Gevrey micro-regularity for solutions to first order nonlinear PDE
type: Item
instance of: scholarly article
title: Gevrey micro-regularity for solutions to first order nonlinear PDE
zbMATH Open document ID: 1180.35024
author: Gerson Petronilho
author: Rafael F. Barostichi
published in: Journal of Differential Equations
publication date: 2009-09-28T00:00:00Z
review text: In this interesting paper, the authors study the microlocal Gevrey regularity of \(C^2\) solutions to a first-order nonlinear PDE. More precisely, let \(f=f(x,t,\zeta_0,\zeta)\) be Gevrey of order \(s>1\) and holomorphic in the variables \((\zeta_0,\zeta)\). Consider a solution \(u\in C^2\) to \(u_t=f(x,t,u,u_x)\). Let \(L^u\) be the linearized operator \[ L^u=\frac{\partial}{\partial t}-\sum_{j=1}^m\frac{\partial f}{\partial\zeta_j} (x,t,u,u_x)\frac{\partial}{\partial x_j}. \] It is known after \textit{N. Hanges} and \textit{F. Trèves} [Trans. Am. Math. Soc. 331, No. 2, 627--638 (1992; Zbl 0758.35018)], that the analytic wave-front set of such \(u\) is contained in the characteristic set \(\mathrm{Char}(L^u)\) of \(L^u\). More recently, \textit{S. Berhanu} [Ann. Inst. Fourier 59, No. 4, 1267--1290 (2009; Zbl 1195.35011)] has given a precise result about the location in \(\mathrm{Char}(L^u)\) of the analytic wave-front set of a \(C^2\) (and \(C^3\)) solution \(u\). The authors show here that in the Gevrey case, any given \(C^2\) solution has Gevrey wave-front set contained in \(\mathrm{Char}(L^u)\). They achieve this result through the study of the notion of \textit{Gevrey approximate solutions}, a tool that very likely will be useful to deal with a broader range of situations.
reviewed by: Alberto Parmeggiani
Mathematics Subject Classification ID: 35A27
Mathematics Subject Classification ID: 35A18
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35F20
zbMATH DE Number: 5608032
zbMATH Keywords: Gevrey approximate solutions
zbMATH Keywords: Gevrey wave-front set
zbMATH Keywords: nonlinear PDE of first order
MaRDI profile type: Publication
full work available at URL: j.jde.2009.06.021
OpenAlex ID: W1968212862
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DOI: 10.1016/J.JDE.2009.06.021
description: scientific article; zbMATH DE number 5608032