A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations

TL;DR

This paper explores a deep correspondence between the dynamics of vector field differential equations and scalar parabolic equations across various dimensions and domains, offering insights for future research in dynamical systems.

Contribution

It establishes a detailed analogy between finite-dimensional vector field dynamics and infinite-dimensional parabolic PDEs in different spatial settings.

Findings

Identifies similarities in dynamics for 1D, 2D, and higher dimensions.

Provides a framework linking vector field flows to parabolic PDE behavior.

Suggests this correspondence can guide future research in dynamical systems.

Abstract

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations $\dot y(t)=g(y(t))$ on $\mathbb{R}^d$ and those of the parabolic equations $\dot u=\Delta u +f(x,u,\nabla u)$ on a bounded domain $\Omega$. We give details on the similarities of these dynamics in the cases $d=1$, $d=2$ and $d\geq 3$ and in the corresponding cases $\Omega=(0,1)$, $\Omega=\mathbb{T}^1$ and dim($\Omega$)$\geq 2$ respectively. In addition to the beauty of such a correspondence, this could serve as a guideline for future research on the dynamics of parabolic equations.