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

Ralf W. Wittenberg

doi:10.3934/dcds.2014.34.5325

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 & 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. 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), 1-6. 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), 301-316. 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), 1362-1368. 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), 701-707. 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), 2023-2039. 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), 017204. 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), 112-115. 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), 203-214. 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), 321-326. 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), 056202, 11pp. 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), 161-173. 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), 243-266. 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), 1631-1634. 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), 197-226. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems, Commun. Pure Appl. Math., 47 (1994), 293-306. doi: 10.1002/cpa.3160470304. Google Scholar
[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. 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), 557-580. 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), 191-194. 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), 265-292. 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), 585-615. 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), 31-66. 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), 760-790. 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), 356-369. 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), 391-394. doi: 10.1103/PhysRevLett.34.391. Google Scholar
[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. 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), R1473-R1476. doi: 10.1103/PhysRevE.62.R1473. Google Scholar
[30]	D. Michelson, Steady solutions of the Kuramoto-Sivashinsky equation, Physica D, 19 (1986), 89-111. 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), 166-183. 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), 533-556. 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), 155-183. 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), 403-410. 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), 2188-2245. 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), 117-136. 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), 296-298. 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), 515-528. 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), 336-347. 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), 1177-1206. 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), 605-635, Erratum, J. Evol. Equ. 11 (2011), 637-639. doi: 10.1007/s00028-011-0103-5. Google Scholar
[42]	D. Tanaka, Chemical turbulence equivalent to Nikolavskii turbulence, Phys. Rev. E, 70 (2004), 015202(R). doi: 10.1103/PhysRevE.70.015202. Google Scholar
[43]	D. Tanaka, Critical exponents of Nikolaevskii turbulence, Phys. Rev. E, 71 (2005), 025203(R). doi: 10.1103/PhysRevE.71.025203. Google Scholar
[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. 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), 159-190. doi: 10.1070/PU1997v040n02ABEH000193. Google Scholar
[46]	M. I. Tribelsky and K. Tsuboi, New scenario for transition to turbulence?, Phys. Rev. Lett., 76 (1996), 1631-1634. 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), 4973-4981. 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), 677-703. 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), 407-416. 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), 452-465. 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), 056225. 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), 7408-7419. 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), 1-6. 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), 301-316. 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), 1362-1368. 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), 701-707. 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), 2023-2039. 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), 017204. 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), 112-115. 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), 203-214. 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), 321-326. 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), 056202, 11pp. 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), 161-173. 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), 243-266. 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), 1631-1634. 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), 197-226. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems, Commun. Pure Appl. Math., 47 (1994), 293-306. doi: 10.1002/cpa.3160470304. Google Scholar
[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. 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), 557-580. 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), 191-194. 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), 265-292. 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), 585-615. 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), 31-66. 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), 760-790. 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), 356-369. 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), 391-394. doi: 10.1103/PhysRevLett.34.391. Google Scholar
[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. 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), R1473-R1476. doi: 10.1103/PhysRevE.62.R1473. Google Scholar
[30]	D. Michelson, Steady solutions of the Kuramoto-Sivashinsky equation, Physica D, 19 (1986), 89-111. 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), 166-183. 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), 533-556. 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), 155-183. 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), 403-410. 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), 2188-2245. 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), 117-136. 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), 296-298. 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), 515-528. 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), 336-347. 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), 1177-1206. 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), 605-635, Erratum, J. Evol. Equ. 11 (2011), 637-639. doi: 10.1007/s00028-011-0103-5. Google Scholar
[42]	D. Tanaka, Chemical turbulence equivalent to Nikolavskii turbulence, Phys. Rev. E, 70 (2004), 015202(R). doi: 10.1103/PhysRevE.70.015202. Google Scholar
[43]	D. Tanaka, Critical exponents of Nikolaevskii turbulence, Phys. Rev. E, 71 (2005), 025203(R). doi: 10.1103/PhysRevE.71.025203. Google Scholar
[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. 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), 159-190. doi: 10.1070/PU1997v040n02ABEH000193. Google Scholar
[46]	M. I. Tribelsky and K. Tsuboi, New scenario for transition to turbulence?, Phys. Rev. Lett., 76 (1996), 1631-1634. 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), 4973-4981. 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), 677-703. 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), 407-416. 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), 452-465. 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), 056225. doi: 10.1103/PhysRevE.79.056225. Google Scholar

[1]	D. Hilhorst, L. A. Peletier, A. I. Rotariu, G. Sivashinsky. Global attractor and inertial sets for a nonlocal Kuramoto-Sivashinsky equation. Discrete & 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 & 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 & Control Theory, 2020, 9 (1) : 181-191. doi: 10.3934/eect.2020002
[5]	Kiah Wah Ong. Dynamic transitions of generalized Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1225-1236. doi: 10.3934/dcdsb.2016.21.1225
[6]	Shuting Chen, Zengji Du, Jiang Liu, Ke Wang. The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021098
[7]	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
[8]	Jared C. Bronski, Razvan C. Fetecau, Thomas N. Gambill. A note on a non-local Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 701-707. doi: 10.3934/dcds.2007.18.701
[9]	Peng Gao. Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5649-5684. doi: 10.3934/dcds.2018247
[10]	Eduardo Cerpa. Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation. Communications on Pure & Applied Analysis, 2010, 9 (1) : 91-102. doi: 10.3934/cpaa.2010.9.91
[11]	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
[12]	Yuncherl Choi, Jongmin Han, Chun-Hsiung Hsia. Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1933-1957. doi: 10.3934/dcdsb.2015.20.1933
[13]	Peng Gao. Null controllability with constraints on the state for the 1-D Kuramoto-Sivashinsky equation. Evolution Equations & 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 & Continuous Dynamical Systems, 1999, 5 (1) : 117-136. doi: 10.3934/dcds.1999.5.117
[15]	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 & Continuous Dynamical Systems - S, 2018, 11 (4) : 691-705. doi: 10.3934/dcdss.2018043
[16]	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
[17]	Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345
[18]	Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6207-6228. doi: 10.3934/dcdsb.2021015
[19]	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
[20]	Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060

2020 Impact Factor: 1.392

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors