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 - A, 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. doi: doi:10.4171/IFB/45.

[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. doi: doi:10.4171/IFB/145.

[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. doi: doi:10.4171/IFB/117.

[4]

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

[5]

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

[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. doi: doi:10.1007/BF02097064.

[7]

P. Constantin, C. Foias, B. Nicolaenko and R. Temam, "Integral and Inertial Manifolds for Dissipative Partial Differential Equations,", Springer-Verlag, 70 (1989).

[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. doi: doi:10.1007/BF01048790.

[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.

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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.

[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. doi: doi:10.1512/iumj.1993.42.42048.

[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. doi: doi:10.1016/S0764-4442(00)00224-X.

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems,", Cambridge University Press, (2001).

[28]

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

[29]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", Springer-Verlag, 68 (1988).

[30]

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

[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.

[32]

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

[33]

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

[34]

J. H. Wells and L. R. Williams, "Embeddings and Extensions in Analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete,", Springer Verlag, (1975).

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. doi: doi:10.4171/IFB/45.

[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. doi: doi:10.4171/IFB/145.

[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. doi: doi:10.4171/IFB/117.

[4]

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

[5]

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

[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. doi: doi:10.1007/BF02097064.

[7]

P. Constantin, C. Foias, B. Nicolaenko and R. Temam, "Integral and Inertial Manifolds for Dissipative Partial Differential Equations,", Springer-Verlag, 70 (1989).

[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. doi: doi:10.1007/BF01048790.

[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.

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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.

[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. doi: doi:10.1512/iumj.1993.42.42048.

[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. doi: doi:10.1016/S0764-4442(00)00224-X.

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

J. C. Robinson, "Infinite-Dimensional Dynamical Systems,", Cambridge University Press, (2001).

[28]

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

[29]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", Springer-Verlag, 68 (1988).

[30]

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

[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.

[32]

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

[33]

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

[34]

J. H. Wells and L. R. Williams, "Embeddings and Extensions in Analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete,", Springer Verlag, (1975).

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