Quantitative stochastic homogenization and regularity theory of parabolic equations
Quantitative stochastic homogenization and regularity theory of parabolic equations
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DOI：
10.2140/apde.2018.11.1945
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发表时间：
2017-05
期刊：
影响因子：
2.2
通讯作者：
S. Armstrong;A. Bordas;J. Mourrat
中科院分区：
文献类型：
--
作者：
S. Armstrong;A. Bordas;J. Mourrat
我们发展了线性均匀抛物方程随机均匀化的定量理论，这些方程的系数取决于空间和时间。受椭圆设置的最新工作的启发，我们的分析集中在从抛物线方程的变分解释中导出的某些次加性量。这些次加性量与解的通量和梯度的空间平均值密切相关。我们实现了一个重整型格式来获得它们收敛的代数速率，这本质上是它们的梯度和通量的弱收敛的量化。由此，我们得到了Cauchy-Dirichlet问题随机可积性最优的均匀化误差估计。我们也发展了非均质方程解的高正则性理论，包括一致的$C^{0,1}$型估计和每有限阶的Liouville定理。
We develop a quantitative theory of stochastic homogenization for linear, uniformly parabolic equations with coefficients depending on space and time. Inspired by recent works in the elliptic setting, our analysis is focused on certain subadditive quantities derived from a variational interpretation of parabolic equations. These subadditive quantities are intimately connected to spatial averages of the fluxes and gradients of solutions. We implement a renormalization-type scheme to obtain an algebraic rate for their convergence, which is essentially a quantification of the weak convergence of the gradients and fluxes of solutions to their homogenized limits. As a consequence, we obtain estimates of the homogenization error for the Cauchy-Dirichlet problem which are optimal in stochastic integrability. We also develop a higher regularity theory for solutions of the heterogeneous equation, including a uniform $C^{0,1}$-type estimate and a Liouville theorem of every finite order.