# American Institute of Mathematical Sciences

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

 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.
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.  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.  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, (1998), 139.  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.   Google Scholar [5] S. C. Brenner, Korn's inequalities for piecewise $H^1$ vector fields,, Mathematics of Computation, 73 (2004), 1067.  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.  doi: 10.1007/BF01212349.  Google Scholar [7] G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics,, Springer-Verlag, (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.  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, (2005), 1.  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.   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, (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.  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.  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.  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.   Google Scholar [17] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 9 (1962), 187.   Google Scholar [18] M. Struwe, On partial regularity results for the Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 437.  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.  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.  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.  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.  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.  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, (1998), 139.  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.   Google Scholar [5] S. C. Brenner, Korn's inequalities for piecewise $H^1$ vector fields,, Mathematics of Computation, 73 (2004), 1067.  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.  doi: 10.1007/BF01212349.  Google Scholar [7] G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics,, Springer-Verlag, (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.  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, (2005), 1.  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.   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, (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.  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.  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.  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.   Google Scholar [17] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 9 (1962), 187.   Google Scholar [18] M. Struwe, On partial regularity results for the Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 437.  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.  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.  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.  doi: 10.1016/j.jmaa.2011.05.042.  Google Scholar
 [1] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 [2] Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 [3] Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 [4] Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352 [5] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [6] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110 [7] Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 [8] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 [9] Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325 [10] Tomáš Roubíček. Cahn-Hilliard equation with capillarity in actual deforming configurations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 41-55. doi: 10.3934/dcdss.2020303 [11] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [12] Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054 [13] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467 [14] Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213 [15] Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303 [16] Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457 [17] Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464 [18] Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366 [19] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [20] Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

2019 Impact Factor: 1.27

## Metrics

• HTML views (0)
• Cited by (5)

• on AIMS