arXiv:2608.15337math.APcs.LG2026-08

物理截断无法恢复均匀化,燃烧速度随相位变化而波动。

The Physical Cutoff Does Not Restore Homogenization: Phase-Dependent Burning in the Strain G-Equation

论文配图:The Physical Cutoff Does Not Restore Homogenization: Phase-Dependent Burning in the Strain G-Equation
图 1 · 摘自论文原文
  • 通过哈密顿夹逼法证明周期流中燃烧界面存在线性时间振荡。
  • 在特定参数下,解在水平通道上保持有界,但某点衰减速率不低于 $CA//log A$。
  • 结果对机器学习中的鲁棒决策有启示,需额外全局稳定性条件。

我们推翻了Xin、Yu和Ronney的预期:在二维标准细胞流 $V_A(x_1,x_2)=A(-\ sin x_1\cos x_2,\cos x_1\sin x_2)$ 下,物理正部应具有有效燃烧速度。当 $0 < d < 20/399$ 且 $\sqrt{1+4d^2} < Ad \le 1 + d/10$ 时,对任意单位平面斜率,周期修正项至少以线性速度产生振荡。解在过 $(π,0)$ 的显式水平通道上保持有界,而在 $(π/2,0)$ 处下降速率不低于 $CA/\log A$($C>0$ 为普适常数)。对任意连续周期扰动的平面初值同样成立。在物理标度 $V_A(x/\varepsilon)$ 与 $d_\varepsilon = \varepsilon d$ 下,任意宏观正时间,距离 $O(\varepsilon)$ 的点间仍存在一阶量级差异,故归一化解无局部一致收敛子列。证明采用哈密顿夹逼 $H_{\mathrm{unc}} \le H_+ \le \widehat H$,其中上界 $\widehat H$ 为矩形支撑函数,等价于状态相关可信集上的上期望,其反向控制动力学具有不变比较通道。此外,对任意 $C^2$ 不可压缩周期流,所有 $\varepsilon$-向外屏障证书的覆盖半径对充分小的 $\varepsilon$ 至多为 $2d\varepsilon$。我们还讨论了对统计与机器学习的影响:矩形、时间一致的局部不确定性未必导致长期遗忘初始状态,因此鲁棒序列决策需额外全局稳定性或遍历性条件。两个 Lean 4 附录记录了 $p=e_1$ 子情形的充分条件形式化及屏障证书刚性定理的逻辑组装。

原文摘要 · Abstract (English)

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. Under the physical scaling $V_A(x/\varepsilon)$ and $d_\varepsilon=\varepsilon d$, an order one value gap persists between points at distance $O(\varepsilon)$ at every positive macroscopic time, so the rescaled solutions have no locally uniformly convergent subsequence. The proof uses the Hamiltonian sandwich $H_{\mathrm{unc}}\le H_+\le\widehat H$. The upper comparator $\widehat H$ is a rectangular support function, equivalently an upper expectation over a state-dependent credal set, whose reversed control dynamics possess an invariant comparison channel. We also prove that for any $C^2$ incompressible periodic flow, every $\varepsilon$-outward barrier certificate has covering radius at most $2d\varepsilon$ for all sufficiently small $\varepsilon$. We further discuss implications for statistics and machine learning: rectangular, time-consistent local uncertainty need not imply forgetting of the initial state in the long run, so additional global stability or ergodicity conditions are needed in robust sequential decision making. Two Lean 4 appendices record conditional formalizations of a sufficient $p=e_1$ subregime and of the logical assembly of the rigidity theorem for barrier certificates.

燃烧模型偏微分方程动态系统机器学习

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