PDE-Based Optimization for Advection Diffusion Equation in 2D Domain

The paper projects the 2D advection–diffusion equation onto a sine–cosine basis, defines F^{mn}_{ij} and G^{mn}_{ij} as in (43), applies Cauchy–Schwarz to obtain (45)–(47), and concludes the advection matrix entries satisfy |A_{(mn),(ij)}(t)| ≤ 8√2π K with K = max K^{mn}_{ij}, together with a linear-in-N bound K ≤ γN; these steps are standard and consistent with (29)–(31), (35)–(39), (43)–(50) . Repeating the argument under the ∥∇v∥_{L^2}-normalization yields (51)–(55) with K̂^{mn}_{ij} as in (53) . However, the paper then asserts “It is easy to show that there exists γ̂ > 0 such that 0 < K̂ ≤ γ̂,” independent of N (equation (56)) . This is false as stated: since each A^{mn}_{kℓij}, B^{mn}_{kℓij} has at most four nonzero contributions and one can take i = m+1, j = n+1 so that one contributing pair is (k,ℓ) = (1,1), the sum in K̂^{mn}_{ij} includes a term 1/(256·2), implying K̂^{mn}_{ij} ≥ i/(16√2), hence K̂ grows at least linearly with N. The candidate solution correctly identifies this flaw and provides a concrete lower bound showing K̂ cannot be bounded independently of N. Additionally, in the gradient-normalized case, the correct entrywise bound from (51)–(54) is |A_{(mn),(ij)}(t)| ≤ 8√2 K̂; the paper’s (55) contains an extra factor of π (a weaker, likely typographical over-bound) .