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November  2018, 17(6): 2455-2477. doi: 10.3934/cpaa.2018117

## Well-posedness for a non-isothermal flow of two viscous incompressible fluids

 Departamento de Matemática, Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Rua Sergio Buarque de Holanda, 651, 13083-859, Campinas - SP, Brazil

* Corresponding author

Received  October 2017 Revised  January 2018 Published  June 2018

Fund Project: G. Planas was partially supported by CNPq-Brazil, grants 306646/2015-3 and 402388/2016-0. J. H. Lopes was supported by Capes-Brazil and CNPq-Brazil, grant 143214/2015-2.

This work is concerned with a non-isothermal diffuse-interface model which describes the motion of a mixture of two viscous incompressible fluids. The model consists of modified Navier-Stokes equations coupled with a phase-field equation given by a convective Allen-Cahn equation, and energy transport equation for the temperature. This model admits a dissipative energy inequality. It is investigated the well-posedness of the problem in the two and three dimensional cases without any restriction on the size of the initial data. Moreover, regular and singular potentials for the phase-field equation are considered.

Citation: Juliana Honda Lopes, Gabriela Planas. Well-posedness for a non-isothermal flow of two viscous incompressible fluids. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2455-2477. doi: 10.3934/cpaa.2018117
##### 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] D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30 (1998), 139-165.  doi: 10.1146/annurev.fluid.30.1.139. [3] T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flows, J. Phys. D (Applied Physics), 32 (1999), 1119-1123.  doi: 10.1088/0022-3727/32/10/307. [4] J. L. Boldrini and G. Planas, Weak solutions of a phase-field model for phase change of an alloy with thermal properties, Math. Methods Appl. Sci., 25 (2002), 1177-1193.  doi: 10.1002/mma.334. [5] J. W. Cahn and J. E. Hillard, Free energy of a non-uniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1063/1.1744102. [6] L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.  doi: 10.1007/s10492-009-0008-6. [7] B. Climent-Ezquerra, F. Guillén-González and M. J. Moreno-Iraberte, Regularity and time-periodicity for a nematic liquid crystal model, Nonlinear Anal., 71 (2009), 539-549.  doi: 10.1016/j.na.2008.10.092. [8] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, 1986. [9] Q. Du, M. Li and C. Liu, Analysis of a phase field Navier-Stokes vesicle-fluid interaction model, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 539-556.  doi: 10.3934/dcdsb.2007.8.539. [10] M. Eleuteri, E. Rocca and G. Schimperna, On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids, Discrete Contin. Dyn. Syst., 35 (2015), 2497-2522.  doi: 10.3934/dcds.2015.35.2497. [11] M. Eleuteri, E. Rocca and G. Schimperna, Existence of solutions to a two-dimensional model for nonisothermal two-phase flows of incompressible fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1431-1454.  doi: 10.1016/j.anihpc.2015.05.006. [12] 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. [13] J. J. Feng, C. Liu, J. Shen and P. Yue, A energetic variational formulation with phase field methods for interfacial dynamics of complex fluids: advantages and challenges, Modeling of Soft Matter (M. T. Calderer, E. M. Terentjev eds. ), vol. IMA 141, Springer, New York, 2005, 1–26. doi: 10.1007/0-387-32153-5_1. [14] X. Feng, Y. He and C. Liu, Analysis of finite element approximations of a phase field model for two-phase fluids, Math. Comp., 76 (2007), 539-571.  doi: 10.1090/S0025-5718-06-01915-6. [15] S. Forest and M. Amestoy, Hypertemperature in thermoelastic solids, C. R. Mecanique, 336 (2008), 347-353.  doi: 10.1016/j.crme.2008.01.007. [16] M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. [17] A. Friedman, Partial Differential Equations, Dover Books on Mathematics, New York, 2008. [18] C. G. 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. [19] C. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows, Discrete Cont. Dyn. Sys., 28 (2010), 1-39.  doi: 10.3934/dcds.2010.28.1. [20] C. 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. [21] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Classics in Applied Mathematics, Boston, 1985. doi: 10.1137/1.9781611972030. [22] P. Ireman and Q-S. Nguyen, Using the gradients of temperature and internal parameters in Continuum Thermodynamics, C. R. Mecanique, 332 (2004), 249-255.  doi: 10.1016/j.crme.2004.01.012. [23] J. Jiang, Y. Li and C. Liu, Two-phase incompressible flows with variable density: An energetic variational approach, Discrete Cont. Dyn. Sys., 37 (2017), 3243-3284.  doi: 10.3934/dcds.2017138. [24] Y. Li, S. Ding and M. Huang, Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities, Discrete Cont. Dyn. Sys. Ser. B, 21 (2016), 1507-1523.  doi: 10.3934/dcdsb.2016009. [25] F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537.  doi: 10.1002/cpa.3160480503. [26] C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Phys. D., 179 (2003), 211-228.  doi: 10.1016/S0167-2789(03)00030-7. [27] C. Liu and N. J. Walkington, An eulerian description of fluids containing visco-elastic particles, Arch. Ration. Mech. Anal, 159 (2001), 229-252.  doi: 10.1007/s002050100158. [28] S. A. Lorca and J. L. Boldrini, The initial value problem for a generalized Boussinesq model, Nonlinear Anal., 36 (1999), 457-480.  doi: 10.1016/S0362-546X(97)00635-4. [29] P. Marín-Rubio, G. Planas and J. Real, Asymptotic behaviour of a phase-field model with three coupled equations without uniqueness, J. Differential Equations, 246 (2009), 4632-4652.  doi: 10.1016/j.jde.2009.01.021. [30] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications 53, Springer-Science+Business Media, 1991. doi: 10.1007/978-94-011-3562-7. [31] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Super. Pisa Ser. 3, 13 (1959), 115-162.  doi: 10.1007/978-3-642-10926-3_1. [32] J. Simon, Compacts sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [33] P. Sun, C. Liu and J. Xu, Phase field model of thermo-induced Marangoni effects in the mixtures and its numerical simulations with mixed finite element method, Commun. Comput. Phys., 6 (2009), 1095-1117.  doi: 10.4208/cicp.2009.v6.p1095. [34] Y. Sun and Z. Zhang, Global regularity for the initial-boundary value problem of 2-D Boussinesq system with variable viscosity and thermal diffusivity, J. Differential Equations, 255 (2013), 1069-1085.  doi: 10.1016/j.jde.2013.04.032. [35] M. E. Taylor, Partial Differential Equations I, Applied Mathematical Sciences, 115, 2011. doi: 10.1007/978-1-4419-7055-8. [36] R. Temam, Navier-Stokes Equations, Studies in Mathematics and its Applications 2, North-Holland, Amsterdam, 1977. [37] H. Wu, Well-posedness of a diffuse-interface model for two-phase incompressible flows with thermo-induced Marangoni effect, European J. Appl. Math., 28 (2017), 380-434.  doi: 10.1017/S0956792516000322. [38] H. Wu and X. Xu, Analysis of a diffuse-interface model for the binary viscous incompressible fluids with thermo-induced marangoni effects, Comunn. Math. Sci., 11 (2013), 603-633.  doi: 10.4310/CMS.2013.v11.n2.a15. [39] 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. [40] 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, J. Comput. Phys., 218 (2006), 417-428.  doi: 10.1016/j.jcp.2006.02.021.

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] D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30 (1998), 139-165.  doi: 10.1146/annurev.fluid.30.1.139. [3] T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flows, J. Phys. D (Applied Physics), 32 (1999), 1119-1123.  doi: 10.1088/0022-3727/32/10/307. [4] J. L. Boldrini and G. Planas, Weak solutions of a phase-field model for phase change of an alloy with thermal properties, Math. Methods Appl. Sci., 25 (2002), 1177-1193.  doi: 10.1002/mma.334. [5] J. W. Cahn and J. E. Hillard, Free energy of a non-uniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1063/1.1744102. [6] L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.  doi: 10.1007/s10492-009-0008-6. [7] B. Climent-Ezquerra, F. Guillén-González and M. J. Moreno-Iraberte, Regularity and time-periodicity for a nematic liquid crystal model, Nonlinear Anal., 71 (2009), 539-549.  doi: 10.1016/j.na.2008.10.092. [8] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, 1986. [9] Q. Du, M. Li and C. Liu, Analysis of a phase field Navier-Stokes vesicle-fluid interaction model, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 539-556.  doi: 10.3934/dcdsb.2007.8.539. [10] M. Eleuteri, E. Rocca and G. Schimperna, On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids, Discrete Contin. Dyn. Syst., 35 (2015), 2497-2522.  doi: 10.3934/dcds.2015.35.2497. [11] M. Eleuteri, E. Rocca and G. Schimperna, Existence of solutions to a two-dimensional model for nonisothermal two-phase flows of incompressible fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1431-1454.  doi: 10.1016/j.anihpc.2015.05.006. [12] 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. [13] J. J. Feng, C. Liu, J. Shen and P. Yue, A energetic variational formulation with phase field methods for interfacial dynamics of complex fluids: advantages and challenges, Modeling of Soft Matter (M. T. Calderer, E. M. Terentjev eds. ), vol. IMA 141, Springer, New York, 2005, 1–26. doi: 10.1007/0-387-32153-5_1. [14] X. Feng, Y. He and C. Liu, Analysis of finite element approximations of a phase field model for two-phase fluids, Math. Comp., 76 (2007), 539-571.  doi: 10.1090/S0025-5718-06-01915-6. [15] S. Forest and M. Amestoy, Hypertemperature in thermoelastic solids, C. R. Mecanique, 336 (2008), 347-353.  doi: 10.1016/j.crme.2008.01.007. [16] M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. [17] A. Friedman, Partial Differential Equations, Dover Books on Mathematics, New York, 2008. [18] C. G. 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. [19] C. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows, Discrete Cont. Dyn. Sys., 28 (2010), 1-39.  doi: 10.3934/dcds.2010.28.1. [20] C. 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. [21] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Classics in Applied Mathematics, Boston, 1985. doi: 10.1137/1.9781611972030. [22] P. Ireman and Q-S. Nguyen, Using the gradients of temperature and internal parameters in Continuum Thermodynamics, C. R. Mecanique, 332 (2004), 249-255.  doi: 10.1016/j.crme.2004.01.012. [23] J. Jiang, Y. Li and C. Liu, Two-phase incompressible flows with variable density: An energetic variational approach, Discrete Cont. Dyn. Sys., 37 (2017), 3243-3284.  doi: 10.3934/dcds.2017138. [24] Y. Li, S. Ding and M. Huang, Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities, Discrete Cont. Dyn. Sys. Ser. B, 21 (2016), 1507-1523.  doi: 10.3934/dcdsb.2016009. [25] F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537.  doi: 10.1002/cpa.3160480503. [26] C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Phys. D., 179 (2003), 211-228.  doi: 10.1016/S0167-2789(03)00030-7. [27] C. Liu and N. J. Walkington, An eulerian description of fluids containing visco-elastic particles, Arch. Ration. Mech. Anal, 159 (2001), 229-252.  doi: 10.1007/s002050100158. [28] S. A. Lorca and J. L. Boldrini, The initial value problem for a generalized Boussinesq model, Nonlinear Anal., 36 (1999), 457-480.  doi: 10.1016/S0362-546X(97)00635-4. [29] P. Marín-Rubio, G. Planas and J. Real, Asymptotic behaviour of a phase-field model with three coupled equations without uniqueness, J. Differential Equations, 246 (2009), 4632-4652.  doi: 10.1016/j.jde.2009.01.021. [30] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications 53, Springer-Science+Business Media, 1991. doi: 10.1007/978-94-011-3562-7. [31] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Super. Pisa Ser. 3, 13 (1959), 115-162.  doi: 10.1007/978-3-642-10926-3_1. [32] J. Simon, Compacts sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [33] P. Sun, C. Liu and J. Xu, Phase field model of thermo-induced Marangoni effects in the mixtures and its numerical simulations with mixed finite element method, Commun. Comput. Phys., 6 (2009), 1095-1117.  doi: 10.4208/cicp.2009.v6.p1095. [34] Y. Sun and Z. Zhang, Global regularity for the initial-boundary value problem of 2-D Boussinesq system with variable viscosity and thermal diffusivity, J. Differential Equations, 255 (2013), 1069-1085.  doi: 10.1016/j.jde.2013.04.032. [35] M. E. Taylor, Partial Differential Equations I, Applied Mathematical Sciences, 115, 2011. doi: 10.1007/978-1-4419-7055-8. [36] R. Temam, Navier-Stokes Equations, Studies in Mathematics and its Applications 2, North-Holland, Amsterdam, 1977. [37] H. Wu, Well-posedness of a diffuse-interface model for two-phase incompressible flows with thermo-induced Marangoni effect, European J. Appl. Math., 28 (2017), 380-434.  doi: 10.1017/S0956792516000322. [38] H. Wu and X. Xu, Analysis of a diffuse-interface model for the binary viscous incompressible fluids with thermo-induced marangoni effects, Comunn. Math. Sci., 11 (2013), 603-633.  doi: 10.4310/CMS.2013.v11.n2.a15. [39] 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. [40] 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, J. Comput. Phys., 218 (2006), 417-428.  doi: 10.1016/j.jcp.2006.02.021.
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