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 & 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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

show all references

References:
[1]

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

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

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