Uniqueness and inviscid limits of solutions for the complex Ginzburg-Landau equation in a two-dimensional domain
type: Item
instance of: scholarly article
title: Uniqueness and inviscid limits of solutions for the complex Ginzburg-Landau equation in a two-dimensional domain
zbMATH Open document ID: 1072.35171
author: Takayoshi Ogawa
author: Tomomi Yokota
published in: Communications in Mathematical Physics
publication date: 2004-02-17T00:00:00Z
review text: The authors discuss the uniqueness and inviscid limit of weak solutions to the initial-boundary value problem (I-BVP) for the complex Ginzburg-Landau equation \[ \begin{matrix} \partial_tu-(\lambda+i\alpha)\Delta u+(a+i\beta)|u|^{p-1}u-\gamma u=0,\quad t>0,\quad x\in\Omega;\\ u(t,x)=0,\quad t>0,\quad x\in\partial\Omega;\qquad u(0,x)=u_0(x), \quad t=0,\quad x\in\Omega, \end{matrix} \] \(\Omega\subset \mathbb R^2\) is not necessary having compact smooth boundaries, \(p>1\), \(\lambda\geq 0\), \(a\), \(\alpha\), \(\beta\), \(\gamma\) are real constants. The first aim of the article is to prove the uniqueness of weak solutions of the problem without the restrictive condition \(|\beta|\leq\frac{2\sqrt{p}}{p-1}a\) of previous authors, provided \(p<3\), which includes the physically important situation under \(n=2\). Two kinds of uniqueness theorem are proved for different classes of weak solutions. The first is for a class with \(L^2\) initial data, the second is for a class with \(H^1\) initial data. The other purpose of the article is to discuss the inviscid limit for the I-BVP in a general domain \(\Omega\subset \mathbb R^2\). The case \(\lambda=0\) gives the I-BVP for the nonlinear Schrödinger equation.
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 82D55
Mathematics Subject Classification ID: 35D10
zbMATH DE Number: 2044318
zbMATH Keywords: Ginzburg-Landau equation
zbMATH Keywords: initial-boundary value problem
zbMATH Keywords: uniqueness
zbMATH Keywords: inviscid limit
zbMATH Keywords: weak solutions
zbMATH Keywords: nonlinear Schrödinger equation
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
full work available at URL: s00220-003-1004-4
OpenAlex ID: W2079088752
DOI: 10.1007/S00220-003-1004-4
Recommended article: The inviscid limit for the complex Ginzburg--Landau equation
Recommended article: Inviscid limits of the complex Ginzburg-Landau equation
Recommended article: The inviscid limit of the complex Ginzburg-Landau equation
Recommended article: The inviscid limit of the derivative complex Ginzburg-Landau equation
Recommended article: Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation
Recommended article: Inviscid limit for the energy-critical complex Ginzburg-Landau equation
Recommended article: On the Cauchy problem for the 1+2 complex Ginzburg-Landau equation
Recommended article: The limit behavior of solutions for the Cauchy problem of the complex Ginzburg‐Landau equation
Recommended article: Q4454827
Recommended article: The inviscid limit of the fractional complex Ginzburg-Landau equation
description: scientific article; zbMATH DE number 2044318