Blow-up results of virial type for Zakharov equations
type: Item
instance of: scholarly article
title: Blow-up results of virial type for Zakharov equations
zbMATH Open document ID: 0858.35117
DOI: 10.1007/BF02102415
author: Frank Merle
published in: Communications in Mathematical Physics
publication date: 1997-03-23T00:00:00Z
review text: The author considers the collapse (blow-up) problem, i.e., appearance of a singularity in the solution to the Cauchy problem, for the multidimensional Zakharov's system: \[ iu_t+\Delta u-nu=0, \qquad c_0^{-2}n_{tt}= \Delta(n+|u|^2), \] where \(u\) and \(n\) are complex and real variables, and \(\Delta\) stands for the Laplacian. Numerical simulations of this system strongly suggest that a singularity develops in a finite time from a generic initial condition. However, the classical virial identities, that help to rigorously prove existence of the collapse in the nonlinear Schrödinger equation, which is the limit case of this system when \(c^2_0\to\infty\), does not allow to obtain the proof for the present system, because the corresponding virial quantity lacks the necessary positive definiteness. Therefore, instead of treating the general case, the author concentrates on the particular case of the radially symmetric initial condition. In this special case, the virial identity, together with the Hamiltonian representation of the system, allows him to produce a proof of existence of the singularity in the generic solution. However, although it seems very feasible that the singularity will always develop in a finite time, the available proof formally admits a possibility that the singularity may also set in at \(t\to\infty\).
reviewed by: B. A. Malomed
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 878044
zbMATH Keywords: blow-up problem
zbMATH Keywords: multidimensional Zakharov's system
zbMATH Keywords: radially symmetric initial condition
zbMATH Keywords: virial identity
zbMATH Keywords: Hamiltonian representation
zbMATH Keywords: existence of the singularity
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02102415
OpenAlex ID: W2067951748
description: scientific article; zbMATH DE number 878044