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Lagrangian, Game Theoretic and PDE Methods for Averaging G-equations in Turbulent Combustion: Existence and Beyond

Jack Xin, Yifeng Yu, Paul Ronney

Abstract

G-equations are popular level set Hamilton-Jacobi nonlinear partial differential equations (PDEs) of first or second order arising in turbulent combustion. Characterizing the effective burning velocity (also known as the turbulent burning velocity) is a fundamental problem there. We review relevant studies of the G-equation models with a focus on both the existence of effective burning velocity (homogenization), and its dependence on physical and geometric parameters (flow intensity and curvature effect) through representative examples. The corresponding physical background is also presented to provide motivations for mathematical problems of interest. The lack of coercivity of Hamiltonian is a hallmark of G-equations. When either the curvature of the level set or the strain effect of fluid flows is accounted for, the Hamiltonian becomes highly non-convex and nonlinear. In the absence of coercivity and convexity, PDE (Eulerian) approach suffers from insufficient compactness to establish averaging (homogenization). We review and illustrate a suite of Lagrangian tools, most notably min-max (max-min) game representations of curvature and strain G-equations, working in tandem with analysis of streamline structures of fluid flows and PDEs. We discuss open problems for future development in this emerging area of dynamic game analysis for averaging non-coercive, non-convex, and nonlinear PDEs such as geometric (curvature-dependent) PDEs with advection.

Lagrangian, Game Theoretic and PDE Methods for Averaging G-equations in Turbulent Combustion: Existence and Beyond

Abstract

G-equations are popular level set Hamilton-Jacobi nonlinear partial differential equations (PDEs) of first or second order arising in turbulent combustion. Characterizing the effective burning velocity (also known as the turbulent burning velocity) is a fundamental problem there. We review relevant studies of the G-equation models with a focus on both the existence of effective burning velocity (homogenization), and its dependence on physical and geometric parameters (flow intensity and curvature effect) through representative examples. The corresponding physical background is also presented to provide motivations for mathematical problems of interest. The lack of coercivity of Hamiltonian is a hallmark of G-equations. When either the curvature of the level set or the strain effect of fluid flows is accounted for, the Hamiltonian becomes highly non-convex and nonlinear. In the absence of coercivity and convexity, PDE (Eulerian) approach suffers from insufficient compactness to establish averaging (homogenization). We review and illustrate a suite of Lagrangian tools, most notably min-max (max-min) game representations of curvature and strain G-equations, working in tandem with analysis of streamline structures of fluid flows and PDEs. We discuss open problems for future development in this emerging area of dynamic game analysis for averaging non-coercive, non-convex, and nonlinear PDEs such as geometric (curvature-dependent) PDEs with advection.
Paper Structure (33 sections, 13 theorems, 202 equations, 10 figures)

This paper contains 33 sections, 13 theorems, 202 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Different types of G-equations
  3. Inviscid G-equation ($d=0$)
  4. Curvature G-equation
  5. Strain G-equation
  6. Turbulent burning velocity and homogenization of G-equation
  7. Preliminary: Viscosity Solutions and Homogenization
  8. Inviscid G-equation: the Basic Case
  9. Homogenization of invicid G-equation
  10. Sketch of proof.
  11. Optimal Convergence Rate $O(\epsilon)$
  12. Explicit formula of $\overline H(p)$ in 2D shear flows
  13. Growth law of $\overline H(p)$ in 2D cellular flows
  14. Growth law of $\overline H(p)$ in the ABC flow and Kolmogorov flow
  15. Curvature G-equation
  16. ...and 18 more sections

Key Result

Lemma 3.2

Suppose that $u$ is a viscosity subsolution to for some fixed constant $C$. If $y$ is reachable from $x$ within time $T$, then

Figures (10)

  • Figure 1: Level set modeling of flame propagation.
  • Figure 2: Curvature effect: faster burn at B (bending towards hot region) than C.
  • Figure 3: A classical (left) vs. an effective (right, dashed) planar front by averaging an oscillatory flame front (right, solid).
  • Figure 4: Averaging fluctuations around a general flame front.
  • Figure 5: Ballistic orbits, periodic (modulo $2\pi$) along $x$-direction, in 1-1-1 ABC (left) and Kolmogorov (right) flows.
  • ...and 5 more figures

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Remark 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 3.7
  • ...and 10 more