scholarly journals Inviscid limit for the energy-critical complex Ginzburg–Landau equation

2008 ◽  
Vol 255 (3) ◽  
pp. 681-725 ◽  
Author(s):  
Chunyan Huang ◽  
Baoxiang Wang
Keyword(s):  
Landau Equation ◽  
Inviscid Limit ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation

The Inviscid Limit of the Fractional Complex Ginzburg–Landau Equation

2016 ◽  
Vol 17 (6) ◽  
pp. 333-341
Author(s):  
Lijun Wang ◽  
Jingna Li ◽  
Li Xia
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Convergence Rate ◽  
Schrodinger Equation ◽  
Landau Equation ◽  
Limit Behavior ◽  
Inviscid Limit ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation

AbstractIn this paper, the inviscid limit behavior of solution of the fractional complex Ginzburg–Landau (FCGL) equation$${\partial _t}u + (a + i\nu){\Lambda ^{2\alpha}}u + (b + i\mu){\left| u \right|^{2\sigma}}u = 0, \quad (x, t) \in {{\Cal T}^n} \times (0, \infty)$$is considered. It is shown that the solution of the FCGL equation converges to the solution of nonlinear fractional complex Schrödinger equation, while the initial data${u_0}$is taken in${L^2}, $${H^\alpha}$, and${L^{2\sigma + 2}}$as$a,\, b$tends to zero, and the convergence rate is also obtained.

Well-posedness for the nonlinear fractional Schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg-Landau equation

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Boling Guo ◽  
Zhaohui Huo
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Landau Equation ◽  
Inviscid Limit ◽  
Well Posedness ◽  
Ginzburg Landau ◽  
Fractional Schrödinger Equation ◽  
Ginzburg Landau Equation ◽  
Nonlinear Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation

AbstractThe well-posedness for the Cauchy problem of the nonlinear fractional Schrödinger equation $u_t + i( - \Delta )^\alpha u + i|u|^2 u = 0,(x,t) \in \mathbb{R}^n \times \mathbb{R},\frac{1} {2} < \alpha < 1 $ is considered. The local well-posedness in subcritical space H s with s > n/2 -α is obtained. Moreover, the inviscid limit behavior of solution for the fractional Ginzburg-Landau equation $u_t + (\nu + i)( - \Delta )^\alpha u + i|u|^2 u = 0$ is also considered. It is shown that the solution of the fractional Ginzburg-Landau equation converges to the solution of nonlinear fractional Schrödinger equation in the natural space C([0, T];H)s) with s > n/2 — α if ν tends to zero.

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