Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 8 (2018), 1945-2014.

Quantitative stochastic homogenization and regularity theory of parabolic equations

Scott Armstrong, Alexandre Bordas, and Jean-Christophe Mourrat

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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 C0,1-type estimate and a Liouville theorem of every finite order.

Article information

Source
Anal. PDE, Volume 11, Number 8 (2018), 1945-2014.

Dates
Received: 22 May 2017
Revised: 15 February 2018
Accepted: 9 April 2018
First available in Project Euclid: 15 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.apde/1547521444

Digital Object Identifier
doi:10.2140/apde.2018.11.1945

Mathematical Reviews number (MathSciNet)
MR3812862

Zentralblatt MATH identifier
1388.60103

Subjects
Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 35B45: A priori estimates
Secondary: 60K37: Processes in random environments 60F05: Central limit and other weak theorems

Keywords
stochastic homogenization parabolic equation large-scale regularity variational methods

Citation

Armstrong, Scott; Bordas, Alexandre; Mourrat, Jean-Christophe. Quantitative stochastic homogenization and regularity theory of parabolic equations. Anal. PDE 11 (2018), no. 8, 1945--2014. doi:10.2140/apde.2018.11.1945. https://projecteuclid.org/euclid.apde/1547521444


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