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

Rayleigh–Type Renormalized Flows in Weighted Geometries Energy dissipation, invariance, and PDE avatars in weighted geometries Mateus R. de Maria∗ Universidade Federal do Ceará (UFC) Fortaleza–CE, Brazil Abstract We introduce a class of Rayleigh–type normalized flows associated with weighted geometries and non–uniform density structures. The dynamics is generated by a weighted Dirichlet operator Aκ=D∗ κDκon L2(dµ), where dµ =w dx and Dκis the renormalized derivative induced by a strictly positive weight w=κ+ 1. The flow preserves the L2(dµ)constraint and dissipates the Rayleigh energy. We establish invariance and positivity properties, derive a sharp Lyapunov dissipation identity, and characterize stationary states as eigenfunctions of Aκ. We also discuss localization tools and connections with weighted diffusion and PDE avatars. Keywords. Rayleigh flow; constrained gradient dynamics; weighted Dirichlet forms; energy dissipation; eigenmode selection. MSC (2020). 35K55; 35B35; 47J35; 58J35. ∗Corresponding author. Email: [email protected]. 1 1 Introduction 1.1 Motivation: Rayleigh quotients, normalization, and weighted geometries 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 strategy to extract low modes is to evolve fby a dissipative dynamics that decreases ⟨Af, f⟩while preventing the trivial decay of amplitude. This leads to normalized (or constrained) flows on a constraint manifold (typically a sphere), where dissipation is projected onto the tangent space. In heterogeneous media, however, the operator itself should encode the geometry induced by the density/weight of the medium. The purpose of this paper is to present a clean, reusable formulation of a Rayleigh–type normalized flow in a one–dimensional weighted geometry, emphasizing the structural identities that make invariance, dissipation, and mode selection transparent. Concretely, we: •encode heterogeneity at the differential level via a renormalized derivative Dκ; •construct the nonlinear flow canonically by normalizing the linear semigroup; •prove a sharp Lyapunov dissipation identity with a rigid equality case (eigenstate selection); •record stability/selection statements under spectral gap and positivity hypotheses, and discuss PDE avatars and extensions. 1.2 Main results Setting. Let Ibe a compact one–dimensional domain (e.g. T). Fix a strictly positive weight w(x) = κ(x) + 1 and the weighted measure dµ =w dx. We work in H=L2(I, dµ) with inner product ⟨·,·⟩µand norm ∥·∥µ. Define the renormalized derivative Dκf:= 1 w(wf)′ and the associated dissipation operator Aκ:= D∗ κDκon H. Standing analytic hypotheses (self–adjointness, smoothing, positivity when used) are recorded in Section 2.4. We consider the Rayleigh–type normalized flow ∂tf=−Aκf+ρ(f)f, ρ(f) = ⟨Aκf, f⟩µ ⟨f, f⟩µ ,(1.1) 2 which preserves the L2(dµ)constraint. A key feature is that (1.1) admits a canonical construction: if g(t) = e−tAκf0is the linear orbit, then f(t) = g(t) ∥g(t)∥µ (1.2) solves the nonlinear flow for t > 0(Chapter 5). Theorem A (constraint, dissipation, and rigidity). The flow preserves the L2(dµ) constraint and admits an intrinsic Lyapunov structure whose equality case is rigid. Theorem 1.1 (Constraint and sharp dissipation).Assume (H1)–(H3) from Section 2.4. Let f: (0, T)→Hbe a nontrivial solution of (1.1) such that f∈C1((0, T); H)and f(t)∈ D(Aκ)for all t∈(0, T ). Then the constraint is invariant: d dt ∥f(t)∥2 µ= 0. Moreover, the energy E(f) = ⟨Aκf, f⟩µ=∥Dκf∥2 µsatisfies the sharp dissipation identity d dt E(f(t)) = −2∥Aκf(t)−ρ(f(t))f(t)∥2 µ≤0. In particular, if E(f(t)) is constant on a time interval, then fis stationary there and Aκf(t) = ρ(f(t))f(t), i.e. the trajectory is an eigenstate. Interpretation. The decay rate of Eis exactly the squared defect from the eigenvalue equation. Theorem B (canonical global well–posedness). A distinguished global solution is obtained by normalizing the linear semigroup orbit; this produces a strong solution for t > 0and fixes the dynamics without auxiliary regularization. Theorem 1.2 (Well–posedness via normalized semigroups).Assume (H1)–(H4). For every f0∈H\ {0}the normalized semigroup formula (1.2) defines a global trajectory f: [0,∞)→H. For every t > 0one has f(t)∈ D(Aκ)and f∈C1((0,∞); H), and the trajectory satisfies (1.1) for all t > 0(hence it is a strong solution on (0,∞)). Moreover, among sufficiently regular normalized solutions, the flow is unique forward in time. If, in addition, the semigroup e−tAκis positivity preserving and f0≥0a.e., then f(t)≥0a.e. for all t≥0. See Theorem 5.3 (PDE for t > 0), Proposition 5.5 (uniqueness on (0,∞)), and Proposition 5.6 (positivity for normalized semigroup solutions). Theorem C (mode selection under a spectral gap / positivity). When Aκhas discrete spectrum, the normalized semigroup dynamics selects the slowest decaying mode; positivity assumptions single out the ground state. Theorem 1.3 (Selection and stability).Assume (H1)–(H4) and that Aκhas discrete spectrum 0≤λ1< λ2≤ · · · with an orthonormal eigenbasis in H. Let P1denote the orthogonal projection onto the λ1–eigenspace. If P1f0= 0, then the normalized semigroup solution satisfies   f(t)−P1f0 ∥P1f0∥µ  µ≤C(f0)e−(λ2−λ1)t(t≥0), 3 and in particular converges in Hto a unit eigenfunction in the λ1–eigenspace. If, furthermore, e−tAκis positivity improving and Iis connected, then for any f0≥0,f0≡ 0, the solution converges to the unique (up to sign) strictly positive ground state (normalized in L2(dµ)). See Proposition 6.4 (spectral gap stability) and Proposition 6.5 (positivity–based ground state selection). 1.3 Related work and conceptual positioning Rayleigh quotient dynamics and normalized gradient flows are classical tools in spectral analysis and PDE, and they also fit naturally within the broader gradient-flow/Lyapunov perspective; see, e.g., [4, 1]. Our weighted setting is aligned with Dirichlet-form and diffusion frameworks [6, 5]. The present note emphasizes a concrete differential encoding of heterogeneity through Dκ, and a canonical normalization–of–semigroup viewpoint that makes invariance, dissipation, and selection statements essentially “one line” once the operator framework is fixed. 1.4 Organization The paper is organized as follows. Chapter 2 introduces Dκ, the operator Aκ, and standing hypotheses. Chapter 3 defines the Rayleigh flow and its invariant sets, including the normalized–semigroup representation. Chapter 4 proves the sharp Lyapunov dissipation identity. Chapter 5 develops well–posedness via normalized semigroups and recovers the PDE for t > 0. Chapter 6 characterizes equilibria as eigenfunctions and records stability/selection mechanisms. Chapters 7–8 present model examples, PDE avatars, and a computational viewpoint. The appendices collect technical lemmas and variational derivations. 4 2 Weighted operators and renormalized derivatives In this section: We introduce the renormalized derivative Dκ, its adjoint structure, and the 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. (2.1) 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∗ κDκon 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−tAκis 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). Bridge to Chapter 3. With the operator framework fixed, we can now define the normalized Rayleigh flow and begin with its basic invariants. 5 3 Definition of the renormalized Rayleigh flow In this section: This chapter sets the weighted Hilbert framework and defines the normalized Rayleigh flow together with its basic invariants. 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}.(3.1) 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.(3.2) 3.3 Invariant sets: L2(µ)constraint and positivity Proposition 3.2 (L2(µ)norm is preserved).Let fbe a sufficiently regular solution of (3.2). 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 (3.2) with f: 1 2 d dt∥f∥2 µ=⟨∂tf, f⟩µ=−⟨Aκf, f⟩µ+ρ(f)⟨f, f⟩µ. By (3.1), 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 (3.2) satisfies f(t)≥0a.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. 6 3.4 Normalization of the linear semigroup Let gsolve ∂tg=−Aκg, g(0) = f0∈H, f0≡ 0.(3.3) Define f(t) := g(t) ∥g(t)∥µ .(3.4) Then f(t)∈Sµfor all t≥0, and for t > 0it solves (3.2) (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. Bridge to Chapter 4. With the dynamics fixed, we next identify the natural Lyapunov functional and prove the sharp dissipation identity that drives convergence and mode selection. 7 4 Energy functionals and monotonicity In this section: This chapter establishes the Lyapunov structure and proves the sharp dissipation identity for the flow. We retain the setting and notation of Chapter 3. The key point is that the normalized dynamics is the L2(dµ)–constrained steepest descent of a quadratic energy, and the decay rate is exactly the squared defect from the eigenvalue equation. 4.1 Natural energy and Rayleigh quotients We keep the notation of Chapter 3, in particular H=L2(I, dµ)and Aκ=D∗ κDκ. Define E(f) := ∥Dκf∥2 µ=⟨Aκf, f⟩µ, f ∈ D(A1/2 κ).(4.1) Recall ρfrom (3.1). 4.2 Lyapunov structure and sharp dissipation identity Consider the Rayleigh flow ∂tf=−Aκf+ρ(f)f. (4.2) Proposition 4.1 (Sharp dissipation identity).Let f(t)be a sufficiently regular solution of (4.2). Then d dtE(f(t)) = −2∥Aκf(t)−ρ(f(t))f(t)∥2 µ≤0.(4.3) In particular, E(f(t)) is non–increasing along the flow. Proof. Differentiate E(f) = ⟨Aκf, f⟩µ: d dtE(f(t)) = 2⟨Aκf(t), ∂tf(t)⟩µ. Insert (4.2): d dtE=−2∥Aκf∥2 µ+ 2ρ(f)⟨Aκf, f⟩µ. Using ρ(f)⟨f, f⟩µ=⟨Aκf, f⟩µfrom (3.1), expand ∥Aκf−ρ(f)f∥2 µ=∥Aκf∥2 µ−2ρ(f)⟨Aκf, f⟩µ+ρ(f)2∥f∥2 µ, and note ρ(f)2∥f∥2 µ=ρ(f)⟨Aκf, f⟩µ. Rearranging yields (4.3). Corollary 4.2 (Finite dissipation).If f(t)is global and E(f(t)) is bounded below, then for any τ > 0, Z∞ τ ∥Aκf(t)−ρ(f(t))f(t)∥2 µdt < ∞. In particular, there exists tn→ ∞ such that ∥Aκf(tn)−ρ(f(tn))f(tn)∥µ→0. 4.3 Interpretation as constrained gradient dynamics On Sµ, the driving force is the projection of Aκfonto TfSµ={u:⟨u, f⟩µ= 0}, namely Aκf−ρ(f)f. Thus (4.2) is steepest descent of Eunder the constraint ∥f∥µ≡1. Remark 4.3 (Stationary points).Stationary points satisfy Aκf=ρ(f)f, i.e. eigenfunctions (Chapter 6). Bridge to Chapter 5. The dissipation identity is purely structural and does not by itself construct solutions. We now build global trajectories canonically by normalizing the linear semigroup and justify the PDE for t > 0. 8 5 Well–posedness and regularity In this section: This chapter constructs global solutions via normalized semigroups, recovers the PDE for t > 0, and proves uniqueness and positivity. We implement the canonical “evolve + renormalize” construction suggested by Chapter 3: start from the linear semigroup orbit g(t) = e−tAκf0and normalize it in L2(dµ). This produces a global trajectory in H, and for t > 0the smoothing hypothesis upgrades it to a strong solution of the PDE. 5.1 Canonical construction via semigroup normalization Assume (H1)–(H4) from Section 2.4. For f0∈H\ {0}define g(t) := e−tAκf0, f(t) := g(t) ∥g(t)∥µ . Then f(t)∈Sµfor all t≥0. The key point is that g(t)= 0 for all t > 0(see Lemma 5.2 below or Appendix A). Definition 5.1 (Normalized semigroup solution).We call f(t)defined by f(t) = e−tAκf0 ∥e−tAκf0∥µ , f0∈H\ {0},(5.1) the normalized semigroup solution of the Rayleigh flow. We set f(0) := f0/∥f0∥µ. Lemma 5.2 (Non–vanishing of the linear orbit).If f0= 0, then e−tAκf0= 0 for every t > 0. Proof. Since Aκis self–adjoint and non–negative, e−tAκis defined by spectral calculus. Because exp(−tλ)>0for all λ≥0, the operator e−tAκhas trivial kernel. Hence e−tAκf0= 0implies f0= 0. 5.2 Derivation of the PDE for t > 0 Theorem 5.3 (PDE for t > 0).Let f0∈H\{0}and let f(t)be the normalized semigroup solution (5.1). Then for every t > 0we have f(t)∈ D(Aκ)and f∈C1((0,∞); H). Moreover, for t > 0the trajectory satisfies the Rayleigh flow equation (3.2): ∂tf(t) = −Aκf(t) + ρ(f(t)) f(t), ρ(f) = ⟨Aκf, f⟩µ ⟨f, f⟩µ . Proof. By (H4), for each t > 0the semigroup orbit satisfies g(t)∈ D(Aκ)and g′(t) = −Aκg(t)in H. Set n(t) := ∥g(t)∥µ>0for t > 0and f(t) = g(t)/n(t). Then fis differentiable for t > 0and f′(t) = g′(t) n(t)−n′(t) n(t)f(t). Since d dt 1 2n(t)2=⟨g′(t), g(t)⟩µ=−⟨Aκg(t), g(t)⟩µ, we get n′(t) n(t)=−⟨Aκg(t), g(t)⟩µ ∥g(t)∥2 µ =−⟨Aκf(t), f(t)⟩µ ∥f(t)∥2 µ =−ρ(f(t)), 9 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 5 for 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. 16 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. 17 Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Data availability No data were generated or analyzed in this study. 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