José M. Valdovinos | Searcharxiv

arXiv2026

Generalized Gaussian Estimates and Local Limit Theorems for Discrete Convolution Powers of Complex Functions: The $d$-dimensional case

We establish generalized Gaussian estimates and local limit theorems with cumulants, up to any order of accuracy, with sharp Gaussian-type error for the convolution powers of certain complex-valued functions on $\mathbb{Z}^d$. These global space-time estimates/error are written in terms of the Legendre-Fenchel transforms of positive-homogeneous polynomials and are mirrored by estimates satisfied by the heat kernels associated to a related class of partial differential operators. The results obtained here enjoy applications to the analysis and stability of numerical difference schemes to partial differential equations. This work extends several recent results, pertaining to one and several dimensions, of P. Diaconis, L. Saloff-Coste, J.-F. Coulombel, G. Faye, L. Coeuret, and the second author.

math.CA↗

arXiv2023

Global decay of perturbations of equilibrium states for one-dimensional heat conducting compressible fluids of Korteweg type

This paper studies the one dimensional Navier-Stokes-Fourier-Korteweg system of equations describing the evolution of a heat-conducting compressible fluid that exhibits viscosity and capillarity. The main goal of the present analysis is to examine the dissipative structure of the system and to prove the global existence and the asymptotic decay of perturbations of equilibrium states. For that purpose, a novel nonlinear change of perturbed state variables, which takes into account that the conserved quantities contain density gradients, is introduced. These new perturbation variables satisfy a partially symmetric system whose linearization fulfills the generalized genuine coupling condition of Humpherys (J. Hyperbolic Differ. Equ. 2, 2005, no. 4, 963-974) for higher order systems. It is shown that the linearized system is symbol symmetrizable and an appropriate compensating matrix is constructed. This procedure allows to obtain linear decay rates which underlie a dissipative mechanism of regularity-gain type. This linear dissipative structure implies, in turn, the global decay of small perturbations to constant equilibrium states as solutions to the full nonlinear system.

math.AP↗