5 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
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
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