Global-in-time Gevrey regularity solution for a class of bistable gradient flows

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August 2016, 21(6): 1689-1711. doi: 10.3934/dcdsb.2016018

Global-in-time Gevrey regularity solution for a class of bistable gradient flows

Nan Chen ^1, , Cheng Wang ^2, and Steven Wise ^3,

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

Received August 2015 Revised March 2016 Published June 2016

Abstract
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In this paper, we prove the existence and uniqueness of a Gevrey regularity solution for a class of nonlinear bistable gradient flows, where with the energy may be decomposed into purely convex and concave parts. Example equations include certain epitaxial thin film growth models and phase field crystal models. The energy dissipation law implies a bound in the leading Sobolev norm. The polynomial structure of the nonlinear terms in the chemical potential enables us to derive a local-in-time solution with Gevrey regularity, with the existence time interval length dependent on a certain $H^m$ norm of the initial data. A detailed Sobolev estimate for the gradient equations results in a uniform-in-time-bound of that $H^m$ norm, which in turn establishes the existence of a global-in-time solution with Gevrey regularity.

Keywords: Gevrey regularity solution, global-in-time existence., Bistable gradient flow.

Mathematics Subject Classification: 35G25, 35K3.

Citation: Nan Chen, Cheng Wang, Steven Wise. Global-in-time Gevrey regularity solution for a class of bistable gradient flows. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1689-1711. doi: 10.3934/dcdsb.2016018

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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