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

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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

Abstract
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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 and 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.
[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.
[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.
[4]	F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212.
[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.
[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.
[7]	G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	Y. Li, S. Ding and M. Huang, Strong solutions for an incompressible Navier-Stokes/Allen-Cahn system with Different Densities, preprint.
[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.
[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.
[17]	J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.
[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.
[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.
[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.
[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.

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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.
[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.
[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.
[4]	F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212.
[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.
[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.
[7]	G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	Y. Li, S. Ding and M. Huang, Strong solutions for an incompressible Navier-Stokes/Allen-Cahn system with Different Densities, preprint.
[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.
[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.
[17]	J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.
[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.
[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.
[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.
[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.

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