Michelson–Sivashinsky equation

In combustion, Michelson–Sivashinsky equation describes the evolution of a premixed flame front, subjected to the Darrieus–Landau instability, in the small heat release approximation. The equation was derived by Gregory Sivashinsky in 1977, who along the Daniel M. Michelson, presented the numerical solutions of the equation in the same year. Let the planar flame front, in a uitable frame of reference be on the ${\displaystyle xy}$-plane, then the evolution of this planar front is described by the amplitude function ${\displaystyle u(\mathbf {x} ,t)}$ (where ${\displaystyle \mathbf {x} =(x,y)}$) describing the deviation from the planar shape. The Michelson–Sivashinsky equation, reads as

${\displaystyle {\frac {\partial u}{\partial t}}+{\frac {1}{2}}(\nabla u)^{2}-\nu \nabla ^{2}u-{\frac {1}{8\pi ^{2}}}\int |\mathbf {k} |e^{i\mathbf {k} \cdot (\mathbf {x} -\mathbf {x} ')}u(\mathbf {x} ,t)d\mathbf {k} d\mathbf {x} '=0,}$

where ${\displaystyle \nu }$ is a constant. Incorporating also the Rayleigh–Taylor instability of the flame, one obtains the Rakib–Sivashinsky equation (named after Z. Rakib and Gregory Sivashinsky),

${\displaystyle {\frac {\partial u}{\partial t}}+{\frac {1}{2}}(\nabla u)^{2}-\nu \nabla ^{2}u-{\frac {1}{8\pi ^{2}}}\int |\mathbf {k} |e^{i\mathbf {k} \cdot (\mathbf {x} -\mathbf {x} ')}u(\mathbf {x} ,t)d\mathbf {k} d\mathbf {x} '+\gamma \left(u-\langle u\rangle \right)=0,\quad }$

where ${\displaystyle \langle u\rangle (t)}$ denotes the spatial average of ${\displaystyle u}$, which is a time-dependent function and ${\displaystyle \gamma }$ is another constant.