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Sharp Convergence of Wasserstein Gradient Flows for Spectrally Nonnegative Interaction Energies

Zhengjiang Lin, Philippe Rigollet

arXiv:2609.27008v1math.APcs.LGmath.OC

TL;DR

For nondiffusive Wasserstein flows without geodesic convexity, standard gradient-flow theory does not ensure convergence. The paper uses entropy and spectral structure to prove global energy decay, then constructs linearized and nonlinear examples showing how slowly that decay can occur.

  • Problem

    Without geodesic convexity, standard Wasserstein gradient-flow theory does not ensure qualitative convergence, while density-weighted dissipation may lose control where the density is small or vanishes.

    Source: §1 · p. 2

  • Method

    The authors use spectral compatibility between the kernel and the Laplacian to make entropy control the full energy gap, and use dyadic Fourier-block estimates to construct slow-decay solutions.

    Source: §1 · p. 3

  • Results

    For nonnegative initial densities in Lp(M), p > 1, the energy gap is o(t^-1); selected linearized and exact nonlinear solutions attain decay comparable to t^-1-delta.

    Source: §1 · p. 3

  • Takeaways & Limitations

    When the uniform measure is the unique minimizer, the flow converges weakly to it, without requiring a positive lower bound on the initial density.

    Source: §1 · p. 3

    The nonlinear slow-decay construction uses a torus-specific Fourier-series estimate; extending it to a closed manifold requires an analogue of that estimate.

    Source: §1.1.2 · p. 8

Abstract

from arXiv · show

We study the long-time behavior of Wasserstein gradient flows for interaction energies \[ \mathsf E[\mu] = \frac12\iint_{M\times M}K(x,y)\,\mathrm d\mu(x)\,\mathrm d\mu(y) \] on a closed manifold $M$. For kernels diagonal in a Laplace eigenbasis with nonnegative spectral coefficients, we prove a differential inequality relating the relative entropy to the energy gap. Consequently, for any nonnegative initial density $u_0\in L^p(M)$, $p>1$, the energy gap is integrable in time and satisfies \[ \mathsf E[\mu_t]-\mathsf E_{\min}=o(t^{-1}). \] If all spectral coefficients are positive, the flow converges weakly to the constant measure. These interaction energies need not be geodesically convex in Wasserstein space, and the associated flows contain no diffusion; their global convergence therefore does not follow from standard Wasserstein gradient flow theory. The kernels covered by our results include zonal kernels on spheres, kernels arising in transformer models, regularized Riesz kernels, and inverse fractional Laplacian kernels. We also investigate the sharpness of the $o(t^{-1})$ rate. For any smooth kernel in this class with infinitely many positive spectral coefficients and any $\delta>0$, we construct a solution of the linearized flow whose energy is comparable to $t^{-1-\delta}$ along a sequence of times tending to infinity. Moreover, for any $\delta>0$, by choosing a suitable inverse fractional Laplacian kernel on the flat torus, we construct an exact solution of the nonlinear Wasserstein gradient flow whose energy is comparable to $t^{-1-\delta}$. The nonlinear construction is based on uniform-in-time estimates for the evolution of the dyadic Fourier coefficient blocks and a blockwise energy-persistence argument. These estimates also yield a uniform-in-time quantitative comparison between the nonlinear Wasserstein gradient flow and its linearization.

1. Introduction

The section studies global convergence and sharp energy-gap decay for nondiffusive Wasserstein gradient flows with nonnegative spectral coefficients, using entropy control and linearized and nonlinear constructions. It develops blockwise persistence and nonlinear-to-linear estimates, extends the sharpness construction under additional manifold estimates, and illustrates the spectral assumptions with spherical and inverse fractional Laplacian kernels.

1. Introduction

  • Spectral nonnegativity makes the uniform measure a global energy minimizer, but the pure transport flow lacks diffusion and need not be geodesically convex.The framework includes smooth zonal kernels on spheres, regularized Riesz kernels, and inverse fractional Laplacian kernels.Source: §1 · pp. 1, 2, 4

1.1.1. Global convergence and entropy.

  • Entropy supplies an additional Lyapunov inequality that controls the full energy gap through spectral compatibility, without a positive lower bound on the density.Since entropy is nonnegative and the energy gap is nonincreasing, the inequality gives integrability of the gap and hence its o(t^-1) decay.Source: §1 · p. 3, §1.1.1 · pp. 5–6

1.1. Main results.

  • For every nonnegative initial density in Lp(M), p > 1, the energy gap is o(t^-1), and if the uniform measure is the unique minimizer, the flow converges weakly to it.No strict positivity, smallness, or high regularity of the initial density is required; vacuum regions persist at every finite time.Source: §1.1 · p. 4, §1.1.1 · p. 5

1.1.2. Sharpness and uniform-in-time nonlinear perturbation theory.

  • For kernels with infinitely many positive modes, a linearized solution has energy comparable to t^-1-δ along times tending to infinity; a suitable smooth kernel also realizes this rate for all large times.For general kernels the sequence-of-times qualification is unavoidable; some smooth kernels preclude two-sided polynomial decay for all sufficiently large times.Source: §1.1.2 · pp. 6–7
  • On the flat torus, inverse fractional Laplacian kernels admit an exact nonlinear solution with energy comparable to t^-1-δ, alongside a uniform-in-time quantitative comparison with its linearization.The construction uses uniform-in-time perturbation estimates and persistence of dyadic Fourier coefficient blocks; the resulting solution has a continuous density.Source: §1 · p. 3, §1.1.2 · p. 7
  • Dyadic block mass persists through the nonlinear transport until its linear decay time scale, yielding a matching lower bound; uniform-in-time Besov bounds also control the nonlinear-to-linear difference.The nonlinear error remains small when ε is sufficiently small; the comparison loses one derivative because the nonlinear source contains ∇f_t.Source: §1.1.2 · p. 8
  • The sharpness construction extends to a closed manifold if an analogue of the dyadic transport estimate and the required well-posedness and regularity-propagation results are available.The spectral construction, blockwise energy persistence, bootstrap, and energy interpolation use only the Laplacian spectral decomposition.Source: §1.1.2 · pp. 8–9

1.2.1. Zonal kernels on the sphere.

  • Spherical zonal examples include exponential dot-product kernels with strictly positive coefficients, linked to self-attention models, and singular Riesz kernels whose spherical harmonic coefficients are strictly positive.Regularized Riesz kernels retain strictly positive coefficients, satisfy the stated assumptions, and their energies converge to the Riesz energy.Source: §1.2.1 · p. 10

1.2.2. Inverse fractional Laplacian.

  • For inverse fractional Laplacian energies, Theorem 1.2 permits any nonnegative initial density in Lp(M), p > 1, without requiring initial closeness to the uniform measure.The C2 assumption supports well-posedness and the entropy calculation; extending the entropy method to singular regimes would require suitable weak-solution theory and an entropy identity.Source: §1.2.2 · p. 11

2. Monotonicity of entropy: proof of Theorem 1.2 and Theorem 1.3

The proof establishes entropy monotonicity through the kernel’s spectral structure and uses it to derive energy convergence. Characteristic-flow estimates ensure the entropy argument applies along the evolution, while compactness identifies weak limits as energy minimizers.

2. Monotonicity of entropy: proof of Theorem 1.2 and Theorem 1.3

  • Integrating the entropy inequality gives a finite time integral of the energy gap; its monotonicity then implies convergence of the gap to zero.Entropy is bounded below by Jensen’s inequality, with equality only for the uniform density.Source: §2 · p. 11
  • Compactness gives weakly convergent subsequences, and continuity of the energy makes every weak limit an energy minimizer.When the minimizer is unique, the flow converges weakly to it.Source: §2 · pp. 11–12
  • The characteristic flow and Jacobian estimates preserve Lp integrability at every finite time, ensuring the entropy remains finite.The proof uses the continuity equation and the Lp assumption with p > 1.Source: §2 · p. 12
  • The Laplacian eigenbasis expansion and nonnegative spectral coefficients yield an entropy differential inequality controlling the energy gap.The proof passes the inequality to the limit using Lp convergence.Source: §2 · p. 13

3. Linearized flows around the uniform measure: proof of Theorem 1.4 and Theorem 1.5

This section establishes decay and sharpness results for linearized flows, including a universal $o(t^{-1})$ bound, and develops dyadic Fourier–Besov estimates to control nonlinear perturbations and preserve spectral blocks. It applies these estimates to construct a positive-density nonlinear flow with matching polynomial energy bounds and to complete the stated corollary.

3. Linearized flows around the uniform measure: proof of Theorem 1.4 and Theorem 1.5

  • The linearized flow decays at rates determined by the spectral products a_nλ_n.The corresponding linearized energy is the quadratic term in the energy gap.Source: §3 · p. 14
  • For any δ > 0, a smooth kernel and smooth initial datum yield linearized energy decay comparable to t^-(1+δ).This establishes sharpness of the o(t^-1) rate for a constructed kernel.Source: §3 · pp. 14–15
  • For any δ > 0, every kernel with infinitely many positive coefficients admits H^-1 initial data whose energy is comparable to t_k^-(1+δ) along times t_k → ∞.With finitely many positive coefficients, the linearized energy is a finite sum of exponentially decaying terms.Source: §3 · pp. 15–17
  • Every mean-zero H^-1 initial datum has linearized energy o(t^-1), but for general kernels the sharp two-sided estimate need not hold at all sufficiently large times.A separate smooth-kernel example establishes failure of uniform-in-time polynomial asymptotics.Source: §3 · p. 17
  • For s > d/2, the inverse fractional Laplacian flow from δ_x0 − σ, which lies outside H^-1, can decay strictly slower than t^-1.The proposition covers d ≥ 2; the text notes the resulting exponent is less than 1 when d ≥ 3.Source: §3 · p. 19

4.1. Preliminaries for Theorem 1.6 and Proposition 4.1.

  • For nonlinear perturbations, the evolution is ∂_t f + A[f] = ε div(fQ[f]); mass is preserved and the energy is equivalent to the homogeneous H^-s norm.The transport derivative causes a one-derivative loss in the uniform comparison with the linearization, motivating evolution estimates for Littlewood–Paley blocks and energy persistence across blocks.Source: §4.1 · p. 21
  • For M = T^d, s > 1, and σ > d/2, Proposition 4.1 establishes global solutions to equation (4.5), uniform bounded Besov norms, and persistence of dyadic blocks.Source: §4.1 · p. 23

4.2. Auxiliary lemmas and proofs for Proposition 4.1.

  • Product, dyadic Fourier commutator, and transport-remainder lemmas provide the harmonic-analysis estimates used to prove Proposition 4.1.The lemmas assume σ > d/2 where specified, and the dyadic transport estimate is formulated on the flat torus.Source: §4.2 · pp. 23, 25, 26

4.3. Proof of Proposition 4.1.

  • The proof closes global control by a first-exit argument, using small initial Besov size to keep the solution bounded and extend it for all time.Local existence is obtained after ensuring the initial density 1 + εf0 is a probability density; the proof then takes a subcritical limit to establish the uniform estimates.Source: §4.3 · pp. 28, 31

4.4. Proof of Theorem 1.6 and Corollary 1.7.

  • The proof of Corollary 1.7 combines Proposition 4.1 with the stated semigroup estimate for the inhomogeneous linearized equation, as described in the cited passage.Lemma 4.5 applies for T ≥ 0, s ≥ 1, and h ∈ Bρ(M); its constant Cs,M depends only on s and M.Source: §4.4 · p. 31
  • The proof of Corollary 1.7 establishes estimate (1.18) with a constant independent of T.Source: §4.4 · p. 33
  • Weyl’s law selects eigenfunctions to build the initial datum on any closed connected manifold, while the nonlinear construction is carried out on the flat torus.The datum is scaled so its Bσ norm is one, and the resulting global solution is used to establish the energy bounds.Source: §4.4 · pp. 33–34
  • A fixed small ε ensures the constructed density remains positive for all time, and its energy gap is ε² times the perturbation energy.The Bσ-to-L∞ embedding and the uniform bound on the perturbation ensure ut ≥ 1/2.Source: §4.4 · p. 34

Appendix A. Auxiliary lemmas in harmonic analysis

Appendix A states Bernstein-type bounds for dyadic spectral pieces and an almost-orthogonality property for the smooth Littlewood–Paley projectors.

Appendix A. Auxiliary lemmas in harmonic analysis

  • Bernstein-type estimates bound derivatives of dyadic spectral pieces in terms of their frequency scale and norms.The appendix gives forms adapted to its notation, including estimates for Sℓh and Pℓh.Source: Appendix A · p. 35
  • The appendix records a direct corollary of the Bernstein bounds for dyadic pieces.The corollary is stated for each ℓ ≥ −1.Source: Appendix A · p. 35
  • The smooth Littlewood–Paley projectors Pk satisfy an almost-orthogonality estimate.The lemma is stated for h and g in L2(M) and dyadic indices k, j, and ℓ.Source: Appendix A · p. 35
  • The appendix proves the auxiliary estimates using spectral expansion, the Leibniz rule, and combinations of preceding bounds.The proof fragments describe these steps and conclude the proof of Lemma A.3.Source: Appendix A · p. 36

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