WEAK AND STRONG SOLUTIONS OF THE COMPLEX GINZBURG-LANDAU EQUATION

The generalized complex Ginzburg-Landau equation, partial derivative(t )A = RA + (1 + inu)DELTAA - (1 + imu)\A\(2sigma)A, has been proposed a nd 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 nonlin ear Schrodinger equation possessing solutions that form localized sing ularities in finite time. In this paper we investigate existence and r egularity of solutions to this equation subject to periodic boundary c onditions in various spatial dimensions. Appropriately defined weak so lutions are established globally in time, and unique strong solutions are found locally. A new collection of a priori estimates are presente d, and we discuss the relationship of our results for the complex Ginz burg-Landau equation to analogous issues for fluid turbulence describe d by the incompressible Navier-Stokes equations.