Operator theory in the complex Ginzburg-Landau equation

Abstract:

Strong well-posedness for the complex Ginzburg-Landau equation \[ \partial u/\partial t + (\lambda +i \alpha )(-\Delta )u + (\kappa +i \beta )|u|^{q-2}u-\gamma u = 0 \] is discussed from the viewpoint of operator theory. It is concluded that the solution operators from $L^{2}(\Omega )$, $\Omega \subset \mathbb {R}^{N}$, into itself form a semigroup of quasi-contractions when $\kappa ^{-1}|\beta | \le c_{q}^{-1} := 2\sqrt {q-1}/|q-2|$ (without any upper bound on $q \ge 2$), and a non-contraction semigroup of Lipschitz operators when $\kappa ^{-1}|\beta | > c_{q}^{-1}$ ($2 \le q \le 2+4/N$). The assertion is proved by energy methods based on monotonicity methods. Also compactness methods are valid for the Cauchy problem when the initial value belongs to $H^{1}(\mathbb {R}^N) \cap L^{q}(\mathbb {R}^N)$ in addition to the restriction $(\alpha /\lambda ,\beta /\kappa ) \in CGL(c_{q}^{-1})$; in this case solutions are unique under sub-critical condition: $q \in [2,2+4/(N-2)_{+}]$.