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

Romain Joly (1) , Geneviève Raugel (2)

1 IF - Institut Fourier
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations $\dot y(t)=g(y(t))$ on $\Rm^d$ and those of the parabolic equations $\dot u=\Delta u +f(x,u,\grad u)$ 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 $\Omega=(0,1)$, $\Omega=\Tm^1$ 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.

Mots clés: Kupka-Smale property; genericity; finite- and infinite-dimensional dynamical systems; vector fields; scalar parabolic equation; finite-and infinite-dimensional dynamical systems; vector fields; scalar; parabolic equation; Kupka-Smale property; genericity; AMS Subject Classification: 35-02; 37-02; 35B05; 35B41; 35K57; 37C10; 37C20

Domaines: Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]

Romain Joly, Geneviève Raugel. A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations. Confluentes Mathematici, 2011, 3 (3), p. 471-493. ⟨10.1142/S1793744211000369⟩. ⟨hal-00482562v3⟩