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 $$h_t = \nabla \cdot (\frac{1}{|\nabla h|} \nabla e^{\frac{\delta E}{\delta h}}) =\nabla \cdot (\frac{1}{|\nabla h|}\nabla e^{- \nabla \cdot (\frac{\nabla h}{|\nabla h|})})$$ where total energy $E=\int |\nabla h|$ is the total variation of $h$. Using a logarithmic correction $E=\int |\nabla h|\ln|\nabla h| d x$ and gradient flow structure with a suitable defined functional, we prove the evolution variational inequality solution preserves a positive gradient $h_x$ 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 $h_{xx}^+$ happens.Comment: 15 page

Existence theorems for a crystal surface model involving the p-Laplace operator

The manufacturing of crystal films lies at the heart of modern nanotechnology. How to accurately predict the motion of a crystal surface is of fundamental importance. Many continuum models have been developed for this purpose, including a number of PDE models, which are often obtained as the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. In this paper we offer an analytical perspective into some of these models. To be specific, we study the existence of a weak solution to the boundary value problem for the equation - \Delta e^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right)}+au=f, where $p>1, a>0$ are given numbers and $f$ is a given function. This problem is derived from a crystal surface model proposed by J.L.~Marzuola and J.~Weare (2013 Physical Review, E 88, 032403). The mathematical challenge is due to the fact that the principal term in our equation is an exponential function of a p-Laplacian. Existence of a suitably-defined weak solution is established under the assumptions that $p\in(1,2], \ N\leq 4$, and $f\in W^{1,p}$. Our investigations reveal that the key to our existence assertion is how to control the set where -\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right) is $\pm\infty$

Gradient flow approach to an exponential thin film equation: global existence and latent singularity

In this work, we study a fourth order exponential equation, $u_t=\Delta e^{-\Delta u},$ 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, $u_t = -u^2(u^3+\alpha u)_{hhhh}$, 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.Comment: 18 pages, 7 figure

Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model

In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface corrugation in 1+1 dimensions via a continuum model below the roughening transition temperature. The model invokes a highly degenerate parabolic partial differential equation (PDE) for surface diffusion, which is related to the weighted-$H^{-1}$ (nonlinear) gradient flow of a convex, singular surface free energy in homoepitaxy. The PDE is motivated both by an atomistic broken-bond model and a mesoscale model for steps. By constructing an explicit solution to the PDE, we demonstrate the lack of symmetry in the evolution of top and bottom facets in periodic surface profiles. Our explicit, analytical solution is compared to numerical simulations of the PDE via a regularized surface free energy.Comment: 23 pages, 5 figures, comments welcome! Text slightly modified, references updated in Version 2. Referee comments addresse

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

SINGULAR NEUMANN PROBLEMS AND LARGE-TIME\ud BEHAVIOR OF SOLUTIONS OF NONCOERCIVE\ud HAMILTON-JACOBI EQUATIONS

We investigate the large-time behavior of viscosity solutions of Hamilton- Jacobi equations with noncoercive Hamiltonian in a multidimensional Euclidean space. Our motivation comes from a model describing growing faceted crystals recently discussed by E. Yokoyama, Y. Giga and P. Rybka. Surprisingly, growth rates of viscosity solutions of these equations depend on x-variable. In a part of the space called the effective domain, growth rates are constant but outside of this domain, they seem to be unstable. Moreover, on the boundary of the effective domain, the gradient with respect to x-variable of solutions blows up as time goes to infinity. Therefore, we are naturally led to study singular Neumann problems for stationary Hamilton-Jacobi equations. We establish the existence, stability and comparison results for singular Neumann problems and apply the results for a large-time asymptotic profile on the effective domain of viscosity solutions of Hamilton-Jacobi equations with noncoercive Hamiltonian