[Paper Review] Homogenization and enhancement of the $G-$equation in random environments

Pierre Cardaliaguet, Panagiotis E. Souganidis|arXiv (Cornell University)|Oct 8, 2011

Advanced Mathematical Modeling in Engineering24 references3 citations

TL;DR

This paper establishes the homogenization of the $G$-equation in random, stationary, and ergodic environments with divergence-free advection, proving convergence to a deterministic anisotropic $G$-equation with a convex, positively homogeneous Hamiltonian. It identifies conditions for velocity enhancement—where the averaged front speed exceeds the baseline—by overcoming non-coercivity and lack of uniform integrability via novel reachability estimates and a constructed random sequence with asymptotic limits.

ABSTRACT

We study the homogenization of a $G$-equation which is advected by a divergence free stationary vector field in a general ergodic random environment. We prove that the averaged equation is an anisotropic deterministic G-equation and we give necessary and sufficient conditions in order to have enhancement. Since the problem is not assumed to be coercive it is not possible to have uniform bounds for the solutions. In addition, as we show, the associated minimal (first passage) time function does not satisfy, in general, the uniform integrability condition which is necessary to apply the sub-additive ergodic theorem. We overcome these obstacles by (i) establishing a new reachability (controllability) estimate for the minimal function and (ii) constructing, for each direction and almost surely, a random sequence which has both a long time averaged limit (due to the sub-additive ergodic theorem) and stays (in the same sense) asymptotically close to the minimal time.

Motivation & Objective

• To resolve the long-standing open problem of homogenizing the $G$-equation in non-coercive, random, stationary, and ergodic environments.
• To establish the existence of a deterministic effective Hamiltonian $\overline{H}$ governing the averaged front propagation.
• To characterize the conditions under which the averaged front speed is strictly greater than the baseline (i.e., enhancement occurs).
• To overcome the absence of uniform bounds and failure of uniform integrability in the minimal time function, which invalidates standard sub-additive ergodic theory.
• To develop a new methodological framework for non-coercive Hamilton-Jacobi equations in random media.

Proposed method

• Introduce a new reachability (controllability) estimate for the minimal time function to bypass lack of uniform bounds.
• Construct a random sequence independent of the original probability space that asymptotically tracks the minimal time function in direction $a$.
• Use the sub-additive ergodic theorem on the constructed sequence to establish almost sure long-time limits.
• Prove that the limit of the rescaled minimal time function exists almost surely and equals the time constant $Z^a(\omega)$.
• Leverage the existence of a time constant to derive the effective Hamiltonian $\overline{H}$ via duality and convexity arguments.
• Establish equivalence between the limit of $t^{-1}\theta(0,ta,\omega)$ and the time constant, proving almost sure convergence in 2D.

Experimental results

Research questions

• RQ1 Under what conditions does the $G$-equation in a random, divergence-free environment homogenize to a deterministic $G$-equation?
• RQ2 When is the effective front speed strictly greater than one (i.e., enhancement)?
• RQ3 How can homogenization be achieved when the equation is non-coercive and the minimal time function lacks uniform integrability?
• RQ4 Can a time constant be defined almost surely for the minimal time function in 2D under general ergodic conditions?
• RQ5 What is the role of the vector field's correlation with the direction of propagation in determining enhancement?

Key findings

• The solution $u^\varepsilon$ of the $G$-equation converges locally uniformly in $(x,t)$ and almost surely in $\omega$ to the solution $\overline{u}$ of the deterministic $G$-equation with effective Hamiltonian $\overline{H}$.
• The effective Hamiltonian $\overline{H}$ is convex, positively homogeneous of degree one, and satisfies $|\overline{H}(p)| \geq |p| + \langle \mathbb{E}[V], p \rangle$ for all $p \in \mathbb{R}^N$.
• Enhancement occurs if and only if $V(y,\omega)$ is not almost surely orthogonal to $p$; otherwise $\overline{H}(p) = |p| + \langle \mathbb{E}[V], p \rangle$.
• In two dimensions, the limit $\lim_{t\to\infty} t^{-1}\theta(0,ta,\omega)$ exists almost surely for any $a \in B$, due to the alignment of the time constant with the direction $a$.
• The constructed random sequence ensures that the minimal time function stays asymptotically close to the time constant, enabling application of the sub-additive ergodic theorem despite non-uniform integrability.
• The proof establishes that $W^a \subset \mathbb{R}a$ in 2D, implying the velocity vector aligns with the direction $a$, and that $\lambda a \in W^a$ for some $\lambda \geq 1$ almost surely.