Rayleigh–Type Renormalized Flows in Weighted Geometries Energy Dissipation, Invariance, and Applications to PDE Dynamics

We introduce a class of Rayleigh–type renormalized flows driven by weighted geometries and non–uniform density structures. Given a strictly positive weight w = κ + 1 and the renormalized derivative 

Dκf := 1 w d dx(wf),

we consider the self–adjoint dissipation operator Aκ := D∗κDκ on L2 (dµ) with dµ = w dx. Our main object is the normalized Rayleigh flow

∂tf = −Aκf + ρ(f) f, ρ(f) = ⟨Aκf, f⟩µ, ∥f(t)∥L2(dµ) ≡ 1,

which can be defined canonically as a normalization of the linear semigroup e−tAκ . We prove invariance of the L 2(dµ) constraint, positivity preservation under standard Markovian assumptions, and a sharp Lyapunov dissipation identity

d/dtE(f(t)) = −2∥Aκf(t) − ρ(f(t))f(t)∥2µ ≤ 0, E(f) = ∥Dκf∥2µ.

Stationary states are characterized as eigenfunctions of Aκ, and spectral gap mechanisms yield stability and mode selection in basic compact settings. The framework provides a unified viewpoint connecting weighted PDE dissipation, constrained gradient dynamics, and eigen mode extractionunder heterogeneous densities. All statements are formulated for compact one-dimensional domains; higher-dimensional extensions are discussed in the conclusion.