American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Heteroclinic connections for multidimensional bistable reaction-diffusion equations
  • DCDS-S Home
  • This Issue
  • Next Article
    Some remarks on the qualitative properties of solutions to a predator-prey model posed on non coincident spatial domains
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 & 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

[1]

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
[2]

Fred C. Pinto. Nonlinear stability and dynamical properties for a Kuramoto-Sivashinsky equation in space dimension two. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 117-136. doi: 10.3934/dcds.1999.5.117
[3]

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
[4]

Jared C. Bronski, Razvan C. Fetecau, Thomas N. Gambill. A note on a non-local Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 701-707. doi: 10.3934/dcds.2007.18.701
[5]

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
[6]

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 - A, 2004, 10 (1&2) : 557-580. doi: 10.3934/dcds.2004.10.557
[7]

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
[8]

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
[9]

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
[10]

Peng Gao. Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5649-5684. doi: 10.3934/dcds.2018247
[11]

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
[12]

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
[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]

Markus Böhm, Björn Schmalfuss. Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3115-3138. doi: 10.3934/dcdsb.2018303
[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]

Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235
[17]

Panayotis Panayotaros, Felipe Rivero. Multistability and localized attractors in a dissipative discrete NLS equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1137-1154. doi: 10.3934/dcdsb.2014.19.1137
[18]

Manuel Fernández-Martínez, Miguel Ángel López Guerrero. Generating pre-fractals to approach real IFS-attractors with a fixed Hausdorff dimension. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1129-1137. doi: 10.3934/dcdss.2015.8.1129
[19]

Wenqiang Zhao. Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3395-3438. doi: 10.3934/dcdsb.2018326
[20]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences