Min-Max Variational Principle and Front Speeds in Random Shear Flows
James Nolen, Jack Xin
TL;DR
The paper studies front speeds of bistable (combustion) reaction-diffusion fronts in mean-zero stationary Gaussian shear flows inside two- and three-dimensional channels, using a min-max variational principle and novel multi-scale test functions to characterize the speed c(δ) for each realization. In the small root-mean-square regime, the approach yields speed asymptotics and a quadratic speed enhancement that holds with probability arbitrarily close to one under the regularity assumptions (almost-sure Hölder in 2D and mean-square Hölder in 3D). For deterministic zero-mean Hölder shear b, the authors construct a multi-scale test function with v(x) = U(ξ) + δ^2 ũ(ξ, x̃) and obtain the small-δ expansion $c(δ) = c_0 + \frac{c_0 δ^2}{2|Ω|}⟨|∇χ|^2⟩ + O(δ^3)$, where χ solves $-Δχ=b$ with Neumann boundary conditions, and they show linear growth $c(δ) ∼ κ δ$ for large δ. In the Gaussian setting b = b̄ + b1, they derive probabilistic bounds showing $c(δ, ω) ≈ c_0 − δ b̄ − \frac{δ^2}{2} ⟨(∫ b_1)^2⟩$ with high probability in 2D (via Borell's inequality and Karhunen-Loeve) and analogous results in 3D under Hölder control; as δ → ∞ the limit c(δ, ω)/δ exists a.s. with finite E[c]/δ when E||b||∞ < ∞. The quadratic speed enhancement thus holds with probability arbitrarily close to one in both 2D and 3D under the stated regularity, with suggested extensions to other cross sections and higher dimensions.
Abstract
Speed ensemble of bistable (combustion) fronts in mean zero stationary Gaussian shear flows inside two and three dimensional channels is studied with a min-max variational principle. In the small root mean square regime of shear flows, a new class of multi-scale test functions are found to yield speed asymptotics. The quadratic speed enhancement law holds with probability arbitrarily close to one under the almost sure continuity (dimension two) and mean square H\"older regularity (dimension three) of the shear flows. Remarks are made on the conditions for the linear growth of front speed expectation in the large root mean square regime.