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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

 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.
Citation: Michael Frankel, Victor Roytburd, Gregory I. Sivashinsky. Dissipativity for a semi-linearized system modeling cellular flames. Discrete & 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. doi: doi:10.1007/s002050000099. Google Scholar [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. doi: doi:10.4171/IFB/117. 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. doi: doi:10.4171/IFB/145. Google Scholar [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. Google Scholar [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. Google Scholar [6] M. L. Frankel and G. I. Sivashinsky, On the nonlinear thermal-diffusive theory of curved flames,, J. Physique, 48 (1987), 25. Google Scholar [7] 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. Google Scholar [8] M. Frankel and V.Roytburd, Numerical study of the semi-linearized system modeling cellular flames,, In preparation (2009)., (2009). Google Scholar [9] J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems,, Comm. Pure Appl. Math., 47 (1994), 293. doi: doi:10.1002/cpa.3160470304. Google Scholar [10] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981). Google Scholar [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. Google Scholar [12] Y. Kuramoto, Diffusion induced chaos in reactions systems,, Progr. Theoret. Phys. Suppl., 64 (1978), 346. doi: doi:10.1143/PTPS.64.346. Google Scholar [13] B. Malomed, Bao-Feng Feng and T. Kawahara, Stabilized Kuramoto-Sivashinsky system,, Phys. Rev. E, 64 (2001). Google Scholar [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. doi: doi:10.1137/0137051. Google Scholar [15] A. A. Nepomnyashchy, Stability of wave regimes in a film flowing down on inclined plane,, Fluid Dyn., 9 (1974), 354. doi: doi:10.1007/BF01025515. Google Scholar [16] G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames, Part I. Derivation of basic equations,, Acta Astronaut. 4 (1977), 4 (1977), 1177. doi: doi:10.1016/0094-5765(77)90096-0. Google Scholar [17] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition,, Springer-Verlag, (1997). Google Scholar [18] J. Topper and T. Kawahara, Approximate equations for long nonlinear waves on a viscous film,, J. Phys. Soc. Japan, 44 (1978), 663. doi: doi:10.1143/JPSJ.44.663. Google Scholar

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. doi: doi:10.1007/s002050000099. Google Scholar [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. doi: doi:10.4171/IFB/117. 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. doi: doi:10.4171/IFB/145. Google Scholar [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. Google Scholar [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. Google Scholar [6] M. L. Frankel and G. I. Sivashinsky, On the nonlinear thermal-diffusive theory of curved flames,, J. Physique, 48 (1987), 25. Google Scholar [7] 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. Google Scholar [8] M. Frankel and V.Roytburd, Numerical study of the semi-linearized system modeling cellular flames,, In preparation (2009)., (2009). Google Scholar [9] J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems,, Comm. Pure Appl. Math., 47 (1994), 293. doi: doi:10.1002/cpa.3160470304. Google Scholar [10] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981). Google Scholar [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. Google Scholar [12] Y. Kuramoto, Diffusion induced chaos in reactions systems,, Progr. Theoret. Phys. Suppl., 64 (1978), 346. doi: doi:10.1143/PTPS.64.346. Google Scholar [13] B. Malomed, Bao-Feng Feng and T. Kawahara, Stabilized Kuramoto-Sivashinsky system,, Phys. Rev. E, 64 (2001). Google Scholar [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. doi: doi:10.1137/0137051. Google Scholar [15] A. A. Nepomnyashchy, Stability of wave regimes in a film flowing down on inclined plane,, Fluid Dyn., 9 (1974), 354. doi: doi:10.1007/BF01025515. Google Scholar [16] G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames, Part I. Derivation of basic equations,, Acta Astronaut. 4 (1977), 4 (1977), 1177. doi: doi:10.1016/0094-5765(77)90096-0. Google Scholar [17] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition,, Springer-Verlag, (1997). Google Scholar [18] J. Topper and T. Kawahara, Approximate equations for long nonlinear waves on a viscous film,, J. Phys. Soc. Japan, 44 (1978), 663. doi: doi:10.1143/JPSJ.44.663. Google Scholar
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