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

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

Ballistic Orbits and Front Speed Enhancement for ABC Flows

We study the two main types of trajectories of the ABC flow in the near-integrable regime: spiral orbits and edge orbits. The former are helical orbits which are perturbations of similar orbits that exist in the integrable regime, while the latter exist only in the non-integrable regime. We prove existence of ballistic (i.e., linearly growing) spiral orbits by using the contraction mapping principle in the Hamiltonian formulation, and we also find and analyze ballistic edge orbits. We discuss the relationship of existence of these orbits with questions concerning front propagation in the presence of flows, in particular, the question of linear (i.e., maximal possible) front speed enhancement rate for ABC flows.Comment: 39 pages, 26 figure