On Koopman Operators for Burgers Equation

The paper proves, for small L2 data u0 in ΩB, that (a) l_n(u0)=√2∫ H(u0)cos(nπs)ds are Koopman eigen-observables with eigenvalues e^{-n^2π^2 t}, (b) finite products give all eigen-observables indexed by ν with λ_ν=π^2∑ n_k^2, (c) u(t,x)=∑_ν e^{-λ_ν t} φ_ν(u0) a_ν(x), and (d) the series converges uniformly on [0,∞)×[0,1] for u0∈ωB, and yields the static expansion at t=0; see Proposition 1, Corollary 1, Proposition 2, Proposition 3, and Corollaries 2–3 in the PDF (equations (9)–(16)) . The candidate solution uses the same Cole–Hopf conjugacy and cosine-series argument, reproducing the same objects a_ν(x) and φ_ν, and the same spectrum, with a slightly different bound via a maximum principle for v−1 rather than the paper’s L2-gradient lemmas; it also sketches a uniform convergence proof by grouping terms by α and using the Weierstrass M-test. Aside from a minor omission—one sentence justifying the uniform (in t) bound on ∥v_x(t,·)∥_∞ (true since v_x solves the heat equation with homogeneous Dirichlet boundary)—the model’s proof aligns with the paper’s results and logic (cf. the Cole–Hopf conjugacy identities (21) and the detailed convergence estimates in Section 5) .