Existence of Gevrey approximate solutions for certain systems of linear vector fields applied to involutive systems of first-order nonlinear PDEs
type: Item
instance of: scholarly article
title: Existence of Gevrey approximate solutions for certain systems of linear vector fields applied to involutive systems of first-order nonlinear PDEs
zbMATH Open document ID: 1228.35278
author: Rafael F. Barostichi
author: Gerson Petronilho
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-07-18T00:00:00Z
review text: The authors consider a \(G^s\)-involutive structure of rank \(n\) in a manifold of dimension \(m+n\). In local coordinates: \[ L_j= \partial/\partial t_j+ \sum^m_{k=1} a_{jk}(x, t)\partial/\partial x_k,\quad j= 1,\dots, n, \] where \(a_{jk}\) belong to the Gevrey space \(G^s\), \(s> 1\), and \([L_i,L_j]= 0\) for \(0\leq i\), \(j\leq n\). A formal solution \(u(x,t)\in G^s\) is constructed for the local Gevrey problem \(u(x,0)= f(x)\in G^s\), \(L_ju= 0\), \(j= 1,\dots, n\). Other interesting results concern the related nonlinear systems: \[ u_{t_j}= F_j(x,t,u,u_x),\quad j= 1,\dots, n, \] for which the authors prove results of propagation of Gevrey wave front sets. For a counterpart of these results in the analytic category see [\textit{N. Hanges} and \textit{F. Treves}, Trans. Am. Math. Soc. 331, No. 2, 627--638 (1992; Zbl 0758.35018)].
reviewed by: Luigi Rodino
Mathematics Subject Classification ID: 35S05
zbMATH DE Number: 5925553
zbMATH Keywords: Gevrey wave front sets
zbMATH Keywords: overdetermined systems
zbMATH Keywords: involutive structures
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.04.041
OpenAlex ID: W2051443583
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DOI: 10.1016/J.JMAA.2011.04.041
description: scientific article; zbMATH DE number 5925553