Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attractors. (English) Zbl 0592.35013

Nicolaenko, B.; Scheurer, B.; Temam, R.

Physica D 16, 155-183 (1985).

The Kuramoto-Sivashinsky equations model pattern formations on unstable flame fronts and thin hydrodynamic films. They are characterized by the coexistence of coherent spatial structures with temporal chaos. We investigate some global dynamical properties, including nonlinear stability. We demonstrate their low modal behavior, in terms of determining modes; and that the fractal dimension of all attractors is bounded by a universal constant times \(\tilde L^{13/8}\), where \(\tilde L\) is a dimensionless pattern cell size (in the one-dimensional case). Such equations are indeed a paradigm of low-dimensional behavior for infinite-dimensional systems.

Reviewer: M.Biroli

MSC:

35B35	Stability in context of PDEs
35K55	Nonlinear parabolic equations
35Q99	Partial differential equations of mathematical physics and other areas of application

Keywords:

Kuramoto-Sivashinsky equations; unstable flame fronts; temporal chaos; nonlinear stability

References:

[1]	Agmon, Sh., Lectures on elliptic boundary value problems, (1965), Van Nostrand New York
[2]	Aimar, M.T.; Aimar, M.T.; Penel, P., Quelques aspects numériques nouveaux pour le modèle de front de flamme, (), See also
[3]	Babchin, A.J.; Frenkel, A.L.; Levich, B.G.; Sivashinsky, G.I., Nonlinear saturation of Rayleigh-Taylor instability in thin films, Phys. fluids, 26, 3159-3161, (1983) · Zbl 0521.76046
[4]	Bardos, C.; Tartar, L., Sur l’unicité rétrograde des equations paraboliques et quelques questions voisines, Arch. rat. mech. anal., 50, 10-25, (1973) · Zbl 0258.35039
[5]	Benney, D.J., Long waves in liquid film, J. math. and phys., 45, 150-155, (1966) · Zbl 0148.23003
[6]	Clavin, P., Dynamic behavior of premixed flame fronts in laminar and turbulent flows, ()
[7]	P. Constantin and C. Foias, “Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations”, Comm. Pure Appl. Math., to appear. · Zbl 0582.35092
[8]	P. Constantin, C. Foias and R. Temam, “Attractors representing turbulent flows,” to appear, Memoirs of the A.M.S. · Zbl 0567.35070
[9]	Foias, C.; Temam, R., Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J. math. pures et appl., 58, 339-368, (1979) · Zbl 0454.35073
[10]	Foias, C.; Manley, O.P.; Temam, R.; Tréve, Y.M., Asymptotic analysis of the Navier-Stokes equations, Physica, 9D, 157-188, (1983) · Zbl 0584.35007
[11]	J.M. Hyman, B. Nicolaenko, B. Scheurer and R. Temam, to appear.
[12]	Kuramoto, Y.; Tsuzuki, T., On the formation of dissipative structures in reaction-diffusion systems, Prog. theor. phys., 54, 687-699, (1975)
[13]	Kuramoto, Y.; Tsuzuki, T., Persistent propatation of concentration waves in dissipative media far from thermal equilibrium, Prog. theor. phys., 55, 356-369, (1976)
[14]	Kuramoto, Y., Diffusion-induced chaos in reactions systems, Suppl. prog. theor. phys., 64, 346-367, (1978)
[15]	Lions, J.L.; Magenes, E., ()
[16]	Lin, S.P., Finite amplitude side-band stability of a viscous fluid, J. fluid mech., 63, 417-429, (1974) · Zbl 0283.76035
[17]	Michelson, D.M.; Sivashinsky, G.I., Nonlinear analysis of hydrodynamic instability in laminar flames II, Numerical experiments acta astronautica, 4, 1207-1221, (1977) · Zbl 0427.76048
[18]	Nicolaenko, B.; Scheurer, B.; Temam, R., Quelques proprietés des attracteurs pour l’equation de Kuramoto-Sivashinsky, C.R. acad. sc. Paris, 298, 23-25, (1984) · Zbl 0555.58017
[19]	Nicolaenko, B.; Scheurer, B., Remarks on the Kuramoto-Sivashinsky equation, (), 391-395 · Zbl 0576.35058
[20]	Pumir, A., Structure localisées et turbulence, Thése 3 éme cycle, (1982), Paris
[21]	Pumir, A.; Manneville, P.; Pomeau, Y., On solitary waves running down an inclined plane, J. fluid mech., 135, 27-50, (1983) · Zbl 0525.76016
[22]	Y. Pomeau, A. Pumir and P. Pelce, “Intrinsic Stochasticity with Many Degrees of Freedom”, preprint C.E.A.-S.P.T., Saclay, France.
[23]	Sivashinsky, G., Nonlinear analysis of hydrodynamic instability in laminar flames, part I. derivation of basic equations, Acta astronautica, 4, 1177-1206, (1977) · Zbl 0427.76047
[24]	Sivashinsky, G., On flame propagation under conditionsof stoichiometry, SIAM J. appl. math., 39, 67-82, (1980) · Zbl 0464.76055
[25]	Sivashinsky, G.I.; Michelson, D.M., On irregular way flow of a liquid down a vertical plane, Prog. theor. phys., 63, 2112-2114, (1980)
[26]	Temam, R., Behaviour at time t=0 of the solutions of semilinear evolution equations, J. diff. equ., 43, 73-92, (1982) · Zbl 0446.35057