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Weak and strong solutions of the complex Ginzburg-Landau equation. (English) Zbl 0810.35119
Summary: The generalized complex Ginzburg-Landau equation, \[ \partial_ t A= RA+ (1+ i\nu) \Delta A-(1+ i\mu)| A|^{2\sigma}A, \] has been proposed and studied as a model for “turbulent” dynamics in nonlinear partial differential equations. It is a particularly interesting model in this respect because it is a dissipative version of the Hamiltonian nonlinear Schrödinger equation possessing solutions that form localized singularities in finite time.
In this paper we investigate existence and regularity of solutions to this equation subject to periodic boundary conditions in various spatial dimensions. Appropriately defined weak solutions are established globally in time, and unique strong solutions are found locally. A new collection of a priori estimates are presented, and we discuss the relationship of our results for the complex Ginzburg-Landau equation to analogous issues for fluid turbulence described by the incompressible Navier-Stokes equations.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q35 PDEs in connection with fluid mechanics
76F05 Isotropic turbulence; homogeneous turbulence
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