Pattern formation in a Swift–Hohenberg equation with spatially periodic coefficients

The paper proves an O(ε^2)–accurate Ginzburg–Landau (GL) approximation on times t ∈ [0, T0/ε^2] for the periodically forced Swift–Hohenberg equation using a Bloch-based ansatz with an O(ε^2) corrector and shows a residual of order O(ε^4); the amplitude equation coefficients are r, ½∂^2_ℓλ1(ℓ0), and −3∫|ψ1(ℓ0,x)|^4ρ(x)dx (Theorem 1.1 and Lemma 3.4). These match the candidate solution’s derivation and estimates, including the non-resonance condition 3ℓ0 ≠ ℓ0 (mod kf) and use of uniformly local Sobolev spaces Y = H^ϑ_ul for the error estimate; see the paper’s statement of Theorem 1.1 and its setup via the Bloch transform and function spaces. The overlap in method and results is substantive and detailed, not merely thematic. Key points are explicitly present in the paper: the PDE and periodic setting, the spectral assumptions (including gap and non-resonance), the Bloch formulation and ansatz, the residual bound, and the final H^ϑ_ul error estimate.