December  2014, 34(12): 5325-5357. doi: 10.3934/dcds.2014.34.5325

Optimal parameter-dependent bounds for Kuramoto-Sivashinsky-type equations

1. 

Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada

Received  August 2010 Revised  May 2013 Published  June 2014

We derive a priori estimates on the absorbing ball in $L^2$ for the stabilized and destabilized Kuramoto-Sivashinsky (KS) equations, and for a sixth-order analog, the Nikolaevskiy equation, and in each case obtain bounds whose parameter dependence is demonstrably optimal. This is done by extending a Lyapunov function construction developed by Bronski and Gambill (Nonlinearity 19 , 2023--2039 (2006)) to take into account the dependence on both large and small parameters in the system. In the case of the destabilized KS equation, the rigorous bound lim $\sup_{t \to \infty}|| u || \leq K \alpha L^{3/2}$ is sharp in both the large parameter $\alpha$ and the system size $L$. We also apply our methods to improve previous estimates on a nonlocal variant of the KS equation.
Citation: Ralf W. Wittenberg. Optimal parameter-dependent bounds for Kuramoto-Sivashinsky-type equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5325-5357. doi: 10.3934/dcds.2014.34.5325
References:
[1]

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[2]

I. A. Beresnev and V. N. Nikolaevskiy, A model for nonlinear seismic-waves in a medium with instability,, Physica D, 66 (1993), 1.  doi: 10.1016/0167-2789(93)90217-O.  Google Scholar

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J. C. Bronski and R. C. Fetecau, An alternative energy bound derivation for a generalized Hasegawa-Mima equation,, Nonlinear Anal.: Real World Appl., 13 (2012), 1362.  doi: 10.1016/j.nonrwa.2011.10.012.  Google Scholar

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J. C. Bronski, R. C. Fetecau and T. N. Gambill, A note on a non-local Kuramoto-Sivashinsky equation,, Discrete Contin. Dyn. Syst. Ser. A, 18 (2007), 701.  doi: 10.3934/dcds.2007.18.701.  Google Scholar

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J. C. Bronski and T. N. Gambill, Uncertainty estimates and $L_2$ bounds for the Kuramoto-Sivashinsky equation,, Nonlinearity, 19 (2006), 2023.  doi: 10.1088/0951-7715/19/9/002.  Google Scholar

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P. Brunet, Stabilized Kuramoto-Sivashinsky equation: A useful model for secondary instabilities and related dynamics of experimental one-dimensional cellular flows,, Phys. Rev. E, 76 (2007).  doi: 10.1103/PhysRevE.76.017204.  Google Scholar

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H. Chaté and P. Manneville, Transition to turbulence via spatiotemporal intermittency,, Phys. Rev. Lett., 58 (1987), 112.  doi: 10.1103/PhysRevLett.58.112.  Google Scholar

[9]

P. Collet, J.-P. Eckmann, H. Epstein and J. Stubbe, A global attracting set for the Kuramoto-Sivashinsky equation,, Commun. Math. Phys., 152 (1993), 203.  doi: 10.1007/BF02097064.  Google Scholar

[10]

P. Collet, J.-P. Eckmann, H. Epstein and J. Stubbe, Analyticity for the Kuramoto-Sivashinsky equation,, Physica D, 67 (1993), 321.  doi: 10.1016/0167-2789(93)90168-Z.  Google Scholar

[11]

S. M. Cox and P. C. Matthews, Pattern formation in the damped Nikolaevskiy equation,, Phys. Rev. E, 76 (2007).  doi: 10.1103/PhysRevE.76.056202.  Google Scholar

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A. Demirkaya and M. Stanislavova, Long time behavior for radially symmetric solutions of the Kuramoto-Sivashinsky equation,, Dynamics of PDE, 7 (2010), 161.  doi: 10.4310/DPDE.2010.v7.n2.a2.  Google Scholar

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K. R. Elder, J. D. Gunton and N. Goldenfeld, Transition to spatiotemporal chaos in the damped Kuramoto-Sivashinsky equation,, Phys. Rev. E, 56 (1997), 1631.  doi: 10.1103/PhysRevE.56.1631.  Google Scholar

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C. Foias, B. Nicolaenko, G. R. Sell and R. Temam, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension,, J. Math. Pures Appl., 67 (1988), 197.   Google Scholar

[16]

M. Frankel and V. Roytburd, Stability for a class of nonlinear pseudo-differential equations,, Appl. Math. Lett., 21 (2008), 425.  doi: 10.1016/j.aml.2007.03.023.  Google Scholar

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L. Giacomelli and F. Otto, New bounds for the Kuramoto-Sivashinsky equation,, Commun. Pure Appl. Math., 58 (2005), 297.  doi: 10.1002/cpa.20031.  Google Scholar

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R. Grauer, An energy estimate for a perturbed Hasegawa-Mima equation,, Nonlinearity, 11 (1998), 659.  doi: 10.1088/0951-7715/11/3/014.  Google Scholar

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D. Hilhorst, L. A. Peletier, A. I. Rotariu and G. Sivashinsky, Global attractor and inertial sets for a nonlocal Kuramoto-Sivashinsky equation,, Discrete Contin. Dynam. Systems, 10 (2004), 557.  doi: 10.3934/dcds.2004.10.557.  Google Scholar

[21]

G. M. Homsy, Model equations for wavy viscous film flow,, Lect. Appl. Math., 15 (1974), 191.   Google Scholar

[22]

J. M. Hyman, B. Nicolaenko and S. Zaleski, Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interfaces,, Physica D, 23 (1986), 265.  doi: 10.1016/0167-2789(86)90136-3.  Google Scholar

[23]

Y. S. Il'yashenko, Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation,, J. Dyn. Diff. Eq., 4 (1992), 585.  doi: 10.1007/BF01048261.  Google Scholar

[24]

M. S. Jolly, R. Rosa and R. Temam, Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky equation,, Adv. Differential Equations, 5 (2000), 31.   Google Scholar

[25]

I. G. Kevrekidis, B. Nicolaenko and J. C. Scovel, Back in the saddle again: A computer assisted study of the Kuramoto-Sivashinsky equation,, SIAM J. Appl. Math., 50 (1990), 760.  doi: 10.1137/0150045.  Google Scholar

[26]

Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium,, Prog. Theor. Phys., 55 (1976), 356.  doi: 10.1143/PTP.55.356.  Google Scholar

[27]

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[28]

P. Manneville, Liapounov exponents for the Kuramoto-Sivashinsky model,, in Macroscopic Modelling of Turbulent Flows, 230 (1985), 319.  doi: 10.1007/3-540-15644-5_26.  Google Scholar

[29]

P. C. Matthews and S. M. Cox, One-dimensional pattern formation with Galilean invariance near a stationary bifurcation,, Phys. Rev. E, 62 (2000).  doi: 10.1103/PhysRevE.62.R1473.  Google Scholar

[30]

D. Michelson, Steady solutions of the Kuramoto-Sivashinsky equation,, Physica D, 19 (1986), 89.  doi: 10.1016/0167-2789(86)90055-2.  Google Scholar

[31]

C. Misbah and A. Valance, Secondary instabilities in the stabilized Kuramoto-Sivashinsky equation,, Phys. Rev. E, 49 (1994), 166.  doi: 10.1103/PhysRevE.49.166.  Google Scholar

[32]

L. Molinet, Local dissipativity in $l^2$ for the Kuramoto-Sivashinsky equation in spatial dimension 2,, J. Dyn. Diff. Eqns., 12 (2000), 533.  doi: 10.1023/A:1026459527446.  Google Scholar

[33]

B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of the Kuramoto-Sivashinsky equations: Nonlinear stability and attractors,, Physica D, 16 (1985), 155.  doi: 10.1016/0167-2789(85)90056-9.  Google Scholar

[34]

A. Novick-Cohen, Interfacial instabilities in directional solidification of dilute binary alloys: The Kuramoto-Sivashinsky equation,, Physica D, 26 (1987), 403.  doi: 10.1016/0167-2789(87)90240-5.  Google Scholar

[35]

F. Otto, Optimal bounds on the Kuramoto-Sivashinsky equation,, J. Funct. Anal., 257 (2009), 2188.  doi: 10.1016/j.jfa.2009.01.034.  Google Scholar

[36]

F. C. Pinto, Nonlinear stability and dynamical properties for a Kuramoto-Sivashinsky equation in space dimension two,, Discrete Contin. Dynam. Systems, 5 (1999), 117.  doi: 10.3934/dcds.1999.5.117.  Google Scholar

[37]

Y. Pomeau and P. Manneville, Wavelength selection in cellular flows,, Phys. Lett. A, 75 (1980), 296.  doi: 10.1016/0375-9601(80)90568-X.  Google Scholar

[38]

Y. Pomeau and S. Zaleski, Wavelength selection in one-dimensional cellular structures,, J. Physique, 42 (1981), 515.  doi: 10.1051/jphys:01981004204051500.  Google Scholar

[39]

J. D. M. Rademacher and R. W. Wittenberg, Viscous shocks in the destabilized Kuramoto-Sivashinsky equation,, ASME J. Comput. Nonlinear Dynamics, 1 (2006), 336.  doi: 10.1115/1.2338656.  Google Scholar

[40]

G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames-I. Derivation of basic equations,, Acta Astron., 4 (1977), 1177.  doi: 10.1016/0094-5765(77)90096-0.  Google Scholar

[41]

M. Stanislavova and A. Stefanov, Asymptotic estimates and stability analysis of Kuramoto-Sivashinsky type models,, J. Evol. Equ., 11 (2011), 605.  doi: 10.1007/s00028-011-0103-5.  Google Scholar

[42]

D. Tanaka, Chemical turbulence equivalent to Nikolavskii turbulence,, Phys. Rev. E, 70 (2004).  doi: 10.1103/PhysRevE.70.015202.  Google Scholar

[43]

D. Tanaka, Critical exponents of Nikolaevskii turbulence,, Phys. Rev. E, 71 (2005).  doi: 10.1103/PhysRevE.71.025203.  Google Scholar

[44]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, 2nd edition, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[45]

M. I. Tribel'skiĭ, Short-wavelength instability and transition to chaos in distributed systems with additional symmetry,, Usp. Fiz. Nauk, 40 (1997), 159.  doi: 10.1070/PU1997v040n02ABEH000193.  Google Scholar

[46]

M. I. Tribelsky and K. Tsuboi, New scenario for transition to turbulence?,, Phys. Rev. Lett., 76 (1996), 1631.  doi: 10.1103/PhysRevLett.76.1631.  Google Scholar

[47]

M. I. Tribelsky and M. G. Velarde, Short-wavelength instability in systems with slow long-wavelength dynamics,, Phys. Rev. E, 54 (1996), 4973.  doi: 10.1103/PhysRevE.54.4973.  Google Scholar

[48]

D. Tseliuko and D. T. Papageorgiou, A global attracting set for nonlocal Kuramoto-Sivashinsky equations arising in interfacial electrohydrodynamics,, Euro. Jnl of Applied Mathematics, 17 (2006), 677.  doi: 10.1017/S0956792506006760.  Google Scholar

[49]

R. W. Wittenberg, Dissipativity, analyticity and viscous shocks in the (de)stabilized Kuramoto-Sivashinsky equation,, Phys. Lett. A, 300 (2002), 407.  doi: 10.1016/S0375-9601(02)00861-7.  Google Scholar

[50]

R. W. Wittenberg and P. Holmes, Scale and space localization in the Kuramoto-Sivashinsky equation,, Chaos, 9 (1999), 452.  doi: 10.1063/1.166419.  Google Scholar

[51]

R. W. Wittenberg and K.-F. Poon, Anomalous scaling on a spatiotemporally chaotic attractor,, Phys. Rev. E, 79 (2009).  doi: 10.1103/PhysRevE.79.056225.  Google Scholar

show all references

References:
[1]

I. Bena, C. Misbah and A. Valance, Nonlinear evolution of a terrace edge during step-flow growth,, Phys. Rev. B, 47 (1993), 7408.  doi: 10.1103/PhysRevB.47.7408.  Google Scholar

[2]

I. A. Beresnev and V. N. Nikolaevskiy, A model for nonlinear seismic-waves in a medium with instability,, Physica D, 66 (1993), 1.  doi: 10.1016/0167-2789(93)90217-O.  Google Scholar

[3]

C.-M. Brauner, M. Frankel, J. Hulshof and V. Roytburd, Stability and attractors for the quasi-steady equation of cellular flames,, Interfaces Free Bound., 8 (2006), 301.  doi: 10.4171/IFB/145.  Google Scholar

[4]

J. C. Bronski and R. C. Fetecau, An alternative energy bound derivation for a generalized Hasegawa-Mima equation,, Nonlinear Anal.: Real World Appl., 13 (2012), 1362.  doi: 10.1016/j.nonrwa.2011.10.012.  Google Scholar

[5]

J. C. Bronski, R. C. Fetecau and T. N. Gambill, A note on a non-local Kuramoto-Sivashinsky equation,, Discrete Contin. Dyn. Syst. Ser. A, 18 (2007), 701.  doi: 10.3934/dcds.2007.18.701.  Google Scholar

[6]

J. C. Bronski and T. N. Gambill, Uncertainty estimates and $L_2$ bounds for the Kuramoto-Sivashinsky equation,, Nonlinearity, 19 (2006), 2023.  doi: 10.1088/0951-7715/19/9/002.  Google Scholar

[7]

P. Brunet, Stabilized Kuramoto-Sivashinsky equation: A useful model for secondary instabilities and related dynamics of experimental one-dimensional cellular flows,, Phys. Rev. E, 76 (2007).  doi: 10.1103/PhysRevE.76.017204.  Google Scholar

[8]

H. Chaté and P. Manneville, Transition to turbulence via spatiotemporal intermittency,, Phys. Rev. Lett., 58 (1987), 112.  doi: 10.1103/PhysRevLett.58.112.  Google Scholar

[9]

P. Collet, J.-P. Eckmann, H. Epstein and J. Stubbe, A global attracting set for the Kuramoto-Sivashinsky equation,, Commun. Math. Phys., 152 (1993), 203.  doi: 10.1007/BF02097064.  Google Scholar

[10]

P. Collet, J.-P. Eckmann, H. Epstein and J. Stubbe, Analyticity for the Kuramoto-Sivashinsky equation,, Physica D, 67 (1993), 321.  doi: 10.1016/0167-2789(93)90168-Z.  Google Scholar

[11]

S. M. Cox and P. C. Matthews, Pattern formation in the damped Nikolaevskiy equation,, Phys. Rev. E, 76 (2007).  doi: 10.1103/PhysRevE.76.056202.  Google Scholar

[12]

A. Demirkaya and M. Stanislavova, Long time behavior for radially symmetric solutions of the Kuramoto-Sivashinsky equation,, Dynamics of PDE, 7 (2010), 161.  doi: 10.4310/DPDE.2010.v7.n2.a2.  Google Scholar

[13]

J. Duan and V. J. Ervin, Dynamics of a nonlocal Kuramoto-Sivashinsky equation,, J. Differential Equations, 143 (1998), 243.  doi: 10.1006/jdeq.1997.3371.  Google Scholar

[14]

K. R. Elder, J. D. Gunton and N. Goldenfeld, Transition to spatiotemporal chaos in the damped Kuramoto-Sivashinsky equation,, Phys. Rev. E, 56 (1997), 1631.  doi: 10.1103/PhysRevE.56.1631.  Google Scholar

[15]

C. Foias, B. Nicolaenko, G. R. Sell and R. Temam, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension,, J. Math. Pures Appl., 67 (1988), 197.   Google Scholar

[16]

M. Frankel and V. Roytburd, Stability for a class of nonlinear pseudo-differential equations,, Appl. Math. Lett., 21 (2008), 425.  doi: 10.1016/j.aml.2007.03.023.  Google Scholar

[17]

L. Giacomelli and F. Otto, New bounds for the Kuramoto-Sivashinsky equation,, Commun. Pure Appl. Math., 58 (2005), 297.  doi: 10.1002/cpa.20031.  Google Scholar

[18]

J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems,, Commun. Pure Appl. Math., 47 (1994), 293.  doi: 10.1002/cpa.3160470304.  Google Scholar

[19]

R. Grauer, An energy estimate for a perturbed Hasegawa-Mima equation,, Nonlinearity, 11 (1998), 659.  doi: 10.1088/0951-7715/11/3/014.  Google Scholar

[20]

D. Hilhorst, L. A. Peletier, A. I. Rotariu and G. Sivashinsky, Global attractor and inertial sets for a nonlocal Kuramoto-Sivashinsky equation,, Discrete Contin. Dynam. Systems, 10 (2004), 557.  doi: 10.3934/dcds.2004.10.557.  Google Scholar

[21]

G. M. Homsy, Model equations for wavy viscous film flow,, Lect. Appl. Math., 15 (1974), 191.   Google Scholar

[22]

J. M. Hyman, B. Nicolaenko and S. Zaleski, Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interfaces,, Physica D, 23 (1986), 265.  doi: 10.1016/0167-2789(86)90136-3.  Google Scholar

[23]

Y. S. Il'yashenko, Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation,, J. Dyn. Diff. Eq., 4 (1992), 585.  doi: 10.1007/BF01048261.  Google Scholar

[24]

M. S. Jolly, R. Rosa and R. Temam, Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky equation,, Adv. Differential Equations, 5 (2000), 31.   Google Scholar

[25]

I. G. Kevrekidis, B. Nicolaenko and J. C. Scovel, Back in the saddle again: A computer assisted study of the Kuramoto-Sivashinsky equation,, SIAM J. Appl. Math., 50 (1990), 760.  doi: 10.1137/0150045.  Google Scholar

[26]

Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium,, Prog. Theor. Phys., 55 (1976), 356.  doi: 10.1143/PTP.55.356.  Google Scholar

[27]

R. E. LaQuey, S. M. Mahajan, P. H. Rutherford and W. M. Tang, Nonlinear saturation of the trapped-ion mode,, Phys. Rev. Lett., 34 (1975), 391.  doi: 10.1103/PhysRevLett.34.391.  Google Scholar

[28]

P. Manneville, Liapounov exponents for the Kuramoto-Sivashinsky model,, in Macroscopic Modelling of Turbulent Flows, 230 (1985), 319.  doi: 10.1007/3-540-15644-5_26.  Google Scholar

[29]

P. C. Matthews and S. M. Cox, One-dimensional pattern formation with Galilean invariance near a stationary bifurcation,, Phys. Rev. E, 62 (2000).  doi: 10.1103/PhysRevE.62.R1473.  Google Scholar

[30]

D. Michelson, Steady solutions of the Kuramoto-Sivashinsky equation,, Physica D, 19 (1986), 89.  doi: 10.1016/0167-2789(86)90055-2.  Google Scholar

[31]

C. Misbah and A. Valance, Secondary instabilities in the stabilized Kuramoto-Sivashinsky equation,, Phys. Rev. E, 49 (1994), 166.  doi: 10.1103/PhysRevE.49.166.  Google Scholar

[32]

L. Molinet, Local dissipativity in $l^2$ for the Kuramoto-Sivashinsky equation in spatial dimension 2,, J. Dyn. Diff. Eqns., 12 (2000), 533.  doi: 10.1023/A:1026459527446.  Google Scholar

[33]

B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of the Kuramoto-Sivashinsky equations: Nonlinear stability and attractors,, Physica D, 16 (1985), 155.  doi: 10.1016/0167-2789(85)90056-9.  Google Scholar

[34]

A. Novick-Cohen, Interfacial instabilities in directional solidification of dilute binary alloys: The Kuramoto-Sivashinsky equation,, Physica D, 26 (1987), 403.  doi: 10.1016/0167-2789(87)90240-5.  Google Scholar

[35]

F. Otto, Optimal bounds on the Kuramoto-Sivashinsky equation,, J. Funct. Anal., 257 (2009), 2188.  doi: 10.1016/j.jfa.2009.01.034.  Google Scholar

[36]

F. C. Pinto, Nonlinear stability and dynamical properties for a Kuramoto-Sivashinsky equation in space dimension two,, Discrete Contin. Dynam. Systems, 5 (1999), 117.  doi: 10.3934/dcds.1999.5.117.  Google Scholar

[37]

Y. Pomeau and P. Manneville, Wavelength selection in cellular flows,, Phys. Lett. A, 75 (1980), 296.  doi: 10.1016/0375-9601(80)90568-X.  Google Scholar

[38]

Y. Pomeau and S. Zaleski, Wavelength selection in one-dimensional cellular structures,, J. Physique, 42 (1981), 515.  doi: 10.1051/jphys:01981004204051500.  Google Scholar

[39]

J. D. M. Rademacher and R. W. Wittenberg, Viscous shocks in the destabilized Kuramoto-Sivashinsky equation,, ASME J. Comput. Nonlinear Dynamics, 1 (2006), 336.  doi: 10.1115/1.2338656.  Google Scholar

[40]

G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames-I. Derivation of basic equations,, Acta Astron., 4 (1977), 1177.  doi: 10.1016/0094-5765(77)90096-0.  Google Scholar

[41]

M. Stanislavova and A. Stefanov, Asymptotic estimates and stability analysis of Kuramoto-Sivashinsky type models,, J. Evol. Equ., 11 (2011), 605.  doi: 10.1007/s00028-011-0103-5.  Google Scholar

[42]

D. Tanaka, Chemical turbulence equivalent to Nikolavskii turbulence,, Phys. Rev. E, 70 (2004).  doi: 10.1103/PhysRevE.70.015202.  Google Scholar

[43]

D. Tanaka, Critical exponents of Nikolaevskii turbulence,, Phys. Rev. E, 71 (2005).  doi: 10.1103/PhysRevE.71.025203.  Google Scholar

[44]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, 2nd edition, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[45]

M. I. Tribel'skiĭ, Short-wavelength instability and transition to chaos in distributed systems with additional symmetry,, Usp. Fiz. Nauk, 40 (1997), 159.  doi: 10.1070/PU1997v040n02ABEH000193.  Google Scholar

[46]

M. I. Tribelsky and K. Tsuboi, New scenario for transition to turbulence?,, Phys. Rev. Lett., 76 (1996), 1631.  doi: 10.1103/PhysRevLett.76.1631.  Google Scholar

[47]

M. I. Tribelsky and M. G. Velarde, Short-wavelength instability in systems with slow long-wavelength dynamics,, Phys. Rev. E, 54 (1996), 4973.  doi: 10.1103/PhysRevE.54.4973.  Google Scholar

[48]

D. Tseliuko and D. T. Papageorgiou, A global attracting set for nonlocal Kuramoto-Sivashinsky equations arising in interfacial electrohydrodynamics,, Euro. Jnl of Applied Mathematics, 17 (2006), 677.  doi: 10.1017/S0956792506006760.  Google Scholar

[49]

R. W. Wittenberg, Dissipativity, analyticity and viscous shocks in the (de)stabilized Kuramoto-Sivashinsky equation,, Phys. Lett. A, 300 (2002), 407.  doi: 10.1016/S0375-9601(02)00861-7.  Google Scholar

[50]

R. W. Wittenberg and P. Holmes, Scale and space localization in the Kuramoto-Sivashinsky equation,, Chaos, 9 (1999), 452.  doi: 10.1063/1.166419.  Google Scholar

[51]

R. W. Wittenberg and K.-F. Poon, Anomalous scaling on a spatiotemporally chaotic attractor,, Phys. Rev. E, 79 (2009).  doi: 10.1103/PhysRevE.79.056225.  Google Scholar

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