1, , Shijin Ding 2, and  Mingxia Huang 1,

1. 

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China, China
2. 

Department of Mathematics, South China Normal University, Guangzhou, Guangdong 510631

Received  March 2013 Revised  September 2013 Published  April 2016

This paper is concerned with a coupled Navier-Stokes/Allen-Cahn system describing a diffuse interface model for two-phase flow of viscous incompressible fluids with different densities in a bounded domain $\Omega\subset\mathbb R^N$($N=2,3$). We establish a criterion for possible break down of such solutions at finite time in terms of the temporal integral of both the maximum norm of the deformation tensor of velocity gradient and the square of maximum norm of gradient of phase field variable in 2D. In 3D, the temporal integral of the square of maximum norm of velocity is also needed. Here, we suppose the initial density function $\rho_0$ has a positive lower bound.
Keywords: blow-up criterion., Allen-Cahn equation, Diffuse interface model, nonhomogeneous Navier-Stokes equations.
Mathematics Subject Classification: Primary: 76T10, 35Q30; Secondary: 35B4.
Citation: 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
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]

H. Abels and E. Feireisl, On a diffuse interface model for a two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57 (2008), 659-698. doi: 10.1512/iumj.2008.57.3391.  Google Scholar

[3]

D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, in Annual review of fluid mechanics, volume 30 of Annu. Rev. Fluid Mech., 139-165. Annual Reviews, Palo Alto, CA, 1998. doi: 10.1146/annurev.fluid.30.1.139.  Google Scholar

[4]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212.  Google Scholar

[5]

S. C. Brenner, Korn's inequalities for piecewise $H^1$ vector fields, Mathematics of Computation, 73 (2004), 1067-1087. doi: 10.1090/S0025-5718-03-01579-5.  Google Scholar

[6]

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equation, Comm. Math. Phys., 94 (1984), 61-66. doi: 10.1007/BF01212349.  Google Scholar

[7]

G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.  Google Scholar

[8]

S. Ding, Y. Li and W. Luo, Global solutions for a coupled compressible Navier-Stokes/Allen-Cahn system in 1-D, J. Math Fluid Mech., 15 (2013), 335-360. doi: 10.1007/s00021-012-0104-3.  Google Scholar

[9]

J. J. Feng, C. Liu, J. Shen and P. Yue, An energetic variational formulation with phase field methods for interfacial dynamics of complex fluids: Advantages and challenges, in Modeling of soft matter, volume 141 of IMA Vol. Math. Appl., 1-26. Springer, New York, 2005. doi: 10.1007/0-387-32153-5_1.  Google Scholar

[10]

E. Feireisl, H. Petzeltová, E. Rocca and G. Schimperna, Ayalysis of a phase-field model for two-phase compressible fluids, Math. Meth. Appl. Sci., 31 (2008), 1972-1995. Google Scholar

[11]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Linearized Steady Problems, Vol. 1, in: Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[12]

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

[13]

H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37 (2006), 1417-1434. doi: 10.1137/S0036141004442197.  Google Scholar

[14]

Y. Li, S. Ding and M. Huang, Strong solutions for an incompressible Navier-Stokes/Allen-Cahn system with Different Densities,, preprint., ().   Google Scholar

[15]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc. R. Soc. Lond. A, 454 (1998), 2617-2654. doi: 10.1098/rspa.1998.0273.  Google Scholar

[16]

G. Ponce, Remarks on a paper: "Remarks on the breakdown of smooth solutions for the 3-D Euler equations", Comm. Math. Phys., 98 (1985), 349-353.  Google Scholar

[17]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.  Google Scholar

[18]

M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458. doi: 10.1002/cpa.3160410404.  Google Scholar

[19]

X. Xu, L. Zhao and C. Liu, Axisymmetric solutions to coupled Navier-Stokes/Allen-Cahn equations, SIAM J. Math. Anal., 41 (2010), 2246-2282. doi: 10.1137/090754698.  Google Scholar

[20]

X. Yang, J. J. Feng, C. Liu and J. Shen, Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method, Journal of Computational Physics, 218 (2006), 417-428. doi: 10.1016/j.jcp.2006.02.021.  Google Scholar

[21]

L. Zhao, B. Guo and H. Huang, Vanishing visosity limit for a coupled Navier-Stokes/Allen-Cahn system, J. Math. Anal. Appl., 384 (2011), 232-245. doi: 10.1016/j.jmaa.2011.05.042.  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]

H. Abels and E. Feireisl, On a diffuse interface model for a two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57 (2008), 659-698. doi: 10.1512/iumj.2008.57.3391.  Google Scholar

[3]

D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, in Annual review of fluid mechanics, volume 30 of Annu. Rev. Fluid Mech., 139-165. Annual Reviews, Palo Alto, CA, 1998. doi: 10.1146/annurev.fluid.30.1.139.  Google Scholar

[4]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212.  Google Scholar

[5]

S. C. Brenner, Korn's inequalities for piecewise $H^1$ vector fields, Mathematics of Computation, 73 (2004), 1067-1087. doi: 10.1090/S0025-5718-03-01579-5.  Google Scholar

[6]

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equation, Comm. Math. Phys., 94 (1984), 61-66. doi: 10.1007/BF01212349.  Google Scholar

[7]

G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.  Google Scholar

[8]

S. Ding, Y. Li and W. Luo, Global solutions for a coupled compressible Navier-Stokes/Allen-Cahn system in 1-D, J. Math Fluid Mech., 15 (2013), 335-360. doi: 10.1007/s00021-012-0104-3.  Google Scholar

[9]

J. J. Feng, C. Liu, J. Shen and P. Yue, An energetic variational formulation with phase field methods for interfacial dynamics of complex fluids: Advantages and challenges, in Modeling of soft matter, volume 141 of IMA Vol. Math. Appl., 1-26. Springer, New York, 2005. doi: 10.1007/0-387-32153-5_1.  Google Scholar

[10]

E. Feireisl, H. Petzeltová, E. Rocca and G. Schimperna, Ayalysis of a phase-field model for two-phase compressible fluids, Math. Meth. Appl. Sci., 31 (2008), 1972-1995. Google Scholar

[11]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Linearized Steady Problems, Vol. 1, in: Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[12]

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

[13]

H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37 (2006), 1417-1434. doi: 10.1137/S0036141004442197.  Google Scholar

[14]

Y. Li, S. Ding and M. Huang, Strong solutions for an incompressible Navier-Stokes/Allen-Cahn system with Different Densities,, preprint., ().   Google Scholar

[15]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc. R. Soc. Lond. A, 454 (1998), 2617-2654. doi: 10.1098/rspa.1998.0273.  Google Scholar

[16]

G. Ponce, Remarks on a paper: "Remarks on the breakdown of smooth solutions for the 3-D Euler equations", Comm. Math. Phys., 98 (1985), 349-353.  Google Scholar

[17]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.  Google Scholar

[18]

M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458. doi: 10.1002/cpa.3160410404.  Google Scholar

[19]

X. Xu, L. Zhao and C. Liu, Axisymmetric solutions to coupled Navier-Stokes/Allen-Cahn equations, SIAM J. Math. Anal., 41 (2010), 2246-2282. doi: 10.1137/090754698.  Google Scholar

[20]

X. Yang, J. J. Feng, C. Liu and J. Shen, Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method, Journal of Computational Physics, 218 (2006), 417-428. doi: 10.1016/j.jcp.2006.02.021.  Google Scholar

[21]

L. Zhao, B. Guo and H. Huang, Vanishing visosity limit for a coupled Navier-Stokes/Allen-Cahn system, J. Math. Anal. Appl., 384 (2011), 232-245. doi: 10.1016/j.jmaa.2011.05.042.  Google Scholar

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