# American Institute of Mathematical Sciences

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

## 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

Received  August 2015 Revised  March 2016 Published  June 2016

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.
Citation: Nan Chen, Cheng Wang, Steven Wise. Global-in-time Gevrey regularity solution for a class of bistable gradient flows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1689-1711. doi: 10.3934/dcdsb.2016018
##### References:
 [1] R. A. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar [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. doi: 10.1016/j.jde.2007.05.022. Google Scholar [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. doi: 10.1512/iumj.2007.56.2891. Google Scholar [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. doi: 10.1016/j.jde.2015.05.009. Google Scholar [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. doi: 10.1023/A:1009072526324. Google Scholar [6] W. Chen, S. Conde, C. 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Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals,, Phys. Rev. E, 70 (2004). doi: 10.1103/PhysRevE.70.051605. Google Scholar [12] A. Ferrari and E. Titi, Gevrey regularity for nonlinear analytic parabolic equations,, Comm. Partial Differential Equations, 23 (1998), 1. Google Scholar [13] C. Foias and R. Temam, Gevrey class regularity for the solution of the Navier-Stokes equations,, J. Funct. Anal., 87 (1989), 359. doi: 10.1016/0022-1236(89)90015-3. Google Scholar [14] K. B. Glasner, Grain boundary motion arising from the gradient flow of the Aviles-Giga functional,, Physica D, 215 (2006), 80. doi: 10.1016/j.physd.2006.01.013. Google Scholar [15] A. A. Golovin and A. A. Nepomnyashchy, Disclinations in square and hexagonal patterns,, Phys. Rev. E, 67 (2003). doi: 10.1103/PhysRevE.67.056202. Google Scholar [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. doi: 10.1006/jfan.1997.3167. Google Scholar [17] Z. Grujic and I. Kukavica, Space analyticity for the nonlinear heat equation in a bounded domain,, J. Differential Equations, 154 (1999), 42. doi: 10.1006/jdeq.1998.3562. Google Scholar [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. doi: 10.1007/s00332-008-9029-7. Google Scholar [19] R. V. Kohn and F. Otto, Upper bound on coarsening rate,, Commun. Math. Phys., 229 (2002), 375. doi: 10.1007/s00220-002-0693-4. Google Scholar [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. doi: 10.1002/cpa.10103. Google Scholar [21] J. Krug, Four lectures on the physics of crystal growth,, Physica A, 313 (2002), 47. doi: 10.1016/S0378-4371(02)01034-8. Google Scholar [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. doi: 10.1007/BF01049102. Google Scholar [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. doi: 10.1016/j.crma.2010.03.023. Google Scholar [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. doi: 10.1090/S0002-9939-08-09693-7. Google Scholar [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. doi: 10.3934/dcds.2011.29.285. Google Scholar [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. doi: 10.1088/0951-7715/24/3/004. Google Scholar [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. doi: 10.4310/CMS.2013.v11.n1.a8. Google Scholar [28] A. Levandovsky and L. Golubovic, Epitaxial growth and erosion on (001) crystal surfaces: Far-from-equilibrium transitions,, Phys. Rev. B, 65 (2004). Google Scholar [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). Google Scholar [30] B. Li and J. G. Liu, Thin film epitaxy with or without slope selection,, Euro. J. Appl. Math., 14 (2003), 713. doi: 10.1017/S095679250300528X. Google Scholar [31] B. Li and J. G. Liu, Epitaxial growth without slope selection: Energetics, coarsening, and dynamic scaling,, J. Nonlinear Sci., 14 (2004), 429. doi: 10.1007/s00332-004-0634-9. Google Scholar [32] J. S. Lowengrub, E. Titi and K. Zhao, Analysis of a mixture model of tumor growth,, Euro. J. Appl. Math., 24 (2013), 691. doi: 10.1017/S0956792513000144. Google Scholar [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. doi: 10.1007/s003329900073. Google Scholar [34] D. Moldovan and L. Golubovic, Interfacial coarsening dynamics in epitaxial growth with slope selection,, Phys. Rev. E, 61 (2000). doi: 10.1103/PhysRevE.61.6190. Google Scholar [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. doi: 10.1016/S0022-5096(98)00102-1. Google Scholar [36] K. Promislow, Time analyticity and Gevrey regularity for solutions of a class of dissipative partial differential equations,, Nonlinear Anal., 16 (1991), 959. doi: 10.1016/0362-546X(91)90100-F. Google Scholar [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). Google Scholar [38] J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge University Press, (2001). Google Scholar [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. doi: 10.1137/110822839. Google Scholar [40] D. Swanson, Gevrey regularity of certain solutions to the Cahn-Hilliard equation with rough initial data,, Methods Appl. Anal., 18 (2011), 417. doi: 10.4310/MAA.2011.v18.n4.a4. Google Scholar [41] J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability,, Phys. Rev. A, 15 (1977). doi: 10.1103/PhysRevA.15.319. Google Scholar [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. doi: 10.3934/dcds.2010.28.405. Google Scholar [43] K. A. Wu, M. Plapp and P. W. Voorhees, Controlling crystal symmetries in phase-field crystal models,, J. Phys.: Condensed Matter, 22 (2010). doi: 10.1088/0953-8984/22/36/364102. Google Scholar [44] C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models,, SIAM J. Numer. Anal., 44 (2006), 1759. doi: 10.1137/050628143. Google Scholar

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##### References:
 [1] R. A. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar [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. doi: 10.1016/j.jde.2007.05.022. Google Scholar [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. doi: 10.1512/iumj.2007.56.2891. Google Scholar [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. doi: 10.1016/j.jde.2015.05.009. Google Scholar [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. doi: 10.1023/A:1009072526324. Google Scholar [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. doi: 10.1007/s10915-011-9559-2. Google Scholar [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. doi: 10.1007/s10915-013-9774-0. Google Scholar [8] M. C. Cross and A. C. Newell, Convection patterns in large aspect ratio systems,, Physica D, 10 (1984), 299. doi: 10.1016/0167-2789(84)90181-7. Google Scholar [9] A. Eden and V. Kalantarov, The convective Cahn-Hilliard equation,, Appl. Math. Lett., 20 (2007), 455. doi: 10.1016/j.aml.2006.05.014. Google Scholar [10] K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth,, Phys. Rev. Lett., 88 (2002). doi: 10.1103/PhysRevLett.88.245701. Google Scholar [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). doi: 10.1103/PhysRevE.70.051605. Google Scholar [12] A. Ferrari and E. Titi, Gevrey regularity for nonlinear analytic parabolic equations,, Comm. Partial Differential Equations, 23 (1998), 1. Google Scholar [13] C. Foias and R. Temam, Gevrey class regularity for the solution of the Navier-Stokes equations,, J. Funct. Anal., 87 (1989), 359. doi: 10.1016/0022-1236(89)90015-3. Google Scholar [14] K. B. Glasner, Grain boundary motion arising from the gradient flow of the Aviles-Giga functional,, Physica D, 215 (2006), 80. doi: 10.1016/j.physd.2006.01.013. Google Scholar [15] A. A. Golovin and A. A. Nepomnyashchy, Disclinations in square and hexagonal patterns,, Phys. Rev. E, 67 (2003). doi: 10.1103/PhysRevE.67.056202. Google Scholar [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. doi: 10.1006/jfan.1997.3167. Google Scholar [17] Z. Grujic and I. Kukavica, Space analyticity for the nonlinear heat equation in a bounded domain,, J. Differential Equations, 154 (1999), 42. doi: 10.1006/jdeq.1998.3562. Google Scholar [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. doi: 10.1007/s00332-008-9029-7. Google Scholar [19] R. V. Kohn and F. Otto, Upper bound on coarsening rate,, Commun. Math. Phys., 229 (2002), 375. doi: 10.1007/s00220-002-0693-4. Google Scholar [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. doi: 10.1002/cpa.10103. Google Scholar [21] J. Krug, Four lectures on the physics of crystal growth,, Physica A, 313 (2002), 47. doi: 10.1016/S0378-4371(02)01034-8. Google Scholar [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. doi: 10.1007/BF01049102. Google Scholar [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. doi: 10.1016/j.crma.2010.03.023. Google Scholar [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. doi: 10.1090/S0002-9939-08-09693-7. Google Scholar [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. doi: 10.3934/dcds.2011.29.285. Google Scholar [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. doi: 10.1088/0951-7715/24/3/004. Google Scholar [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. doi: 10.4310/CMS.2013.v11.n1.a8. Google Scholar [28] A. Levandovsky and L. Golubovic, Epitaxial growth and erosion on (001) crystal surfaces: Far-from-equilibrium transitions,, Phys. Rev. B, 65 (2004). Google Scholar [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). Google Scholar [30] B. Li and J. G. Liu, Thin film epitaxy with or without slope selection,, Euro. J. Appl. Math., 14 (2003), 713. doi: 10.1017/S095679250300528X. Google Scholar [31] B. Li and J. G. Liu, Epitaxial growth without slope selection: Energetics, coarsening, and dynamic scaling,, J. Nonlinear Sci., 14 (2004), 429. doi: 10.1007/s00332-004-0634-9. Google Scholar [32] J. S. Lowengrub, E. Titi and K. Zhao, Analysis of a mixture model of tumor growth,, Euro. J. Appl. Math., 24 (2013), 691. doi: 10.1017/S0956792513000144. Google Scholar [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. doi: 10.1007/s003329900073. Google Scholar [34] D. Moldovan and L. Golubovic, Interfacial coarsening dynamics in epitaxial growth with slope selection,, Phys. Rev. E, 61 (2000). doi: 10.1103/PhysRevE.61.6190. Google Scholar [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. doi: 10.1016/S0022-5096(98)00102-1. Google Scholar [36] K. Promislow, Time analyticity and Gevrey regularity for solutions of a class of dissipative partial differential equations,, Nonlinear Anal., 16 (1991), 959. doi: 10.1016/0362-546X(91)90100-F. Google Scholar [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). Google Scholar [38] J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge University Press, (2001). Google Scholar [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. doi: 10.1137/110822839. Google Scholar [40] D. Swanson, Gevrey regularity of certain solutions to the Cahn-Hilliard equation with rough initial data,, Methods Appl. Anal., 18 (2011), 417. doi: 10.4310/MAA.2011.v18.n4.a4. Google Scholar [41] J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability,, Phys. Rev. A, 15 (1977). doi: 10.1103/PhysRevA.15.319. Google Scholar [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. doi: 10.3934/dcds.2010.28.405. Google Scholar [43] K. A. Wu, M. Plapp and P. W. Voorhees, Controlling crystal symmetries in phase-field crystal models,, J. Phys.: Condensed Matter, 22 (2010). doi: 10.1088/0953-8984/22/36/364102. Google Scholar [44] C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models,, SIAM J. Numer. Anal., 44 (2006), 1759. doi: 10.1137/050628143. Google Scholar
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