2001 Quantitative homogenization of analytic semigroups and reaction-diffusion equations with Diophantine spatial frequencies
Bernold Fiedler, Mark I. Vishik
Adv. Differential Equations 6(11): 1377-1408 (2001). DOI: 10.57262/ade/1357139965


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.


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Bernold Fiedler. Mark I. Vishik. "Quantitative homogenization of analytic semigroups and reaction-diffusion equations with Diophantine spatial frequencies." Adv. Differential Equations 6 (11) 1377 - 1408, 2001. https://doi.org/10.57262/ade/1357139965


Published: 2001
First available in Project Euclid: 2 January 2013

zbMATH: 1036.35022
MathSciNet: MR1859352
Digital Object Identifier: 10.57262/ade/1357139965

Primary: 35B27
Secondary: 35B42 , 35K55 , 35K90 , 35Q30

Rights: Copyright © 2001 Khayyam Publishing, Inc.


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Vol.6 • No. 11 • 2001
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