Publication Cover
Applicable Analysis
An International Journal
Volume 98, 2019 - Issue 14
313
Views
7
CrossRef citations to date
0
Altmetric
Articles

Well-posedness of the fractional Ginzburg–Landau equation

, &
Pages 2545-2558 | Received 18 Oct 2017, Accepted 11 Apr 2018, Published online: 30 May 2018

References

  • Cazenave T , Dickstein F , Weissler FB . Finite-time blowup for a complex Ginzburg-Landau equation. SIAM J Math Anal. 2013;45:244–266.
  • Guo B , Huang H , Jiang M . Ginzburg-Landau equation. Chinese ed. Beijing: Science Press; 2002.
  • Mitidieri E , Pokhozhaev SI . Nonexistence of weak solutions for some degenrate elliptic and parabolic problems on RN . J Evol Equ. 2001;1:189–220.
  • Zaslavsky G . Hamiltonian chaos and fractional dynamics. Oxford: Oxford University Press; 2005.
  • Tarasov V . Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media. Berlin: Springer-Verlag, jointly with Higher Education Press, Beijing; 2011.
  • Wilk G , Wlodarczyk Z . Do we observe Levy flights in cosmic rays? Nucl Phys. 1999;75:191–193.
  • Mainardi F , Gorenflo R . On Mittag-Leffler-type functions in fractional evolution processes. J Comput Appl Math. 2000;118:283–299.
  • Metzler R , Klafter J . The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep. 2000;339:1–77.
  • Weitzner H , Zaslavdky G . Some applications of fractional derivatives. Commun Nonlinear Sci Numer Simul. 2003;8:273–281.
  • Tarasov V , Zaslavsky G . Fractional dynamics of coupled oscillators with long-range interaction. Chaos. 2006;16:23–110.
  • Nec Y , Nepomnyashchy A , Golovin A . Oscillatory instability in super-diffusive reaction diffusion systems: fractional amplitude and phase diffusion equations. Phys Rev E. 2008;78:60–102.
  • Li M , Huang C , Wang N . Galerkin finite element method for the nonlinear fractional Ginzburg-Landau equation. Appl Numer Math. 2017;118:131–149.
  • Mitidieri E , Pokhozhaev SI . A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities. Proc Steklov Inst Math. 2001;234:1–362.
  • Milovanov A , Rasmussen J . Fractional generalization of the Ginzburg-Landau equation: an unconventional approach to critical phenomena in complex media. Phys Lett A. 2005;337:75–80.
  • Pu X , Guo B . Well-posedness and dynamics for the fractional Ginzburg-Landau equation. Appl Anal. 2013;92:318–334.
  • Tarasov V , Zaslavsky G . Fractional Ginzburg-Landau equation for fractal media. Physica A. 2005;354:249–261.
  • Tarasov V . Psi-series solution of fractional Ginzburg-Landau equation. J Phys A Math Gen. 2006;39:8395–8407.
  • Li J , Xia L . Well-posedness of fractional Ginzburg-Landau equation in sobolev spaces. Appl Anal. 2013;5:1074–1084.
  • Li J , Xia L . The fractional Ginzburg-Landau equation with distributional initial data. Commun Pure Appl Anal. 2013;5:2173–2187.
  • Wu J . Dissipative quasi-geostrophic equations with Lp data. Electron J Differ Equ. 2001;2001:1–13.
  • Kato T , Ponce G . The Navier-Stokes equations with weak initial data. Int Math Res Notices. 1994;10:435–444.
  • Wu J . Well-posedness of a semi-linear heat equation with weak initial data. J Fourier Anal Appl. 1998;4:629–642.
  • Pazy A . Semigroups of linear operators and applications to partial differential equations. New York (NY): Springer-Verlag; 1983.
  • Kato T . Strong Lp solutions of the Navier-Stokes equation in ℝ m with applications to weak solutions. Math Z. 1984;187:471–480.
  • Kato T . The Navier-Stokes solutions for an incompressible fluid in R2 with measure as the initial vorticity. Differ Integr Equ. 1994;7:949–966.
  • Dix DB . Nonuniqueness and uniqueness in the initial value problem for Burgers’ equation. SIAM J Math Anal. 1996;27:708–724.
  • Taylor M . Remarks on fractional diffusion equations. Lecture Notes. Department of Mathematics, University of North Carolina. [cited 2007 Oct 3]:[44 p]. Available from: http://www.unc.edu/math/Faculty/met/fdif.pdf.
  • Fino A , Karch G . Decay of mass for nonlinear equation with fractional Laplacian. Monatsh Math. 2010;160:375–384.
  • Cordoba A , Cordoba D . A maximum principle applied to quasi-geostrophic equations. Commun Math Phys. 2004;249:511–528.
  • Ju N . The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations. Commun Math Phys. 2005;255:161–181.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.