随机介质中燃烧的长时间动力学
Long Time Dynamics for Combustion in Random Media
摘要：
本文研究了随机介质中燃烧过程的长时间动力学，模型采用具有随机点火反应的反应扩散方程。人们期望，在合理的假设下的随机性，大的空间-时间尺度动力学的解决方案，这些方程几乎肯定是由一个不同的有效PDE，这应该是一个齐次Hamilton-Jacobi方程。虽然这在以前已经在一个维度上以及对于在几个维度上的各向同性反应（即，与径向对称的法律），我们在这里提供的第一个证明，这种现象在一般非各向同性的多维设置。我们的结果适用于具有有限依赖范围的反应（即，它们的值在空间中足够远的点处是独立的），以及对于具有无限范围依赖性的一些，并且基于证明这些反应在所有方向上的确定性前沿（传播）速度的存在。
英文摘要：
We study long time dynamics of combustive processes in random media, modeled by reaction-diffusion equations with random ignition reactions. One expects that under reasonable hypotheses on the randomness, large space-time scale dynamics of solutions to these equations is almost surely governed by a different effective PDE, which should be a homogeneous Hamilton-Jacobi equation. While this was previously proved in one dimension as well as for isotropic reactions in several dimensions (i.e., with radially symmetric laws), we provide here the first proof of this phenomenon in the general non-isotropic multidimensional setting. Our results hold for reactions that have finite ranges of dependence (i.e., their values are independent at sufficiently distant points in space) as well as for some with infinite ranges of dependence, and are based on proving existence of deterministic front (propagation) speeds in all directions for these reactions.
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