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April  2019, 39(4): 2101-2131. doi: 10.3934/dcds.2019088

Global existence and decay to equilibrium for some crystal surface models

Rafael Granero-Belinchón 1, and  Martina Magliocca 2,

1. 

Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria. Avda. Los Castros s/n, Santander, Spain
2. 

Dipartimento di Matematica, Università degli Studi Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy

Received  May 2018 Revised  September 2018 Published  January 2019

In this paper we study the large time behavior of the solutions to the following nonlinear fourth-order equations
$ \ \ \ \ \ {\partial _t} u = \Delta e^{-\Delta u}, \\ {\partial _t} u = -u^2\Delta ^2(u^3). $
These two PDE were proposed as models of the evolution of crystal surfaces by J. Krug, H.T. Dobbs, and S. Majaniemi (Z. Phys. B, 97,281-291, 1995) and H. Al Hajj Shehadeh, R. V. Kohn, and J. Weare (Phys. D, 240, 1771-1784, 2011), respectively. In particular, we find explicitly computable conditions on the size of the initial data (measured in terms of the norm in a critical space) guaranteeing the global existence and exponential decay to equilibrium in the Wiener algebra and in Sobolev spaces.
Keywords: Crystal surface model, nonlinear fourth-order parabolic equations, global existence, decay to equilibrium.
Mathematics Subject Classification: 35A01, 35B40, 35G25, 35K30, 35K55.
Citation: Rafael Granero-Belinchón, Martina Magliocca. Global existence and decay to equilibrium for some crystal surface models. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2101-2131. doi: 10.3934/dcds.2019088
References:
[1]

D. Ambrose, The radius of analyticity for solutions to a problem in epitaxial growth on the torus, arXiv preprint, arXiv: 1807.01740, 2018. Google Scholar

[2]

H. Bae, R. Granero-Belinchón and O. Lazar, Global existence of weak solutions to dissipative transport equations with nonlocal velocity, Nonlinearity, 31 (2018), 1484-1515.  doi: 10.1088/1361-6544/aaa2e0.  Google Scholar

[3]

G. Bruell and R. Granero-Belinchón, On the thin film Muskat and the thin film Stokes equations, arXiv preprint, 2018, arXiv: 1802.05509 [math.AP]. Google Scholar

[4]

J. Burczak and R. Granero-Belinchón, On a generalized doubly parabolic keller–segel system in one spatial dimension, Mathematical Models and Methods in Applied Sciences, 26 (2016), 111-160.  doi: 10.1142/S0218202516500044.  Google Scholar

[5]

P. Constantin, D. Córdoba, F. Gancedo, L. Rodriguez-Piazza and R. M. Strain, On the muskat problem: Global in time results in 2d and 3d, American Journal of Mathematics, 138 (2016), 1455-1494.  doi: 10.1353/ajm.2016.0044.  Google Scholar

[6]

D. Córdoba and F. Gancedo, Contour dynamics of incompressible 3-d fluids in a porous medium with different densities, Communications in Mathematical Physics, 273 (2007), 445-471.  doi: 10.1007/s00220-007-0246-y.  Google Scholar

[7]

F. Gancedo, E. Garcia-Juarez, N. Patel and R. M. Strain, On the muskat problem with viscosity jump: Global in time results, arXiv preprint, 2017, arXiv: 1710.11604. Google Scholar

[8]

Y. Gao, H. Ji, J.-G. Liu and T. P. Witelski, A vicinal surface model for epitaxial growth with logarithmic free energy, AIMS, 23 (2018), 4433-4453.  doi: 10.3934/dcdsb.2018170.  Google Scholar

[9]

Y. Gao, J.-G. Liu and J. Lu, Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime, SIAM Journal on Mathematical Analysis, 49 (2017), 1705-1731.  doi: 10.1137/16M1094543.  Google Scholar

[10]

Y. Gao, J.-G. Liu and X. Y. Lu, Gradient Flow Approach to an Exponential Thin Film Equation: Global Existence and Latent Singularity, ESAIM: COCV, Forthcoming article, 2018, arXiv: 1710.06995. doi: 10.1051/cocv/2018037.  Google Scholar

[11]

Y. Giga and R. V. Kohn, Scale-invariant extinction time estimates for some singular diffusion equations, Discrete Contin. Dyn. Syst., 30 (2011), 509-535.  doi: 10.3934/dcds.2011.30.509.  Google Scholar

[12]

J. Krug, H. T. Dobbs and S. Majaniemi, Adatom mobility for the solid-on-solid model, Zeitschrift für Physik B Condensed Matter, 97 (1995), 281–291. doi: 10.1007/BF01307478.  Google Scholar

[13]

J.-G. Liu and X. Xu, Existence theorems for a multidimensional crystal surface model, SIAM Journal on Mathematical Analysis, 48 (2016), 3667-3687.  doi: 10.1137/16M1059400.  Google Scholar

[14]

J. G. Liu and X. Xu, Analytical validation of a continuum model for the evolution of a crystal surface in multiple space dimensions, SIAM Journal on Mathematical Analysis, 49 (2017), 2220-2245.  doi: 10.1137/16M1098474.  Google Scholar

[15]

Jian-Guo Liu and Robert M. Strain., Global stability for solutions to the exponential PDE describing epitaxial growth., arXiv preprint arXiv: 1805.02246, 2018. Google Scholar

[16]

J. L. Marzuola and J. Weare, Relaxation of a family of broken-bond crystal-surface models, Physical Review E, 88 (2013), 032403. doi: 10.1103/PhysRevE.88.032403.  Google Scholar

[17]

N. Patel and R. M. Strain, Large time decay estimates for the muskat equation, Communications in Partial Differential Equations, 42 (2017), 977-999.  doi: 10.1080/03605302.2017.1321661.  Google Scholar

[18]

H. A. H. Shehadeh, R. V. Kohn and J. Weare, The evolution of a crystal surface: Analysis of a one-dimensional step train connecting two facets in the adl regime, Physica D: Nonlinear Phenomena, 240 (2011), 1771-1784.  doi: 10.1016/j.physd.2011.07.016.  Google Scholar

[19]

J. Simon, Compact sets in the space $L^{p}(O, T; B)$, Annali di Matematica Pura ed Applicata, 146 (1986), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[20]

X. Xu, Existence theorems for a crystal surface model involving the p-laplacian operator, arXiv preprint, 2017, arXiv: 1711.07405. Google Scholar

show all references

References:
[1]

D. Ambrose, The radius of analyticity for solutions to a problem in epitaxial growth on the torus, arXiv preprint, arXiv: 1807.01740, 2018. Google Scholar

[2]

H. Bae, R. Granero-Belinchón and O. Lazar, Global existence of weak solutions to dissipative transport equations with nonlocal velocity, Nonlinearity, 31 (2018), 1484-1515.  doi: 10.1088/1361-6544/aaa2e0.  Google Scholar

[3]

G. Bruell and R. Granero-Belinchón, On the thin film Muskat and the thin film Stokes equations, arXiv preprint, 2018, arXiv: 1802.05509 [math.AP]. Google Scholar

[4]

J. Burczak and R. Granero-Belinchón, On a generalized doubly parabolic keller–segel system in one spatial dimension, Mathematical Models and Methods in Applied Sciences, 26 (2016), 111-160.  doi: 10.1142/S0218202516500044.  Google Scholar

[5]

P. Constantin, D. Córdoba, F. Gancedo, L. Rodriguez-Piazza and R. M. Strain, On the muskat problem: Global in time results in 2d and 3d, American Journal of Mathematics, 138 (2016), 1455-1494.  doi: 10.1353/ajm.2016.0044.  Google Scholar

[6]

D. Córdoba and F. Gancedo, Contour dynamics of incompressible 3-d fluids in a porous medium with different densities, Communications in Mathematical Physics, 273 (2007), 445-471.  doi: 10.1007/s00220-007-0246-y.  Google Scholar

[7]

F. Gancedo, E. Garcia-Juarez, N. Patel and R. M. Strain, On the muskat problem with viscosity jump: Global in time results, arXiv preprint, 2017, arXiv: 1710.11604. Google Scholar

[8]

Y. Gao, H. Ji, J.-G. Liu and T. P. Witelski, A vicinal surface model for epitaxial growth with logarithmic free energy, AIMS, 23 (2018), 4433-4453.  doi: 10.3934/dcdsb.2018170.  Google Scholar

[9]

Y. Gao, J.-G. Liu and J. Lu, Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime, SIAM Journal on Mathematical Analysis, 49 (2017), 1705-1731.  doi: 10.1137/16M1094543.  Google Scholar

[10]

Y. Gao, J.-G. Liu and X. Y. Lu, Gradient Flow Approach to an Exponential Thin Film Equation: Global Existence and Latent Singularity, ESAIM: COCV, Forthcoming article, 2018, arXiv: 1710.06995. doi: 10.1051/cocv/2018037.  Google Scholar

[11]

Y. Giga and R. V. Kohn, Scale-invariant extinction time estimates for some singular diffusion equations, Discrete Contin. Dyn. Syst., 30 (2011), 509-535.  doi: 10.3934/dcds.2011.30.509.  Google Scholar

[12]

J. Krug, H. T. Dobbs and S. Majaniemi, Adatom mobility for the solid-on-solid model, Zeitschrift für Physik B Condensed Matter, 97 (1995), 281–291. doi: 10.1007/BF01307478.  Google Scholar

[13]

J.-G. Liu and X. Xu, Existence theorems for a multidimensional crystal surface model, SIAM Journal on Mathematical Analysis, 48 (2016), 3667-3687.  doi: 10.1137/16M1059400.  Google Scholar

[14]

J. G. Liu and X. Xu, Analytical validation of a continuum model for the evolution of a crystal surface in multiple space dimensions, SIAM Journal on Mathematical Analysis, 49 (2017), 2220-2245.  doi: 10.1137/16M1098474.  Google Scholar

[15]

Jian-Guo Liu and Robert M. Strain., Global stability for solutions to the exponential PDE describing epitaxial growth., arXiv preprint arXiv: 1805.02246, 2018. Google Scholar

[16]

J. L. Marzuola and J. Weare, Relaxation of a family of broken-bond crystal-surface models, Physical Review E, 88 (2013), 032403. doi: 10.1103/PhysRevE.88.032403.  Google Scholar

[17]

N. Patel and R. M. Strain, Large time decay estimates for the muskat equation, Communications in Partial Differential Equations, 42 (2017), 977-999.  doi: 10.1080/03605302.2017.1321661.  Google Scholar

[18]

H. A. H. Shehadeh, R. V. Kohn and J. Weare, The evolution of a crystal surface: Analysis of a one-dimensional step train connecting two facets in the adl regime, Physica D: Nonlinear Phenomena, 240 (2011), 1771-1784.  doi: 10.1016/j.physd.2011.07.016.  Google Scholar

[19]

J. Simon, Compact sets in the space $L^{p}(O, T; B)$, Annali di Matematica Pura ed Applicata, 146 (1986), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[20]

X. Xu, Existence theorems for a crystal surface model involving the p-laplacian operator, arXiv preprint, 2017, arXiv: 1711.07405. Google Scholar

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