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Computing complete Lyapunov functions for discrete-time dynamical systems
The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures
1. | Department of Mathematics & The Institute for Scientific Computing, and Applied Mathematics, Indiana University, Bloomington, Indiana 47405, USA |
2. | Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607, USA |
We study the well-posedness for the mildly compressible Navier-Stokes-Cahn-Hilliard system with non-constant viscosity and Landau potential in two and three dimensional domains.
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,
Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Comm. Math. Phys., 289 (2009), 45-73.
doi: 10.1007/s00220-009-0806-4. |
[3] |
H. Abels,
Strong well-posedness of a diffuse interface model for a viscous, quasi-incompressible two-phase flow, SIAM J. Math. Anal., 44 (2012), 316-340.
doi: 10.1137/110829246. |
[4] |
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. |
[5] |
H. Abels and E. Feireisl,
On a diffuse interface model for two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57 (2008), 659-698.
doi: 10.1512/iumj.2008.57.3391. |
[6] |
H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22 (2012), 1150013, 40 pp.
doi: 10.1142/S0218202511500138. |
[7] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[8] |
A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications., Applied Mathematical Sciences, 151. Springer-Verlag, Berlin, 2002.
doi: 10.1007/978-3-662-12905-0. |
[9] |
F. Boyer,
Mathematical study of multi-phase flow under shear through order parameter formulation, Asympt. Anal., 20 (1999), 175-212.
|
[10] |
F. Boyer,
A theoretical and numerical model for the study of incompressible mixture flows, Comput. Fluids, 31 (2002), 41-68.
doi: 10.1016/S0045-7930(00)00031-1. |
[11] |
F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183. Springer, New York, 2013.
doi: 10.1007/978-1-4614-5975-0. |
[12] |
C. S. Cao and C. G. Gal,
Global solutions for the 2D NS-CH model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility, Nonlinearity, 25 (2012), 3211-3234.
doi: 10.1088/0951-7715/25/11/3211. |
[13] |
L. Cherfils, E. Feireisl, M. Michálek, A. Miranville, M. Petcu and D. Pražák,
The compressible Navier-Stokes-Cahn-Hilliard equations with dynamic boundary conditions, Math. Models Methods Appl. Sci., 29 (2019), 2557-2584.
doi: 10.1142/S0218202519500544. |
[14] |
F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Universitext. Springer, London, EDP Sciences, Les Ulis, 2012.
doi: 10.1007/978-1-4471-2807-6. |
[15] |
H. Ding, P. D. M. Spelt and C. Shu, Diffuse interface model for incompressible two-phase flows with large density ratios, J. Comput. Phys., 226 (2007), 2078-2095. Google Scholar |
[16] |
C. G. Gal and M. Grasselli,
Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436.
doi: 10.1016/j.anihpc.2009.11.013. |
[17] |
C. G. Gal and M. Grasselli,
Trajectory attractors for binary fluid mixtures in 3D, Chinese Ann. Math. Ser. B, 31 (2010), 655-678.
doi: 10.1007/s11401-010-0603-6. |
[18] |
C. G. Gal, M. Grasselli and A. Miranville, Cahn-Hilliard-Navier-Stokes systems with moving contact lines, Calc. Var. Partial Differential Equations, 55 (2016), Art. 50, 47 pp.
doi: 10.1007/s00526-016-0992-9. |
[19] |
C. G. Gal, M. Grasselli and H. Wu,
Global weak solutions to a diffuse interface model for incompressible two-phase flows with moving contact lines and different densities, Arch. Ration. Mech. Anal., 234 (2019), 1-56.
doi: 10.1007/s00205-019-01383-8. |
[20] |
M.-H. Giga, A. Kirshtein and C. Liu, Variational modeling and complex fluids, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2018), 73–113.
doi: 10.1007/978-3-319-13344-7_2. |
[21] |
A. Giorgini, A. Miranville and R. Temam,
Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system, SIAM J. Math. Anal., 51 (2019), 2535-2574.
doi: 10.1137/18M1223459. |
[22] |
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. |
[23] |
P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016.
doi: 10.1201/b19556.![]() ![]() |
[24] |
J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972. |
[25] |
J. Lowengrub and L. Truskinovsky,
Quasi-incompressible Cahn–Hilliard fluids and topological transitions, Proc. Roy. Soc. Lond. A, 454 (1998), 2617-2654.
doi: 10.1098/rspa.1998.0273. |
[26] |
V. N. Starovoĭtov,
Model of the motion of a two-component liquid with allowance of capillary forces, J. Appl. Mech. Tech. Phys., 35 (1994), 891-897.
doi: 10.1007/BF02369582. |
[27] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[28] |
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, 2001.
doi: 10.1090/chel/343. |
[29] |
G. Tomassetti, An interpretation of Temam's stabilization term in the quasi-incompressible Navier-Stokes system, arXiv: 1909.11168. Google Scholar |
[30] |
K. E. Wardle and T. Lee,
Finite element lattice Boltzmann simulations of free surface flow in a concentric cylinder, Comput. Math. Appl., 65 (2013), 230-238.
doi: 10.1016/j.camwa.2011.09.020. |
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,
Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Comm. Math. Phys., 289 (2009), 45-73.
doi: 10.1007/s00220-009-0806-4. |
[3] |
H. Abels,
Strong well-posedness of a diffuse interface model for a viscous, quasi-incompressible two-phase flow, SIAM J. Math. Anal., 44 (2012), 316-340.
doi: 10.1137/110829246. |
[4] |
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. |
[5] |
H. Abels and E. Feireisl,
On a diffuse interface model for two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57 (2008), 659-698.
doi: 10.1512/iumj.2008.57.3391. |
[6] |
H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22 (2012), 1150013, 40 pp.
doi: 10.1142/S0218202511500138. |
[7] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[8] |
A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications., Applied Mathematical Sciences, 151. Springer-Verlag, Berlin, 2002.
doi: 10.1007/978-3-662-12905-0. |
[9] |
F. Boyer,
Mathematical study of multi-phase flow under shear through order parameter formulation, Asympt. Anal., 20 (1999), 175-212.
|
[10] |
F. Boyer,
A theoretical and numerical model for the study of incompressible mixture flows, Comput. Fluids, 31 (2002), 41-68.
doi: 10.1016/S0045-7930(00)00031-1. |
[11] |
F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183. Springer, New York, 2013.
doi: 10.1007/978-1-4614-5975-0. |
[12] |
C. S. Cao and C. G. Gal,
Global solutions for the 2D NS-CH model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility, Nonlinearity, 25 (2012), 3211-3234.
doi: 10.1088/0951-7715/25/11/3211. |
[13] |
L. Cherfils, E. Feireisl, M. Michálek, A. Miranville, M. Petcu and D. Pražák,
The compressible Navier-Stokes-Cahn-Hilliard equations with dynamic boundary conditions, Math. Models Methods Appl. Sci., 29 (2019), 2557-2584.
doi: 10.1142/S0218202519500544. |
[14] |
F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Universitext. Springer, London, EDP Sciences, Les Ulis, 2012.
doi: 10.1007/978-1-4471-2807-6. |
[15] |
H. Ding, P. D. M. Spelt and C. Shu, Diffuse interface model for incompressible two-phase flows with large density ratios, J. Comput. Phys., 226 (2007), 2078-2095. Google Scholar |
[16] |
C. G. Gal and M. Grasselli,
Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436.
doi: 10.1016/j.anihpc.2009.11.013. |
[17] |
C. G. Gal and M. Grasselli,
Trajectory attractors for binary fluid mixtures in 3D, Chinese Ann. Math. Ser. B, 31 (2010), 655-678.
doi: 10.1007/s11401-010-0603-6. |
[18] |
C. G. Gal, M. Grasselli and A. Miranville, Cahn-Hilliard-Navier-Stokes systems with moving contact lines, Calc. Var. Partial Differential Equations, 55 (2016), Art. 50, 47 pp.
doi: 10.1007/s00526-016-0992-9. |
[19] |
C. G. Gal, M. Grasselli and H. Wu,
Global weak solutions to a diffuse interface model for incompressible two-phase flows with moving contact lines and different densities, Arch. Ration. Mech. Anal., 234 (2019), 1-56.
doi: 10.1007/s00205-019-01383-8. |
[20] |
M.-H. Giga, A. Kirshtein and C. Liu, Variational modeling and complex fluids, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2018), 73–113.
doi: 10.1007/978-3-319-13344-7_2. |
[21] |
A. Giorgini, A. Miranville and R. Temam,
Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system, SIAM J. Math. Anal., 51 (2019), 2535-2574.
doi: 10.1137/18M1223459. |
[22] |
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. |
[23] |
P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016.
doi: 10.1201/b19556.![]() ![]() |
[24] |
J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972. |
[25] |
J. Lowengrub and L. Truskinovsky,
Quasi-incompressible Cahn–Hilliard fluids and topological transitions, Proc. Roy. Soc. Lond. A, 454 (1998), 2617-2654.
doi: 10.1098/rspa.1998.0273. |
[26] |
V. N. Starovoĭtov,
Model of the motion of a two-component liquid with allowance of capillary forces, J. Appl. Mech. Tech. Phys., 35 (1994), 891-897.
doi: 10.1007/BF02369582. |
[27] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[28] |
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, 2001.
doi: 10.1090/chel/343. |
[29] |
G. Tomassetti, An interpretation of Temam's stabilization term in the quasi-incompressible Navier-Stokes system, arXiv: 1909.11168. Google Scholar |
[30] |
K. E. Wardle and T. Lee,
Finite element lattice Boltzmann simulations of free surface flow in a concentric cylinder, Comput. Math. Appl., 65 (2013), 230-238.
doi: 10.1016/j.camwa.2011.09.020. |
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