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Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities
1. | School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China, China |
2. | Department of Mathematics, South China Normal University, Guangzhou, Guangdong 510631 |
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. |
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. |
[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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