##
**Randomly forced CGL equation: stationary measures and the inviscid limit.**
*(English)*
Zbl 1047.35061

The authors consider the randomly forced complex Ginzburg-Landau (CGL) equation
\[
\dot u-(\nu+ i)\Delta u+ i\lambda| u|^2u= \eta(t,x),\quad u= u(t,x),\tag{1}
\]
where \(0<\nu\leq 1\), \(\lambda>0\), \(\dim x\leq 4\) and either \(x\in\Omega\Subset\mathbb R^d\), or \(x\in\mathbb R^d\). In the first case (1) is supplemented with the Dirichlet boundary condition, in the second case in (1) the odd-periodic boundary conditions are imposed. The force in (1) is a random field white in time and sufficiently smooth in \(x\). Under some suitable assumptions on the data of (1), the authors show that (1) has a unique solution defined for \(t\geq 0\) and derive various a priori estimates. Based on this a priori estimates they prove that equation (1) has stationary in time solutions. The main results of the authors are devoted to the study the CGL equation perturbed by a random force of order \(\sqrt{\nu}\):
\[
\dot u-(\nu+ i)\Delta u+ i\lambda| u|^2 u= \sqrt{\nu} \eta(t,x)
\]
and deals with the behaviour of stationary solutions as the \(\nu\to 0\).

Reviewer: Messoud A. Efendiev (Berlin)

### MSC:

35K55 | Nonlinear parabolic equations |

35B45 | A priori estimates in context of PDEs |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35R60 | PDEs with randomness, stochastic partial differential equations |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76M35 | Stochastic analysis applied to problems in fluid mechanics |