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The Fractional Ginzburg-Landau equation with distributional initial data
1. | Department of Mathematics, Jinan University, Guangzhou 510632, China |
2. | Department of Mathematics, Shenzhen University, Shenzhen 518060, China |
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., ().
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., ().
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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