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Global-in-time Gevrey regularity solution for a class of bistable gradient flows
1. | Department of Mathematics and Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, United States |
2. | Department of Mathematics, The University of Massachusetts, North Dartmouth, MA 02747-2300 |
3. | Mathematics Department, University of Tennessee, Knoxville, TN 37996, United States |
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. |
[2] |
A. Biswas and D. Swanson, Existence and generalized Gevrey regularity of solutions to the Kuramoto-Sivashinsky equation in $R^n$, J. Differential Equations, 240 (2007), 145-163.
doi: 10.1016/j.jde.2007.05.022. |
[3] |
A. Biswas and D. Swanson, Gevrey regularity of solutions to the 3-D Navier-Stokes equations with weighted $l_p$ initial data, Indiana Univ. Math. J., 56 (2007), 1157-1188.
doi: 10.1512/iumj.2007.56.2891. |
[4] |
Z. Bradshaw, Z. Grujic and I. Kukavica, Local analyticity radii of solutions to the 3d Navier-Stokes equations with locally analytic forcing, J. Differential Equations, 259 (2015), 3955-3975.
doi: 10.1016/j.jde.2015.05.009. |
[5] |
C. Cao, M. Rammaha and E. Titi, Gevrey regularity for nonlinear analytic parabolic equations on the sphere, J. Dynam. Differential Equations, 12 (2000), 411-433.
doi: 10.1023/A:1009072526324. |
[6] |
W. Chen, S. Conde, C. Wang, X. Wang and S. M. Wise, A linear energy stable scheme for a thin film model without slope selection, J. Sci. Comput., 52 (2012), 546-562.
doi: 10.1007/s10915-011-9559-2. |
[7] |
W. Chen, C. Wang, X. Wang and S. M. Wise, A linear iteration algorithm for a second-order energy stable scheme for a thin film model without slope selection, J. Sci. Comput., 59 (2014), 574-601.
doi: 10.1007/s10915-013-9774-0. |
[8] |
M. C. Cross and A. C. Newell, Convection patterns in large aspect ratio systems, Physica D, 10 (1984), 299-328.
doi: 10.1016/0167-2789(84)90181-7. |
[9] |
A. Eden and V. Kalantarov, The convective Cahn-Hilliard equation, Appl. Math. Lett., 20 (2007), 455-461.
doi: 10.1016/j.aml.2006.05.014. |
[10] |
K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth, Phys. Rev. Lett., 88 (2002), 245701.
doi: 10.1103/PhysRevLett.88.245701. |
[11] |
K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals, Phys. Rev. E, 70 (2004), 051605.
doi: 10.1103/PhysRevE.70.051605. |
[12] |
A. Ferrari and E. Titi, Gevrey regularity for nonlinear analytic parabolic equations, Comm. Partial Differential Equations, 23 (1998), 1-16. |
[13] |
C. Foias and R. Temam, Gevrey class regularity for the solution of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.
doi: 10.1016/0022-1236(89)90015-3. |
[14] |
K. B. Glasner, Grain boundary motion arising from the gradient flow of the Aviles-Giga functional, Physica D, 215 (2006), 80-98.
doi: 10.1016/j.physd.2006.01.013. |
[15] |
A. A. Golovin and A. A. Nepomnyashchy, Disclinations in square and hexagonal patterns, Phys. Rev. E, 67 (2003), 056202, 7pp.
doi: 10.1103/PhysRevE.67.056202. |
[16] |
Z. Grujic and I. Kukavica, Space analyticity for the Navier-Stokes and related equations with initial data in $L^p$, J. Funct. Anal., 152 (1998), 447-466.
doi: 10.1006/jfan.1997.3167. |
[17] |
Z. Grujic and I. Kukavica, Space analyticity for the nonlinear heat equation in a bounded domain, J. Differential Equations, 154 (1999), 42-54.
doi: 10.1006/jdeq.1998.3562. |
[18] |
V. Kalantarov, B. Levant and E. Titi, Gevrey regularity for the attractor of the {3D Navier-Stokes-Voight} equations, J. Nonlinear Sci., 19 (2009), 133-152.
doi: 10.1007/s00332-008-9029-7. |
[19] |
R. V. Kohn and F. Otto, Upper bound on coarsening rate, Commun. Math. Phys., 229 (2002), 375-395.
doi: 10.1007/s00220-002-0693-4. |
[20] |
R. V. Kohn and X. Yan, Upper bound on the coarsening rate for an epitaxial growth model, Comm. Pure Appl. Math., 56 (2003), 1549-1564.
doi: 10.1002/cpa.10103. |
[21] |
J. Krug, Four lectures on the physics of crystal growth, Physica A, 313 (2002), 47-82.
doi: 10.1016/S0378-4371(02)01034-8. |
[22] |
I. Kukavica, R. Temam, V. Vlad, and M. Ziane, On the time analyticity radius of the solutions of the two-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 3 (1991), 611-618.
doi: 10.1007/BF01049102. |
[23] |
I. Kukavica, R. Temam, V. Vlad and M. Ziane, Existence and uniqueness of solutions for the hydrostatic Euler equations on a bounded domain with analytic data, C. R. Math. Acad. Sci. Paris, 348 (2010), 639-645.
doi: 10.1016/j.crma.2010.03.023. |
[24] |
I. Kukavica and V. Vlad, On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Amer. Math. Soc., 137 (2009), 669-677.
doi: 10.1090/S0002-9939-08-09693-7. |
[25] |
I. Kukavica and V. Vlad, The domain of analyticity of solutions to the three-dimensional Euler equations in a half space, Discrete Contin. Dyn. Syst., 29 (2011), 285-303.
doi: 10.3934/dcds.2011.29.285. |
[26] |
I. Kukavica and V. Vlad, On the analyticity and Gevrey-class regularity up to the boundary for the Euler equations, Nonlinearity, 24 (2011), 765-796.
doi: 10.1088/0951-7715/24/3/004. |
[27] |
I. Kukavica and V. Vlad, On the local existence of analytic solutions to the Prandtl boundary layer equations, Commun. Math. Sci., 11 (2013), 269-292.
doi: 10.4310/CMS.2013.v11.n1.a8. |
[28] |
A. Levandovsky and L. Golubovic, Epitaxial growth and erosion on (001) crystal surfaces: Far-from-equilibrium transitions, Phys. Rev. B, 65 (2004), 241402. |
[29] |
A. Levandovsky, L. Golubovic and D. Moldovan, Interfacial states and far-from-equilibrium transitions in the epitaxial growth and erosion on (110) crystal surfaces, Phys. Rev. E, 74 (2006), 061601. |
[30] |
B. Li and J. G. Liu, Thin film epitaxy with or without slope selection, Euro. J. Appl. Math., 14 (2003), 713-743.
doi: 10.1017/S095679250300528X. |
[31] |
B. Li and J. G. Liu, Epitaxial growth without slope selection: Energetics, coarsening, and dynamic scaling, J. Nonlinear Sci., 14 (2004), 429-451.
doi: 10.1007/s00332-004-0634-9. |
[32] |
J. S. Lowengrub, E. Titi and K. Zhao, Analysis of a mixture model of tumor growth, Euro. J. Appl. Math., 24 (2013), 691-734.
doi: 10.1017/S0956792513000144. |
[33] |
H. Ly and E. Titi, Global Gevrey regularity for the bénard convection in a porous medium with zero Darcy-Prandtl number, J. Nonlinear Sci., 9 (1999), 333-362.
doi: 10.1007/s003329900073. |
[34] |
D. Moldovan and L. Golubovic, Interfacial coarsening dynamics in epitaxial growth with slope selection, Phys. Rev. E, 61 (2000), 6190.
doi: 10.1103/PhysRevE.61.6190. |
[35] |
M. Ortiz, E. A. Repetto and H. Si, A continuum model of kinetic roughening and coarsening in thin films, J. Mech. Phys. Solids, 47 (1999), 697-730.
doi: 10.1016/S0022-5096(98)00102-1. |
[36] |
K. Promislow, Time analyticity and Gevrey regularity for solutions of a class of dissipative partial differential equations, Nonlinear Anal., 16 (1991), 959-980.
doi: 10.1016/0362-546X(91)90100-F. |
[37] |
N. Provatas, J. A. Dantzig, B. Athreya, P. Chan, P. Stefanovic, N. Goldenfeld and K. R. Elder, Using the phase-field crystal method in the multiscale modeling of microstructure evolution, JOM, 59 (2007), p83. |
[38] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, UK, 2001. |
[39] |
J. Shen, C. Wang, X. Wang and S. M. Wise, Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: Application to thin film epitaxy, SIAM J. Numer. Anal., 50 (2012), 105-125.
doi: 10.1137/110822839. |
[40] |
D. Swanson, Gevrey regularity of certain solutions to the Cahn-Hilliard equation with rough initial data, Methods Appl. Anal., 18 (2011), 417-426.
doi: 10.4310/MAA.2011.v18.n4.a4. |
[41] |
J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), p319.
doi: 10.1103/PhysRevA.15.319. |
[42] |
C. Wang, X. Wang and S. M. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete Contin. Dyn. Sys. A, 28 (2010), 405-423.
doi: 10.3934/dcds.2010.28.405. |
[43] |
K. A. Wu, M. Plapp and P. W. Voorhees, Controlling crystal symmetries in phase-field crystal models, J. Phys.: Condensed Matter, 22 (2010), 364102.
doi: 10.1088/0953-8984/22/36/364102. |
[44] |
C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal., 44 (2006), 1759-1779.
doi: 10.1137/050628143. |
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. |
[2] |
A. Biswas and D. Swanson, Existence and generalized Gevrey regularity of solutions to the Kuramoto-Sivashinsky equation in $R^n$, J. Differential Equations, 240 (2007), 145-163.
doi: 10.1016/j.jde.2007.05.022. |
[3] |
A. Biswas and D. Swanson, Gevrey regularity of solutions to the 3-D Navier-Stokes equations with weighted $l_p$ initial data, Indiana Univ. Math. J., 56 (2007), 1157-1188.
doi: 10.1512/iumj.2007.56.2891. |
[4] |
Z. Bradshaw, Z. Grujic and I. Kukavica, Local analyticity radii of solutions to the 3d Navier-Stokes equations with locally analytic forcing, J. Differential Equations, 259 (2015), 3955-3975.
doi: 10.1016/j.jde.2015.05.009. |
[5] |
C. Cao, M. Rammaha and E. Titi, Gevrey regularity for nonlinear analytic parabolic equations on the sphere, J. Dynam. Differential Equations, 12 (2000), 411-433.
doi: 10.1023/A:1009072526324. |
[6] |
W. Chen, S. Conde, C. Wang, X. Wang and S. M. Wise, A linear energy stable scheme for a thin film model without slope selection, J. Sci. Comput., 52 (2012), 546-562.
doi: 10.1007/s10915-011-9559-2. |
[7] |
W. Chen, C. Wang, X. Wang and S. M. Wise, A linear iteration algorithm for a second-order energy stable scheme for a thin film model without slope selection, J. Sci. Comput., 59 (2014), 574-601.
doi: 10.1007/s10915-013-9774-0. |
[8] |
M. C. Cross and A. C. Newell, Convection patterns in large aspect ratio systems, Physica D, 10 (1984), 299-328.
doi: 10.1016/0167-2789(84)90181-7. |
[9] |
A. Eden and V. Kalantarov, The convective Cahn-Hilliard equation, Appl. Math. Lett., 20 (2007), 455-461.
doi: 10.1016/j.aml.2006.05.014. |
[10] |
K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth, Phys. Rev. Lett., 88 (2002), 245701.
doi: 10.1103/PhysRevLett.88.245701. |
[11] |
K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals, Phys. Rev. E, 70 (2004), 051605.
doi: 10.1103/PhysRevE.70.051605. |
[12] |
A. Ferrari and E. Titi, Gevrey regularity for nonlinear analytic parabolic equations, Comm. Partial Differential Equations, 23 (1998), 1-16. |
[13] |
C. Foias and R. Temam, Gevrey class regularity for the solution of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.
doi: 10.1016/0022-1236(89)90015-3. |
[14] |
K. B. Glasner, Grain boundary motion arising from the gradient flow of the Aviles-Giga functional, Physica D, 215 (2006), 80-98.
doi: 10.1016/j.physd.2006.01.013. |
[15] |
A. A. Golovin and A. A. Nepomnyashchy, Disclinations in square and hexagonal patterns, Phys. Rev. E, 67 (2003), 056202, 7pp.
doi: 10.1103/PhysRevE.67.056202. |
[16] |
Z. Grujic and I. Kukavica, Space analyticity for the Navier-Stokes and related equations with initial data in $L^p$, J. Funct. Anal., 152 (1998), 447-466.
doi: 10.1006/jfan.1997.3167. |
[17] |
Z. Grujic and I. Kukavica, Space analyticity for the nonlinear heat equation in a bounded domain, J. Differential Equations, 154 (1999), 42-54.
doi: 10.1006/jdeq.1998.3562. |
[18] |
V. Kalantarov, B. Levant and E. Titi, Gevrey regularity for the attractor of the {3D Navier-Stokes-Voight} equations, J. Nonlinear Sci., 19 (2009), 133-152.
doi: 10.1007/s00332-008-9029-7. |
[19] |
R. V. Kohn and F. Otto, Upper bound on coarsening rate, Commun. Math. Phys., 229 (2002), 375-395.
doi: 10.1007/s00220-002-0693-4. |
[20] |
R. V. Kohn and X. Yan, Upper bound on the coarsening rate for an epitaxial growth model, Comm. Pure Appl. Math., 56 (2003), 1549-1564.
doi: 10.1002/cpa.10103. |
[21] |
J. Krug, Four lectures on the physics of crystal growth, Physica A, 313 (2002), 47-82.
doi: 10.1016/S0378-4371(02)01034-8. |
[22] |
I. Kukavica, R. Temam, V. Vlad, and M. Ziane, On the time analyticity radius of the solutions of the two-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 3 (1991), 611-618.
doi: 10.1007/BF01049102. |
[23] |
I. Kukavica, R. Temam, V. Vlad and M. Ziane, Existence and uniqueness of solutions for the hydrostatic Euler equations on a bounded domain with analytic data, C. R. Math. Acad. Sci. Paris, 348 (2010), 639-645.
doi: 10.1016/j.crma.2010.03.023. |
[24] |
I. Kukavica and V. Vlad, On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Amer. Math. Soc., 137 (2009), 669-677.
doi: 10.1090/S0002-9939-08-09693-7. |
[25] |
I. Kukavica and V. Vlad, The domain of analyticity of solutions to the three-dimensional Euler equations in a half space, Discrete Contin. Dyn. Syst., 29 (2011), 285-303.
doi: 10.3934/dcds.2011.29.285. |
[26] |
I. Kukavica and V. Vlad, On the analyticity and Gevrey-class regularity up to the boundary for the Euler equations, Nonlinearity, 24 (2011), 765-796.
doi: 10.1088/0951-7715/24/3/004. |
[27] |
I. Kukavica and V. Vlad, On the local existence of analytic solutions to the Prandtl boundary layer equations, Commun. Math. Sci., 11 (2013), 269-292.
doi: 10.4310/CMS.2013.v11.n1.a8. |
[28] |
A. Levandovsky and L. Golubovic, Epitaxial growth and erosion on (001) crystal surfaces: Far-from-equilibrium transitions, Phys. Rev. B, 65 (2004), 241402. |
[29] |
A. Levandovsky, L. Golubovic and D. Moldovan, Interfacial states and far-from-equilibrium transitions in the epitaxial growth and erosion on (110) crystal surfaces, Phys. Rev. E, 74 (2006), 061601. |
[30] |
B. Li and J. G. Liu, Thin film epitaxy with or without slope selection, Euro. J. Appl. Math., 14 (2003), 713-743.
doi: 10.1017/S095679250300528X. |
[31] |
B. Li and J. G. Liu, Epitaxial growth without slope selection: Energetics, coarsening, and dynamic scaling, J. Nonlinear Sci., 14 (2004), 429-451.
doi: 10.1007/s00332-004-0634-9. |
[32] |
J. S. Lowengrub, E. Titi and K. Zhao, Analysis of a mixture model of tumor growth, Euro. J. Appl. Math., 24 (2013), 691-734.
doi: 10.1017/S0956792513000144. |
[33] |
H. Ly and E. Titi, Global Gevrey regularity for the bénard convection in a porous medium with zero Darcy-Prandtl number, J. Nonlinear Sci., 9 (1999), 333-362.
doi: 10.1007/s003329900073. |
[34] |
D. Moldovan and L. Golubovic, Interfacial coarsening dynamics in epitaxial growth with slope selection, Phys. Rev. E, 61 (2000), 6190.
doi: 10.1103/PhysRevE.61.6190. |
[35] |
M. Ortiz, E. A. Repetto and H. Si, A continuum model of kinetic roughening and coarsening in thin films, J. Mech. Phys. Solids, 47 (1999), 697-730.
doi: 10.1016/S0022-5096(98)00102-1. |
[36] |
K. Promislow, Time analyticity and Gevrey regularity for solutions of a class of dissipative partial differential equations, Nonlinear Anal., 16 (1991), 959-980.
doi: 10.1016/0362-546X(91)90100-F. |
[37] |
N. Provatas, J. A. Dantzig, B. Athreya, P. Chan, P. Stefanovic, N. Goldenfeld and K. R. Elder, Using the phase-field crystal method in the multiscale modeling of microstructure evolution, JOM, 59 (2007), p83. |
[38] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, UK, 2001. |
[39] |
J. Shen, C. Wang, X. Wang and S. M. Wise, Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: Application to thin film epitaxy, SIAM J. Numer. Anal., 50 (2012), 105-125.
doi: 10.1137/110822839. |
[40] |
D. Swanson, Gevrey regularity of certain solutions to the Cahn-Hilliard equation with rough initial data, Methods Appl. Anal., 18 (2011), 417-426.
doi: 10.4310/MAA.2011.v18.n4.a4. |
[41] |
J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), p319.
doi: 10.1103/PhysRevA.15.319. |
[42] |
C. Wang, X. Wang and S. M. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete Contin. Dyn. Sys. A, 28 (2010), 405-423.
doi: 10.3934/dcds.2010.28.405. |
[43] |
K. A. Wu, M. Plapp and P. W. Voorhees, Controlling crystal symmetries in phase-field crystal models, J. Phys.: Condensed Matter, 22 (2010), 364102.
doi: 10.1088/0953-8984/22/36/364102. |
[44] |
C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal., 44 (2006), 1759-1779.
doi: 10.1137/050628143. |
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