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February  2011, 4(1): 83-99. doi: 10.3934/dcdss.2011.4.83

Dissipativity for a semi-linearized system modeling cellular flames

Michael Frankel 1, , Victor Roytburd 2, and  Gregory I. Sivashinsky 3,

1. 

Department of Mathematical Sciences, IUPUI, Indianapolis, IN 46202-3216, United States
2. 

Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180-3590
3. 

School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Received  November 2008 Revised  September 2009 Published  October 2010

We study a Semi-Linearized System (SLS) of second order PDEs modeling flame front dynamics. SLS is a simplified version of the weak $\kappa\theta$ model of cellular flames which is dynamically similar to the Kuramoto-Sivashinsky (KS) equation [7, 4]. We prove existence of the solutions at large, and their proximity, for finite time, to the solutions of KS. We demonstrate that SLS possesses a universal absorbing set and a compact attractor. Furthermore, we show that the attractor is of finite Hausdorff dimension.
Keywords: Hausdorff dimension., dissipative systems, Kuramoto-Sivashinsky equation, attractors.
Mathematics Subject Classification: Primary: 35K55, 35B25; Secondary: 80A2.
Citation: Michael Frankel, Victor Roytburd, Gregory I. Sivashinsky. Dissipativity for a semi-linearized system modeling cellular flames. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 83-99. doi: 10.3934/dcdss.2011.4.83
References:
[1]

C.-M. Brauner and A. Lunardi, Instabilities in a two-dimensional combustion model with free boundary, Arch. Rational. Mech. Anal., 154 (2000), 157-182. doi: doi:10.1007/s002050000099.

[2]

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.

[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: doi:10.4171/IFB/145.

[4]

C.-M. Brauner, M. Frankel, J. Hulshof, A. Lunardy and G. I. Sivashinsky, On the model of cellular flames: Existence in the large and asymptotics, DCDS-S, 1 (2008), 27-39.

[5]

C. Foias and G. Prodi, Sur le comportement global des solutions non stationnaire des équations Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.

[6]

M. L. Frankel and G. I. Sivashinsky, On the nonlinear thermal-diffusive theory of curved flames, J. Physique, 48 (1987), 25-28.

[7]

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.

[8]

M. Frankel and V.Roytburd, Numerical study of the semi-linearized system modeling cellular flames, In preparation (2009).

[9]

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

[10]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, New York 1981.

[11]

D. Hilhorst, L. A. Peletier, A. I. Rotariu and G. I. Sivashinsky, Global attractor and inertial sets for a non-local Kuramoto-Sivashinsky equation, DCDS-A, 10 (2004), 557-580.

[12]

Y. Kuramoto, Diffusion induced chaos in reactions systems, Progr. Theoret. Phys. Suppl., 64 (1978), 346-367. doi: doi:10.1143/PTPS.64.346.

[13]

B. Malomed, Bao-Feng Feng and T. Kawahara, Stabilized Kuramoto-Sivashinsky system, Phys. Rev. E, 64 (2001), 046304.

[14]

B. J. Matkowsky and G. I. Sivashinsky, An asymptotic derivation of two models in flame theory associated with the constant density approximation, SIAM J. Appl. Math., 37 (1979), 696-699. doi: doi:10.1137/0137051.

[15]

A. A. Nepomnyashchy, Stability of wave regimes in a film flowing down on inclined plane, Fluid Dyn., 9, (1974) 354-359. doi: doi:10.1007/BF01025515.

[16]

G. I. Sivashinsky, Nonlinear analysis of hydrodynamic 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.

[17]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Springer-Verlag, New York, 1997.

[18]

J. Topper and T. Kawahara, Approximate equations for long nonlinear waves on a viscous film, J. Phys. Soc. Japan, 44 (1978), 663-666. doi: doi:10.1143/JPSJ.44.663.

show all references

References:
[1]

C.-M. Brauner and A. Lunardi, Instabilities in a two-dimensional combustion model with free boundary, Arch. Rational. Mech. Anal., 154 (2000), 157-182. doi: doi:10.1007/s002050000099.

[2]

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.

[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: doi:10.4171/IFB/145.

[4]

C.-M. Brauner, M. Frankel, J. Hulshof, A. Lunardy and G. I. Sivashinsky, On the model of cellular flames: Existence in the large and asymptotics, DCDS-S, 1 (2008), 27-39.

[5]

C. Foias and G. Prodi, Sur le comportement global des solutions non stationnaire des équations Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.

[6]

M. L. Frankel and G. I. Sivashinsky, On the nonlinear thermal-diffusive theory of curved flames, J. Physique, 48 (1987), 25-28.

[7]

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.

[8]

M. Frankel and V.Roytburd, Numerical study of the semi-linearized system modeling cellular flames, In preparation (2009).

[9]

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

[10]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, New York 1981.

[11]

D. Hilhorst, L. A. Peletier, A. I. Rotariu and G. I. Sivashinsky, Global attractor and inertial sets for a non-local Kuramoto-Sivashinsky equation, DCDS-A, 10 (2004), 557-580.

[12]

Y. Kuramoto, Diffusion induced chaos in reactions systems, Progr. Theoret. Phys. Suppl., 64 (1978), 346-367. doi: doi:10.1143/PTPS.64.346.

[13]

B. Malomed, Bao-Feng Feng and T. Kawahara, Stabilized Kuramoto-Sivashinsky system, Phys. Rev. E, 64 (2001), 046304.

[14]

B. J. Matkowsky and G. I. Sivashinsky, An asymptotic derivation of two models in flame theory associated with the constant density approximation, SIAM J. Appl. Math., 37 (1979), 696-699. doi: doi:10.1137/0137051.

[15]

A. A. Nepomnyashchy, Stability of wave regimes in a film flowing down on inclined plane, Fluid Dyn., 9, (1974) 354-359. doi: doi:10.1007/BF01025515.

[16]

G. I. Sivashinsky, Nonlinear analysis of hydrodynamic 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.

[17]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Springer-Verlag, New York, 1997.

[18]

J. Topper and T. Kawahara, Approximate equations for long nonlinear waves on a viscous film, J. Phys. Soc. Japan, 44 (1978), 663-666. doi: doi:10.1143/JPSJ.44.663.

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