论文标题
通过游戏分析存在有效的燃烧速度，可用于曲率G-方程
Existence of an effective burning velocity in cellular flow for curvature G-equation via game analysis
论文作者
论文摘要
G-方程是湍流燃烧中的流行水平集模型，当考虑流体流动流中移动火焰的曲率时：$$ g_t+\ left（1-d \，\ Mathrm {div} {\ frac {\ frac {dg} {dg} {| dg | cd） DG = 0。 $$这里$ d> 0 $是Markstein号码，正式零件$（）_+$是为了避免非物理负层状火焰速度。为了简化演示，我们主要关注$ v：\ mathbb {r}^2 \ to \ mathbb {r}^2 $是带有Hamiltonian $ h = \ sin x_1 \，\ sin x_2 $和Amplitude $ a $的二维蜂窝流量。我们的主要结果是，对于任何单位向量$ p \ in \ mathbb {r}^2 $，存在一个正数$ \ edline h（p）$，这样，如果$ g（x，x，0）= p \ cdot x $，则$ g \ cdot \ weft | g（x，x，x，t） $ \ mathbb {r}^2 \ times [0，\ infty）$} $} $} $} $} $} $} $} $} $}仅取决于Markstein Number $ d $和蜂窝流量振幅$ a $。数字$ \叠加h（p）$对应于物理文献中有效的燃烧速度。此处遇到的非重新矫正性是平均曲率方程均匀化的主要困难之一。为了克服它，我们引入了一种新方法，该方法将PDE方法与使用细胞流的流线结构的曲率G-方程式的Kohn-Serfaty确定性游戏表征进行了动态分析。还讨论了将一般二维不可压缩流的扩展。在三维不可压缩的流动中，当流量强度超过分叉值时，$ \叠加h（p）$的存在可能会失败[32]。
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].
