A STRIKING CORRESPONDENCE BETWEEN THE DYNAMICS GENERATED BY THE VECTOR FIELDS AND BY THE SCALAR PARABOLIC EQUATIONS

Confluentes Mathematici

RAUGEL, GENEVIÈVE¹  JOLY, ROMAIN² 

[1] CNRS, Laboratoire de Mathématiques d'Orsay, Orsay Cedex, F-91405, France;Institut Fourier, UMR CNRS 5582, Université de Grenoble, B.P. 74, F-38402 Saint-Martin-d'Hères, France

关键词: Finite- and infinite-dimensional dynamical systems;    vector fields;    scalar parabolic equation;    Kupka–Smale property;    genericity;   

DOI  :  10.1142/S1793744211000369

学科分类：数学（综合）

来源: World Scientific Publishing Co. Pte. Ltd.

【 摘 要 】

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations ẏ(t) = g(y(t)) on ℝd and those of the parabolic equations $\dot u = \Delta u + f(x, u, \nabla u)$ on a bounded domain Ω. We give details on the similarities of these dynamics in the cases d = 1, d = 2 and d ≥ 3 and in the corresponding cases Ω = (0, 1), Ω = 𝕋1 and dim(Ω) ≥ 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.

【 授权许可】

Unknown