Two-phase incompressible flows with variable density: An energetic variational approach

Jie Jiang; Yinghua Li; Chun Liu

doi:10.3934/dcds.2017138

Previous Article
Local well-posedness of the Camassa-Holm equation on the real line
DCDS Home
This Issue
Next Article
Sharp estimation for the solutions of delay differential and Halanay type inequalities

June 2017, 37(6): 3243-3284. doi: 10.3934/dcds.2017138

Two-phase incompressible flows with variable density: An energetic variational approach

Jie Jiang ^1,, , Yinghua Li ^2, and Chun Liu ^3,

1.	Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, Hubei Province, China
2.	School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
3.	Department of Mathematics, Penn State University, University Park, PA 16802, USA

* Corresponding author

Received May 2016 Revised January 2017 Published February 2017

In this paper, we study a diffuse-interface model for two-phase incompressible flows with different densities. First, we present a derivation of the model using an energetic variational approach. Our model allows large density ratio between the two phases and moreover, it is thermodynamically consistent and admits a dissipative energy law. Under suitable assumptions on the average density function, we establish the global existence of a weak solution in the 3D case as well as the global well-posedness of strong solutions in the 2D case to an initial-boundary problem for the resulting Allen-Cahn-Navier-Stokes system. Furthermore, we investigate the longtime behavior of the 2D strong solutions. In particular, we obtain existence of a maximal compact attractor and prove that the solution will converge to an equilibrium as time goes to infinity.

Keywords: Two-phase flow, incompressible Navier-Stokes, variable density, global existence, longtime behavior.

Mathematics Subject Classification: Primary: 35Q30, 35Q35, 35B40; Secondary: 76T99.

Citation: Jie Jiang, Yinghua Li, Chun Liu. Two-phase incompressible flows with variable density: An energetic variational approach. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3243-3284. doi: 10.3934/dcds.2017138

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, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Commun. Math. Phys., 289 (2009), 45-73. doi: 10.1007/s00220-009-0806-4. Google Scholar
[3]	H. Abels, H. Garcke and G. Grün, Thermodynamically consistent diffuse interface models for incompressible two-phase flows with different densities, preprint, arXiv: 1011.00528 (2010). Google Scholar
[4]	H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities Math. Mod. Meth. Appl. Sic. 22 (2012), 1150013, 40pp. doi: 10.1142/S0218202511500138. Google Scholar
[5]	H. Abels, D. Depner and H. Garcke, Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities, J. Math. Fluid Mech., 15 (2013), 453-480. doi: 10.1007/s00021-012-0118-x. Google Scholar
[6]	H. Abels, D. Depner and H. Garcke, On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility, Ann. I. H. Poincaré-AN, 30 (2013), 1175-1190. doi: 10.1016/j.anihpc.2013.01.002. Google Scholar
[7]	H. W. Alt, The entropy principle for interfaces. Fluids and solids, Adv. Math. Sci. Appl., 19 (2009), 585-663. Google Scholar
[8]	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. Google Scholar
[9]	S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids Elsevier, 1989. Google Scholar
[10]	V. I. Arnold, Mathematical Methods of Classical Mechanics Springer-Verlag, New York-Heidelberg, 1978. Google Scholar
[11]	F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212. Google Scholar
[12]	F. Boyer, Nonhomogeneous Cahn-Hilliard fluids, Ann. Inst. H. Poincaré -AN, 18 (2001), 225-259. doi: 10.1016/S0294-1449(00)00063-9. Google Scholar
[13]	P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics Cambridge, 1995. Google Scholar
[14]	H. Ding, P. D. M. Spelt and C. Shu, Diffusive interface model for incompressible two-phase flows with large density ratios, J. Comp. Phys., 22 (2007), 2078-2095. Google Scholar
[15]	S. Ding, Y. Li and W. Luo, Global solutions for a coupled compressible Navier-Stokes/Allen-Cahn system in 1D, J. Math. Fluid Mech., 15 (2013), 335-360. doi: 10.1007/s00021-012-0104-3. Google Scholar
[16]	D. A. Edwards, H. Brenner and D. T. Wasan, Interfacial Transport Process and Rheology Butterworths/Heinemann, London, 1991. Google Scholar
[17]	E. FEreisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Different. Equations, 12 (2000), 647-673. doi: 10.1023/A:1026467729263. Google Scholar
[18]	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
[19]	C. G. Gal and M. Grasselli, Longtime behavior of a model for homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst, 28 (2010), 1-39. doi: 10.3934/dcds.2010.28.1. Google Scholar
[20]	M. E. Gurtin, D. Polignone and J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831. doi: 10.1142/S0218202596000341. Google Scholar
[21]	A. Haraux, Systémes Dynamiques Dissipatifs et Applications Masson, Paris, 1991. Google Scholar
[22]	J. U. Kim, Weak solutions of an initial boundary value problem for an incompressible viscous fluid with nonnegative density, SIAM J. Math. Anal., 18 (1987), 89-96. doi: 10.1137/0518007. Google Scholar
[23]	R. Kubo, The fluctuation-dissipation theorem, Report on Progress in Physics, 1966. Google Scholar
[24]	J. L. Lions, On some questions in boundary value problems of mathematical physics, Contemporary developments in continuum mechanics and partial differential equations: Proceedings of the International Symposium on Continuum Mechanics and Partial Differential Equations, vol. 30 of Mathematics Studies, New York, 1977, North-Holland, 285–346. Google Scholar
[25]	C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D, 179 (2003), 211-228. doi: 10.1016/S0167-2789(03)00030-7. Google Scholar
[26]	C. Liu, J. Shen and X. F. Yang, Decoupled energy stable schemes for a phase-field model of two-phase incompressible flows with variable density, J. Sci. Comput., 62 (2015), 601-622. doi: 10.1007/s10915-014-9867-4. Google Scholar
[27]	J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, P. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 2617-2654. doi: 10.1098/rspa.1998.0273. Google Scholar
[28]	L. Onsager, Reciprocal relations in irreversible processes-Ⅰ, Phys. Rev., 37 (1931), 405. Google Scholar
[29]	L. Onsager, Reciprocal relations in irreversible processes-Ⅱ, Phys. Rev., 38 (1931), 2265. Google Scholar
[30]	L. Rayleigh, Some general theorem relating to vibrations, Proc. Lond. Math. Soc., 4 (1873), 357-368. Google Scholar
[31]	J. Shen and X. F. Yang, A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities, SIAM J. Sci. Comput., 32 (2010), 1159-1179. doi: 10.1137/09075860X. Google Scholar
[32]	J. Shen, X. Yang and Q. Wang, On mass conservation in phase field models for binary fluids, Commun. Comput. Phys., 13 (2013), 1045-1065. doi: 10.4208/cicp.300711.160212a. Google Scholar
[33]	W. Shen and S. Zheng, Maximal attractor for the coupled Cahn-Hilliard equations, Nonlinear Anal., 49 (2002), 21-34. doi: 10.1016/S0362-546X(00)00246-7. Google Scholar
[34]	J. Simon, Compact sets in the space $L^p(0, T; B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar
[35]	J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. doi: 10.1137/0521061. Google Scholar
[36]	K. Taira, Analytic Semigroups and Semilinear Initial Boundary Value Problems Cambridge Univ. Press, Cambridge, New York, 1995. doi: 10.1017/CBO9780511662362. Google Scholar
[37]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Applied Mathematics and Sciences, Vol. 68 Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8. Google Scholar
[38]	H. Wu and X. Xu, Analysis of a diffuse-interface model for the binary viscous incompressible fluids with thermo-induced Marangoni effects, Commun. Math. Sci., 11 (2013), 603-633. doi: 10.4310/CMS.2013.v11.n2.a15. Google Scholar
[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. Google Scholar
[40]	S. Zheng, Nonlinear Evolution Equations Pitman series Monographs and Survey in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, Florida, 2004. doi: 10.1201/9780203492222. 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, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Commun. Math. Phys., 289 (2009), 45-73. doi: 10.1007/s00220-009-0806-4. Google Scholar
[3]	H. Abels, H. Garcke and G. Grün, Thermodynamically consistent diffuse interface models for incompressible two-phase flows with different densities, preprint, arXiv: 1011.00528 (2010). Google Scholar
[4]	H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities Math. Mod. Meth. Appl. Sic. 22 (2012), 1150013, 40pp. doi: 10.1142/S0218202511500138. Google Scholar
[5]	H. Abels, D. Depner and H. Garcke, Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities, J. Math. Fluid Mech., 15 (2013), 453-480. doi: 10.1007/s00021-012-0118-x. Google Scholar
[6]	H. Abels, D. Depner and H. Garcke, On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility, Ann. I. H. Poincaré-AN, 30 (2013), 1175-1190. doi: 10.1016/j.anihpc.2013.01.002. Google Scholar
[7]	H. W. Alt, The entropy principle for interfaces. Fluids and solids, Adv. Math. Sci. Appl., 19 (2009), 585-663. Google Scholar
[8]	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. Google Scholar
[9]	S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids Elsevier, 1989. Google Scholar
[10]	V. I. Arnold, Mathematical Methods of Classical Mechanics Springer-Verlag, New York-Heidelberg, 1978. Google Scholar
[11]	F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212. Google Scholar
[12]	F. Boyer, Nonhomogeneous Cahn-Hilliard fluids, Ann. Inst. H. Poincaré -AN, 18 (2001), 225-259. doi: 10.1016/S0294-1449(00)00063-9. Google Scholar
[13]	P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics Cambridge, 1995. Google Scholar
[14]	H. Ding, P. D. M. Spelt and C. Shu, Diffusive interface model for incompressible two-phase flows with large density ratios, J. Comp. Phys., 22 (2007), 2078-2095. Google Scholar
[15]	S. Ding, Y. Li and W. Luo, Global solutions for a coupled compressible Navier-Stokes/Allen-Cahn system in 1D, J. Math. Fluid Mech., 15 (2013), 335-360. doi: 10.1007/s00021-012-0104-3. Google Scholar
[16]	D. A. Edwards, H. Brenner and D. T. Wasan, Interfacial Transport Process and Rheology Butterworths/Heinemann, London, 1991. Google Scholar
[17]	E. FEreisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Different. Equations, 12 (2000), 647-673. doi: 10.1023/A:1026467729263. Google Scholar
[18]	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
[19]	C. G. Gal and M. Grasselli, Longtime behavior of a model for homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst, 28 (2010), 1-39. doi: 10.3934/dcds.2010.28.1. Google Scholar
[20]	M. E. Gurtin, D. Polignone and J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831. doi: 10.1142/S0218202596000341. Google Scholar
[21]	A. Haraux, Systémes Dynamiques Dissipatifs et Applications Masson, Paris, 1991. Google Scholar
[22]	J. U. Kim, Weak solutions of an initial boundary value problem for an incompressible viscous fluid with nonnegative density, SIAM J. Math. Anal., 18 (1987), 89-96. doi: 10.1137/0518007. Google Scholar
[23]	R. Kubo, The fluctuation-dissipation theorem, Report on Progress in Physics, 1966. Google Scholar
[24]	J. L. Lions, On some questions in boundary value problems of mathematical physics, Contemporary developments in continuum mechanics and partial differential equations: Proceedings of the International Symposium on Continuum Mechanics and Partial Differential Equations, vol. 30 of Mathematics Studies, New York, 1977, North-Holland, 285–346. Google Scholar
[25]	C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D, 179 (2003), 211-228. doi: 10.1016/S0167-2789(03)00030-7. Google Scholar
[26]	C. Liu, J. Shen and X. F. Yang, Decoupled energy stable schemes for a phase-field model of two-phase incompressible flows with variable density, J. Sci. Comput., 62 (2015), 601-622. doi: 10.1007/s10915-014-9867-4. Google Scholar
[27]	J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, P. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 2617-2654. doi: 10.1098/rspa.1998.0273. Google Scholar
[28]	L. Onsager, Reciprocal relations in irreversible processes-Ⅰ, Phys. Rev., 37 (1931), 405. Google Scholar
[29]	L. Onsager, Reciprocal relations in irreversible processes-Ⅱ, Phys. Rev., 38 (1931), 2265. Google Scholar
[30]	L. Rayleigh, Some general theorem relating to vibrations, Proc. Lond. Math. Soc., 4 (1873), 357-368. Google Scholar
[31]	J. Shen and X. F. Yang, A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities, SIAM J. Sci. Comput., 32 (2010), 1159-1179. doi: 10.1137/09075860X. Google Scholar
[32]	J. Shen, X. Yang and Q. Wang, On mass conservation in phase field models for binary fluids, Commun. Comput. Phys., 13 (2013), 1045-1065. doi: 10.4208/cicp.300711.160212a. Google Scholar
[33]	W. Shen and S. Zheng, Maximal attractor for the coupled Cahn-Hilliard equations, Nonlinear Anal., 49 (2002), 21-34. doi: 10.1016/S0362-546X(00)00246-7. Google Scholar
[34]	J. Simon, Compact sets in the space $L^p(0, T; B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar
[35]	J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. doi: 10.1137/0521061. Google Scholar
[36]	K. Taira, Analytic Semigroups and Semilinear Initial Boundary Value Problems Cambridge Univ. Press, Cambridge, New York, 1995. doi: 10.1017/CBO9780511662362. Google Scholar
[37]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Applied Mathematics and Sciences, Vol. 68 Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8. Google Scholar
[38]	H. Wu and X. Xu, Analysis of a diffuse-interface model for the binary viscous incompressible fluids with thermo-induced Marangoni effects, Commun. Math. Sci., 11 (2013), 603-633. doi: 10.4310/CMS.2013.v11.n2.a15. Google Scholar
[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. Google Scholar
[40]	S. Zheng, Nonlinear Evolution Equations Pitman series Monographs and Survey in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, Florida, 2004. doi: 10.1201/9780203492222. Google Scholar

[1]	Ciprian G. Gal, Maurizio Grasselli. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 1-39. doi: 10.3934/dcds.2010.28.1
[2]	Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198
[3]	J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647
[4]	Changyan Li, Hui Li. Well-posedness of the two-phase flow problem in incompressible MHD. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5609-5632. doi: 10.3934/dcds.2021090
[5]	Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021
[6]	Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651
[7]	Yongfu Wang. Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows with vacuum. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4317-4333. doi: 10.3934/dcdsb.2020099
[8]	Stefano Scrobogna. Global existence and convergence of nondimensionalized incompressible Navier-Stokes equations in low Froude number regime. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5471-5511. doi: 10.3934/dcds.2020235
[9]	Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215
[10]	Hakima Bessaih, María J. Garrido-Atienza. Longtime behavior for 3D Navier-Stokes equations with constant delays. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1931-1948. doi: 10.3934/cpaa.2020085
[11]	Shuguang Shao, Shu Wang, Wen-Qing Xu, Bin Han. Global existence for the 2D Navier-Stokes flow in the exterior of a moving or rotating obstacle. Kinetic & Related Models, 2016, 9 (4) : 767-776. doi: 10.3934/krm.2016015
[12]	Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067
[13]	Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541
[14]	Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163
[15]	Brahim Amaziane, Mladen Jurak, Leonid Pankratov, Anja Vrbaški. Some remarks on the homogenization of immiscible incompressible two-phase flow in double porosity media. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 629-665. doi: 10.3934/dcdsb.2018037
[16]	Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045
[17]	Theodore Tachim Medjo. A two-phase flow model with delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3273-3294. doi: 10.3934/dcdsb.2017137
[18]	Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure & Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609
[19]	Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
[20]	Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217

American Institute of Mathematical Sciences

Two-phase incompressible flows with variable density: An energetic variational approach

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Two-phase incompressible flows with variable density: An energetic variational approach

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors