Existence of an effective burning velocity in cellular flow for curvature G-equation via game analysis

Abstract

G-equation is a popular level set model in turbulent combustion, and becomes an advective mean curvature type evolution equation when curvature of a moving flame in a fluid flow is considered: $$ G_t + \left(1-d\, \mathrm{Div}{\frac{DG}{|DG|}}\right)_+|DG|+V(x)\cdot DG=0. $$ Here $d>0$ is the Markstein number and the positive part $()_+$ is imposed to avoid a non-physical negative laminar flame speed. For simplicity of presentation, we focus mainly on the case when $V:\mathbb{R}^2\to \mathbb{R}^2$ is the two dimensional cellular flow with Hamiltonian $H = \sin x_1 \, \sin x_2$ and amplitude $A$. Our main result is that for any unit vector $p\in \mathbb{R}^2$, there exists a positive number $\overline H(p)$ such that if $G(x,0)=p\cdot x$, then $$ \left|G(x,t)-p\cdot x+\overline H(p)t\right|\leq C \quad \text{in $\mathbb{R}^2\times [0,\infty)$} $$ for a constant $C$ depending only on the Markstein number $d$ and the cellular flow amplitude $A$. The number $\overline H(p)$ corresponds to the effective burning velocity in the physics literature. The non-coercivity encountered here is one of the major difficulties for homogenization of the mean curvature-type equations. To overcome it, we introduce a new approach that combines PDE methods with a dynamical analysis of the Kohn-Serfaty deterministic game characterization of the curvature G-equation utilizing the streamline structure of cellular flows. Extension to general two-dimensional incompressible flows is also discussed. In three dimensional incompressible flows, the existence of $\overline H(p)$ might fail when the flow intensity exceeds a bifurcation value even for simple shear flows [32].