In combustion theory, a thin flame zone is usually replaced by a free interface. A very challenging problem is the derivation of a self-consistent equation for the flame front which yields a reduction of the dimensionality of the system. A paradigm is the Kuramoto-Sivashinsky (K-S) equation, which models cellular instabilities and turbulence phenomena. In this survey paper, we browse through a series of models in which one reaches a fully nonlinear parabolic equation for the free interface, involving pseudo-differential operators. The K-S equation appears to be asymptotically the lowest order of approximation near the threshold of stability.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v49n2.16.01}, url = {http://global-sci.org/intro/article_detail/jms/993.html} }