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

Journal of Differential Equations Rayleigh–Type Renormalized Flows in Weighted Geometries --Manuscript Draft-- Manuscript Number: Article Type: Article Keywords: Partial Differential Equations; Dynamical Systems; Functional Analysis; Rayleigh Flows; Spectral Theory Corresponding Author: Mateus Rodrigues de Maria Universidade Federal do Ceará Fortaleza, Ceará BRAZIL First Author: Mateus Rodrigues de Maria Order of Authors: Mateus Rodrigues de Maria Abstract: We introduce a class of Rayleigh–type renormalized flows driven by weighted geometries and non–uniform density structures. Given a strictly positive weight [[EQUATION]]w=κ+1 and the renormalized derivative [[EQUATION]]Dκf:=w1dxd(wf), we consider the self–adjoint dissipation operator [[EQUATION]]Aκ:=Dκ∗Dκ on [[EQUATION]]L2(dμ) with [[EQUATION]]dμ=wdx. Our main object is the normalized Rayleigh flow[[EQUATION]]∂tf=−Aκf+ρ(f)f, [[EQUATION]]ρ(f)=Aκf,fμ, [[EQUATION]]∥f(t)∥L2(dμ)≡1,defined canonically as a normalization of the linear semigroup [[EQUATION]]e−tAκ. We prove invariance of the [[EQUATION]]L2(dμ) constraint, positivity preservation under standard Markovian assumptions, and a sharp Lyapunov dissipation identity[[EQUATION]]dtdE(f(t))=−2∥Aκf(t)−ρ(f(t))f(t)∥μ2≤0, [[EQUATION]]E(f)=∥Dκf∥μ2.Stationary states are eigenfunctions of [[EQUATION]]Aκ, and spectral gap mechanisms yield stability and mode selection in compact settings. The framework connects weighted PDE dissipation, constrained gradient dynamics, and eigenmode extraction under heterogeneous densities; higher–dimensional extensions are discussed. Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation Rayleigh–Type Renormalized Flows in Weighted Geometries Energy Dissipation, Invariance, and Applications to PDE Dynamics Mateus R. de Maria Abstract 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∗ κDκon 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 L2(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)∥2 µ≤0,E(f)=∥Dκf∥2 µ. 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 eigenmode extraction under heterogeneous densities. All statements are formulated for compact one-dimensional domains; higher-dimensional extensions are discussed in the conclusion. 1 Manuscript Click here to view linked References Contents 1 Introduction and motivation 4 1.1 Rayleigh quotients, normalization, and dissipation .............. 4 1.2 Weighted geometries and renormalized derivatives .............. 4 1.3 Main object: the normalized Rayleigh flow .................. 4 1.4 What is new (and what is not) ......................... 5 1.5 Outline of the paper .............................. 5 2 Weighted operators and renormalized derivatives 6 2.1 Definition of the renormalized derivative ................... 6 2.2 Basic properties ................................. 6 2.3 Integration by parts and adjoint structure .................. 6 2.4 Standing hypotheses .............................. 6 3 Definition of the renormalized Rayleigh flow 7 3.1 Phase space and admissible class ....................... 7 3.2 Rayleigh quotient and flow equation ...................... 7 3.3 Invariant sets: L2(µ)constraint and positivity ................ 7 3.4 Normalization of the linear semigroup ..................... 8 4 Energy functionals and monotonicity 9 4.1 Natural energy and Rayleigh quotients .................... 9 4.2 Lyapunov structure and sharp dissipation identity .............. 9 4.3 Interpretation as constrained gradient dynamics ............... 9 5 Well–posedness and regularity 10 5.1 Canonical construction via semigroup normalization ............. 10 5.2 Derivation of the PDE for t > 0........................ 10 5.3 Uniqueness and positivity ........................... 11 6 Stationary states and stability 12 6.1 Stationary states are eigenfunctions ...................... 12 6.2 Omega–limit points are stationary (compactness route) ........... 12 6.3 Spectral gap stability .............................. 12 6.4 Positivity and mode selection ......................... 13 7 Model examples 14 7.1 One–dimensional weighted diffusion ...................... 14 7.2 A smooth periodic weight ........................... 14 7.3 Piecewise constant weights ........................... 14 7.4 Takeaway .................................... 14 8 Informational and computational perspectives 15 8.1 Rayleigh flows as constrained optimization dynamics ............ 15 8.2 Discrete schemes and spectral extraction ................... 15 8.3 Links with replicator–type dynamics (finite dimensional intuition) ..... 15 8.4 Outlook: information geometry and mirror–type normalizations ...... 16 2 9 Summary and future directions 17 9.1 What was established .............................. 17 9.2 Conceptual interpretation ........................... 17 9.3 Directions for extension ............................. 17 A Appendix A: Technical lemmas and canonical well–posedness 18 A.1 Semigroup smoothing .............................. 18 A.2 Canonical well–posedness for the normalized Rayleigh flow ......... 18 B Appendix B: Variational derivations and alternative formulations 19 B.1 Constrained gradient flow on the L2(µ)–sphere ................ 19 B.2 Lagrange multiplier derivation ......................... 19 B.3 Alternative normalizations ........................... 19 3 1 Introduction and motivation In this section: (i) we recall Rayleigh quotients and normalized dissipation, (ii) we motivate the weighted differential geometry behind Dκ, (iii) we summarize the scope and organization of the paper. 1.1 Rayleigh quotients, normalization, and dissipation Rayleigh quotients sit at the intersection of spectral theory, variational principles, and dissipative PDE. Given a non–negative self–adjoint operator Aon a Hilbert space, the quantity R(f) = ⟨Af, f⟩ ⟨f, f⟩ measures the energy level of a state relative to its size, and extremizers are eigenfunctions. A classical idea is to evolve fby a dissipative dynamics that decreases ⟨Af, f⟩while preventing trivial decay of amplitude. This leads to normalized flows on constraint manifolds (typically spheres), where dissipation is geometrically projected onto the tangent space. In heterogeneous media, however, the operator Aitself should encode the geometry induced by the density/weight of the medium. This is the setting of the present paper. 1.2 Weighted geometries and renormalized derivatives Let w(x)=κ(x) + 1 be a strictly positive weight and define the weighted measure dµ = w dx. A natural derivative adapted to this geometry is Dκf:= 1 w(wf)′, which is the one–dimensional prototype of a weighted divergence structure. We associate to it the dissipation operator Aκ:= D∗ κDκon L2(I, dµ), so that the energy E(f)=∥Dκf∥2 µis intrinsic to the weighted geometry. 1.3 Main object: the normalized Rayleigh flow Our main object is the Rayleigh–type normalized flow ∂tf=−Aκf+ρ(f)f, ρ(f)=⟨Aκf, f⟩µ,∥f(t)∥µ≡1.(1) This is the steepest descent of Econstrained to the L2(dµ)–sphere, hence it keeps the trajectory on a fixed–norm manifold while dissipating energy. A key feature is that (1) admits a canonical construction: if g(t) = e−tAκf0solves the linear dissipation, then f(t) = g(t) ∥g(t)∥µ solves (1) for t > 0(see Chapter 5and Appendix A). 4 1.4 What is new (and what is not) Not new. Normalized Rayleigh flows and constrained gradient dynamics are classical tools in spectral analysis and PDE. Weighted self–adjoint operators and Dirichlet forms are also standard in analysis on weighted spaces. New contribution. The contribution of this paper is structural and unifying: •we encode the heterogeneity at the differential level via the renormalized derivative Dκ, so that dissipation, normalization, and invariances become transparent and compatible with the weighted geometry; •we emphasize the normalized–semigroup representation as the canonical definition of the nonlinear Rayleigh flow, yielding a clean bridge between weighted PDE dissipation and eigenmode extraction under heterogeneous densities. 1.5 Outline of the paper Chapter 3defines the flow and its invariant sets. Chapter 4establishes the Lyapunov structure and sharp dissipation identity. Chapter 5develops well–posedness via normalized semigroups and recovers the PDE for t>0. Chapter 6characterizes equilibria as eigenfunctions and discusses stability via spectral gaps. Chapters 7–8present model examples and computational viewpoints. Appendices collect technical lemmas and variational derivations. 5 2 Weighted operators and renormalized derivatives In this section: (i) we define the renormalized derivative Dκand its basic calculus, (ii) we record weighted integration by parts and the adjoint structure, (iii) we state standing analytic hypotheses used throughout. 2.1 Definition of the renormalized derivative Let κ=κ(x)be a non–negative weight function and set w(x) := κ(x) + 1. We define Dκf:= 1 w d dxwf. 2.2 Basic properties For sufficiently regular f, g, Dκ(fg) = g Dκf+fdg dx. When κ≡0(hence w≡1), Dκreduces to the classical derivative. Also Dκf=f′+w′ wf. 2.3 Integration by parts and adjoint structure Let dµ =w dx. Under periodic or Neumann–type boundary conditions (no boundary terms), ZI (Dκf)g dµ =−ZI fdg dx dµ. (2) Hence, formally on L2(dµ), D∗ κ=−d dx, Aκ:= D∗ κDκ=−d dxw−1(w·)′. 2.4 Standing hypotheses (H1) (Weight)w=κ+ 1 ∈W1,∞(I)and 0< wmin ≤w(x)≤wmax <∞. We set dµ =w dx and H=L2(I, dµ). (H2) (Boundary conditions) Periodic (or Neumann–type) boundary conditions so that integration by parts yields no boundary terms for the form aκ(f, g) = ⟨Dκf, Dκg⟩µ. (H3) (Self–adjoint dissipation) The quadratic form aκwith domain H1(I)is closed and non–negative, hence defines a non–negative self–adjoint operator Aκ:= D∗ κDκ on H, and −Aκgenerates a contraction semigroup e−tAκ. (H4) (Smoothing) For each t>0,e−tAκ(H)⊂ D(Aκ)and Aκe−tAκis bounded (Lemma A.1). (H5) (Positivity, when used) When positivity statements are invoked, assume e−tAκ is positivity preserving on H(and positivity improving when mode selection is claimed). Under (H1)–(H4) the normalized Rayleigh flow is well-defined for all f0∈H\ {0}via normalized semigroup trajectories (Definition 5.1). 6 3 Definition of the renormalized Rayleigh flow In this section: (i) we set the weighted phase space and the operator Aκ, (ii) we define the Rayleigh quotient and the renormalized Rayleigh flow, (iii) we record invariant sets and the normalized–semigroup representation. 3.1 Phase space and admissible class Let Ibe a compact one–dimensional domain (e.g. I=T). Assume the standing hypotheses of Section 2.4. Set w=κ+ 1,dµ =w dx, and H:= L2(I, dµ),⟨f, g⟩µ:= ZI fg dµ, ∥f∥2 µ:= ⟨f, f⟩µ. Let Dκf=w−1(wf)′and Aκ:= D∗ κDκ. Constraint manifold. Sµ:= {f∈H:∥f∥µ= 1}. When positivity is desired, we restrict further to S+ µ:= {f∈Sµ:f≥0a.e.}. 3.2 Rayleigh quotient and flow equation We use the global Rayleigh quotient ρ(f) := ⟨Aκf, f⟩µ ⟨f, f⟩µ , f ∈ D(A1/2 κ)\ {0}.(3) On Sµ,ρ(f)=⟨Aκf, f⟩µ. Definition 3.1 (Renormalized Rayleigh flow).The renormalized Rayleigh flow associated with Aκis ∂tf=−Aκf+ρ(f)f, f(0) = f0∈H, f0≡ 0.(4) 3.3 Invariant sets: L2(µ)constraint and positivity Proposition 3.2 (L2(µ)norm is preserved).Let fbe a sufficiently regular solution of (4). Then d dt∥f(t)∥2 µ= 0. In particular, if ∥f0∥µ= 1 then f(t)∈Sµfor all tin the interval of existence. Proof. Take the H–inner product of (4) with f: 1 2 d dt∥f∥2 µ=⟨∂tf, f⟩µ=−⟨Aκf, f⟩µ+ρ(f)⟨f, f⟩µ. By (3), ρ(f)⟨f, f⟩µ=⟨Aκf, f⟩µ. Proposition 3.3 (Positivity preservation (structural)).Assume e−tAκis positivity preserving on H. If f0≥0a.e., then any sufficiently regular solution of (4)satisfies f(t)≥0 a.e. for all t. Idea. Let α(t) = Rt 0ρ(f(s)) ds and set g(t)=e−α(t)f(t). Then ∂tg=−Aκg. Positivity of the semigroup implies g(t)≥0, hence f(t)≥0. 7 3.4 Normalization of the linear semigroup Let gsolve ∂tg=−Aκg, g(0) = f0∈H, f0≡ 0.(5) Define f(t) := g(t) ∥g(t)∥µ .(6) Then f(t)∈Sµfor all t≥0, and for t > 0it solves (4) (see Theorem 5.3). This gives the canonical construction of solutions. Remark 3.4 (Alternative constraints).If one requires preservation of Rf dµ, a different normalization is needed. We record such variants in Appendix B. 8 8 Informational and computational perspectives In this section: (i) we interpret the flow as constrained optimization, (ii) we present discrete “evolve + renormalize” schemes, (iii) we comment on links with replicator/mirror analogies. This chapter emphasizes a practical interpretation: Rayleigh–type renormalized flows are continuous–time algorithms for eigenmode extraction and energy optimization under a geometry dictated by the weight. Remark 8.1 (Status of this chapter).The statements below are interpretative and algorithmic. Parallels with replicator dynamics or mirror normalizations should be read as analogies rather than as new theorems in this note. 8.1 Rayleigh flows as constrained optimization dynamics On Sµ={f:∥f∥µ= 1}, the energy E(f)=⟨Aκf, f⟩µcoincides with the Rayleigh quotient. Projecting the gradient direction onto the tangent space yields Aκf−ρ(f)f. Hence ∂tf=−(Aκf−ρ(f)f) is steepest descent of Eon Sµ. The sharp dissipation identity is Proposition 4.1. 8.2 Discrete schemes and spectral extraction A robust viewpoint is: evolve linearly and renormalize. For example, a backward Euler step for the linear orbit reads gn+1 = (I+ ∆t Aκ)−1gn, g0=f0, and we then normalize fn+1 := gn+1 ∥gn+1∥µ . Since Aκis self–adjoint and nonnegative, (I+ ∆t Aκ)−1is a contraction on H. This is a continuous analogue of inverse iteration / power methods, with the geometry encoded at the differential level via Dκ. 8.3 Links with replicator–type dynamics (finite dimensional intuition) In finite dimensions with a symmetric positive semidefinite matrix A, ˙x=−(Ax −(x⊤Ax)x) is a canonical Rayleigh flow on the sphere converging to eigenvectors. Variants with positivity/simplex constraints lead to replicator–like dynamics. The present framework can be viewed as an infinite dimensional weighted analogue of this spectral descent. 15 8.4 Outlook: information geometry and mirror–type normalizations One may replace the L2(µ)constraint by alternative normalizations, e.g. Rf dµ = 1 or entropy constraints, leading to different projection geometries. We keep these extensions as future work (Appendix B). 16 9 Summary and future directions In this section: (i) we summarize the core structural results, (ii) we give a conceptual reading of the framework, (iii) we list natural extensions. 9.1 What was established We introduced a class of renormalized Rayleigh–type flows driven by a weighted differential structure. The central ingredients are: •A weighted derivative Dκf=w−1(wf)′with w=κ+1, naturally tied to the measure dµ =w dx and to weighted integration by parts. •A non–negative self–adjoint operator Aκ=D∗ κDκand the normalized flow ∂tf= −Aκf+ρ(f)f. •Structural invariances (positivity, L2(µ)–mass preservation), and a sharp Lyapunov dissipation identity for the weighted energy E(f)=∥Dκf∥2 µ. •A characterization of stationary states as eigenfunctions of Aκ, together with spectral gap stability mechanisms and positivity–based mode selection. 9.2 Conceptual interpretation The flow may be read simultaneously as: •a dissipative PDE in a weighted geometry; •a constrained gradient descent for a quadratic energy; •a spectral extraction dynamics adapted to non–uniform densities. 9.3 Directions for extension 1. Higher dimensional geometries: define Dκas a weighted divergence/gradient pair on manifolds and study Aκas a weighted Laplacian–type operator. 2. Alternative normalizations: L1(µ)constraints, entropy constraints, or information– geometric constraints leading to mirror–type flows. 3. Nonlinear operators: replace Aκby quasilinear or nonlocal dissipations while preserving an intrinsic Rayleigh quotient structure. 4. Applications: stability selection in reaction–diffusion systems, weighted filtering, and robust mode extraction in heterogeneous media. The present paper isolates the minimal robust core. The expectation is that, once the geometry is encoded at the derivative level, many classical dissipative mechanisms acquire new invariance and normalization options without losing analytic transparency. 17 A Appendix A: Technical lemmas and canonical well– posedness In this section: (i) we record a smoothing estimate for the semigroup, (ii) we state a clean canonical well–posedness theorem for the normalized flow, (iii) we give brief proofs/sketches for reuse across the paper. A.1 Semigroup smoothing Lemma A.1 (Semigroup smoothing).Assume (H1)–(H3). Then −Aκgenerates a contraction semigroup e−tAκon H. Moreover, for each t > 0,e−tAκ(H)⊂ D(Aκ)and there exists C > 0such that ∥Aκe−tAκ∥L(H)≤C t, t ∈(0,1]. Sketch. This is standard for non–negative self–adjoint operators: use functional calculus ∥Ae−tA∥= supλ≥0λe−tλ ≲t−1. A.2 Canonical well–posedness for the normalized Rayleigh flow Theorem A.2 (Canonical well–posedness via normalization).Assume (H1)–(H4). For any f0∈H\ {0}define g(t) = e−tAκf0, f(t) = g(t) ∥g(t)∥µ , f(0) = f0 ∥f0∥µ . Then: 1. f∈C([0,∞); H)and ∥f(t)∥µ≡1. 2. f∈C1((0,∞); H)and f(t)∈ D(Aκ)for all t > 0. 3. For every t>0,fsatisfies the Rayleigh flow equation ∂tf=−Aκf+ρ(f)f, ρ(f) = ⟨Aκf, f⟩µ ⟨f, f⟩µ . 4. (Uniqueness for t > 0) If ˜ fis another solution on (0,∞)with ∥˜ f(t)∥µ≡1and ˜ f(τ) = f(τ)for some τ > 0, then ˜ f(t) = f(t)for all t≥τ. 5. If, in addition, e−tAκis positivity preserving and f0≥0a.e., then f(t)≥0a.e. for all t≥0. Sketch. Items (1)–(3) follow from the quotient rule and Lemma A.1 (see Chapter 5for the full derivation). Uniqueness on [τ, ∞)follows by transforming the nonlinear equation into the linear one via an integrating factor. Positivity follows since normalization preserves the cone. 18 B Appendix B: Variational derivations and alternative formulations In this section: (i) we derive the flow as constrained gradient descent on Sµ, (ii) we recover the eigenvalue condition via Lagrange multipliers, (iii) we comment on alternative normalizations. B.1 Constrained gradient flow on the L2(µ)–sphere Let E(f) = ⟨Aκf, f⟩µon H=L2(dµ)and consider the constraint ∥f∥2 µ= 1. The unconstrained H–gradient is ∇E(f)=2Aκf. The tangent space at f∈Sµis TfSµ={u∈H:⟨u, f⟩µ= 0}. Projecting Aκfonto TfSµyields ΠTf(Aκf) = Aκf− ⟨Aκf, f⟩µf=Aκf−ρ(f)f. Therefore the steepest descent dynamics of Eon Sµis ∂tf=−ΠTf(Aκf) = −(Aκf−ρ(f)f), which is equivalent to the normalized Rayleigh flow. B.2 Lagrange multiplier derivation Consider minimizing E(f)under ∥f∥2 µ= 1. A stationary point satisfies δE(f)−λ(∥f∥2 µ−1)= 0 ⇒Aκf=λf, so equilibria are eigenfunctions, with λ=ρ(f)on the constraint. B.3 Alternative normalizations One may impose instead an L1(µ)constraint Rf dµ = 1 (when positivity is enforced). In that case, a normalized flow may be written as ∂tf=−Aκf+ Λ(t)f, where Λ(t)enforces d dt Rf dµ = 0. This differs from the Rayleigh quotient normalization and generally does not yield the same quadratic dissipation identity. Another option is to normalize by entropy or other convex constraints, leading to mirror–type projections. These variants connect to information geometry but are outside the minimal core of this note. 19 Declaration on Generative AI and AI-assisted Technologies During the preparation of this manuscript, the author used ChatGPT (OpenAI) for language polishing, organization suggestions, and editorial clarification. After using this tool, the author reviewed, edited, and validated all content, and takes full responsibility for the scientific accuracy, originality, and conclusions of the work. References [1] L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser, 2008. [2] L. C. Evans, Partial Differential Equations, 2nd ed., AMS, 2010. [3] C. Villani, Topics in Optimal Transportation, AMS, 2003. [4] F. Otto, The geometry of dissipative evolution equations, Comm. PDE 26 (2001), 101–174. [5] K.-T. Sturm, Analysis on local Dirichlet spaces, in From Stochastic Analysis to Mathematical Finance, Springer, 2006. [6] D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de probabilités XIX, Springer, 1985. [7] M. R. de Maria, Renormalized Calculus and Weighted Fundamental Theorems, preprint, 2025. [8] M. R. de Maria, Rayleigh–Type Renormalized Flows in Weighted Geometries, preprint, 2025. 20 Click here to access/download LaTeX Source Files main.tex