Sharp estimates for analytic pseudodifferential operators and application to Cauchy problems
type: Item
instance of: scholarly article
title: Sharp estimates for analytic pseudodifferential operators and application to Cauchy problems
zbMATH Open document ID: 0535.35082
DOI: 10.1016/0022-0396(83)90051-7
published in: Journal of Differential Equations
publication date: 1983-01-01T00:00:00Z
review text: This paper deals with sharp estimates for analytic pseudodifferential operators in complex tubular neighbourhoods of a compact analytic manifold \(\Gamma\). The operators mentioned above are defined by means of their distribution kernels and \(L^{\infty}\) and Hölder estimates in the complex domain are established. In order to study some classes of nonlinear Cauchy problems for analytic pseudodifferential operators the authors use Nishida's theorem in the framework of decreasing scale of Banach spaces as well as the inequalities just obtained. Results concerning existence and uniqueness of the analytic solutions are proved. Theorem III.1 can be considered as a specific generalization of the nonlinear Cauchy-Kovalevsky theorem for analytic pseudodifferential operators P. To illustrate the main statements consider the following simple example (III.5) \(\partial u_ 1/\partial t = f(t,x,u_1,u_2)\), \(\partial u_ 2/\partial t=Pf(t,x,u_ 1,u_ 2), u_ 1(0,x)=u_ 0(x),u_ 2(0,x)=Pu_ 0(x),\) where \(f_ 1,u_ 0\) are holomorphic functions. Then (III.5) possesses a unique analytic solution. - The authors study Euler's equations for nonviscous incompressible fluid at the end of the paper.
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 35S10
Mathematics Subject Classification ID: 35A10
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 3849750
zbMATH Keywords: sharp estimates
zbMATH Keywords: analytic pseudodifferential operators
zbMATH Keywords: Cauchy problems
zbMATH Keywords: compact analytic manifold
zbMATH Keywords: Hölder estimates
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: Euler's equations
zbMATH Keywords: nonviscous incompressible fluid
author: Mohamed Salah Baouendi
author: Charles Goulaouic
reviewed by: Peter R. Popivanov
MaRDI profile type: Publication
full work available at URL: 0022-0396(83)90051-7
OpenAlex ID: W2037515667
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cites work: A note on a theorem of Nirenberg
description: scientific article; zbMATH DE number 3849750