相关论文: Continuum models for surface growth

We analyse the singularity formation of congruences of solutions of systems of second order PDEs via the construction of \emph{shape maps}. The trace of such maps represents a congruence volume whose collapse we study through an appropriate…

We propose a computer-assisted approach to studying the effective continuum behavior of spatially discrete evolution equations. The advantage of the approach is that the "coarse model" (the continuum, effective equation) need not be…

The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in…

The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to…

A one-dimensional cellular automaton with a probabilistic evolution rule can generate stochastic surface growth in $(1 + 1)$ dimensions. Two such discrete models of surface growth are constructed from a probabilistic cellular automaton…

A method based on the kinetics of adatoms on a growing surface under epitaxial growth at low temperature in (1+1) dimensions is proposed to obtain a closed form of local growth equation. It can be generalized to any growth problem as long…

Partial differential equations (PDEs) are at the heart of many mathematical and scientific advances. While great progress has been made on the theory of PDEs of standard types during the last eight decades, the analysis of nonlinear PDEs of…

The paper studies a finite element method for computing transport and diffusion along evolving surfaces. The method does not require a parametrization of a surface or an extension of a PDE from a surface into a bulk outer domain. The…

Numerous processes across both the physical and biological sciences are driven by diffusion. Partial differential equations (PDEs) are a popular tool for modelling such phenomena deterministically, but it is often necessary to use…

We investigate connections between the continuum and atomistic descriptions of deformable crystals, using certain interesting results from number theory. The energy of a deformed crystal is calculated in the context of a lattice model with…

We consider a partial differential equation model for the growth of heterogeneous cell populations subdivided into multiple distinct discrete phenotypes. In this model, cells preferentially move towards regions where they feel less…

A solid-on-solid model of epitaxial growth in 1+1 dimensions is investigated in which slope dependent upward and downward particle currents compete on the surface. The microscopic mechanisms which give rise to these currents are the…

Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…

This paper presents a general theory and isogeometric finite element implementation for studying mass conserving phase transitions on deforming surfaces. The mathematical problem is governed by two coupled fourth-order nonlinear partial…

From the smallest biological systems to the largest cosmological structures, spatial domains undergo expansion and contraction. Within these growing domains, diffusive transport is a common phenomenon. Mathematical models have been widely…

Pattern formation is a widely observed phenomenon in diverse fields including materials physics, developmental biology and ecology, among many others. The physics underlying the patterns is specific to the mechanisms, and is encoded by…

The relaxation of axisymmetric crystal surfaces with a single facet below the roughening transition is studied via a continuum approach that accounts for step energy g_1 and step-step interaction energy g_3>0. For diffusion-limited…

We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We…

Volume conserving surface (VCS) models without deposition and evaporation, as well as ideal molecular-beam epitaxy models, are prototypes to study the symmetries of conserved dynamics. In this work we study two similar VCS models with…

We introduce suitable coordinate systems for pipes and their variants that allow us to transform partial differential equations (PDEs) on the pipe surfaces or in the solid pipes into computational domains with fixed limits/ranges. Such a…