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

Previous Article
Realization of tangent perturbations in discrete and continuous time conservative systems
DCDS Home
This Issue
Next Article
Segregated peak solutions of coupled Schrödinger systems with Neumann boundary conditions

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

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

Ralf W. Wittenberg ^1,

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

Received August 2010 Revised May 2013 Published June 2014

Abstract
Related Papers

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.

Keywords: global attractor, Lyapunov function., Nikolaevskiy equation, a priori bounds, Kuramoto-Sivashinsky equation.

Mathematics Subject Classification: Primary: 35B45, 35K30; Secondary: 37L3.

Citation: Ralf W. Wittenberg. Optimal parameter-dependent bounds for Kuramoto-Sivashinsky-type equations. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5325-5357. doi: 10.3934/dcds.2014.34.5325

References:

[1]	I. Bena, C. Misbah and A. Valance, Nonlinear evolution of a terrace edge during step-flow growth, Phys. Rev. B, 47 (1993), 7408-7419. doi: 10.1103/PhysRevB.47.7408.
[2]	I. A. Beresnev and V. N. Nikolaevskiy, A model for nonlinear seismic-waves in a medium with instability, Physica D, 66 (1993), 1-6. doi: 10.1016/0167-2789(93)90217-O.
[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), 301-316. doi: 10.4171/IFB/145.
[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), 1362-1368. doi: 10.1016/j.nonrwa.2011.10.012.
[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), 701-707. doi: 10.3934/dcds.2007.18.701.
[6]	J. C. Bronski and T. N. Gambill, Uncertainty estimates and $L_2$ bounds for the Kuramoto-Sivashinsky equation, Nonlinearity, 19 (2006), 2023-2039. doi: 10.1088/0951-7715/19/9/002.
[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), 017204. doi: 10.1103/PhysRevE.76.017204.
[8]	H. Chaté and P. Manneville, Transition to turbulence via spatiotemporal intermittency, Phys. Rev. Lett., 58 (1987), 112-115. doi: 10.1103/PhysRevLett.58.112.
[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), 203-214. doi: 10.1007/BF02097064.
[10]	P. Collet, J.-P. Eckmann, H. Epstein and J. Stubbe, Analyticity for the Kuramoto-Sivashinsky equation, Physica D, 67 (1993), 321-326. doi: 10.1016/0167-2789(93)90168-Z.
[11]	S. M. Cox and P. C. Matthews, Pattern formation in the damped Nikolaevskiy equation, Phys. Rev. E, 76 (2007), 056202, 11pp. doi: 10.1103/PhysRevE.76.056202.
[12]	A. Demirkaya and M. Stanislavova, Long time behavior for radially symmetric solutions of the Kuramoto-Sivashinsky equation, Dynamics of PDE, 7 (2010), 161-173. doi: 10.4310/DPDE.2010.v7.n2.a2.
[13]	J. Duan and V. J. Ervin, Dynamics of a nonlocal Kuramoto-Sivashinsky equation, J. Differential Equations, 143 (1998), 243-266. doi: 10.1006/jdeq.1997.3371.
[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), 1631-1634. doi: 10.1103/PhysRevE.56.1631.
[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), 197-226.
[16]	M. Frankel and V. Roytburd, Stability for a class of nonlinear pseudo-differential equations, Appl. Math. Lett., 21 (2008), 425-430. doi: 10.1016/j.aml.2007.03.023.
[17]	L. Giacomelli and F. Otto, New bounds for the Kuramoto-Sivashinsky equation, Commun. Pure Appl. Math., 58 (2005), 297-318. doi: 10.1002/cpa.20031.
[18]	J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems, Commun. Pure Appl. Math., 47 (1994), 293-306. doi: 10.1002/cpa.3160470304.
[19]	R. Grauer, An energy estimate for a perturbed Hasegawa-Mima equation, Nonlinearity, 11 (1998), 659-666. doi: 10.1088/0951-7715/11/3/014.
[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), 557-580. doi: 10.3934/dcds.2004.10.557.
[21]	G. M. Homsy, Model equations for wavy viscous film flow, Lect. Appl. Math., 15 (1974), 191-194.
[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), 265-292. doi: 10.1016/0167-2789(86)90136-3.
[23]	Y. S. Il'yashenko, Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation, J. Dyn. Diff. Eq., 4 (1992), 585-615. doi: 10.1007/BF01048261.
[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), 31-66.
[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), 760-790. doi: 10.1137/0150045.
[26]	Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356-369. doi: 10.1143/PTP.55.356.
[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), 391-394. doi: 10.1103/PhysRevLett.34.391.
[28]	P. Manneville, Liapounov exponents for the Kuramoto-Sivashinsky model, in Macroscopic Modelling of Turbulent Flows, (eds. U. Frisch, J. Keller, G. Papanicolaou and O. Pironneau), vol. 230 of Lecture Notes in Physics, Springer-Verlag, Berlin Heidelberg, (1985), 319-326. doi: 10.1007/3-540-15644-5_26.
[29]	P. C. Matthews and S. M. Cox, One-dimensional pattern formation with Galilean invariance near a stationary bifurcation, Phys. Rev. E, 62 (2000), R1473-R1476. doi: 10.1103/PhysRevE.62.R1473.
[30]	D. Michelson, Steady solutions of the Kuramoto-Sivashinsky equation, Physica D, 19 (1986), 89-111. doi: 10.1016/0167-2789(86)90055-2.
[31]	C. Misbah and A. Valance, Secondary instabilities in the stabilized Kuramoto-Sivashinsky equation, Phys. Rev. E, 49 (1994), 166-183. doi: 10.1103/PhysRevE.49.166.
[32]	L. Molinet, Local dissipativity in $l^2$ for the Kuramoto-Sivashinsky equation in spatial dimension 2, J. Dyn. Diff. Eqns., 12 (2000), 533-556. doi: 10.1023/A:1026459527446.
[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), 155-183. doi: 10.1016/0167-2789(85)90056-9.
[34]	A. Novick-Cohen, Interfacial instabilities in directional solidification of dilute binary alloys: The Kuramoto-Sivashinsky equation, Physica D, 26 (1987), 403-410. doi: 10.1016/0167-2789(87)90240-5.
[35]	F. Otto, Optimal bounds on the Kuramoto-Sivashinsky equation, J. Funct. Anal., 257 (2009), 2188-2245. doi: 10.1016/j.jfa.2009.01.034.
[36]	F. C. Pinto, Nonlinear stability and dynamical properties for a Kuramoto-Sivashinsky equation in space dimension two, Discrete Contin. Dynam. Systems, 5 (1999), 117-136. doi: 10.3934/dcds.1999.5.117.
[37]	Y. Pomeau and P. Manneville, Wavelength selection in cellular flows, Phys. Lett. A, 75 (1980), 296-298. doi: 10.1016/0375-9601(80)90568-X.
[38]	Y. Pomeau and S. Zaleski, Wavelength selection in one-dimensional cellular structures, J. Physique, 42 (1981), 515-528. doi: 10.1051/jphys:01981004204051500.
[39]	J. D. M. Rademacher and R. W. Wittenberg, Viscous shocks in the destabilized Kuramoto-Sivashinsky equation, ASME J. Comput. Nonlinear Dynamics, 1 (2006), 336-347. doi: 10.1115/1.2338656.
[40]	G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames-I. Derivation of basic equations, Acta Astron., 4 (1977), 1177-1206. doi: 10.1016/0094-5765(77)90096-0.
[41]	M. Stanislavova and A. Stefanov, Asymptotic estimates and stability analysis of Kuramoto-Sivashinsky type models, J. Evol. Equ., 11 (2011), 605-635, Erratum, J. Evol. Equ. 11 (2011), 637-639. doi: 10.1007/s00028-011-0103-5.
[42]	D. Tanaka, Chemical turbulence equivalent to Nikolavskii turbulence, Phys. Rev. E, 70 (2004), 015202(R). doi: 10.1103/PhysRevE.70.015202.
[43]	D. Tanaka, Critical exponents of Nikolaevskii turbulence, Phys. Rev. E, 71 (2005), 025203(R). doi: 10.1103/PhysRevE.71.025203.
[44]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, no. 68 in Applied Mathematical Sciences, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[45]	M. I. Tribel'skiĭ, Short-wavelength instability and transition to chaos in distributed systems with additional symmetry, Usp. Fiz. Nauk, 40 (1997), 159-190. doi: 10.1070/PU1997v040n02ABEH000193.
[46]	M. I. Tribelsky and K. Tsuboi, New scenario for transition to turbulence?, Phys. Rev. Lett., 76 (1996), 1631-1634. doi: 10.1103/PhysRevLett.76.1631.
[47]	M. I. Tribelsky and M. G. Velarde, Short-wavelength instability in systems with slow long-wavelength dynamics, Phys. Rev. E, 54 (1996), 4973-4981. doi: 10.1103/PhysRevE.54.4973.
[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), 677-703. doi: 10.1017/S0956792506006760.
[49]	R. W. Wittenberg, Dissipativity, analyticity and viscous shocks in the (de)stabilized Kuramoto-Sivashinsky equation, Phys. Lett. A, 300 (2002), 407-416. doi: 10.1016/S0375-9601(02)00861-7.
[50]	R. W. Wittenberg and P. Holmes, Scale and space localization in the Kuramoto-Sivashinsky equation, Chaos, 9 (1999), 452-465. doi: 10.1063/1.166419.
[51]	R. W. Wittenberg and K.-F. Poon, Anomalous scaling on a spatiotemporally chaotic attractor, Phys. Rev. E, 79 (2009), 056225. doi: 10.1103/PhysRevE.79.056225.

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), 7408-7419. doi: 10.1103/PhysRevB.47.7408.
[2]	I. A. Beresnev and V. N. Nikolaevskiy, A model for nonlinear seismic-waves in a medium with instability, Physica D, 66 (1993), 1-6. doi: 10.1016/0167-2789(93)90217-O.
[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), 301-316. doi: 10.4171/IFB/145.
[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), 1362-1368. doi: 10.1016/j.nonrwa.2011.10.012.
[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), 701-707. doi: 10.3934/dcds.2007.18.701.
[6]	J. C. Bronski and T. N. Gambill, Uncertainty estimates and $L_2$ bounds for the Kuramoto-Sivashinsky equation, Nonlinearity, 19 (2006), 2023-2039. doi: 10.1088/0951-7715/19/9/002.
[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), 017204. doi: 10.1103/PhysRevE.76.017204.
[8]	H. Chaté and P. Manneville, Transition to turbulence via spatiotemporal intermittency, Phys. Rev. Lett., 58 (1987), 112-115. doi: 10.1103/PhysRevLett.58.112.
[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), 203-214. doi: 10.1007/BF02097064.
[10]	P. Collet, J.-P. Eckmann, H. Epstein and J. Stubbe, Analyticity for the Kuramoto-Sivashinsky equation, Physica D, 67 (1993), 321-326. doi: 10.1016/0167-2789(93)90168-Z.
[11]	S. M. Cox and P. C. Matthews, Pattern formation in the damped Nikolaevskiy equation, Phys. Rev. E, 76 (2007), 056202, 11pp. doi: 10.1103/PhysRevE.76.056202.
[12]	A. Demirkaya and M. Stanislavova, Long time behavior for radially symmetric solutions of the Kuramoto-Sivashinsky equation, Dynamics of PDE, 7 (2010), 161-173. doi: 10.4310/DPDE.2010.v7.n2.a2.
[13]	J. Duan and V. J. Ervin, Dynamics of a nonlocal Kuramoto-Sivashinsky equation, J. Differential Equations, 143 (1998), 243-266. doi: 10.1006/jdeq.1997.3371.
[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), 1631-1634. doi: 10.1103/PhysRevE.56.1631.
[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), 197-226.
[16]	M. Frankel and V. Roytburd, Stability for a class of nonlinear pseudo-differential equations, Appl. Math. Lett., 21 (2008), 425-430. doi: 10.1016/j.aml.2007.03.023.
[17]	L. Giacomelli and F. Otto, New bounds for the Kuramoto-Sivashinsky equation, Commun. Pure Appl. Math., 58 (2005), 297-318. doi: 10.1002/cpa.20031.
[18]	J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems, Commun. Pure Appl. Math., 47 (1994), 293-306. doi: 10.1002/cpa.3160470304.
[19]	R. Grauer, An energy estimate for a perturbed Hasegawa-Mima equation, Nonlinearity, 11 (1998), 659-666. doi: 10.1088/0951-7715/11/3/014.
[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), 557-580. doi: 10.3934/dcds.2004.10.557.
[21]	G. M. Homsy, Model equations for wavy viscous film flow, Lect. Appl. Math., 15 (1974), 191-194.
[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), 265-292. doi: 10.1016/0167-2789(86)90136-3.
[23]	Y. S. Il'yashenko, Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation, J. Dyn. Diff. Eq., 4 (1992), 585-615. doi: 10.1007/BF01048261.
[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), 31-66.
[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), 760-790. doi: 10.1137/0150045.
[26]	Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356-369. doi: 10.1143/PTP.55.356.
[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), 391-394. doi: 10.1103/PhysRevLett.34.391.
[28]	P. Manneville, Liapounov exponents for the Kuramoto-Sivashinsky model, in Macroscopic Modelling of Turbulent Flows, (eds. U. Frisch, J. Keller, G. Papanicolaou and O. Pironneau), vol. 230 of Lecture Notes in Physics, Springer-Verlag, Berlin Heidelberg, (1985), 319-326. doi: 10.1007/3-540-15644-5_26.
[29]	P. C. Matthews and S. M. Cox, One-dimensional pattern formation with Galilean invariance near a stationary bifurcation, Phys. Rev. E, 62 (2000), R1473-R1476. doi: 10.1103/PhysRevE.62.R1473.
[30]	D. Michelson, Steady solutions of the Kuramoto-Sivashinsky equation, Physica D, 19 (1986), 89-111. doi: 10.1016/0167-2789(86)90055-2.
[31]	C. Misbah and A. Valance, Secondary instabilities in the stabilized Kuramoto-Sivashinsky equation, Phys. Rev. E, 49 (1994), 166-183. doi: 10.1103/PhysRevE.49.166.
[32]	L. Molinet, Local dissipativity in $l^2$ for the Kuramoto-Sivashinsky equation in spatial dimension 2, J. Dyn. Diff. Eqns., 12 (2000), 533-556. doi: 10.1023/A:1026459527446.
[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), 155-183. doi: 10.1016/0167-2789(85)90056-9.
[34]	A. Novick-Cohen, Interfacial instabilities in directional solidification of dilute binary alloys: The Kuramoto-Sivashinsky equation, Physica D, 26 (1987), 403-410. doi: 10.1016/0167-2789(87)90240-5.
[35]	F. Otto, Optimal bounds on the Kuramoto-Sivashinsky equation, J. Funct. Anal., 257 (2009), 2188-2245. doi: 10.1016/j.jfa.2009.01.034.
[36]	F. C. Pinto, Nonlinear stability and dynamical properties for a Kuramoto-Sivashinsky equation in space dimension two, Discrete Contin. Dynam. Systems, 5 (1999), 117-136. doi: 10.3934/dcds.1999.5.117.
[37]	Y. Pomeau and P. Manneville, Wavelength selection in cellular flows, Phys. Lett. A, 75 (1980), 296-298. doi: 10.1016/0375-9601(80)90568-X.
[38]	Y. Pomeau and S. Zaleski, Wavelength selection in one-dimensional cellular structures, J. Physique, 42 (1981), 515-528. doi: 10.1051/jphys:01981004204051500.
[39]	J. D. M. Rademacher and R. W. Wittenberg, Viscous shocks in the destabilized Kuramoto-Sivashinsky equation, ASME J. Comput. Nonlinear Dynamics, 1 (2006), 336-347. doi: 10.1115/1.2338656.
[40]	G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames-I. Derivation of basic equations, Acta Astron., 4 (1977), 1177-1206. doi: 10.1016/0094-5765(77)90096-0.
[41]	M. Stanislavova and A. Stefanov, Asymptotic estimates and stability analysis of Kuramoto-Sivashinsky type models, J. Evol. Equ., 11 (2011), 605-635, Erratum, J. Evol. Equ. 11 (2011), 637-639. doi: 10.1007/s00028-011-0103-5.
[42]	D. Tanaka, Chemical turbulence equivalent to Nikolavskii turbulence, Phys. Rev. E, 70 (2004), 015202(R). doi: 10.1103/PhysRevE.70.015202.
[43]	D. Tanaka, Critical exponents of Nikolaevskii turbulence, Phys. Rev. E, 71 (2005), 025203(R). doi: 10.1103/PhysRevE.71.025203.
[44]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, no. 68 in Applied Mathematical Sciences, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[45]	M. I. Tribel'skiĭ, Short-wavelength instability and transition to chaos in distributed systems with additional symmetry, Usp. Fiz. Nauk, 40 (1997), 159-190. doi: 10.1070/PU1997v040n02ABEH000193.
[46]	M. I. Tribelsky and K. Tsuboi, New scenario for transition to turbulence?, Phys. Rev. Lett., 76 (1996), 1631-1634. doi: 10.1103/PhysRevLett.76.1631.
[47]	M. I. Tribelsky and M. G. Velarde, Short-wavelength instability in systems with slow long-wavelength dynamics, Phys. Rev. E, 54 (1996), 4973-4981. doi: 10.1103/PhysRevE.54.4973.
[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), 677-703. doi: 10.1017/S0956792506006760.
[49]	R. W. Wittenberg, Dissipativity, analyticity and viscous shocks in the (de)stabilized Kuramoto-Sivashinsky equation, Phys. Lett. A, 300 (2002), 407-416. doi: 10.1016/S0375-9601(02)00861-7.
[50]	R. W. Wittenberg and P. Holmes, Scale and space localization in the Kuramoto-Sivashinsky equation, Chaos, 9 (1999), 452-465. doi: 10.1063/1.166419.
[51]	R. W. Wittenberg and K.-F. Poon, Anomalous scaling on a spatiotemporally chaotic attractor, Phys. Rev. E, 79 (2009), 056225. doi: 10.1103/PhysRevE.79.056225.

[1]	D. Hilhorst, L. A. Peletier, A. I. Rotariu, G. Sivashinsky. Global attractor and inertial sets for a nonlocal Kuramoto-Sivashinsky equation. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 557-580. doi: 10.3934/dcds.2004.10.557
[2]	L. Dieci, M. S Jolly, Ricardo Rosa, E. S. Van Vleck. Error in approximation of Lyapunov exponents on inertial manifolds: The Kuramoto-Sivashinsky equation. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 555-580. doi: 10.3934/dcdsb.2008.9.555
[3]	Aslihan Demirkaya. The existence of a global attractor for a Kuramoto-Sivashinsky type equation in 2D. Conference Publications, 2009, 2009 (Special) : 198-207. doi: 10.3934/proc.2009.2009.198
[4]	Peng Gao. Global exact controllability to the trajectories of the Kuramoto-Sivashinsky equation. Evolution Equations and Control Theory, 2020, 9 (1) : 181-191. doi: 10.3934/eect.2020002
[5]	Kiah Wah Ong. Dynamic transitions of generalized Kuramoto-Sivashinsky equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1225-1236. doi: 10.3934/dcdsb.2016.21.1225
[6]	Milena Stanislavova, Atanas Stefanov. Effective estimates of the higher Sobolev norms for the Kuramoto-Sivashinsky equation. Conference Publications, 2009, 2009 (Special) : 729-738. doi: 10.3934/proc.2009.2009.729
[7]	Jared C. Bronski, Razvan C. Fetecau, Thomas N. Gambill. A note on a non-local Kuramoto-Sivashinsky equation. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 701-707. doi: 10.3934/dcds.2007.18.701
[8]	Peng Gao. Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5649-5684. doi: 10.3934/dcds.2018247
[9]	Eduardo Cerpa. Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation. Communications on Pure and Applied Analysis, 2010, 9 (1) : 91-102. doi: 10.3934/cpaa.2010.9.91
[10]	Piotr Zgliczyński. Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof. Journal of Computational Dynamics, 2015, 2 (1) : 95-142. doi: 10.3934/jcd.2015.2.95
[11]	Yuncherl Choi, Jongmin Han, Chun-Hsiung Hsia. Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1933-1957. doi: 10.3934/dcdsb.2015.20.1933
[12]	Shuting Chen, Zengji Du, Jiang Liu, Ke Wang. The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1471-1496. doi: 10.3934/dcdsb.2021098
[13]	Peng Gao. Null controllability with constraints on the state for the 1-D Kuramoto-Sivashinsky equation. Evolution Equations and Control Theory, 2015, 4 (3) : 281-296. doi: 10.3934/eect.2015.4.281
[14]	Fred C. Pinto. Nonlinear stability and dynamical properties for a Kuramoto-Sivashinsky equation in space dimension two. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 117-136. doi: 10.3934/dcds.1999.5.117
[15]	David Massatt. On the well-posedness of the anisotropically-reduced two-dimensional Kuramoto-Sivashinsky Equation. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2021305
[16]	Mudassar Imran, Youssef Raffoul, Muhammad Usman, Chi Zhang. A study of bifurcation parameters in travelling wave solutions of a damped forced Korteweg de Vries-Kuramoto Sivashinsky type equation. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 691-705. doi: 10.3934/dcdss.2018043
[17]	Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080
[18]	Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345
[19]	Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6207-6228. doi: 10.3934/dcdsb.2021015
[20]	Milena Stanislavova. On the global attractor for the damped Benjamin-Bona-Mahony equation. Conference Publications, 2005, 2005 (Special) : 824-832. doi: 10.3934/proc.2005.2005.824

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Ralf W. Wittenberg
on Google Scholar
Ralf W. Wittenberg

[Back to Top]