January  2011, 29(1): 327-341. doi: 10.3934/dcds.2011.29.327

Global dissipativity and inertial manifolds for diffusive burgers equations with low-wavenumber instability

1. 

Department of Mathematics, 1S-208, The City University of New York – CSI, 2800 Victory Boulevard, Staten Island, NY 10314, United States

Received  August 2009 Revised  April 2010 Published  September 2010

The global well-posedness, the existence of globally absorbing sets and the existence of inertial manifolds are investigated in a class of diffusive (viscous) Burgers equations. The class includes diffusive Burgers equation with nontrivial forcing, Burgers-Sivashinsky equation and Quasi-Steady equation of cellular flames. Global dissipativity is proven in two space dimensions for periodic boundary conditions. For the proof of the existence of inertial manifolds, the spectral-gap condition, which Burgers-type equations do not satisfy in their original form, is circumvented by employing the Cole-Hopf transform. The procedure is valid in both one and two space dimensions.
Citation: Jesenko Vukadinovic. Global dissipativity and inertial manifolds for diffusive burgers equations with low-wavenumber instability. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 327-341. doi: 10.3934/dcds.2011.29.327
References:
[1]

H. Berestycki, S. Kamin and G. I. Sivashinsky, Metastability in a flame front evolution equation, Interfaces Free Bound., 3 (2001), 361-392. doi: doi:10.4171/IFB/45.  Google Scholar

[2]

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: doi:10.4171/IFB/145.  Google Scholar

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C.-M. Brauner, M. Frankel, J. Hulshof and G. I. Sivashinsky, Weakly nonlinear asymptotics of the $\kappa-\theta$ model of cellular flames: The QS equation, Interfaces Free Bound., 7 (2005), 131-146. doi: doi:10.4171/IFB/117.  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), 2023-2039. doi: doi:10.1088/0951-7715/19/9/002.  Google Scholar

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S. N. Chow, K. Lu and G. R. Sell, Smoothness of inertial manifolds, J. Math. Anal. Appl., 169 (1992), 283-312. doi: doi:10.1016/0022-247X(92)90115-T.  Google Scholar

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P. Collet, J. P. Eckmann, H. Epstein and J. Stubbe, Globally attracting set for the Kuramoto-Sivashinsky equation, Commun. Math. Phys., 152 (1993), 203-214. doi: doi:10.1007/BF02097064.  Google Scholar

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P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Spectral barriers and inertial manifolds for dissipative partial differential equations, J. Dynam. Differential Equations, 1 (1988), 45-73. doi: doi:10.1007/BF01048790.  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), 197-226.  Google Scholar

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C. Foias, G. R. Sell and R. Temam, Variétés inertielles des équations differeéntielles dissipatives, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 285-288.  Google Scholar

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C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Diff. Eq., 73 (1988) 309-353. doi: doi:10.1016/0022-0396(88)90110-6.  Google Scholar

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M. Frankel, P. V. Gordon and G. I. Sivashinsky, On disintegration of near-limit cellular flames, Phys. Lett. A, 310 (2003), 389-392. doi: doi:10.1016/S0375-9601(03)00385-2.  Google Scholar

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M. Frankel and V. Roytburd, Dissipative dynamics for a class of nonlinear pseudo-differential equations, J. Evol. Equ., 8 (2008), 491-512. doi: doi:10.1007/s00028-008-0373-8.  Google Scholar

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L. Giacomelli and F. Otto, New bounds for the Kuramoto-Sivashinsky equation, Commun. Pure Appl. Math., 58 (2005), 297-318. doi: doi:10.1002/cpa.20031.  Google Scholar

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I. Kukavica, On Fourier parameterization of global attractors for equations in one space dimension, Discrete Cont. Dyn. Syst., 13 (2005), 553-560. doi: doi:10.3934/dcds.2005.13.553.  Google Scholar

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M. Kwak, Finite dimensional description of convective reaction-diffusion equations, J. Dynam. Differential Equations, 4 (1992), 515-543. doi: doi:10.1007/BF01053808.  Google Scholar

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J. Mallet-Paret and G. R. Sell, Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Amer. Math. Soc., 1 (1988), 805-866.  Google Scholar

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J. Mallet-Paret, G. R. Sell and Z. Shao, Obstructions to the existence of normally hyperbolic inertial manifolds, Indiana Univ. Math. J., 42 (1993), 1027-1055. doi: doi:10.1512/iumj.1993.42.42048.  Google Scholar

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L. Molinet, A bounded global absorbing set for the Burgers-Sivashinsky equation in space dimension two, C.R. Acad. Sci. Paris Sér. I Math., 330 (2000), 635-640. doi: doi:10.1016/S0764-4442(00)00224-X.  Google Scholar

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L. Molinet, Local dissipativity in $L^2$ for the Kuramoto-Sivashinsky equation in spatial dimension 2, J. Dynam. Differential Equations, 12 (2000), 533-556. doi: doi:10.1023/A:1026459527446.  Google Scholar

[23]

F. Otto, Optimal bounds on the Kuramoto-Sivashinsky equation, J. Funct. Anal., 257 (2009), 2188-2245. doi: doi:10.1016/j.jfa.2009.01.034.  Google Scholar

[24]

I. Richards, On the gaps between numbers which are the sum of two squares, Adv. in Math., 46 (1982), 1-2. doi: doi:10.1016/0001-8708(82)90051-2.  Google Scholar

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J. C. Robinson, Inertial manifolds and the cone condition, Dyn. Systems Appl., 2 (1993), 311-330.  Google Scholar

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J. C. Robinson, A concise proof of the geometric construction of inertial manifolds, Phys. Lett. A, 200 (1995), 415-417. doi: doi:10.1016/0375-9601(95)00231-Q.  Google Scholar

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J. C. Robinson, "Infinite-Dimensional Dynamical Systems," Cambridge University Press, Cambridge Texts in Applied Mathematics, Cambridge, 2001.  Google Scholar

[28]

G. I. Sivashinsky, Nonlinear analysis of hydrodynamics instability in laminar flames Part I. Derivation of basic equations, Acta Astronaut., 4 (1977), 1177-1206. doi: doi:10.1016/0094-5765(77)90096-0.  Google Scholar

[29]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, Applied Math. Sciences 68, New York, 1988.  Google Scholar

[30]

R. Temam, Inertial manifolds, Math. Intelligencer, 12 (1990), 68-73. doi: doi:10.1007/BF03024036.  Google Scholar

[31]

J. Vukadinovic, Finite-dimensional description of the long-term dynamics for the Doi-Hess model for rodlike nematic polymers in shear flows, Commun. Math. Sci., 6 (2008), 975-993.  Google Scholar

[32]

J. Vukadinovic, Inertial manifolds for a Smoluchowski equation on the unit sphere, Commun. Math. Phys., 285 (2009), 975-990. doi: doi:10.1007/s00220-008-0460-2.  Google Scholar

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J. Vukadinovic, Inertial manifolds for a Smoluchowski equation on a circle, Nonlinearity, 21 (2008), 1533-1545. doi: doi:10.1088/0951-7715/21/7/009.  Google Scholar

[34]

J. H. Wells and L. R. Williams, "Embeddings and Extensions in Analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete," Springer Verlag, New York-Hedelberg, 1975.  Google Scholar

show all references

References:
[1]

H. Berestycki, S. Kamin and G. I. Sivashinsky, Metastability in a flame front evolution equation, Interfaces Free Bound., 3 (2001), 361-392. doi: doi:10.4171/IFB/45.  Google Scholar

[2]

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: doi:10.4171/IFB/145.  Google Scholar

[3]

C.-M. Brauner, M. Frankel, J. Hulshof and G. I. Sivashinsky, Weakly nonlinear asymptotics of the $\kappa-\theta$ model of cellular flames: The QS equation, Interfaces Free Bound., 7 (2005), 131-146. doi: doi:10.4171/IFB/117.  Google Scholar

[4]

J. C. Bronski and T. N. Gambill, Uncertainty estimates and $L_2$ bounds for the Kuramoto-Sivashinsky equation, Nonlinearity, 19 (2006), 2023-2039. doi: doi:10.1088/0951-7715/19/9/002.  Google Scholar

[5]

S. N. Chow, K. Lu and G. R. Sell, Smoothness of inertial manifolds, J. Math. Anal. Appl., 169 (1992), 283-312. doi: doi:10.1016/0022-247X(92)90115-T.  Google Scholar

[6]

P. Collet, J. P. Eckmann, H. Epstein and J. Stubbe, Globally attracting set for the Kuramoto-Sivashinsky equation, Commun. Math. Phys., 152 (1993), 203-214. doi: doi:10.1007/BF02097064.  Google Scholar

[7]

P. Constantin, C. Foias, B. Nicolaenko and R. Temam, "Integral and Inertial Manifolds for Dissipative Partial Differential Equations," Springer-Verlag, Applied Math. Sciences 70, New York, 1989.  Google Scholar

[8]

P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Spectral barriers and inertial manifolds for dissipative partial differential equations, J. Dynam. Differential Equations, 1 (1988), 45-73. doi: doi:10.1007/BF01048790.  Google Scholar

[9]

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

[10]

C. Foias, G. R. Sell and R. Temam, Variétés inertielles des équations differeéntielles dissipatives, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 285-288.  Google Scholar

[11]

C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Diff. Eq., 73 (1988) 309-353. doi: doi:10.1016/0022-0396(88)90110-6.  Google Scholar

[12]

M. Frankel, P. V. Gordon and G. I. Sivashinsky, On disintegration of near-limit cellular flames, Phys. Lett. A, 310 (2003), 389-392. doi: doi:10.1016/S0375-9601(03)00385-2.  Google Scholar

[13]

M. Frankel and V. Roytburd, Dissipative dynamics for a class of nonlinear pseudo-differential equations, J. Evol. Equ., 8 (2008), 491-512. doi: doi:10.1007/s00028-008-0373-8.  Google Scholar

[14]

L. Giacomelli and F. Otto, New bounds for the Kuramoto-Sivashinsky equation, Commun. Pure Appl. Math., 58 (2005), 297-318. doi: doi:10.1002/cpa.20031.  Google Scholar

[15]

J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems, Commun. Pure Appl. Math., 47 (1994), 293-306. doi: doi:10.1002/cpa.3160470304.  Google Scholar

[16]

I. Kukavica, On Fourier parameterization of global attractors for equations in one space dimension, Discrete Cont. Dyn. Syst., 13 (2005), 553-560. doi: doi:10.3934/dcds.2005.13.553.  Google Scholar

[17]

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: doi:10.1143/PTP.55.356.  Google Scholar

[18]

M. Kwak, Finite dimensional description of convective reaction-diffusion equations, J. Dynam. Differential Equations, 4 (1992), 515-543. doi: doi:10.1007/BF01053808.  Google Scholar

[19]

J. Mallet-Paret and G. R. Sell, Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Amer. Math. Soc., 1 (1988), 805-866.  Google Scholar

[20]

J. Mallet-Paret, G. R. Sell and Z. Shao, Obstructions to the existence of normally hyperbolic inertial manifolds, Indiana Univ. Math. J., 42 (1993), 1027-1055. doi: doi:10.1512/iumj.1993.42.42048.  Google Scholar

[21]

L. Molinet, A bounded global absorbing set for the Burgers-Sivashinsky equation in space dimension two, C.R. Acad. Sci. Paris Sér. I Math., 330 (2000), 635-640. doi: doi:10.1016/S0764-4442(00)00224-X.  Google Scholar

[22]

L. Molinet, Local dissipativity in $L^2$ for the Kuramoto-Sivashinsky equation in spatial dimension 2, J. Dynam. Differential Equations, 12 (2000), 533-556. doi: doi:10.1023/A:1026459527446.  Google Scholar

[23]

F. Otto, Optimal bounds on the Kuramoto-Sivashinsky equation, J. Funct. Anal., 257 (2009), 2188-2245. doi: doi:10.1016/j.jfa.2009.01.034.  Google Scholar

[24]

I. Richards, On the gaps between numbers which are the sum of two squares, Adv. in Math., 46 (1982), 1-2. doi: doi:10.1016/0001-8708(82)90051-2.  Google Scholar

[25]

J. C. Robinson, Inertial manifolds and the cone condition, Dyn. Systems Appl., 2 (1993), 311-330.  Google Scholar

[26]

J. C. Robinson, A concise proof of the geometric construction of inertial manifolds, Phys. Lett. A, 200 (1995), 415-417. doi: doi:10.1016/0375-9601(95)00231-Q.  Google Scholar

[27]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems," Cambridge University Press, Cambridge Texts in Applied Mathematics, Cambridge, 2001.  Google Scholar

[28]

G. I. Sivashinsky, Nonlinear analysis of hydrodynamics instability in laminar flames Part I. Derivation of basic equations, Acta Astronaut., 4 (1977), 1177-1206. doi: doi:10.1016/0094-5765(77)90096-0.  Google Scholar

[29]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, Applied Math. Sciences 68, New York, 1988.  Google Scholar

[30]

R. Temam, Inertial manifolds, Math. Intelligencer, 12 (1990), 68-73. doi: doi:10.1007/BF03024036.  Google Scholar

[31]

J. Vukadinovic, Finite-dimensional description of the long-term dynamics for the Doi-Hess model for rodlike nematic polymers in shear flows, Commun. Math. Sci., 6 (2008), 975-993.  Google Scholar

[32]

J. Vukadinovic, Inertial manifolds for a Smoluchowski equation on the unit sphere, Commun. Math. Phys., 285 (2009), 975-990. doi: doi:10.1007/s00220-008-0460-2.  Google Scholar

[33]

J. Vukadinovic, Inertial manifolds for a Smoluchowski equation on a circle, Nonlinearity, 21 (2008), 1533-1545. doi: doi:10.1088/0951-7715/21/7/009.  Google Scholar

[34]

J. H. Wells and L. R. Williams, "Embeddings and Extensions in Analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete," Springer Verlag, New York-Hedelberg, 1975.  Google Scholar

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