Analysis and Comparison of Large Time Front Speeds in Turbulent Combustion Models

Predicting turbulent flame speed (the large time front speed) is a fundamental problem in turbulent combustion theory. Several models have been proposed to study the turbulent flame speed, such as the G-equations, the F-equations (Majda-Souganidis model) and reaction-diffusion-advection (RDA) equations. In the first part of this paper, we show that flow induced strain reduces front speeds of G-equations in periodic compressible and shear flows. The F-equations arise in asymptotic analysis of reaction-diffusion-advection equations and are quadratically nonlinear analogues of the G-equations. In the second part of the paper, we compare asymptotic growth rates of the turbulent flame speeds from the G-equations, the F-equations and the RDA equations in the large amplitude ($A$) regime of spatially periodic flows. The F and G equations share the same asymptotic front speed growth rate; in particular, the same sublinear growth law $A\over \log(A)$ holds in cellular flows. Moreover, in two space dimensions, if one of these three models (G-equation, F-equation and the RDA equation) predicts the bending effect (sublinear growth in the large flow), so will the other two. The nonoccurrence of speed bending is characterized by the existence of periodic orbits on the torus and the property of their rotation vectors in the advective flow fields. The cat's eye flow is discussed as a typical example of directional dependence of the front speed bending. The large time front speeds of the viscous F-equation have the same growth rate as those of the inviscid F and G-equations in two dimensional periodic incompressible flows.Comment: 42 page

Periodic orbits of the ABC flow with A=B=C=1A=B=C=1

In this paper, we prove that the ODE system $$\begin{align*} \dot x &=\sin z+\cos y\\ \dot y &= \sin x+\cos z\\ \dot z &=\sin y + \cos x, \end{align*}$$ whose right-hand side is the Arnold-Beltrami-Childress (ABC) flow with parameters $A=B=C=1$, has periodic orbits on $(2\pi\mathbb T)^3$ with rotation vectors parallel to $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. An application of this result is that the well-known G-equation model for turbulent combustion with this ABC flow on $\mathbb R^3$ has a linear (i.e., maximal possible) flame speed enhancement rate as the amplitude of the flow grows.Comment: 9 page

Uniqueness of Values of Aronsson Operators and Running Costs in “tug-of-War” Games

Let $A_H$ be the Aronsson operator associated with a Hamiltonian $H(x,z,p).$ Aronsson operators arise from $L^\infty$ variational problems, two person game theory, control problems, etc. In this paper, we prove, under suitable conditions, that if $u\in W^{1,\infty}_{\rm loc}(\Omega)$ is simultaneously a viscosity solution of both of the equations $A_H(u)=f(x)$ and $A_H(u)=g(x)$ in $\Omega$, where $f, g\in C(\Omega),$ then $f=g.$ The assumption $u\in W_{loc}^{1,\infty}(\Omega)$ can be relaxed to $u\in C(\Omega)$ in many interesting situations. Also, we prove that if $f,g,u\in C(\Omega)$ and $u$ is simultaneously a viscosity solution of the equations ${\Delta_\infty u\over |Du|^2}=-f(x)$ and ${\Delta_{\infty}u\over |Du|^2}=-g(x)$ in $\Omega$ then $f=g.$ This answers a question posed in Peres, Schramm, Scheffield and Wilson [PSSW] concerning whether or not the value function uniquely determines the running cost in the "tug-of-war" game.Comment: To appear in "Ann. Inst. H. Poincare Anal. Non Lineaire

Smart Chinese Reader: A Chinese Language Learning Aid with Web Browser

Smart Chinese Reader is a program based on NLP (natural language processing) technology to help you learn Chinese language through deep reading. It provides Chinese word segmentation, Chinese part of speech tagging, Chinese to English translation, example sentence search, and text to speech conversion. Compared with dictionary apps, it lets you gain more Chinese language knowledge (meanings and usages of Chinese words, patterns and even rhythms of Chinese sentences) from a text, rather than just to get through the text. It makes your Chinese learning more effective