A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations - Archive ouverte HAL
Article Dans Une Revue Confluentes Mathematici Année : 2011

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

Romain Joly
Geneviève Raugel
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Résumé

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.
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Dates et versions

hal-00482562 , version 1 (10-05-2010)
hal-00482562 , version 2 (14-01-2011)
hal-00482562 , version 3 (27-08-2018)

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Citer

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⟩
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