|
|||||
 |
|
|
|
|
|
|
Quantitative
stochastic homogenization and regularity theory of parabolic
equations
Scott Armstrong, Alexandre Bordas and Jean-Christophe Mourrat |
|
Vol. 11 (2018), No. 8, 1945–2014
|
|
DOI: 10.2140/apde.2018.11.1945
|
Abstract |
|
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 -type estimate and a Liouville theorem of every finite order. |
Keywords
stochastic homogenization, parabolic equation, large-scale
regularity, variational methods
|
Mathematical Subject Classification 2010
Primary: 35B27, 35B45
Secondary: 60K37, 60F05
|
Milestones
Received: 22 May 2017
Revised: 15 February 2018
Accepted: 9 April 2018
Published: 6 June 2018
|
Authors |
| Scott Armstrong | |
| Courant Institute of Mathematical
Sciences New York University New York, NY United States |
|
| Alexandre Bordas | |
| Ecole normale supérieure de
Lyon Lyon France |
|
| Jean-Christophe Mourrat | |
| Ecole normale supérieure de
Lyon CNRS Lyon France |
|
