Sharp Convergence of Wasserstein Gradient Flows for Spectrally Nonnegative Interaction Energies

arXiv:2609.27008v1 Announce Type: cross Abstract: We study the long-time behavior of Wasserstein gradient flows for interaction energies [ mathsf E[mu] = frac12iint_{Mimes M}K(x,y),mathrm dmu(x),mathrm dmu(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_0in 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 elta>0, we construct a solution of the linearized flow whose energy is comparable to t^{-1-elta} along a sequence of times tending to infinity. Moreover, for any elta>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-elta}. 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.

Source: arXiv cs.LG | 2026-09-24