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Dynamical bifurcation for the Kuramoto-Sivashinsky equation. (English) Zbl 1209.35017
This study concerns the one-dimensional Kuramoto-Sivashinsky equation with periodic boundary condition, and with the constraint that the mean over a period is 0. The aim is to study the bifurcation from the trivial solution corresponding to eigenvalues of the linearized equation. Using the attractor bifurcation theory of Tian Ma and S. Wang [Chin. Ann. Math., Ser. B 26, No. 2, 185–206 (2005; Zbl 1193.37105)], T. Ma and S. Wang [Commun. Pure Appl. Anal. 2, No. 4, 591–599 (2003; Zbl 1210.37056)], bifurcation of a nontrivial attractor from the trivial attractor 0 is proved when the eigenvalue λ=1 is crossed, and a characterization of its homotopy type is given, both in the general case and when the problem is considered in the space of odd functions.
35B32Bifurcation (PDE)
[1]Hoope, A. P.; Grimshaw, R.: Nonlinear instability at the interface between two viscous fluids, Phys. fluids 28, 37-45 (1985) · Zbl 0565.76051 · doi:10.1063/1.865160
[2]Kuramoto, Y.; Tsuzuki, T.: Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Progr. theoret. Phys. 55, No. 2, 356-369 (1976)
[3]Sivashinsky, G. I.: On flame propagation under condition of stoichiometry, SIAM J. Appl. math. 39, 67-82 (1980) · Zbl 0464.76055 · doi:10.1137/0139007
[4]Sivashinsky, G. I.: Instabilities, pattern-formation and turbulence in flames, Annu. rev. Fluid mech. 15, 179-199 (1983) · Zbl 0538.76053
[5]Grimshaw, R.; Hooper, A. P.: The non-existence of a certain class of travelling wave solutions of the Kuramoto–Sivashinsky equation, Physica D 50, 231-238 (1991) · Zbl 0775.35008 · doi:10.1016/0167-2789(91)90177-B
[6]Foias, C.; Kukavica, I.: Determing nodes for the Kuramoto–Sivashinsky equation, J. dyn. Differ. equ. 7, No. 2, 365-373 (1995) · Zbl 0835.35067 · doi:10.1007/BF02219361
[7]Liu, X.: Gevrey class regularity and approximate inertial manifolds for the Kuramoto–Sivashinsky equation, Physica D 50, 135-151 (1991) · Zbl 0743.35029 · doi:10.1016/0167-2789(91)90085-N
[8]Khater, A. H.; Temsah, R. S.: Numerical solutions of the generalized Kuramoto–Sivashinsky equation by Chebyshev spectral collocation methods, Comput. math. Appl. 56, 1465-1472 (2008) · Zbl 1155.65381 · doi:10.1016/j.camwa.2008.03.013
[9]Nicolaenko, B.; Scheurer, B.; Temam, R.: Some global dynamical properties of the Kuramoto–Sivashinsky equation: nonlinear stability and attractors, Physica D 16, 155-183 (1985) · Zbl 0592.35013 · doi:10.1016/0167-2789(85)90056-9
[10]Tadmor, E.: The well-posedness of the Kuramoto–Sivashinsky equation, SIAM J. Math. anal. 17, No. 4, 884-893 (1986) · Zbl 0606.35073 · doi:10.1137/0517063
[11]Temam, R.; Wang, X. M.: Estimates on the lowest dimension of inertial manifolds for the Kuramoto–Sivashinsky equation in the general case, Differential integral equations 7, No. 3–4, 1095-1108 (1994) · Zbl 0858.35017
[12]Cholewa, J. W.; Dlotko, T.: Global attractor for the Cahn–Hilliard system, Bull. austral. Math. soc. 49, 277-292 (1994) · Zbl 0803.35013 · doi:10.1017/S0004972700016348
[13]Dlotko, T.: Global attractor for the Cahn–Hilliard equation in H2 and H3, J. differential equations 113, 381-393 (1994)
[14]Hale, J. K.: Asymptotic behaviour of dissipative systems, (1988)
[15]Nicolaenko, B.; Scheurer, B.; Temam, R.: Some global dynamical properties of a class of pattern formation equations, Comm. partial differential equations 14, 245-297 (1989) · Zbl 0691.35019 · doi:10.1080/03605308908820597
[16]Novick-Cohen, A.; Segel, L. A.: Nonlinear aspects of the Cahn–Hilliard equation, Physica 10D, 277-298 (1984)
[17]Ma, T.; Wang, S. H.: Dynamic bifurcation of nonlinear evolution equations, Chin. ann. Math. 26, No. 2, 185-206 (2005) · Zbl 1193.37105 · doi:10.1142/S0252959905000166
[18]Ma, T.; Wang, S. H.: Bifurcation theory and applications, (2005)
[19]Wang, Z. P.; Zhong, C. K.: Dynamic bifurcation for the generalized Burgers equations, J. Lanzhou univ. Nat. sci. 45, No. 4, 133-139 (2009) · Zbl 1212.35435
[20]Pazy, A.: Semigroups of linear operators and applications to partial differential equations, Appl. math. Sci. 44 (1983) · Zbl 0516.47023
[21]Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics, Applied mathematical sciences 68 (1997) · Zbl 0871.35001