The Physical Cutoff Does Not Restore Homogenization: Phase-Dependent Burning in the Strain G-Equation
We disprove the expectation stated by Xin, Yu, and Ronney that the physical positive part strain $G$-equation should possess an effective burning velocity in cellular flows. For the standard cellular flow in dimension two $V_A(x_1,x_2)=A(-\sin x_1\cos x_2,\cos x_1\sin x_2)$, if $0<d<20/399$ and $\sqrt{1+4d^2}<Ad\le1+d/10$, then for every unit planar slope the periodic correction develops oscillations at least linearly in time. The solution remains bounded below on an explicit horizontal channel through $(π,0)$, while at $(π/2,0)$ it decreases at rate at least $CA/\log A$, with $C>0$ universal. The same conclusions hold for arbitrary continuous periodic perturbations of planar initial data...
| Authors | Michele Caprio |
|---|---|
| Primary category | math.AP |
| Published | 15 August 2026 |
| Last updated at source | 15 August 2026 |
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