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Optimal parameter-dependent bounds for Kuramoto-Sivashinsky-type equations
1. | Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada |
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 |
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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 |
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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 |
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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 |
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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 |
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