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Homogenization of stokes system using bloch waves
Sharp interface limit in a phase field model of cell motility
1. | Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA |
2. | Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering of National Academy of Sciences of Ukraine, 47 Nauky Ave., 61103 Kharkiv, Ukraine |
We consider a phase field model of cell motility introduced in [
References:
[1] |
M. Alfaro,
Generation, motion and thickness of transition layers for a nonlocal Allen-Cahn equation, Nonlinear Analysis, 72 (2010), 3324-3336.
doi: 10.1016/j.na.2009.12.013. |
[2] |
M. Alfaro and P. Alifrangis,
Convergence of a mass conserving Allen-Cahn equation whose Lagrange multiplier is nonlocal and local, Interfaces Free Bound., 16 (2014), 243-268.
doi: 10.4171/IFB/319. |
[3] |
S. Allen and J. W. Cahn,
A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095.
doi: 10.1016/0001-6160(79)90196-2. |
[4] |
G. Barles and F. D. Lio,
A geometrical approach to front propagation problems in bounded domains with Neumann-type boundary conditions, Interfaces Free Bound., 5 (2016), 239-274.
doi: 10.4171/IFB/79. |
[5] |
E. Barnhart, K. C. Lee, G. M. Allen, J. A. Theriot and A. Mogilner,
Balance between cellsubstrate adhesion and myosin contraction determines the frequency of motility initiation in fish keratocytes, Proceedings of the National Academy of Sciences, 112 (2015), 5045-5050.
doi: 10.1073/pnas.1417257112. |
[6] |
L. Berlyand, M. Potomkin and V. Rybalko,
Phase-field model of cell motility: Traveling waves and sharp interface limit, Comptes Rendus Mathematique, 354 (2016), 986-992.
doi: 10.1016/j.crma.2016.09.001. |
[7] |
K. A. Brakke, The Motion of a Surface by Its Mean Curvature, Mathematical Notes, 20. Princeton University Press, Princeton, N. J. , 1978. i+252 pp. |
[8] |
M. Brassel and E. Bretin,
A modified phase field approximation for mean curvature flow with conservation of the volume, Mathematical Methods in the Applied Sciences, 34 (2011), 1157-1180.
doi: 10.1002/mma.1426. |
[9] |
L. Bronsard and B. Stoth,
Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM J. Math. Anal., 28 (1997), 769-807.
doi: 10.1137/S0036141094279279. |
[10] |
X. Chen, D. Hilhorst and E. Logak,
Asymptotic behavior of solutions of an Allen-Cahn equation with a nonlocal term, Nonlinear Analysis: Theory, Methods & Applications, 28 (1997), 1283-1298.
doi: 10.1016/S0362-546X(97)82875-1. |
[11] |
X. Chen, D. Hilhorst and E. Logak,
Mass conserving Allen-Cahn equation and volume preserving mean curvature flow, Interfaces Free Bound., 12 (2010), 527-549.
doi: 10.4171/IFB/244. |
[12] |
Y. G. Chen, Y. Giga and S. Goto,
Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.
doi: 10.4310/jdg/1214446564. |
[13] |
L. C. Evans and J. Spruck,
Motion of level sets by mean curvature. Ⅰ, J. Differential Geom., 33 (1991), 635-681.
doi: 10.4310/jdg/1214446559. |
[14] |
L. C. Evans, H. M. Soner and P. E. Souganidis,
Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123.
doi: 10.1002/cpa.3160450903. |
[15] |
P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. vi+93 pp.
doi: 10.1137/1.9781611970180. |
[16] |
L. Gearhart,
Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394.
doi: 10.1090/S0002-9947-1978-0461206-1. |
[17] |
D. Golovaty,
The volume preserving motion by mean curvature as an asymptotic limit of reaction-diffusion equations, Q. of Appl. Math., 55 (1997), 243-298.
doi: 10.1090/qam/1447577. |
[18] |
M. Grayson,
The heat equation shrinks embedded plane curves to points, J. Differential Geom., 26 (1987), 285-314.
doi: 10.4310/jdg/1214441371. |
[19] |
R. S. Hamilton,
Three-manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), 255-306.
doi: 10.4310/jdg/1214436922. |
[20] |
M. H. Holmes, Introduction to Perturbation Methods, 2nd edition. Texts in Applied Mathematics, 20. Springer, New York, 2013. xviii+436 pp.
doi: 10.1007/978-1-4614-5477-9. |
[21] |
G. Huisken,
Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266.
doi: 10.4310/jdg/1214438998. |
[22] |
K. Keren, Z. Pincus, G. M. Allen, E. Barnhart, G. Marriott, A. Mogilner and J. A. Theriot,
Mechanism of shape determination in motile cells, Nature, 453 (2008), 475-480.
doi: 10.1038/nature06952. |
[23] |
R. Kohn, F. Otto, M. Reznikoff and E. Vanden-Eijden,
Action minimization and sharpinterface limits for the stochastic Allen-Cahn equation, Comm. Pure Appl. Math., 60 (2007), 393-438.
doi: 10.1002/cpa.20144. |
[24] |
F. D. Lio, C. I. Kim and D. Slepčev,
Nonlocal front propagation problems in bounded domains with Neumann-type boundary conditions and applications, Asymptot. Anal., 37 (2004), 257-292.
|
[25] |
M. Mizuhara, L. Berlyand, V. Rybalko and L. Zhang,
On an evolution equation in a cell motility model, Phys. D, 318/319 (2016), 12-25.
doi: 10.1016/j.physd.2015.10.008. |
[26] |
L. Modica,
The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[27] |
A. Mogilner,
Mathematics of cell motility: Have we got its number?, J. Math. Biol., 58 (2009), 105-134.
doi: 10.1007/s00285-008-0182-2. |
[28] |
P. de Mottoni and M. Schatzman,
Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589.
doi: 10.1090/S0002-9947-1995-1672406-7. |
[29] |
F. Otto, H. Weber and G. Westdickenberg,
Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size, Electron. J. Probab., 19 (2014), 1-76.
doi: 10.1214/EJP.v19-2813. |
[30] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. viii+279 pp.
doi: 10.1007/978-1-4612-5561-1. |
[31] |
J. Prúss,
On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
[32] |
P. Recho, T. Putelat and L. Truskinovsky,
Mechanics of motility initiation and motility arrest in crawling cells, J. Mech. Phys. Solids, 84 (2015), 469-505.
doi: 10.1016/j.jmps.2015.08.006. |
[33] |
P. Recho and L. Truskinovsky, Asymmetry between pushing and pulling for crawling cells, Phys. Rev. E, 87 (2013), 022720.
doi: 10.1103/PhysRevE.87.022720. |
[34] |
B. Rubinstein, K. Jacobson and A. Mogilner,
Multiscale two-dimensional modeling of a motile simple-shaped cell, Multiscale Model. Simul., 3 (2005), 413-439.
doi: 10.1137/04060370X. |
[35] |
J. Rubinstein and P. Sternberg,
Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math., 48 (1992), 249-264.
doi: 10.1093/imamat/48.3.249. |
[36] |
J. Rubinstein, P. Sternberg and J. B. Keller,
Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133.
doi: 10.1137/0149007. |
[37] |
S. Serfaty,
Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst. A, 31 (2011), 1427-1451.
doi: 10.3934/dcds.2011.31.1427. |
[38] |
D. Shao, W. Rappel and H. Levine, Computational model for cell morphodynamics, Physical Review Letters, 105 (2010), 108104.
doi: 10.1103/PhysRevLett.105.108104. |
[39] |
S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview press, 2014, xiii+513 pp. |
[40] |
F. Ziebert, S. Swaminathan and I. S. Aranson,
Model for self-polarization and motility of keratocyte fragments, J. R. Soc. Interface, 9 (2011), 1084-1092.
doi: 10.1098/rsif.2011.0433. |
show all references
References:
[1] |
M. Alfaro,
Generation, motion and thickness of transition layers for a nonlocal Allen-Cahn equation, Nonlinear Analysis, 72 (2010), 3324-3336.
doi: 10.1016/j.na.2009.12.013. |
[2] |
M. Alfaro and P. Alifrangis,
Convergence of a mass conserving Allen-Cahn equation whose Lagrange multiplier is nonlocal and local, Interfaces Free Bound., 16 (2014), 243-268.
doi: 10.4171/IFB/319. |
[3] |
S. Allen and J. W. Cahn,
A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095.
doi: 10.1016/0001-6160(79)90196-2. |
[4] |
G. Barles and F. D. Lio,
A geometrical approach to front propagation problems in bounded domains with Neumann-type boundary conditions, Interfaces Free Bound., 5 (2016), 239-274.
doi: 10.4171/IFB/79. |
[5] |
E. Barnhart, K. C. Lee, G. M. Allen, J. A. Theriot and A. Mogilner,
Balance between cellsubstrate adhesion and myosin contraction determines the frequency of motility initiation in fish keratocytes, Proceedings of the National Academy of Sciences, 112 (2015), 5045-5050.
doi: 10.1073/pnas.1417257112. |
[6] |
L. Berlyand, M. Potomkin and V. Rybalko,
Phase-field model of cell motility: Traveling waves and sharp interface limit, Comptes Rendus Mathematique, 354 (2016), 986-992.
doi: 10.1016/j.crma.2016.09.001. |
[7] |
K. A. Brakke, The Motion of a Surface by Its Mean Curvature, Mathematical Notes, 20. Princeton University Press, Princeton, N. J. , 1978. i+252 pp. |
[8] |
M. Brassel and E. Bretin,
A modified phase field approximation for mean curvature flow with conservation of the volume, Mathematical Methods in the Applied Sciences, 34 (2011), 1157-1180.
doi: 10.1002/mma.1426. |
[9] |
L. Bronsard and B. Stoth,
Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM J. Math. Anal., 28 (1997), 769-807.
doi: 10.1137/S0036141094279279. |
[10] |
X. Chen, D. Hilhorst and E. Logak,
Asymptotic behavior of solutions of an Allen-Cahn equation with a nonlocal term, Nonlinear Analysis: Theory, Methods & Applications, 28 (1997), 1283-1298.
doi: 10.1016/S0362-546X(97)82875-1. |
[11] |
X. Chen, D. Hilhorst and E. Logak,
Mass conserving Allen-Cahn equation and volume preserving mean curvature flow, Interfaces Free Bound., 12 (2010), 527-549.
doi: 10.4171/IFB/244. |
[12] |
Y. G. Chen, Y. Giga and S. Goto,
Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.
doi: 10.4310/jdg/1214446564. |
[13] |
L. C. Evans and J. Spruck,
Motion of level sets by mean curvature. Ⅰ, J. Differential Geom., 33 (1991), 635-681.
doi: 10.4310/jdg/1214446559. |
[14] |
L. C. Evans, H. M. Soner and P. E. Souganidis,
Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123.
doi: 10.1002/cpa.3160450903. |
[15] |
P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. vi+93 pp.
doi: 10.1137/1.9781611970180. |
[16] |
L. Gearhart,
Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394.
doi: 10.1090/S0002-9947-1978-0461206-1. |
[17] |
D. Golovaty,
The volume preserving motion by mean curvature as an asymptotic limit of reaction-diffusion equations, Q. of Appl. Math., 55 (1997), 243-298.
doi: 10.1090/qam/1447577. |
[18] |
M. Grayson,
The heat equation shrinks embedded plane curves to points, J. Differential Geom., 26 (1987), 285-314.
doi: 10.4310/jdg/1214441371. |
[19] |
R. S. Hamilton,
Three-manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), 255-306.
doi: 10.4310/jdg/1214436922. |
[20] |
M. H. Holmes, Introduction to Perturbation Methods, 2nd edition. Texts in Applied Mathematics, 20. Springer, New York, 2013. xviii+436 pp.
doi: 10.1007/978-1-4614-5477-9. |
[21] |
G. Huisken,
Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266.
doi: 10.4310/jdg/1214438998. |
[22] |
K. Keren, Z. Pincus, G. M. Allen, E. Barnhart, G. Marriott, A. Mogilner and J. A. Theriot,
Mechanism of shape determination in motile cells, Nature, 453 (2008), 475-480.
doi: 10.1038/nature06952. |
[23] |
R. Kohn, F. Otto, M. Reznikoff and E. Vanden-Eijden,
Action minimization and sharpinterface limits for the stochastic Allen-Cahn equation, Comm. Pure Appl. Math., 60 (2007), 393-438.
doi: 10.1002/cpa.20144. |
[24] |
F. D. Lio, C. I. Kim and D. Slepčev,
Nonlocal front propagation problems in bounded domains with Neumann-type boundary conditions and applications, Asymptot. Anal., 37 (2004), 257-292.
|
[25] |
M. Mizuhara, L. Berlyand, V. Rybalko and L. Zhang,
On an evolution equation in a cell motility model, Phys. D, 318/319 (2016), 12-25.
doi: 10.1016/j.physd.2015.10.008. |
[26] |
L. Modica,
The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142.
doi: 10.1007/BF00251230. |
[27] |
A. Mogilner,
Mathematics of cell motility: Have we got its number?, J. Math. Biol., 58 (2009), 105-134.
doi: 10.1007/s00285-008-0182-2. |
[28] |
P. de Mottoni and M. Schatzman,
Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589.
doi: 10.1090/S0002-9947-1995-1672406-7. |
[29] |
F. Otto, H. Weber and G. Westdickenberg,
Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size, Electron. J. Probab., 19 (2014), 1-76.
doi: 10.1214/EJP.v19-2813. |
[30] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. viii+279 pp.
doi: 10.1007/978-1-4612-5561-1. |
[31] |
J. Prúss,
On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
[32] |
P. Recho, T. Putelat and L. Truskinovsky,
Mechanics of motility initiation and motility arrest in crawling cells, J. Mech. Phys. Solids, 84 (2015), 469-505.
doi: 10.1016/j.jmps.2015.08.006. |
[33] |
P. Recho and L. Truskinovsky, Asymmetry between pushing and pulling for crawling cells, Phys. Rev. E, 87 (2013), 022720.
doi: 10.1103/PhysRevE.87.022720. |
[34] |
B. Rubinstein, K. Jacobson and A. Mogilner,
Multiscale two-dimensional modeling of a motile simple-shaped cell, Multiscale Model. Simul., 3 (2005), 413-439.
doi: 10.1137/04060370X. |
[35] |
J. Rubinstein and P. Sternberg,
Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math., 48 (1992), 249-264.
doi: 10.1093/imamat/48.3.249. |
[36] |
J. Rubinstein, P. Sternberg and J. B. Keller,
Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133.
doi: 10.1137/0149007. |
[37] |
S. Serfaty,
Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst. A, 31 (2011), 1427-1451.
doi: 10.3934/dcds.2011.31.1427. |
[38] |
D. Shao, W. Rappel and H. Levine, Computational model for cell morphodynamics, Physical Review Letters, 105 (2010), 108104.
doi: 10.1103/PhysRevLett.105.108104. |
[39] |
S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview press, 2014, xiii+513 pp. |
[40] |
F. Ziebert, S. Swaminathan and I. S. Aranson,
Model for self-polarization and motility of keratocyte fragments, J. R. Soc. Interface, 9 (2011), 1084-1092.
doi: 10.1098/rsif.2011.0433. |





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