Quasicrystals in pattern formation, Part I: Local existence and basic properties

The paper’s Theorem 3.1(a–d) for the Swift–Hohenberg equation is proved via two core ingredients: (i) the Fourier–Bohr coefficient calculus (Proposition 3.8) giving ⟨u,u^3⟩=⟨u^2,u^2⟩ and ⟨(Δ+1)^2u,v⟩=⟨(Δ+1)u,(Δ+1)v⟩, which yields the logistic differential inequality for N(t)=∥u(t)∥_2^2, and (ii) a Lyapunov potential Pλ(u)=½∥(Δ+1)u∥_2^2−½λ∥u∥_2^2+¼∥u^2∥_2^2 with d/dt Pλ=−∥F(u,λ)∥_2^2≤0. These steps are explicit in the paper’s proof of (a,b) and in Proposition 3.10 for the Lyapunov structure, and they underpin the construction and bounds in (c,d) (including Theorem 3.6 for (d)) . The candidate solution follows the same route: it derives the same logistic inequality for y(t)=∥u∥_2^2, uses the same potential (up to a factor G=2Pλ) to show monotonic decay of G, and picks orbit-sum initial data on the critical shell. The only notable difference is that the candidate extracts a slightly stronger explicit lower bound for ∥uλ(t)∥_2 (monotonicity of y(t) within the invariant cone G≤0 gives y(t)≥y(0)=λ/D_H), while the paper states a somewhat weaker but sufficient lower bound via a direct inequality from Pλ(u(t))≤−cλ^2. Since Theorem 3.1 only requires some CH∈(0,1], both bounds are consistent with the theorem’s statement. For (d), the paper proves separation from constants via Theorem 3.6, whereas the candidate gives a direct estimate from G≤0 and Jensen; both are valid and compatible with the theorem. Overall, the methods and logic are substantially the same and correct.