9 research outputs found
Global Strong Solution With BV Derivatives to Singular Solid-on-Solid model With Exponential Nonlinearity
In this work, we consider the one dimensional very singular fourth-order
equation for solid-on-solid model in attachment-detachment-limit regime with
exponential nonlinearity where total energy is the total variation of . Using a logarithmic correction
and gradient flow structure with a
suitable defined functional, we prove the evolution variational inequality
solution preserves a positive gradient which has upper and lower bounds
but in BV space. We also obtain the global strong solution to the
solid-on-solid model which allows an asymmetric singularity happens.Comment: 15 page
A vicinal surface model for epitaxial growth with logarithmic free energy
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, , gives the evolution
for the surface slope as a function of the local height 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 converging to a constant that only depends on the initial data
is also investigated both analytically and numerically.Comment: 18 pages, 7 figure
Gradient flow approach to an exponential thin film equation: global existence and latent singularity
In this work, we study a fourth order exponential equation, derived from thin film growth on crystal surface in multiple
space dimensions. We use the gradient flow method in metric space to
characterize the latent singularity in global strong solution, which is
intrinsic due to high degeneration. We define a suitable functional, which
reveals where the singularity happens, and then prove the variational
inequality solution under very weak assumptions for initial data. Moreover, the
existence of global strong solution is established with regular initial data.Comment: latent singularity, curve of maximal slope. arXiv admin note: text
overlap with arXiv:1711.07405 by other author
Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects
We study a nonlocal 4th order degenerate equation deriving from the epitaxial
growth on crystalline materials. We first prove the global existence of
evolution variational inequality solution with a general initial data using the
gradient flow structure. Then with a monotone initial data, we prove the
subdifferential of the associated convex functional is indeed single-valued,
which gives higher regularities of the global solution. Particularly, higher
regularites imply that the strict monotonicity maintains for all time, which
provides rigorous justification for global-in time monotone solution to
epitaxial growth model with nonlocal elastic effects on vicinal surface
Continuum limit of a mesoscopic model with elasticity of step motion on vicinal surfaces
This work considers the rigorous derivation of continuum models of step
motion starting from a mesoscopic Burton-Cabrera-Frank (BCF) type model
following the work [Xiang, SIAM J. Appl. Math. 2002]. We prove that as the
lattice parameter goes to zero, for a finite time interval, a modified discrete
model converges to the strong solution of the limiting PDE with first order
convergence rate.Comment: 52 page