A Comparison Analysis of Sivashinsky's Type Evolution Equations
Describing Flame Propagation in Channels
(in collaboration with Leonardo Guidi)
Resumo: We establish a comparison between Rakib--Sivashinsky and
Michelson--Sivashinsky quasilinear parabolic differential equations
governing the weak thermal limit of flame front propagating in
channels [GM]. For
the former equation, based in [GM1], a complete description of all
steady solutions is given and local and global stability are
presented. These analysis complement the work of Berestycki et
al [BKS]. For the latter,
bi--coalescent and interpolating unstable steady solutions are introduced
and shown to be more numerous than the previous known coalescent
solutions.
These facts are argued to be responsible for the disagreement between
the observed dynamics in numerical experiments and the exact (linear)
stability analysis by Vaynblat and Matalon [VM] and give
ingredients to construct metastable solutions describing parabolic
steadily propagating flame with centered tip.
[BKS] H. Berestycki, S. Kamin and G. Sivashinsky} (2001)
Metastability in a flame front evolution equation,
Interfaces and Free Boundaries 3, 361-392.
[GM] L. F. Guidi and D. H. U. Marchetti} (2003)
A Comparison Analysis of Sivashinsky's Type Evolution Equations
Describing Flame Propagation in Channels, Phys. Lett. A 308, 162-172.
[GM1] L. F. Guidi and D. H. U. Marchetti} (2001)
Renormalization Group Flow of the Two-Dimensional Hierarchical Coulomb
Gas, Commun. Math. Phys. 219, 671-702.
[VM] D. Vaynblat and M. Matalon} (2000)
Stability of POle Solutions for Planar Propagating Flames: I. Exact
Eigenvalues and Eigenfunctions, SIAM J. Appl. Math. 60, 679.