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, (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).

[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).

[4]

T. Biben, K. Kassner and C. Misbah, Phase field approach to three-dimensional vesicle dynamics,, Physical Rev. E, 72 (2005). 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. 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. 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. 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, (2001).

[9]

Q. Du, M. Li and C. Liu, Analysis of a phase field Navier-Stokes vesicle-fluid interation model,, Discrete Contin, 8 (2007), 539. 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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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.

[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. 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.

[23]

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

[24]

R. Lipowsky, The morphology of lipid membranes,, Current Opinion in Structural Biology, 5 (1995), 531. 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. 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. 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, (1999). doi: 10.1142/9789812816856.

[28]

U. Seifert, Configurations of fluid membranes and Vesicles,, Advances in Physics, 46 (1997), 13. 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. 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. 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, 39 (1976), 58. 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. doi: 10.1137/11085952X.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, $2^{nd}$ edition, (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).

[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).

[4]

T. Biben, K. Kassner and C. Misbah, Phase field approach to three-dimensional vesicle dynamics,, Physical Rev. E, 72 (2005). 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. 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. 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. 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, (2001).

[9]

Q. Du, M. Li and C. Liu, Analysis of a phase field Navier-Stokes vesicle-fluid interation model,, Discrete Contin, 8 (2007), 539. 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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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.

[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. 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.

[23]

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

[24]

R. Lipowsky, The morphology of lipid membranes,, Current Opinion in Structural Biology, 5 (1995), 531. 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. 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. 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, (1999). doi: 10.1142/9789812816856.

[28]

U. Seifert, Configurations of fluid membranes and Vesicles,, Advances in Physics, 46 (1997), 13. 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. 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. 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, 39 (1976), 58. 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. doi: 10.1137/11085952X.

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