Global existence and attractors for the two-dimensional Burgers-Ginzburg- Landau equations
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Authors
Changhong Guo
- School of Management, Guangdong University of Technology, Guangzhou 510520, P. R. China.
Shaomei Fang
- Department of Mathematics, South China Agricultural University, Guangzhou 510640, P. R. China.
Abstract
This paper investigates the periodic initial value problem for the two-dimensional Burgers-Ginzburg-Landau (2D Burgers-
GL) equations, which can be derived from the so-called modulated modulation equations (MME) that govern the dynamics of
the modulated amplitudes of some periodic critical modes. The well-posedness of the solutions and the global attractors for the
2D Burgers-GL equations are obtained via delicate a priori estimates, the Galerkin method, and operator semigroup method.
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ISRP Style
Changhong Guo, Shaomei Fang, Global existence and attractors for the two-dimensional Burgers-Ginzburg- Landau equations, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 6, 3123--3135
AMA Style
Guo Changhong, Fang Shaomei, Global existence and attractors for the two-dimensional Burgers-Ginzburg- Landau equations. J. Nonlinear Sci. Appl. (2017); 10(6):3123--3135
Chicago/Turabian Style
Guo, Changhong, Fang, Shaomei. "Global existence and attractors for the two-dimensional Burgers-Ginzburg- Landau equations." Journal of Nonlinear Sciences and Applications, 10, no. 6 (2017): 3123--3135
Keywords
- invex set
- well-posedness
- global attractors
- a priori estimates.
MSC
References
-
[1]
A. J. S. Al-Saif, A. Abdul-Hussein, Generating exact solutions of two-dimensional coupled Burgers equations by the first integral method, Res. J. Phys. Appl. Sci., 1 (2012), 29–33.
-
[2]
A. R. Bahadır, A fully implicit finite-difference scheme for two-dimensional Burgers’ equations, Appl. Math. Comput., 137 (2003), 131–137.
-
[3]
A. J. Bernoff, Slowly varying fully nonlinear wavetrains in the Ginzburg-Landau equation, Phys. D, 30 (1988), 363–381.
-
[4]
J. M. Burgers, A mathematical model illustrating the theory of turbulence, edited by Richard von Mises and Theodore von Kármán, Advances in Applied Mechanics, Academic Press, Inc., New York, N. Y., (1948), 171–199.
-
[5]
A. Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London (1969)
-
[6]
C.-H. Guo, S.-M. Fang, B.-L. Guo, Global smooth solutions of the generalized KS-CGL equations with flames governed by a sequential reaction, Commun. Math. Sci., 12 (2014), 1457–1474.
-
[7]
C.-H. Guo, S.-M. Fang, B.-L. Guo, Long time behavior of solutions to coupled Burgers-complex Ginzbury-Landau (Burgers-CGL) equations for flames governed by sequential reaction, [[Long time behavior of solutions to coupled Burgers-complex Ginzburg-Landau (Burgers-CGL) equations for flames governed by sequential reaction]] Appl. Math. Mech. (English Ed.), 35 (2014), 515–534.
-
[8]
B.-L. Guo, H.-Y. Huang, On the Burgers-Ginzburg-Landau equations, Commun. Nonlinear Sci. Numer. Simul., 3 (1998), 143–147.
-
[9]
B.-L. Guo, H.-Y. Huang, Well-posedness and global attractors for the Burgers-Ginzburg-Landau equations, Acta Math. Appl. Sin. Engl. Ser., 18 (2002), 579–588.
-
[10]
D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin- New York (1981)
-
[11]
H.-Y. Huang, Attractors of the discretized Burgers-Ginzburg-Landau equations, (Chinese) Acta Math. Sci. Ser. A Chin. Ed., 22 (2002), 316–322.
-
[12]
H.-Y. Huang, Existence of periodic solutions of the Burgers-Ginzburg-Landau equations, Acta Math. Sci. Ser. B Engl. Ed., 24 (2004), 613–622.
-
[13]
R. Iturriaga, K. Khanin, Burgers turbulence and random Lagrangian systems, Comm. Math. Phys., 232 (2003), 377–428.
-
[14]
L. Kramer, W. Zimmermann, On the Eckhaus instability for spatially periodic patterns, Phys. D, 16 (1985), 221–232.
-
[15]
A. Mielke, G. Schneider, Derivation and justification of the complex Ginzburg-Landau equation as a modulation equation, Dynamical systems and probabilistic methods in partial differential equations, Berkeley, CA, (1994), Lectures in Appl. Math., Amer. Math. Soc., Providence, RI, 31 (1996), 191–216.
-
[16]
A. A. Nepomnhyashchy, Order parameter equations for long-wavelength instabilities, Phys. D, 86 (1995), 90–95.
-
[17]
K. Nishinari, J. Matsukidaira, D. Takahashi, Two-dimensional Burgers cellular automaton, J. Phys. Soc. Japan, 70 (2001), 2267–2272.
-
[18]
R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Second edition, Applied Mathematical Sciences, Springer-Verlag, New York (1997)
-
[19]
A. van Harten, Modulated modulation equations, Proceedings of the IUTAM/ISIMM Symposium on Structure and Dynamics of Nonlinear Waves in Fluids, Hannover, (1994), Adv. Ser. Nonlinear Dynam., World Sci. Publ., River Edge, NJ, 7 (1995), 117–130.
-
[20]
H.-Q. Zhu, H.-Z. Shu, M.-Y. Ding, Numerical solutions of two-dimensional Burgers’ equations by discrete Adomian decomposition method, Comput. Math. Appl., 60 (2010), 840–848.