The Cauchy problem for the system \(\partial_ tz - \partial_ x(\bar z^ y) = 0\)
type: Item
instance of: scholarly article
title: The Cauchy problem for the system \(\partial_ tz - \partial_ x(\bar z^ y) = 0\)
zbMATH Open document ID: 0819.35093
author: Hermano Frid
author: M. M. Santos
published in: Journal of Differential Equations
publication date: 1994-10-10T00:00:00Z
review text: The authors consider the Cauchy problem \[ \partial_ t z- \partial_ x \overline {z}^ \gamma=0, \;\;z=u+iv, \quad -\infty<x<\infty, \;\;t>0,\qquad z(x,0)= z_ 0(x), \quad -\infty<x<\infty, \tag{1} \] where \(\gamma\) is a constant, \(1<\gamma<2\). The main result is the existence of a global solution if \(z_ 0\in L^ 2\cap L^ \infty\), and \(z_ 0\) takes values only in a closed sector of the plane with vertex at the origin and angle \(\pi/ (\gamma+1)\). The proof using compensated compactness consists of establishing bounds in \(L^ 2\) (rather than the usual \(L^ \infty\) bounds) for the corresponding viscous system, and then using generalized Young measures. The paper uses and extends ideas from earlier work of DiPerna, Majda and Schonbek on generalized Young measures, and more recent work of Serre and Kan specifically aimed at systems of two equations.
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L60
zbMATH DE Number: 635888
zbMATH Keywords: compensated compactness
zbMATH Keywords: generalized Young measures
reviewed by: Michael Shearer
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1085
OpenAlex ID: W1997875084
DOI: 10.1006/JDEQ.1994.1085
description: scientific article; zbMATH DE number 635888