In this paper, the authors consider complex Ginzburg-Landau equation(CGL) in three spatial dimensions ut=ρu+(1+iγ）△u-(1+iμ）|u|^2σu+f,where u is an unknown complex-value function defined in 3+ 1 dimensional spac...In this paper, the authors consider complex Ginzburg-Landau equation(CGL) in three spatial dimensions ut=ρu+(1+iγ）△u-(1+iμ）|u|^2σu+f,where u is an unknown complex-value function defined in 3+ 1 dimensional space-time R^3+1，△ is a Laplacian in R^3, ρ > 0, γ μ are real parameters, Ω∈R^3 is a bounded domain. By using the method of Galeerkin and Faedo-Schauder fix point theorem we prove the existence of approximate solution uN of the problem. By establishing the uniform boundedness of the norm ||uN|| and the standard compactness arguments, the convergence of the approximate solutions is considered. Moreover, the existence of the periodic solution is obtained.展开更多
In this paper,we consider the complex Ginzburg-Landau equation (CGL) in three spatial dimensions ut=ρu+(1+iγ)△u-1+iμ—|u|^2σu,(1) u(0,x)=u0(x),(2) where u is an unknown complex-value function defined in 3+1 di...In this paper,we consider the complex Ginzburg-Landau equation (CGL) in three spatial dimensions ut=ρu+(1+iγ)△u-1+iμ—|u|^2σu,(1) u(0,x)=u0(x),(2) where u is an unknown complex-value function defined in 3+1 dimensional space-time R^3+1,△ is a Laplacian in R^3,ρ>0,γ,μ are real parameters,Ω∈R^3 is a bounded domain,We show that the semigroup S(t) associated with the problem(1),(2) satisfies Lipschitz continuity and the squeezing property for the initial-value problem(1),(2) with the initial-value condition belonging to H^2(Ω）,therefore we obtain the existence of exponential attractor.展开更多
文摘In this paper, the authors consider complex Ginzburg-Landau equation(CGL) in three spatial dimensions ut=ρu+(1+iγ）△u-(1+iμ）|u|^2σu+f,where u is an unknown complex-value function defined in 3+ 1 dimensional space-time R^3+1，△ is a Laplacian in R^3, ρ > 0, γ μ are real parameters, Ω∈R^3 is a bounded domain. By using the method of Galeerkin and Faedo-Schauder fix point theorem we prove the existence of approximate solution uN of the problem. By establishing the uniform boundedness of the norm ||uN|| and the standard compactness arguments, the convergence of the approximate solutions is considered. Moreover, the existence of the periodic solution is obtained.
文摘In this paper,we consider the complex Ginzburg-Landau equation (CGL) in three spatial dimensions ut=ρu+(1+iγ)△u-1+iμ—|u|^2σu,(1) u(0,x)=u0(x),(2) where u is an unknown complex-value function defined in 3+1 dimensional space-time R^3+1,△ is a Laplacian in R^3,ρ>0,γ,μ are real parameters,Ω∈R^3 is a bounded domain,We show that the semigroup S(t) associated with the problem(1),(2) satisfies Lipschitz continuity and the squeezing property for the initial-value problem(1),(2) with the initial-value condition belonging to H^2(Ω）,therefore we obtain the existence of exponential attractor.
