David Buchberger | Searcharxiv

arXiv2026

Numerical Study of a Surface Growth Model with Singular Noise

We study a stochastic model for epitaxial thin-film growth driven by spatially rough additive noise in a regime where the noise is singular and regularization via truncation in Fourier space is used to give a meaning to the solution, leading to a vanishing nonlinearity in the limit. In order to study this phenomenon numerically, the nonlinearity is discretized by a spectral Galerkin projection, while time integration is performed with an exponential Euler scheme. For roughness stronger than space-time white noise, we derive strong error estimates in $L^p(\Omega;C([0,T];\mathcal H^1))$ that display explicitly the interaction between the spatial cut-off, the time step, and the decay of the nonlinear current. We also quantify the growth of the truncated stochastic convolution and the corresponding vanishing rate of the nonlinearity. Numerical experiments illustrate the transition from persistent hill formation to noise-dominated dynamics as the roughness parameter increases.

math.NA↗

arXiv2024

A derivation of the time dependent von K\'arm\'an equations from atomistic models

We derive the time-dependent von K\'arm\'an plate equations from three dimensional, purely atomistic particle models. In particular, we prove that a thin structure of interacting particles whose dynamics is governed by Newton's laws of motion is effectively described by the von K\'arm\'an equations in the limit of vanishing interatomic distance $\eps$ and vanishing plate thickness $h$. While the classical plate equations are obtained for $\eps \ll h \ll 1$, we find new plate equations for finitely many layers in the ultrathin case $\eps \sim h$.

math.AP↗