The Global Attractor Of The Fractional Complex Ginzburg-Landau Equation

The purpose of this dissertation is to study the well-posedness and dynamics of the solutions of the following complex fractional Ginzburg-Landau equation with periodic boundary: where v, μ and R are positive constants, α∈(1/2,1],N≤4α and σ∈[0,∞) is arbitrary.The existence and uniqueness of weak solution in phase space L2(TN) are ob-tained by the usual Fourier-Galerkin method. To obtain the existence of a global attractor in Hα(TN) without the additional assumptions on μ and σ as that re-quired in references, some new a priori estimates are necessary and established. Precisely, we establish firstly some new LP-type a priori estimates for the weak so-lution by the Alikakos iteration, which allow us to obtain further some continuity of S(s):L2(TN)â†'Hα(TN)(s>0), and finally the necessary asymptotical compact-ness for the existence of a global attractor in Hα(TN) is obtained.