Quantitative Homogenization of Analytic Semigroups and Reaction–diffusion Equations with Diophantine Spatial Frequencies

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 ε. Under periodic boundary conditions, we derive quantitative homogenization estimates of order ε on strong Sobolev spaces H in the triangle 0 < γ < min(σ − n/2, 2− σ). 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 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 methods, these results are extended to quantitative homogenization of global attractors in near-gradient reaction–diffusion systems.