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

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.