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Maria Colombo

Publications and source records attributed to Maria Colombo.

At least 19 recordsLinked to original sources
arXiv2026

Effective Lagrangian regularity and the uniqueness threshold for random H\"older velocity fields

We study the behavior of the ordinary differential equations, flow maps, and continuity equations associated to autonomous random velocity fields that admit a natural multiscale finite range decomposition. The velocity fields we consider are only H\"older regular in space---$C^{\alpha-}(\mathbb{T}^d)$ for some $\alpha \in (0,1)$---thus the associated ODE and continuity equation are not a priori well-posed. However, above the critical threshold of $\alpha = 1/2$, due to multiscale stochastic cancellations, we prove well-posedness is almost surely restored away from the zero level set of the velocity field. This threshold marks a genuine transition, as demonstrated by examples lying below the threshold that exhibit robust ill-posedness. We additionally provide effective regularity estimates below the critical threshold and prove analogous results in the related "refreshing" regime.

math.PR↗
arXiv2026

Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow

Stein Variational Gradient Descent (SVGD) is a deterministic interacting-particle method for sampling from a target probability measure given access to its score function. In the mean-field and continuous-time limit, it is known that the flow converges weakly toward the target, but no quantitative rate is known for the last iterate. In this paper, we establish quantitative local convergence in strong norms for this dynamics, when the interaction kernel is of Riesz type on the $d$-dimensional torus. Specifically, assuming that the initial density and the target are smooth and close in $L^2$-norm, we obtain explicit polynomial convergence rates in $L^2$-norm that depend on the dimension and on the regularity parameters of the kernel, the initialization and the target. We further show that these rates are sharp in certain regimes, and support the theory with numerical experiments. In the edge case of kernels with a Coulomb singularity, we recover the global exponential convergence result established in prior work. Our analysis is inspired by recent results on Wasserstein gradient flows of kernel mean discrepancies.

stat.ML↗
arXiv2026

Quantitative Convergence of Wasserstein Gradient Flows of Kernel Mean Discrepancies

We study the quantitative convergence of Wasserstein gradient flows of Kernel Mean Discrepancy (KMD) (also known as Maximum Mean Discrepancy (MMD)) functionals. Our setting covers in particular the training dynamics of shallow neural networks in the infinite-width and continuous time limit, as well as interacting particle systems with pairwise Riesz kernel interaction in the mean-field and overdamped limit. Our main analysis concerns the model case of KMD functionals given by the squared Sobolev distance $ \mathscr{E}^{\nu}_{s}(\mu)= \frac{1}{2}\lVert \mu-\nu\rVert_{\dot H^{-s}}^{2}$ for any $s\geq 1 $ and $\nu$ a fixed probability measure on the $d$-dimensional torus. First, inspired by Yudovich theory for the $2d$-Euler equation, we establish existence and uniqueness in natural weak regularity classes. Next, we show that for $s=1$ the flow converges globally at an exponential rate under minimal assumptions, while for $s>1$ we prove local convergence at polynomial rates that depend explicitly on $s$ and on the Sobolev regularity of $\mu$ and $\nu$. These rates hold both at the energy level and in higher regularity classes and are tight for $\nu$ uniform. We then consider the gradient flow of the population loss for shallow neural networks with ReLU activation, which can be cast as a Wasserstein--Fisher--Rao gradient flow on the space of nonnegative measures on the sphere $\mathbb{S}^d$. Exploiting a correspondence with the Sobolev energy case with $s=(d+3)/2$, we derive an explicit polynomial local convergence rate for this dynamics. Except for the special case $s=1$, even non-quantitative convergence was previously open in all these settings. We also include numerical experiments in dimension $d=1$ using both PDE and particle methods which illustrate our analysis.

math.AP↗
arXiv2026

Instability of two-dimensional Taylor-Green Vortices

For a wide class of linear Hamiltonian operators we develop a general criterion that characterizes the unstable eigenvalues as the zeros of a holomorphic function given by the determinant of a finite-dimensional matrix. We apply the latter result to prove the spectral instability of the Taylor-Green vortex in two-dimensional ideal fluids. The linearized Euler operator at this steady state possesses different invariant subspaces, within which we apply our criterion to rule out or detect instabilities. We show linear stability of odd perturbations, for which the unstable spectrum can appear only on the real axis. We exclude this possibility by applying our stability criterion. Real instabilities, instead, exist and can be detected with the same criterion if we consider suitable rescalings of the Taylor-Green vortex. In the subspace of functions even in both variables, the problem is reduced to finding a single complex root of our stability function. We successfully locate this value by combining our general criterion with a rigorous computer-assisted argument. As a consequence, we fully characterize the unstable spectrum of the Taylor-Green vortex.

math.AP↗
arXiv2025

Lyapunov exponents, entropy and mixing for DiPerna-Lions flows

The main goal of this work is to establish an asymptotic form of Bressan's mixing conjecture. To this end, we develop an ergodic-theoretic framework for incompressible DiPerna-Lions flows. Lyapunov exponents are defined via an Oseledets-type decomposition and related to metric entropy through a version of Ruelle's inequality in this low-regularity setting. These tools yield sharp bounds on asymptotic regularity propagation and mixing rates, leading to our main result.

math.AP↗
arXiv2025

Long-wave instability of periodic shear flows for the 2D Navier-Stokes equations

In 1959, Kolmogorov proposed to study the instability of the shear flow $(\sin(y),0)$ in the vanishing viscosity regime in tori $\mathbb{T}_{\alpha}\times \mathbb{T}$. This question was later resolved by Meshalkin and Sinai. We extend the problem to general shear flows $(U(y),0)$ and show that every $U(y)$ exhibits long-wave instability whenever $\|\partial_y^{-1} U\|_{L^2} > \nu$ and $\alpha\ll \nu$, with $\nu$ being the kinematic viscosity. This instability mechanism confirms previous findings by Yudovich in 1966, supported also by several numerical results, and is established through two independent approaches: one via the construction of Kato's isomorphism and one via normal forms. Unlike in many other applications of the latter methods, both proofs deal with the presence of a delicate term in the linearized operator that becomes singular as $\alpha$ approaches $0$.

math.AP↗
arXiv2025

Vanishing viscosity non-unique solutions to the forced 2D Euler Equations

The forced 2D Euler equations exhibit non-unique solutions with vorticity in $L^p$, $p > 1$, whereas the corresponding Navier-Stokes solutions are unique. We investigate whether the inviscid limit $\nu \to 0^+$ from the forced 2D Navier-Stokes to Euler equations is a selection principle capable of "resolving" the non-uniqueness. We focus on solutions in a neighborhood of the non-uniqueness scenario discovered by Vishik; specifically, we incorporate viscosity $\nu$ and consider $O(\varepsilon)$ size perturbations of the initial datum. We discover a uniqueness threshold $\varepsilon \sim \nu^{\kappa_{\rm c}}$, below which the vanishing viscosity solution is unique and radial, and at which there are viscous solutions converging to non-unique, non-radial solutions.

math.AP↗
arXiv2025

Convergence of drift-diffusion PDEs arising as Wasserstein gradient flows of convex functions

We study the quantitative convergence of drift-diffusion PDEs that arise as Wasserstein gradient flows of linearly convex functions over the space of probability measures on ${\mathbb R}^d$. In this setting, the objective is in general not displacement convex, so it is not clear a priori whether global convergence even holds. Still, our analysis reveals that diffusion {allows} a favorable interaction between Wasserstein geometry and linear convexity, leading to a general quantitative convergence theory, analogous to that of gradient flows in convex settings in the Euclidean space. Specifically, we prove that if the objective is convex and suitably coercive, the suboptimality gap decreases at a rate $O(1/t)$. This improves to a rate faster than any polynomial -- or even exponential in compact settings -- when the objective is strongly convex relative to the entropy. Our results extend the range of mean-field Langevin dynamics that enjoy quantitative convergence guarantees, and enable new applications to optimization over the space of probability measures. To illustrate this, we show quantitative convergence results for the minimization of entropy-regularized nonconvex problems, we propose and study an \emph{approximate Fisher Information} regularization covered by our setting, and we apply our results to an estimator for trajectory inference which involves the minimization of the relative entropy with respect to the Wiener measure in path space.

math.OC↗
arXiv2025

A convex integration scheme for the continuity equation past the Sobolev embedding threshold

We introduce a convex integration scheme for the continuity equation in the context of the Di Perna-Lions theory that allows to build incompressible vector fields in $C_{t}W^{1,p}_x$ and nonunique solutions in $C_{t} L^{q}_x$ for any $p,q$ with $\frac{1}{p} + \frac{1}{q} > 1 + \frac{1}{d}- \delta$ for some $\delta>0$. This improves the previous bound, corresponding to $\delta=0$, or equivalently $q' > p^*$, obtained with convex integration so far, and critical for those schemes in view of the Sobolev embedding that guarantees that solutions are distributional in the opposite range.

math.AP↗
arXiv2024

Flexibility of Two-Dimensional Euler Flows with Integrable Vorticity

We propose a new convex integration scheme in fluid mechanics, and we provide an application to the two-dimensional Euler equations. We prove the flexibility and nonuniqueness of $L^\infty L^2$ weak solutions with vorticity in $L^\infty L^p$ for some $p>1$, surpassing for the first time the critical scaling of the standard convex integration technique. To achieve this, we introduce several new ideas, including: (i) A new family of building blocks built from the Lamb-Chaplygin dipole. (ii) A new method to cancel the error based on time averages and non-periodic, spatially-anisotropic perturbations.

math.AP↗
arXiv2024

On multidimensional nonlocal conservation laws with BV kernels

We establish local-in-time existence and uniqueness results for nonlocal conservation laws in several space dimensions under weak (that is, Sobolev or BV) differentiability assumptions on the convolution kernel. In contrast to the case of a smooth kernel, in general the solution experiences finite-time blow-up. We provide an explicit example showing that solutions corresponding to different smooth approximations of the convolution kernel in general converge to different measures after the blow-up time. This rules out a fairly natural strategy for extending the notion of solution of the nonlocal conservation law after the blow-up time.

math.AP↗
arXiv2024

Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field

Given a divergence-free vector field ${\bf u} \in L^\infty_t W^{1,p}_x(\mathbb R^d)$ and a nonnegative initial datum $\rho_0 \in L^r$, the celebrated DiPerna--Lions theory established the uniqueness of the weak solution in the class of $L^\infty_t L^r_x$ densities for $\frac{1}{p} + \frac{1}{r} \leq 1$. This range was later improved in [BCDL21] to $\frac{1}{p} + \frac{d-1}{dr} \leq 1$. We prove that this range is sharp by providing a counterexample to uniqueness when $\frac{1}{p} + \frac{d-1}{dr} > 1$. To this end, we introduce a novel flow mechanism. It is not based on convex integration, which has provided a non-optimal result in this context, nor on purely self-similar techniques, but shares features of both, such as a local (discrete) self similar nature and an intermittent space-frequency localization.

math.AP↗
arXiv2023

An overview on the local limit of non-local conservation laws, and a new proof of a compactness estimate

Consider a non-local (i.e., involving a convolution term) conservation law: when the convolution term converges to a Dirac delta, in the limit we formally recover a classical (or "local") conservation law. In this note we overview recent progress on this so-called non-local to local limit and in particular we discuss the case of anistropic kernels, which is extremely relevant in view of applications to traffic models. We also provide a new proof of a related compactness estimate.

math.AP↗
arXiv2023

Ole\u{\i}nik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit

We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term $W:=\mathbb{1}_{(-\infty,0]}(\cdot)\exp(\cdot) \ast \rho$ satisfy an Ole\u{\i}nik-type entropy condition. More precisely, under different sets of assumptions on the velocity function $V$, we prove that $W$ satisfies a one-sided Lipschitz condition and that $V'(W) W \partial_x W$ satisfies a one-sided bound, respectively. As a byproduct, we deduce that, as the exponential kernel is rescaled to converge to a Dirac delta distribution, the weak solution of the nonlocal problem converges to the unique entropy-admissible solution of the corresponding local conservation law, under the only assumption that the initial datum is essentially bounded and not necessarily of bounded variation.

math.AP↗
arXiv2022

Onsager critical solutions of the forced Navier-Stokes equations

We answer positively to [BDL22, Question 2.4] by building new examples of solutions to the forced 3d-Navier-Stokes equations with vanishing viscosity, which exhibit anomalous dissipation and which enjoy uniform bounds in the space $L_t^3 C_x^{1/3 - \varepsilon}$, for any fixed $\varepsilon >0$. Our construction combines ideas of [BDL22] and [CCS22].

math.AP↗
arXiv2022

Infinite-width limit of deep linear neural networks

This paper studies the infinite-width limit of deep linear neural networks initialized with random parameters. We obtain that, when the number of neurons diverges, the training dynamics converge (in a precise sense) to the dynamics obtained from a gradient descent on an infinitely wide deterministic linear neural network. Moreover, even if the weights remain random, we get their precise law along the training dynamics, and prove a quantitative convergence result of the linear predictor in terms of the number of neurons. We finally study the continuous-time limit obtained for infinitely wide linear neural networks and show that the linear predictors of the neural network converge at an exponential rate to the minimal $\ell_2$-norm minimizer of the risk.

cs.LG↗
arXiv2022

Non-uniqueness of Leray solutions to the hypodissipative Navier-Stokes equations in two dimensions

We exhibit non-unique Leray solutions of the forced Navier-Stokes equations with hypodissipation in two dimensions. Unlike the solutions constructed in \cite{albritton2021non}, the solutions we construct live at a supercritical scaling, in which the hypodissipation formally becomes negligible as $t \to 0^+$. In this scaling, it is possible to perturb the Euler non-uniqueness scenario of Vishik \cite{Vishik1,Vishik2} to the hypodissipative setting at the nonlinear level. Our perturbation argument is quasilinear in spirit and circumvents the spectral theoretic approach to incorporating the dissipation in \cite{albritton2021non}.

math.AP↗