发表机构
Qiuzhen College, Tsinghua University（清华大学邱班）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究周期均匀化与消失黏性同尺度下 Hamilton-Jacobi 方程的收敛速率，通过校正子展开和黏性比较，在正定 Hessian 条件下建立最优速率 O(ε|log ε|)，无需对有效解求导。
AI 中文摘要
我们研究在 Hamilton-Jacobi 方程中，周期均匀化与消失黏性同时发生时（其动量 Hessian 矩阵在每一点正定）的收敛速率。对于有界 Lipschitz 初始数据及一类光滑 Hamilton 量，我们在固定时间区间上建立了最优速率 $O(\varepsilon|\log\varepsilon|)$。二阶项起着关键作用：椭圆胞元问题为有效 Hamilton 量 $\bar{H}$ 及其 Legendre 对偶 $\bar{L}$ 提供了更高阶导数估计。证明纯粹基于 PDE，采用校正子展开与黏性比较，但其指导思路来源于 Hopf-Lax 公式的控制解释。对于固定目标点 $(x,t)$，极小化起点选取特征速度及其对偶动量，这些引导了光滑比较剖面的构造，其梯度为第一阶校正子提供动量参数。因此，即使在有效解非光滑的点，我们也无需对其求导。
英文摘要
We study convergence rates when periodic homogenization and vanishing viscosity occur at the same scale in Hamilton--Jacobi equations whose momentum Hessians are positive definite at every point. For bounded Lipschitz initial data and a class of smooth Hamiltonians, we establish an optimal rate $O(\varepsilon|\log\varepsilon|)$ on fixed time intervals. The second-order term plays a key role: the elliptic cell problem provides higher derivative bounds for the effective Hamiltonian $\bar{H}$ and its Legendre dual $\bar{L}$. The proof is purely PDE, based on corrector expansions and viscosity comparison, while its guiding idea comes from the control interpretation of the Hopf--Lax formula. For a fixed target $(x,t)$, a minimizing origin selects a characteristic velocity and its dual momentum, these guide the construction of smooth comparison profiles, whose gradients supply the momentum argument of the first-order corrector. Thus we never need to differentiate the effective solution, even at points where it is nonsmooth.
