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 $$\left|G(x,t)-p\cdot x+\overline{H}(p)t\right|\le C\text{in R2×[0,∞)}$$ 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].

In this final version, we have further clarified statements and proofs based on referees' comments