The aim of this article is to consider a three-dimensional Cauchy problem for the parabolic-
elliptic system arising from biological transport networks. For such problem, we first establish …

The construction of loss functions presents a major challenge in data-driven modeling
involving weak-form operators in PDEs and gradient flows, particularly due to the need to …

We analytically and numerically study a fourth-order PDE modeling rough crystal surface
diffusion on the macroscopic level. We discuss existence of solutions globally in time and …

In this paper we prove the global existence of a strong solution to the initial boundary value
problem for the exponential partial differential equation∂ tu− Δ e− Δ u+ e− Δ u− 1= 0. The …

As a counterpoint to recent numerical methods for crystal surface evolution, which agree
well with microscopic dynamics but suffer from significant stiffness that prevents simulation …

We investigate the local equilibrium (LE) distribution of a crystal surface jump process as it
approaches its hydrodynamic (continuum) limit in a nonstandard scaling regime introduced …

We derive the PDE governing the hydrodynamic limit of a Metropolis rate crystal surface
height process in the" rough scaling" regime introduced by Marzuola and Weare. The PDE …

We derive the PDE governing the hydrodynamic limit of a Metropolis rate crystal surface
height process in the “rough scaling” regime introduced by Marzuola and Weare. The PDE …

We study regularity properties of weak solutions to the boundary value problem for the
equation− Δρ+ au= f in a bounded domain ${\Omega}\subset {\mathbb {R}}^{N} $, where …

In Part I, we derive an asymptotic expansion for the log likelihood of Gaussian mixture
models (GMMs) with equal covariance matrices in the low signal-to-noise regime. The …