Effect of curvature diffusivity on homogenized G-equation flame speed
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Must increasing curvature diffusivity slow the homogenized flame speed in the G-equation? The recorded test is a smooth periodic three-dimensional shear flow V=(0,0,f(x₁,x₂)) in direction (2,2,1), including the physical cutoff cell model.
Why it matters and what counts as progress
Why it matters: A speed increase at small curvature would contradict monotone slowdown in a specified three-dimensional flame model.
What counts as progress: Prove monotone slowdown in a specified regime or construct a flow with positive right derivative of speed at zero diffusivity.
How the problem developed
- 2018 · Theorem [1]
Liu, Xin, and Yu Proved monotonicity of the effective flame speed with respect to curvature diffusivity for two-dimensional shear flows.
- 2024 · Open problem [1]
Xu, Xin, and Yu Asked whether curvature diffusivity always decreases the homogenized flame speed in three dimensions.
- 28 Jul 2026 · Counterexample claim [1]
Tran, Xin, and Yu, with ChatGPT 5.5 Plus Constructed a smooth three-dimensional periodic shear flow in which Markstein curvature diffusivity increases the effective G-equation flame speed.
AI contribution
Turbulent Flame Speed Can Increase under Curvature Smoothing
Tran, Xin and Yu construct a smooth periodic three-dimensional shear flow V=(0,0,f(x₁,x₂)) and direction (2,2,1) whose homogenized G-equation flame speed has c(0)=0 and positive right derivative as curvature diffusivity d tends to zero. For small d the cutoff is inactive, so the physical cutoff cell speed increases.
Sources
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Publication dates and editorial changes.
AI-assisted construction shows curvature can raise modeled flame speed