Regularity results of solution uniform in time for complex Ginzburg-Landau equation

Abstract

We provide the H²-regularity result of the solution ψ and its first- order time derivative ψ_t and the second-order time derivative ψ_tt for the complex Ginzburg-Landau equation with the Dirichlet or Neumann boundary conditions. The analysis shows that these regularity results are uniform when t tends to ∞ and 0 and are dependent of the powers of ε⁻¹.

Keywords

Complex Ginzburg-Landau equation (CGL) / H²-regularity / sharp a prioriestimates

References

[1] Adams R A. Sobolev Space. New York: Academic Press, 1975

[2] Bao W Z, Tang Q L. Numerical study of quantized vortex interaction in the Ginzburg- Landau equation on bounded domains. Commun Comput Phys, 2013, 14(3): 819–850

[3] Bao W Z, Tang Q L. Numerical study of quantized vortex interactions in the non- linear Schrödinger equation on bounded domains. Multiscale Model Simul, 2014, 12(2): 411–439

[4] Cross C, Hohenberg C. Pattern formation outside of equilibrium. Rev Modern Phys, 1993, 65: 851–1112

[5] Feng X B, He Y N, Liu C. Analysis of finite element approximations of a phase field model for two-phase fluids. Math Comp, 2007, 76(258): 539–571

[6] Ginzburg L, Landau D. On the theory of superconductivity. Zh Eksp Teor Fiz, 1950, 20: 1064–1082 (in Russian); in: ter Haar D, ed. Collected Papers of L. D. Landau. Oxford: Pergamon Press, 1965, 546–568

[7] Jiang W, Tang Q L. Numerical study of quantized vortex interaction in complex Ginzburg-Landau equation on bounded domains. Appl Math Comput, 2013, 222: 210–230

[8] Lin F H. Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds. Comm Pure Appl Math, 1998, 51(4): 385–441

[9] Nishiura Y. Far-from-Equilibrium Dynamics. Transl Math Monogr, Vol 209. Providence: Amer Math Soc, 2002

[10] Okazawa N, Yokota T. Global existence and smoothing effect for the complex Ginzburg- Landau equation with p-Laplacian. J Differential Equations, 2002, 182(2): 541–576

[11] Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Appl Math Sci, Vol 68. New York: Springer-Verlag, 1988

[12] Yang Y S. On the Ginzburg-Landau wave equation. Bull Lond Math Soc, 1990, 22(2): 167–170

[13] Zhang Q G, Li Y N, Su M L. The local and global existence of solutions for a time fractional complex Ginzburg-Landau equation. J Math Anal Appl, 2019, 469(1): 16–43

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