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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 \(\nu\to 0\).

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
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