【作者】 李心；

【导师】 孙春友；

【作者基本信息】 兰州大学 ， 基础数学， 2013， 硕士

【摘要】 本文主要研究如下分数阶复Ginzburg-Landau方程在周期边值情形下解的适定性,以及方程弱解的动力学行为：这里v,μ,R是正常数,α∈(1/2,1],N≤4α,σ∈[0,∞).本文通过运用Fourier-Galerkin方法证明了弱解的存在性.为了得到在没有μ和α限制条件下Hα(TN)中全局吸引子的存在性,我们建立了一系列先验估计.特别的,在给出对任意2≤p<∞,‖u(t)‖LP≤M这一估计的基础上.本文又建立了连续性定理,即映射(?)(s)：L2(TN)→Hα(TN)(s>0)是连续的.最后证明了方程弱解所诱导的算子半群在Hα中全局吸引子的存在性.更多还原

【Abstract】 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)→Hα(TN)(s>0), and finally the necessary asymptotical compact-ness for the existence of a global attractor in Hα(TN) is obtained.更多还原