Estimates for Parabolic Problems

Abstract

We present quantitative estimates for the homogenization of the parabolic equation

$$\begin{aligned} \partial _t u - \nabla \cdot \mathbf {a}(x) \nabla u = 0 \quad \text{ in } \ I\times U \subseteq \mathbb {R}\times {\mathbb {R}^d}. \end{aligned}$$

The coefficients $\mathbf {a}(x)$ are assumed to depend only on the spatial variable x rather than (t, x) (Quantitative homogenization results for parabolic equations with space-time random coefficients can also be obtained from the ideas presented in this book: see [7]). The main purpose of this chapter is to illustrate that the parabolic equation (8.1) can be treated satisfactorily using the elliptic estimates we have already obtained in earlier chapters. In particular, we present error estimates for general Cauchy–Dirichlet problems in bounded domains, two-scale expansion estimates, and a parabolic large-scale regularity theory. We conclude, in the last two sections, with $L^\infty$-type estimates for the homogenization error and the two-scale expansion error for both the parabolic and elliptic Green functions. The statements of these estimates are given below in Theorem 8.20 and Corollary 8.21. Like the estimates in Chap. 2, these estimates are suboptimal in the scaling of the error but optimal in stochastic integrability (i.e., the scaling of the error is given by a small exponent $\alpha >0$ and the stochastic integrability is $\mathcal {O}_{d-}$-type). In the next chapter, we present complementary estimates which are optimal in the scaling of the error and consistent with the bounds on the first-order correctors proved in Chap. 4. See Theorem 9.11.