March  2015, 20(2): 397-422. doi: 10.3934/dcdsb.2015.20.397

A phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model: Existence and uniqueness of solutions

1. 

Universidade Federal de Viçosa, Departamento de Matemática, Rua P.H.Rolfs, s/n, Viçosa, MG, CEP 36570-000, Brazil

2. 

Universidade Estadual de Campinas, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, Campinas, SP, CEP 13083-859, Brazil

Received  March 2014 Revised  July 2014 Published  January 2015

In this work we analyze a system of nonlinear evolution partial differential equations modeling the fluid-structure interaction associated to the dynamics of an elastic vesicle immersed in a moving incompressible viscous fluid. This system of equations couples an equation for a phase field variable, used to determine the position of vesicle membrane deformed by the action of the fluid, to the $\alpha$-Navier- Stokes equations with an extra nonlinear interaction term. We prove global in time existence and uniqueness of solutions for this system in suitable functional spaces even in the three-dimensional case.
Citation: Ariane Piovezan Entringer, José Luiz Boldrini. A phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model: Existence and uniqueness of solutions. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 397-422. doi: 10.3934/dcdsb.2015.20.397
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, $2^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2003.

[2]

M. Abkarian, C. Lartigue and A. Viallat, Tank treading and unbinding of deformable vesicles in shear flow: Determination of the lift force, Phys. Rev. Lett., 88 (2002), 068103.

[3]

J. Beauncourt, F. Rioual, T. Sion, T. Biben and C. Misbah, Steady to unsteady dynamics of a vesicle in a flow, Phys. Rev. E, 69 (2004), 011906.

[4]

T. Biben, K. Kassner and C. Misbah, Phase field approach to three-dimensional vesicle dynamics, Physical Rev. E, 72 (2005), 041921. doi: 10.1103/PhysRevE.72.041921.

[5]

C. Bjorland and M. E. Schonbek, On questions of decay and existence for the viscous Camassa-Holm equations, Ann. Inst. H. Poincaré Anal Non Linéaire, 25 (2008), 907-936. doi: 10.1016/j.anihpc.2007.07.003.

[6]

A. Çaǧlar, Convergence analysis of the Navier-Stokes alpha model, Numerical Methods for Partial Differential Equations, 26 (2010), 1154-1167. doi: 10.1002/num.20481.

[7]

S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, A connection between Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids, 11 (1999), 2343-2353. doi: 10.1063/1.870096.

[8]

J. A. Domaradzki and D. D. Holm, Navier-Stokes-alpha Model: LES equations with nonlinear dispersion, in Special LES volume of ERCOFTAC Bulletin, Modern Simulations Strategies for turbulent flow (editor B. J. Geurts), Edwards Publising, 2001.

[9]

Q. Du, M. Li and C. Liu, Analysis of a phase field Navier-Stokes vesicle-fluid interation model, Discrete Contin, Dyn. Syst. Ser. B, 8 (2007), 539-556 (electronic). doi: 10.3934/dcdsb.2007.8.539.

[10]

Q. Du, C. Liu and X. Wang, A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, Journal of Computational Physics, 198 (2004), 450-468. doi: 10.1016/j.jcp.2004.01.029.

[11]

Q. Du, C. Liu, R. Ryhan and X. Wang, A phase field formulation of the Willmore problem, Nonlinearity, 18 (2005), 1249-1267. doi: 10.1088/0951-7715/18/3/016.

[12]

Q. Du, C. Liu, R. Ryham and X. Wang, Modeling the spontaneous curvature effects in static cell membrane deformations by a phase field formulation, Commun. Pure Appl. Anal., 4 (2005), 537-548. doi: 10.3934/cpaa.2005.4.537.

[13]

Q. Du, C. Liu, R. Ryham and X. Wang, Retrieving topological information for phase field models, SIAM J. Appl. Math., 65 (2005), 1913-1932 (electronic). doi: 10.1137/040606417.

[14]

Q. Du, C. Liu and X. Wang, Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions, Journal of Computational Physics, 212 (2006), 757-777. doi: 10.1016/j.jcp.2005.07.020.

[15]

C. Foias, D. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. and Differential Equations, 14 (2002), 1-35. doi: 10.1023/A:1012984210582.

[16]

C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Phys. D, 152/153 (2001), 505-519. doi: 10.1016/S0167-2789(01)00191-9.

[17]

B. J. Geurts and D. D. Holm, Regularization modeling for large eddy simulation, Phys. Fluid, 15 (2003), L13-L16. doi: 10.1063/1.1529180.

[18]

B. J. Geurts and D. D. Holm, Leray and LANS-$\alpha$ modeling of turbulent mixing, J. Turbulence, 7 (2006), 1-33 (electronic). doi: 10.1080/14685240500501601.

[19]

L. Guermond, J. T. Oden and S. Prudhomme, An interpretation of the Navier-Stokes alpha model as a frame-indifferent Leray regularization, Phys. D, 177 (2003), 23-30. doi: 10.1016/S0167-2789(02)00748-0.

[20]

W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch. C, 28 (1973), 693-703.

[21]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.

[22]

J. Leray, Essay sur les mouvements plans d'une liquide visqueux que limitent des parois, J. Math Pures Appl., 13 (1934), 331-418.

[23]

J. Leray, Sur les mouviments d'une liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.

[24]

R. Lipowsky, The morphology of lipid membranes, Current Opinion in Structural Biology, 5 (1995), 531-540. doi: 10.1016/0959-440X(95)80040-9.

[25]

Y. Liu, T. Takahashi and M. Tucsnak, Strong solutions for a phase field Navier-Stokes vesicle-fluid interaction model, J. Math. Fluid Mech., 14 (2012), 177-195. doi: 10.1007/s00021-011-0059-9.

[26]

C. J. McConnell, J. B. Carmichael and M. E. DeMont, Modeling blood flow in the aorta, American Biology Teacher, 59 (1997), 586-588. doi: 10.2307/4450389.

[27]

Z. Ou-Yang, J. Liu and Y. Xie, Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases, World Scientific, Singapore, 1999. doi: 10.1142/9789812816856.

[28]

U. Seifert, Configurations of fluid membranes and Vesicles, Advances in Physics, 46 (1997), 13-137. doi: 10.1080/00018739700101488.

[29]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

[30]

A. F. Stalder, A. Frydrychowicz, M. F. Russe, J. G. Korvink, J Hennig, K. Li and M. Markl, Assessment of flow instabilities in the healthy aorta using flow-sensitive MRI, Journal of Magnetic Resonance Imaging, 33 (2011), 839-846. doi: 10.1002/jmri.22512.

[31]

D. Stein and H. N. Sabbah, Turbulent blood flow in the ascending aorta of humans with normal and diseased aortic valves, Circulation Research, Journal of the American Heart Association, 39 (1976), 58-65. doi: 10.1161/01.RES.39.1.58.

[32]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland Publishing Company, 1977.

[33]

H. Wu and X. Xu, Strong solutions, global regularity and stability of a hydrodynamic system modeling vesicle and fluid interactions, SIAM J. Math. Anal., 45 (2013), 181-214. doi: 10.1137/11085952X.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, $2^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2003.

[2]

M. Abkarian, C. Lartigue and A. Viallat, Tank treading and unbinding of deformable vesicles in shear flow: Determination of the lift force, Phys. Rev. Lett., 88 (2002), 068103.

[3]

J. Beauncourt, F. Rioual, T. Sion, T. Biben and C. Misbah, Steady to unsteady dynamics of a vesicle in a flow, Phys. Rev. E, 69 (2004), 011906.

[4]

T. Biben, K. Kassner and C. Misbah, Phase field approach to three-dimensional vesicle dynamics, Physical Rev. E, 72 (2005), 041921. doi: 10.1103/PhysRevE.72.041921.

[5]

C. Bjorland and M. E. Schonbek, On questions of decay and existence for the viscous Camassa-Holm equations, Ann. Inst. H. Poincaré Anal Non Linéaire, 25 (2008), 907-936. doi: 10.1016/j.anihpc.2007.07.003.

[6]

A. Çaǧlar, Convergence analysis of the Navier-Stokes alpha model, Numerical Methods for Partial Differential Equations, 26 (2010), 1154-1167. doi: 10.1002/num.20481.

[7]

S. Chen, C. Foias, D. D. Holm, E. Olson, E. S. Titi and S. Wynne, A connection between Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids, 11 (1999), 2343-2353. doi: 10.1063/1.870096.

[8]

J. A. Domaradzki and D. D. Holm, Navier-Stokes-alpha Model: LES equations with nonlinear dispersion, in Special LES volume of ERCOFTAC Bulletin, Modern Simulations Strategies for turbulent flow (editor B. J. Geurts), Edwards Publising, 2001.

[9]

Q. Du, M. Li and C. Liu, Analysis of a phase field Navier-Stokes vesicle-fluid interation model, Discrete Contin, Dyn. Syst. Ser. B, 8 (2007), 539-556 (electronic). doi: 10.3934/dcdsb.2007.8.539.

[10]

Q. Du, C. Liu and X. Wang, A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, Journal of Computational Physics, 198 (2004), 450-468. doi: 10.1016/j.jcp.2004.01.029.

[11]

Q. Du, C. Liu, R. Ryhan and X. Wang, A phase field formulation of the Willmore problem, Nonlinearity, 18 (2005), 1249-1267. doi: 10.1088/0951-7715/18/3/016.

[12]

Q. Du, C. Liu, R. Ryham and X. Wang, Modeling the spontaneous curvature effects in static cell membrane deformations by a phase field formulation, Commun. Pure Appl. Anal., 4 (2005), 537-548. doi: 10.3934/cpaa.2005.4.537.

[13]

Q. Du, C. Liu, R. Ryham and X. Wang, Retrieving topological information for phase field models, SIAM J. Appl. Math., 65 (2005), 1913-1932 (electronic). doi: 10.1137/040606417.

[14]

Q. Du, C. Liu and X. Wang, Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions, Journal of Computational Physics, 212 (2006), 757-777. doi: 10.1016/j.jcp.2005.07.020.

[15]

C. Foias, D. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. and Differential Equations, 14 (2002), 1-35. doi: 10.1023/A:1012984210582.

[16]

C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Phys. D, 152/153 (2001), 505-519. doi: 10.1016/S0167-2789(01)00191-9.

[17]

B. J. Geurts and D. D. Holm, Regularization modeling for large eddy simulation, Phys. Fluid, 15 (2003), L13-L16. doi: 10.1063/1.1529180.

[18]

B. J. Geurts and D. D. Holm, Leray and LANS-$\alpha$ modeling of turbulent mixing, J. Turbulence, 7 (2006), 1-33 (electronic). doi: 10.1080/14685240500501601.

[19]

L. Guermond, J. T. Oden and S. Prudhomme, An interpretation of the Navier-Stokes alpha model as a frame-indifferent Leray regularization, Phys. D, 177 (2003), 23-30. doi: 10.1016/S0167-2789(02)00748-0.

[20]

W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch. C, 28 (1973), 693-703.

[21]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.

[22]

J. Leray, Essay sur les mouvements plans d'une liquide visqueux que limitent des parois, J. Math Pures Appl., 13 (1934), 331-418.

[23]

J. Leray, Sur les mouviments d'une liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.

[24]

R. Lipowsky, The morphology of lipid membranes, Current Opinion in Structural Biology, 5 (1995), 531-540. doi: 10.1016/0959-440X(95)80040-9.

[25]

Y. Liu, T. Takahashi and M. Tucsnak, Strong solutions for a phase field Navier-Stokes vesicle-fluid interaction model, J. Math. Fluid Mech., 14 (2012), 177-195. doi: 10.1007/s00021-011-0059-9.

[26]

C. J. McConnell, J. B. Carmichael and M. E. DeMont, Modeling blood flow in the aorta, American Biology Teacher, 59 (1997), 586-588. doi: 10.2307/4450389.

[27]

Z. Ou-Yang, J. Liu and Y. Xie, Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases, World Scientific, Singapore, 1999. doi: 10.1142/9789812816856.

[28]

U. Seifert, Configurations of fluid membranes and Vesicles, Advances in Physics, 46 (1997), 13-137. doi: 10.1080/00018739700101488.

[29]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

[30]

A. F. Stalder, A. Frydrychowicz, M. F. Russe, J. G. Korvink, J Hennig, K. Li and M. Markl, Assessment of flow instabilities in the healthy aorta using flow-sensitive MRI, Journal of Magnetic Resonance Imaging, 33 (2011), 839-846. doi: 10.1002/jmri.22512.

[31]

D. Stein and H. N. Sabbah, Turbulent blood flow in the ascending aorta of humans with normal and diseased aortic valves, Circulation Research, Journal of the American Heart Association, 39 (1976), 58-65. doi: 10.1161/01.RES.39.1.58.

[32]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland Publishing Company, 1977.

[33]

H. Wu and X. Xu, Strong solutions, global regularity and stability of a hydrodynamic system modeling vesicle and fluid interactions, SIAM J. Math. Anal., 45 (2013), 181-214. doi: 10.1137/11085952X.

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