Localized estimates and Cauchy problem for the logarithmic complex Ginzburg–Landau equation

1997 ◽  
Vol 38 (5) ◽  
pp. 2475-2482 ◽  
Author(s):  
J. Ginibre ◽  
G. Velo
Keyword(s):  
Cauchy Problem ◽  
Landau Equation ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation

scholarly journals On a coupled system of a Ginzburg–Landau equation with a quasilinear conservation law

2019 ◽  
Vol 22 (07) ◽  
pp. 1950054
Author(s):  
João-Paulo Dias ◽  
Filipe Oliveira ◽  
Hugo Tavares
Keyword(s):  
Cauchy Problem ◽  
Fluid Flow ◽  
Conservation Law ◽  
Landau Equation ◽  
Coupled System ◽  
Global Weak Solutions ◽  
Ginzburg Landau ◽  
Wave Solutions ◽  
Ginzburg Landau Equation ◽  
The World

We study a coupled system of a complex Ginzburg–Landau equation with a quasilinear conservation law [Formula: see text] which can describe the interaction between a laser beam and a fluid flow (see [I.-S. Aranson and L. Kramer, The world of the complex Ginzburg–Landau equation, Rev. Mod. Phys. 74 (2002) 99–143]). We prove the existence of a local in time strong solution for the associated Cauchy problem and, for a certain class of flux functions, the existence of global weak solutions. Furthermore, we prove the existence of standing wave solutions of the form [Formula: see text] in several cases.

Cauchy problem for the Ginzburg-Landau equation for the superconductivity model

1997 ◽  
Vol 127 (6) ◽  
pp. 1181-1192 ◽  
Author(s):  
Boling Guo ◽  
Guangwei Yuan
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Smooth Solution ◽  
Landau Equation ◽  
Ginzburg Landau ◽  
Landau Model ◽  
Lorentz Gauge ◽  
Global Smooth Solution ◽  
Ginzburg Landau Equation

In this paper, the existence and uniqueness of the global smooth solution are proved for an evolutionary Ginzburg–Landau model for superconductivity under the Coulomb and Lorentz gauge.

scholarly journals On the Cauchy problem for the 1+2 complex Ginzburg-Landau equation

1995 ◽  
Vol 36 (3) ◽  
pp. 313-324 ◽  
Author(s):  
Charles Bu
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Analytical Methods ◽  
Critical Case ◽  
Landau Equation ◽  
Local Existence ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Spatial Dimensions ◽  
The Cauchy Problem

AbstractWe present analytical methods to investigate the Cauchy problem for the complex Ginzburg-Landau equation u1 = (v + iα)Δu − (κ + iβ) |u|2qu + γu in 2 spatial dimensions (here all parameters are real). We first obtain the local existence for v > 0, κ ≥ 0. Global existence is established in the critical case q = 1. In addition, we prove the global existence when .

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