2022/09/19 by Gao, Hongwei, Long, Ziang, Xin, Jack +1 · 1 citation
#70H20 #76M45 #76M50 #76N20 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2209.09228
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: Gt + (1-d Div(DG)/(|DG|))+|DG|+V(x)⋅ 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:ℝ2→ ℝ2 is the two dimensional cellular flow with Hamiltonian H = sin x1 sin x2 and amplitude A. Our main result is that for any unit vector p∈ ℝ2, there exists a positive number H(p) such that if G(x,0)=p⋅ x, then |G(x,t)-p⋅ x+ H(p)t|≤ C in ℝ2× [0,∞) for a constant C depending only on the Markstein number d and the cellular flow amplitude A. The number 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 H(p) might fail when the flow intensity exceeds a bifurcation value even for simple shear flows [32].