Advances in Differential Equations

Quantitative homogenization of analytic semigroups and reaction-diffusion equations with Diophantine spatial frequencies

Bernold Fiedler and Mark I. Vishik

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Abstract

Based on an analytic semigroup setting, we first consider semilinear reaction--diffusion equations with spatially quasiperiodic coefficients in the nonlinearity, rapidly varying on spatial scale $\varepsilon$. Under periodic boundary conditions, we derive quantitative homogenization estimates of order $\varepsilon^\gamma$ on strong Sobolev spaces $H^\sigma$ in the triangle $$0 <\gamma < \min (\sigma -n/2,2-\sigma).$$ Here $n$ denotes spatial dimension. The estimates measure the distance to a solution of the homogenized equation with the same initial condition, on bounded time intervals. The same estimates hold for $C^1$ convergence of local stable and unstable manifolds of hyperbolic equilibria. As a second example, we apply our abstract semigroup result to homogenization of the Navier--Stokes equations with spatially rapidly varying quasiperiodic forces in space dimensions 2 and 3. In both examples, a Diophantine condition on the spatial frequencies is crucial to our homogenization results. Our Diophantine condition is satisfied for sets of frequency vectors of full Lebesgue measure. In the companion paper [7], based on $L^2$ methods, these results are extended to quantitative homogenization of global attractors in near-gradient reaction--diffusion systems.

Article information

Source
Adv. Differential Equations Volume 6, Number 11 (2001), 1377-1408.

Dates
First available in Project Euclid: 2 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1357139965

Mathematical Reviews number (MathSciNet)
MR1859352

Zentralblatt MATH identifier
1036.35022

Subjects
Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50]
Secondary: 35B42: Inertial manifolds 35K55: Nonlinear parabolic equations 35K90: Abstract parabolic equations 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

Citation

Fiedler, Bernold; Vishik, Mark I. Quantitative homogenization of analytic semigroups and reaction-diffusion equations with Diophantine spatial frequencies. Adv. Differential Equations 6 (2001), no. 11, 1377--1408. https://projecteuclid.org/euclid.ade/1357139965.


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