发表机构
University of Warwick（华威大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究驳斥了物理正应变G方程在胞流中应具有有效燃烧速度的预期，通过分析其周期修正项的振荡性、值差等特性完成证明，并探讨了对统计学和机器学习的启示。
AI 中文摘要
我们驳斥了Xin、Yu和Ronney提出的预期，即物理正应变G方程在胞流中应具有有效燃烧速度。对于二维标准胞流V_A(x₁,x₂)=A(-sinx₁cosx₂,cosx₁sinx₂)，若0<d<20/399且√(1+4d²)<Ad≤1+d/10，则对于每一个单位平面斜率，周期修正项至少随时间线性振荡；解在经过(π,0)的显式水平通道上保持下方有界，而在(π/2,0)处其下降速率至少为CA/log A（C>0为常数）。对于平面初始数据的任意连续周期扰动，上述结论同样成立。在物理缩放V_A(x/ε)和d_ε=εd下，每一个正宏观时间处，距离为O(ε)的点之间始终存在一阶值差，因此缩放后的解不存在局部一致收敛的子序列。证明采用哈密顿三明治H_unc≤H_+≤Ĥ，上比较器Ĥ为矩形支撑函数，等价于依赖状态的 credal 集上的上期望，其反向控制动力学具有不变比较通道。我们还证明，对于任意C²不可压缩周期流，所有ε-向外障碍证书在ε足够小时，覆盖半径至多为2dε。我们进一步讨论其对统计学和机器学习的启示：矩形、时间一致的局部不确定性并不意味着长期会遗忘初始状态，因此鲁棒序列决策制定需要额外的全局稳定性或遍历性条件。两个Lean 4附录记录了充分p=e₁子区域的条件形式化以及障碍证书刚性定理的逻辑组合。
英文摘要
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.
