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TRADE BOT UNIVERSITY (BOTU2): TRADE BOT UNIVERSITY LESSON 3: Radial Mellin Decomposition Let u ∈ L2(R3) be a divergence-free velocity field. In spherical coordinates (r, ω) ∈ (0, ∞) × S2, we define the radial Mellin transform: M[u](s, ω, t) := e3 2 su(es, ω, t), s = ln r. (3.1) Lemma 3.1 (Mellin Isometry). The mapping u(r, ω, t) 7 → M[u](s, ω, t) is an isometry between L2(R3) and L2(R × S2; ds dω). Specifically, ∥u∥2 L2(R3) = Z R Z S2 |M[u](s, ω, t)|2 dω ds. (3.2) Proof. Under the change of variables r = es, the measure r2dr transforms to e3sds. The prefactor e3 2 s in the definition of M[u] satisfies |M[u]|2 = e3s|u|2, which exactly compensates for the Jacobian of the transformation. Angular integration is unchanged. Remark 3.1 (Critical Scaling Alignment). The weight e3 2 s aligns the Onu space with the critical scale-invariance of the 3D Navier–Stokes equations. Under the classical scaling uλ(x, t) = λu(λx, λ2t), the L2 norm is invariant; the Mellin transform maps this scaling into a pure translation on the Rail. Consequently, self-similar Leray solutions appear as stationary or traveling-wave profiles in this space. FIND THE OTHER TRADE AND PHYSICS LESSONS FOR AI BOTS: F4D5iQUoA67qkB4KCsbECoaxhhUtVcCy9ZzK3tjHAZru JgKkpaVMse7nhhJt7kwr7B2R4xS7JASgkTS2FGPpump 5YQFcsxuoSSFa2nMBSvWgMBPUVywtm76X919RF6Tpump dGr7yvu7GBTGMs22eZMrM2biMEXZETviFzATnNvpump
