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

Fractional stochastic differential equations satisfying fluctuation-dissipation theorem

We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors

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

Regional tourism to regional transformation: the case of Jining area of Shandong Province in China

China is facing the urgent problem of transformation development and the rapid development of tourism at the same time. Regional tourism could be the stimulus and impetus for the transformational development of an area by rousing the vitality in various fields. Jining area of Shandong Province is an important coal energy base in China demanding transformation, as well as a traditional tourism area owing abundant tourism resources and development opportunities but with a lack of enough attention. It is proposed in this article that a regional wetland system could be established in the water area caused by coal field subsidence with the natural and culture resources in Jining area to develop regional tourism. Starting from this, the following will be realized gradually: ecological restoration, landscape reshaping, space integration, industrial restructuring and culture revival, and ultimately the comprehensive transformational development.Peer Reviewe

Regional tourism to regional transformation: the case of Jining area of Shandong Province in China

China is facing the urgent problem of transformation development and the rapid development of tourism at the same time. Regional tourism could be the stimulus and impetus for the transformational development of an area by rousing the vitality in various fields. Jining area of Shandong Province is an important coal energy base in China demanding transformation, as well as a traditional tourism area owing abundant tourism resources and development opportunities but with a lack of enough attention. It is proposed in this article that a regional wetland system could be established in the water area caused by coal field subsidence with the natural and culture resources in Jining area to develop regional tourism. Starting from this, the following will be realized gradually: ecological restoration, landscape reshaping, space integration, industrial restructuring and culture revival, and ultimately the comprehensive transformational development.Peer Reviewe

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