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Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model
A vicinal surface model for epitaxial growth with logarithmic free energy
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong |
Department of Mathematics‡, University of California, Los Angeles, Los Angeles, CA 90095, USA |
Department of Mathematics#, Duke University, Durham, NC 27708, USA |
Department of Physics†, Duke University, Durham, NC 27708, USA |
We study a continuum model for solid films that arises from the modeling of one-dimensional step flows on a vicinal surface in the attachment-detachment-limited regime. The resulting nonlinear partial differential equation, $u_t = -u^2(u^3+α u)_{hhhh}$, gives the evolution for the surface slope $u$ as a function of the local height $h$ in a monotone step train. Subject to periodic boundary conditions and positive initial conditions, we prove the existence, uniqueness and positivity of global strong solutions to this PDE using two Lyapunov energy functions. The long time behavior of $u$ converging to a constant that only depends on the initial data is also investigated both analytically and numerically.
References:
[1] |
H. Al Hajj 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, 240 (2011), 1771-1784.
doi: 10.1016/j.physd.2011.07.016. |
[2] |
F. Bernis and A. Friedman,
Higher order nonlinear degenerate parabolic equations, Journal of Differential Equations, 83 (1990), 179-206.
doi: 10.1016/0022-0396(90)90074-Y. |
[3] |
W. K. Burton, N. Cabrera and F. C. Frank,
The growth of crystals and the equilibrium structure of their surfaces, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 243 (1951), 299-358.
doi: 10.1098/rsta.1951.0006. |
[4] |
W. L. Chan, A. Ramasubramaniam, V. B. Shenoy and E. Chason, Relaxation kinetics of nano-ripples on Cu (001) surface, Physical Review B, 70 (2004), 245403.
doi: 10.1103/PhysRevB.70.245403. |
[5] |
W. E and N. K. Yip,
Continuum theory of epitaxial crystal growth. Ⅰ, Journal of Statistical Physics, 104 (2001), 221-253.
doi: 10.1023/A:1010361711825. |
[6] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19 (1998), American Mathematical Society. |
[7] |
Y. Gao, J.-G. Liu and J. Lu,
Continuum limit of a mesoscopic model with elasticity of step motion on vicinal surfaces, Journal of Nonlinear Science, 27 (2017), 873-926.
doi: 10.1007/s00332-016-9354-1. |
[8] |
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. |
[9] |
N. Israeli and D. Kandel, Decay of one-dimensional surface modulations, Physical Review B, 62 (2000), 13707.
doi: 10.1103/PhysRevB.62.13707. |
[10] |
H.-C. Jeong and E. D. Williams,
Steps on surfaces: Experiment and theory, Surface Science Reports, 34 (1999), 171-294.
doi: 10.1016/S0167-5729(98)00010-7. |
[11] |
R. V. Kohn, Surface relaxation below the roughening temperature: Some recent progress and open questions, Nonlinear Partial Differential Equations, Springer, 7 (2012), 207–221.
doi: 10.1007/978-3-642-25361-4_11. |
[12] |
R. V. Kohn, T. S. Lo and N. K. Yip, Continuum limit of a step flow model of epitaxial growth, MRS Proceedings, vol. 701, Cambridge Univ Press, 2001, 1–7. Google Scholar |
[13] |
R. V. Kohn and H. M. Versieux,
Numerical analysis of a steepest-descent PDE model for surface relaxation below the roughening temperature, SIAM Journal on Numerical Analysis, 48 (2010), 1781-1800.
doi: 10.1137/090750378. |
[14] |
R. Kohn and Y. Giga,
Scale-invariant extinction time estimates for some singular diffusion equations, Discrete and Continuous Dynamical Systems, 30 (2011), 509-535.
doi: 10.3934/dcds.2011.30.509. |
[15] |
B. Li and J.-G. Liu,
Thin film epitaxy with or without slope selection, European Journal of Applied Mathematics, 14 (2003), 713-743.
doi: 10.1017/S095679250300528X. |
[16] |
B. Li and J.-G. Liu,
Epitaxial growth without slope selection: Energetics, coarsening, and dynamic scaling, Journal of Nonlinear Science, 14 (2004), 429-451.
doi: 10.1007/s00332-004-0634-9. |
[17] |
D. Margetis and R. V. Kohn,
Continuum relaxation of interacting steps on crystal surfaces in $2+1$ dimensions, Multiscale Modeling & Simulation, 5 (2006), 729-758.
doi: 10.1137/06065297X. |
[18] |
W. W. Mullins,
Theory of thermal grooving, Journal of Applied Physics, 28 (1957), 333-339.
doi: 10.1063/1.1722742. |
[19] |
P. Nozières, On the motion of steps on a vicinal surface, Journal de Physique, 48 (1987), 1605-1608. Google Scholar |
[20] |
M. Ozdemir and A. Zangwill, Morphological equilibration of a corrugated crystalline surface, Physical Review B 42 (1990), 5013.
doi: 10.1103/PhysRevB.42.5013. |
[21] |
A. Pimpinelli and J. Villain, Physics of Crystal Growth, vol. 19, Cambridge University Press, 1998. Google Scholar |
[22] |
A. A. Rettori and J. Villain,
Flattening of grooves on a crystal surface: A method of investigation of surface roughness, Journal de Physique, 49 (1988), 257-267.
doi: 10.1051/jphys:01988004902025700. |
[23] |
V. B. Shenoy, A. Ramasubramaniam and L. B. Freund,
A variational approach to nonlinear dynamics of nanoscale surface modulations, Surface Science, 529 (2003), 365-383.
doi: 10.1016/S0039-6028(03)00276-0. |
[24] |
V. B. Shenoy, A. Ramasubramaniam, H. Ramanarayan, D. T. Tambe, W. L. Chan and E. Chason, Influence of step-edge barriers on the morphological relaxation of nanoscale ripples on crystal surfaces, Physical Review Letters, 92 (2004), 256101.
doi: 10.1103/PhysRevLett.92.256101. |
[25] |
C. Villani, Topics in Optimal Transportation, 58, American Mathematical Soc., 2003.
doi: 10.1007/b12016. |
[26] |
Y. Xiang,
Derivation of a continuum model for epitaxial growth with elasticity on vicinal surface, SIAM Journal on Applied Mathematics, 63 (2002), 241-258.
doi: 10.1137/S003613990139828X. |
show all references
References:
[1] |
H. Al Hajj 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, 240 (2011), 1771-1784.
doi: 10.1016/j.physd.2011.07.016. |
[2] |
F. Bernis and A. Friedman,
Higher order nonlinear degenerate parabolic equations, Journal of Differential Equations, 83 (1990), 179-206.
doi: 10.1016/0022-0396(90)90074-Y. |
[3] |
W. K. Burton, N. Cabrera and F. C. Frank,
The growth of crystals and the equilibrium structure of their surfaces, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 243 (1951), 299-358.
doi: 10.1098/rsta.1951.0006. |
[4] |
W. L. Chan, A. Ramasubramaniam, V. B. Shenoy and E. Chason, Relaxation kinetics of nano-ripples on Cu (001) surface, Physical Review B, 70 (2004), 245403.
doi: 10.1103/PhysRevB.70.245403. |
[5] |
W. E and N. K. Yip,
Continuum theory of epitaxial crystal growth. Ⅰ, Journal of Statistical Physics, 104 (2001), 221-253.
doi: 10.1023/A:1010361711825. |
[6] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19 (1998), American Mathematical Society. |
[7] |
Y. Gao, J.-G. Liu and J. Lu,
Continuum limit of a mesoscopic model with elasticity of step motion on vicinal surfaces, Journal of Nonlinear Science, 27 (2017), 873-926.
doi: 10.1007/s00332-016-9354-1. |
[8] |
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. |
[9] |
N. Israeli and D. Kandel, Decay of one-dimensional surface modulations, Physical Review B, 62 (2000), 13707.
doi: 10.1103/PhysRevB.62.13707. |
[10] |
H.-C. Jeong and E. D. Williams,
Steps on surfaces: Experiment and theory, Surface Science Reports, 34 (1999), 171-294.
doi: 10.1016/S0167-5729(98)00010-7. |
[11] |
R. V. Kohn, Surface relaxation below the roughening temperature: Some recent progress and open questions, Nonlinear Partial Differential Equations, Springer, 7 (2012), 207–221.
doi: 10.1007/978-3-642-25361-4_11. |
[12] |
R. V. Kohn, T. S. Lo and N. K. Yip, Continuum limit of a step flow model of epitaxial growth, MRS Proceedings, vol. 701, Cambridge Univ Press, 2001, 1–7. Google Scholar |
[13] |
R. V. Kohn and H. M. Versieux,
Numerical analysis of a steepest-descent PDE model for surface relaxation below the roughening temperature, SIAM Journal on Numerical Analysis, 48 (2010), 1781-1800.
doi: 10.1137/090750378. |
[14] |
R. Kohn and Y. Giga,
Scale-invariant extinction time estimates for some singular diffusion equations, Discrete and Continuous Dynamical Systems, 30 (2011), 509-535.
doi: 10.3934/dcds.2011.30.509. |
[15] |
B. Li and J.-G. Liu,
Thin film epitaxy with or without slope selection, European Journal of Applied Mathematics, 14 (2003), 713-743.
doi: 10.1017/S095679250300528X. |
[16] |
B. Li and J.-G. Liu,
Epitaxial growth without slope selection: Energetics, coarsening, and dynamic scaling, Journal of Nonlinear Science, 14 (2004), 429-451.
doi: 10.1007/s00332-004-0634-9. |
[17] |
D. Margetis and R. V. Kohn,
Continuum relaxation of interacting steps on crystal surfaces in $2+1$ dimensions, Multiscale Modeling & Simulation, 5 (2006), 729-758.
doi: 10.1137/06065297X. |
[18] |
W. W. Mullins,
Theory of thermal grooving, Journal of Applied Physics, 28 (1957), 333-339.
doi: 10.1063/1.1722742. |
[19] |
P. Nozières, On the motion of steps on a vicinal surface, Journal de Physique, 48 (1987), 1605-1608. Google Scholar |
[20] |
M. Ozdemir and A. Zangwill, Morphological equilibration of a corrugated crystalline surface, Physical Review B 42 (1990), 5013.
doi: 10.1103/PhysRevB.42.5013. |
[21] |
A. Pimpinelli and J. Villain, Physics of Crystal Growth, vol. 19, Cambridge University Press, 1998. Google Scholar |
[22] |
A. A. Rettori and J. Villain,
Flattening of grooves on a crystal surface: A method of investigation of surface roughness, Journal de Physique, 49 (1988), 257-267.
doi: 10.1051/jphys:01988004902025700. |
[23] |
V. B. Shenoy, A. Ramasubramaniam and L. B. Freund,
A variational approach to nonlinear dynamics of nanoscale surface modulations, Surface Science, 529 (2003), 365-383.
doi: 10.1016/S0039-6028(03)00276-0. |
[24] |
V. B. Shenoy, A. Ramasubramaniam, H. Ramanarayan, D. T. Tambe, W. L. Chan and E. Chason, Influence of step-edge barriers on the morphological relaxation of nanoscale ripples on crystal surfaces, Physical Review Letters, 92 (2004), 256101.
doi: 10.1103/PhysRevLett.92.256101. |
[25] |
C. Villani, Topics in Optimal Transportation, 58, American Mathematical Soc., 2003.
doi: 10.1007/b12016. |
[26] |
Y. Xiang,
Derivation of a continuum model for epitaxial growth with elasticity on vicinal surface, SIAM Journal on Applied Mathematics, 63 (2002), 241-258.
doi: 10.1137/S003613990139828X. |




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