论文标题
具有高度振荡电势的周期性抛物线方程的均质化和收敛速率
Homogenization and Convergence Rates for Periodic Parabolic Equations with Highly Oscillating Potentials
论文作者
论文摘要
本文考虑了一个具有高度振荡电势的二阶周期性抛物线方程的家族，这对于随机同质化的随时间变化电位而被认为是多次。遵循标准的两尺度扩展幻觉，我们可以猜测并成功地确定不同情况下的均质方程，即电势满足相应的假设，基于$ l^2（0，t; h^1（ω））$的合适统一估计，用于解决方案的规范。要处理更单一的情况并获得$ l^\ infty（0，t; l^2（ω））$的收敛速率，我们需要更准确地估算Hessian项以及T衍生术语，这可能取决于$ \ varepsilon $。困难是为$ l^2（0，t; h^1（ω））$ - 标准找到合适的统一估计，并为高阶导数项提供合适的估计。
This paper considers a family of second-order periodic parabolic equations with highly oscillating potentials, which have been considered many times for the time-varying potentials in stochastic homogenization. Following a standard two-scale expansions illusion, we can guess and succeed in determining the homogenized equation in different cases that the potentials satisfy the corresponding assumptions, based on suitable uniform estimates of the $L^2(0,T;H^1(Ω))$-norm for the solutions. To handle the more singular case and obtain the convergence rates in $L^\infty(0,T;L^2(Ω))$, we need to estimate the Hessian term as well as the t-derivative term more exactly, which may be depend on $\varepsilon$. The difficulty is to find suitable uniform estimates for the $L^2(0,T;H^1(Ω))$-norm and suitable estimates for the higher order derivative terms.
