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September  2013, 12(5): 2173-2187. doi: 10.3934/cpaa.2013.12.2173

The Fractional Ginzburg-Landau equation with distributional initial data

Jingna Li 1, and  Li Xia 2,

1. 

Department of Mathematics, Jinan University, Guangzhou 510632, China
2. 

Department of Mathematics, Shenzhen University, Shenzhen 518060, China

Received  June 2012 Revised  October 2012 Published  January 2013

The paper is concerned with real fractional Ginzburg-Landau equation. Existence and uniqueness of local and global mild solution with distributional initial data are obtained by contraction mapping principle and carefully choosing the working space, and Gevrey regularity of mild solution for flat torus case is discussed.
Keywords: distributional initial data, well-posedness, Gevrey regularity., Fractional Ginzburg-Landau equation.
Mathematics Subject Classification: Primary: 35Q99, 35D10; Secondary: 78A9.
Citation: Jingna Li, Li Xia. The Fractional Ginzburg-Landau equation with distributional initial data. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2173-2187. doi: 10.3934/cpaa.2013.12.2173
References:
[1]

G. Zaslavsky, "Hamiltonian Chaos and Fractional Dynamics," Oxford University Press, Oxford, 2005.

[2]

V. Tarasov, "Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media,'' Springer-Verlag, Berlin Heidelberg, jointly with Higher Education Press, Beijing, 2011.

[3]

G. Wilk and Z. Wlodarczyk, Do we observe Levy flights in cosmic rays? Nucl. Phys., B75A (1999), 191-193.

[4]

M. Naber, Time fractional Schrödinger equation, J. Math. Phys., 45 (2004), 3339-3352. doi: 10.1063/1.1769611.

[5]

G. Zaslavsky and A. Edelman, Fractional kinetics: from pseudochaotic dynamics to Maxwell's demon, Physica D., 193 (2004), 128-147. doi: 10.1016/j.physd.2004.01.014.

[6]

F. Mainardi and R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes, J. Comput. Appl. Math., 118 (2000), 283-299. doi: 10.1016/S0377-0427(00)00294-6.

[7]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[8]

H. Weitzner and G. Zaslavdky, Some applications of fractional derivatives, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273-281. doi: 10.1016/S1007-5704(03)00049-2.

[9]

V. Tarasov and G. Zaslavsky, Fractional dynamics of coupled oscillators with long-range interaction, Chaos., 16 (2006), 023110. doi: 10.1063/1.2197167.

[10]

Y. Nec, A. Nepomnyashchy and A. Golovin, Oscillatory instability in super-diffusive reaction-diffusion systems: Fractional amplitude and phase diffusion equations, Phys. Rev. E., 78 (2008), 060102.

[11]

V. Tarasov and G. Zaslavsky, Fractional Ginzburg-Landau equation for fractal media, Physica A, 354 (2005), 249-261. doi: 10.1016/j.physa.2005.02.047.

[12]

A. Milovanov and J. Rasmussen, Fractional generalization of the Ginzburg-Landau equation: An unconventional approach to critical phenomena in complex media, Phys. Lett. A., 337 (2005), 75-80. doi: 10.1016/j.physleta.2005.01.047.

[13]

V. Tarasov, Psi-series solution of fractional Ginzburg-Landau equation, J. Phys. A: Math. Gen., 39 (2006), 8395-8407.

[14]

J. Li and L. Xia, Well-posedness of fractional Ginzburg-Landau equation in sobolev spaces, Appl. Anal., in press, DOI:10.1080/00036811.2011.649733. doi: 10.1080/00036811.2011.649733.

[15]

J. Wu, Well-posedness of a semi-linear heat equation with weak initial data, J. Fourier. Anal. Appl., 4 (1998), 629-642. doi: 10.1007/BF02498228.

[16]

T. Kato and G. Ponce, The Navier-Stokes equations with weak initial data, Int. Math. Res. Not., 10 (1994), 435-444. doi: 10.1155/S1073792894000474.

[17]

J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data, Elect. J. Differ. Equ., 2001 (2001), 1-13.

[18]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differetial Equations," Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[19]

B. Guo, H. Huang and M. Jiang, "Ginzburg-Landau Equation,'' Chinese ed, Science Press, Beijing, China, 2002.

show all references

References:
[1]

G. Zaslavsky, "Hamiltonian Chaos and Fractional Dynamics," Oxford University Press, Oxford, 2005.

[2]

V. Tarasov, "Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media,'' Springer-Verlag, Berlin Heidelberg, jointly with Higher Education Press, Beijing, 2011.

[3]

G. Wilk and Z. Wlodarczyk, Do we observe Levy flights in cosmic rays? Nucl. Phys., B75A (1999), 191-193.

[4]

M. Naber, Time fractional Schrödinger equation, J. Math. Phys., 45 (2004), 3339-3352. doi: 10.1063/1.1769611.

[5]

G. Zaslavsky and A. Edelman, Fractional kinetics: from pseudochaotic dynamics to Maxwell's demon, Physica D., 193 (2004), 128-147. doi: 10.1016/j.physd.2004.01.014.

[6]

F. Mainardi and R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes, J. Comput. Appl. Math., 118 (2000), 283-299. doi: 10.1016/S0377-0427(00)00294-6.

[7]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[8]

H. Weitzner and G. Zaslavdky, Some applications of fractional derivatives, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273-281. doi: 10.1016/S1007-5704(03)00049-2.

[9]

V. Tarasov and G. Zaslavsky, Fractional dynamics of coupled oscillators with long-range interaction, Chaos., 16 (2006), 023110. doi: 10.1063/1.2197167.

[10]

Y. Nec, A. Nepomnyashchy and A. Golovin, Oscillatory instability in super-diffusive reaction-diffusion systems: Fractional amplitude and phase diffusion equations, Phys. Rev. E., 78 (2008), 060102.

[11]

V. Tarasov and G. Zaslavsky, Fractional Ginzburg-Landau equation for fractal media, Physica A, 354 (2005), 249-261. doi: 10.1016/j.physa.2005.02.047.

[12]

A. Milovanov and J. Rasmussen, Fractional generalization of the Ginzburg-Landau equation: An unconventional approach to critical phenomena in complex media, Phys. Lett. A., 337 (2005), 75-80. doi: 10.1016/j.physleta.2005.01.047.

[13]

V. Tarasov, Psi-series solution of fractional Ginzburg-Landau equation, J. Phys. A: Math. Gen., 39 (2006), 8395-8407.

[14]

J. Li and L. Xia, Well-posedness of fractional Ginzburg-Landau equation in sobolev spaces, Appl. Anal., in press, DOI:10.1080/00036811.2011.649733. doi: 10.1080/00036811.2011.649733.

[15]

J. Wu, Well-posedness of a semi-linear heat equation with weak initial data, J. Fourier. Anal. Appl., 4 (1998), 629-642. doi: 10.1007/BF02498228.

[16]

T. Kato and G. Ponce, The Navier-Stokes equations with weak initial data, Int. Math. Res. Not., 10 (1994), 435-444. doi: 10.1155/S1073792894000474.

[17]

J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data, Elect. J. Differ. Equ., 2001 (2001), 1-13.

[18]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differetial Equations," Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[19]

B. Guo, H. Huang and M. Jiang, "Ginzburg-Landau Equation,'' Chinese ed, Science Press, Beijing, China, 2002.

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