Séminaire d'Analyse

Inviscid limit for the derivative Ginzburg–Landau equation with small data in higher spatial dimensions
Baoxiang Wang (Université de Peking (Chine))
Thursday 13 January 2011 11:15 -  Tours -  Salle 2290 (Bât E2)

Résumé :
Applying the frequency-uniform decomposition technique, we study the Cauchy problem for derivative Ginzburg--Landau equation $$ u_t= (\nu + i)\triangle u+\overrightarrow{\lambda_1}\cdot\nabla(|u|^2u) + (\overrightarrow{\lambda_2}\cdot\nabla u)|u|^2+ \alpha |u|^{2\delta}u $$, where $\delta\in \mathbb{N}, \overrightarrow{\lambda_1}, \overrightarrow{\lambda_2}$ are complex constant vectors, $ \nu \in [0, 1]$, $\alpha \in \mathbb{C}$. For $n\geq 3$, we show that it is uniformly global wellposed for all $\nu \in [0,1]$ if initial data $u_0$ in Sobolev spaces $H^{s+n/2}$ ($s>3$) and $\|u_0\|_{L^2}$ is small enough. Moreover, we show that its solution will converge to that of the derivative Schr\"odinger equation in $C(0,T; L^2)$ if $\nu\rightarrow 0$ and $u_0$ in $M^s_{2,1}$ or $H^{s+n/2}$ with $s>4$. For $n= 2$, we obtain the well-posedness results and inviscid limit with the Cauchy data in $M_{1, 1}^{s}$ ($s>3$) and $\|u_0\|_{L^1} \ll 1$.

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