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Umberto Pappalettera

Publications and source records attributed to Umberto Pappalettera.

At least 19 recordsLinked to original sources
arXiv2026

Anomalous properties of 2D active scalars perturbed by rough transport noise

We study SPDEs associated with $2$D active scalars driven by incompressible transport noise of Kraichnan type, with regularity exponent $α\in (0,1)$; our examples include the Euler, SQG and IPM systems. We investigate whether a number of ``turbulent'' phenomenologies, which are well understood in the linear Kraichnan model, persist in this nonlinear setting, uniformly in vanishing viscosity approximations. First, for suitable values of $α$ and initial data in $L^p_x$, we establish anomalous regularization estimates, measured in appropriate endpoint Besov-type spaces of regularity $β=β(α,p)>0$. Remarkably, these results allow for some scaling supercritical regimes of the parameters $α,p$; on the other hand, for (sub)critical parameters, we recover the same regularity exponent $β=1-α$ as in the linear case. Second, in the (sub)critical case, we further prove anomalous integrability, namely solutions becoming instantaneously $L^\infty_x$-valued at positive times, uniformly in the viscosity; moreover, in this case we establish strong existence and pathwise uniqueness of solutions to the inviscid SPDE, which are recovered as the unique vanishing viscosity limit. Finally, in the $2$D Euler case, for $α\in (0,1/2)$, we establish anomalous dissipation of enstrophy and sharpness of anomalous regularization.

math.AP↗
arXiv2026

Failure of the Weak Sard property without Anomalous Dissipation

For every $α\in(0,1)$ we construct an autonomous, divergence-free vector field $u \in C^α_c(\mathbb{R}^2,\mathbb{R}^2)$ that does not have the weak Sard property and, nonetheless, does not induce anomalous dissipation of $L^2(\mathbb{R}^2)$ norm for solutions to the associated advection-diffusion equation. This disproves a conjecture proposed by Bagnara, Boutros, De Lellis and Mayboroda in \cite{BaBoDeMa26}.

math.AP↗
arXiv2026

Spectral instability and non-uniqueness for the Keller-Segel system

We show that the Cauchy problem associated with the parabolic-elliptic Keller-Segel model is locally ill-posed in $L^q(\mathbb{R}^n)$ for dimensions $n \in \{3,\dots,9\}$ and throughout the supercritical range $q\in [1,\frac{n}{2})$. An analog non-uniqueness result is given in the critical space of bounded functions taking values in $L^{n/2,\infty}(\R^n)$. The non-uniqueness is driven by an instability mechanism in self-similarity variables, in the spirit of the program proposed by Jia and Šverák for the three-dimensional Navier-Stokes equations.

math.AP↗
arXiv2026

Dissipative solutions to randomly forced 3D Euler equations

The purpose of this work is twofold. First, we construct probabilistically strong solutions to the three-dimensional Euler equations perturbed by additive noise that are $\mathbb{P}$-almost surely continuous in time, Hölder in space, and satisfy the local energy inequality up to an arbitrarily large stopping time. Second, we prove several non-unique ergodicity results for the forced Euler equations with continuous-in-time external forcing. The solutions we construct are genuinely random and, almost surely, strictly dissipative and not steady states.

math.AP↗
arXiv2025

Anomalous dissipation and regularization in isotropic Gaussian turbulence

In this work we rigorously establish a number of properties of "turbulent" solutions to the stochastic transport and the stochastic continuity equations constructed by Le Jan and Raimond in [Ann. Probab. 30(2): 826-873, 2002]. The advecting velocity field, not necessarily incompressible, is Gaussian and white-in-time, space-homogeneous and isotropic, with $α$-Hölder regularity in space, $α\in (0,1)$. We cover the full range of compressibility ratios giving spontaneous stochasticity of particle trajectories. For the stochastic transport equation, we prove that generic $L^2_x$ data experience anomalous dissipation of the mean energy, and study basic properties of the resulting anomalous dissipation measure. Moreover, we show that starting from such irregular data, the solution immediately gains regularity and enters into a fractional Sobolev space $H^{1-α-}_x$. The proof of the latter is obtained as a consequence of a new sharp regularity result for the degenerate parabolic PDE satisfied by the associated two-point self-correlation function, which is of independent interest. In the incompressible case, a Duchon-Robert-type formula for the anomalous dissipation measure is derived, making a precise connection between this self-regularizing effect and a limit on the flux of energy in the turbulent cascade. Finally, for the stochastic continuity equation, we prove that solutions starting from a Dirac delta initial condition undergo an average squared dispersion growing with respect to time as $t^{1/(1-α)}$, rigorously establishing the analogue of Richardson's law of particle separations in fluid dynamics.

math.PR↗
arXiv2025

Zero-noise selection and Large Deviations in $L^\infty_t L^p_x$ for the stochastic transport equation beyond DiPerna-Lions

We consider $L^\infty_t L^p_x$ solutions of the stochastic transport equation with drift in $L^\infty_t W^{1,q}_x$. We show strong existence and pathwise uniqueness of solutions in a regime of parameters $p,q$ for which non-unique weak solutions of the deterministic transport equation exist. When the intensity of the noise goes to zero, we prove that the solutions of the stochastic transport equation converge to the unique renormalized solution of the transport equation in the sense of DiPerna-Lions. Furthermore, we show that the convergence is governed by a Large Deviations Principle in the space $L^\infty_t L^p_x$. Since the space $L^\infty_t L^p_x$ is not separable, the weak convergence approach to Large Deviations by Budhiraja, Dupuis, and Maroulas is not directly applicable.

math.PR↗
arXiv2025

Collapse and Burst of generalized Surface Quasi-Geostrophic point Vortices

We consider the generalized Surface Quasi-Geostrophic point vortices dynamics, and identify a sufficient condition implying existence of bursts out of (and collapses into) any given initial configuration of vortices. The condition is related to the stability of the linearized dynamics around three vortices evolving in a self-similar fashion.

math.CA↗
arXiv2024

On approximations of stochastic optimal control problems with an application to climate equations

The paper is devoted to the optimal control of a system with two time-scales, in a regime when the limit equation is not of averaging type but, in the spirit of Wong-Zakai principle, it is a stochastic differential equation for the slow variable, with noise emerging from the fast one. It proves that it is possible to control the slow variable by acting only on the fast scales. The concrete problem, of interest for climate research, is embedded into an abstract framework in Hilbert spaces, with a stochastic process driven by an approximation of a given noise. The principle established here is that convergence of the uncontrolled problem is sufficient for convergence of both the optimal costs and the optimal controls. This target is reached using Girsanov transform and the representation of the optimal cost and the optimal controls using a Forward Backward System. A challenge in this program is represented by the generality considered here of unbounded control actions.

math.OC↗
arXiv2024

Kolmogorov $4/5$ law for the forced 3D Navier-Stokes equations

We identify a sufficient condition under which solutions to the 3D forced Navier--Stokes equations satisfy an $L^p$-in-time version of the Kolmogorov 4/5 law for the behavior of the averaged third order longitudinal structure function along the vanishing viscosity limit. The result has a natural probabilistic interpretation: the predicted behavior is observed on average after waiting for some sufficiently generic random time. The sufficient condition is satisfied e.g. by the solutions constructed by Bruè, Colombo, Crippa, De~Lellis, and Sorella. In this particular case, our results can be applied to derive a bound for the exponent of the third order absolute structure function in accordance with the Kolmogorov turbulence theory.

math.AP↗
arXiv2024

Anomalous and total dissipation due to advection by solutions of randomly forced Navier-Stokes equations

We propose a novel approach to induce anomalous dissipation through advection driven by turbulent fluid flows. Specifically, we establish the existence of a velocity field $v$ satisfying randomly forced Navier-Stokes equations, leading to total dissipation of kinetic energy in finite time when advecting a passive scalar. This dissipation phenomenon is uniform across viscosity parameters and initial conditions, representing a case of anomalous dissipation. We further explore dissipation induced by individual realizations of $v$. Our results extend to scenarios where the passive scalar is replaced by solutions to two or three-dimensional deterministic Navier-Stokes equations advected by $v$.

math.AP↗
arXiv2023

On measure-preserving selection of solutions of ODEs

For every $k \in \mathbb{N}$ and $α\in (0,1)$ we construct a divergence-free $u \in C^k([0,T],C^α(\mathbb{T}^d,\mathbb{R}^d))$, $d \geq 2$, such that there is no measurable selection of solutions of the ODE $\dot{X}_t = u(t,X_t)$ that preserves the Lebesgue measure.

math.AP↗
arXiv2023

Global existence and non-uniqueness for the Cauchy problem associated to 3D Navier-Stokes equations perturbed by transport noise

We show global existence and non-uniqueness of probabilistically strong, analytically weak solutions of the three-dimensional Navier-Stokes equations perturbed by Stratonovich transport noise. We can prescribe either: \emph{i}) any divergence-free, square integrable intial condition; or \emph{ii}) the kinetic energy of solutions up to a stopping time, which can be chosen arbitrarily large with high probability. Solutions enjoy some Sobolev regularity in space but are not Leray-Hopf.

math.PR↗
arXiv2023

Global existence and non-uniqueness of 3D Euler equations perturbed by transport noise

We construct Hölder continuous, global-in-time probabilistically strong solutions to 3D Euler equations perturbed by Stratonovich transport noise. Kinetic energy of the solutions can be prescribed a priori up to a stopping time, that can be chosen arbitrarily large with high probability. We also prove that there exist infinitely many Hölder continuous initial conditions leading to non-uniqueness of solutions to the Cauchy problem associated with the system. Our construction relies on a flow transformation reducing the SPDE under investigation to a random PDE, and convex integration techniques introduced in the deterministic setting by De Lellis and Székelyhidi, here adapted to consider the stochastic case. In particular, our novel approach allows to construct probabilistically strong solutions on $[0,\infty)$ directly.

math.PR↗
arXiv2023

Second order perturbation theory of two-scale systems in fluid dynamics

In the present paper we study slow-fast systems of coupled equations from fluid dynamics, where the fast component is perturbed by additive noise. We prove that, under a suitable limit of infinite separation of scales, the slow component of the system converges in law to a solution of the initial equation perturbed with transport noise, and subject to the influence of an additional Itō-Stokes drift. The obtained limit equation is very similar to turbulent models derived heuristically. Our results apply to the Navier-Stokes equations in dimension $d=2,3$; the Surface Quasi-Geostrophic equations in dimension $d=2$; and the Primitive equations in dimension $d=2,3$.

math.PR↗
arXiv2022

Large Deviations for Stochastic equations in Hilbert Spaces with non-Lipschitz drift

We prove a Freidlin-Wentzell result for stochastic differential equations in infinite-dimensional Hilbert spaces perturbed by a cylindrical Wiener process. We do not assume the drift to be Lipschitz continuous, but only continuous with at most linear growth. Our result applies, in particular, to a large class of nonlinear fractional diffusion equations perturbed by a space-time white noise.

math.PR↗
arXiv2022

On the infinite dimension limit of invariant measures and solutions of Zeitlin's 2D Euler equations

In this work we consider a finite dimensional approximation for the 2D Euler equations on the sphere, proposed by V. Zeitlin, and show their convergence towards a solution to Euler equations with marginals distributed as the enstrophy measure. The method relies on nontrivial computations on the structure constants of $\mathbb{S}^2$, that appear to be new. In the last section we discuss the problem of extending our results to Gibbsian measures associated with higher Casimirs.

math.AP↗
arXiv2021

From additive to transport noise in 2D fluid dynamics

Additive noise in Partial Differential equations, in particular those of fluid mechanics, has relatively natural motivations. The aim of this work is showing that suitable multiscale arguments lead rigorously, from a model of fluid with additive noise, to transport type noise. The arguments apply both to small-scale random perturbations of the fluid acting on a large-scale passive scalar and to the action of the former on the large scales of the fluid itself. Our approach consists in studying the (stochastic) characteristics associated to small-scale random perturbations of the fluid, here modelled by stochastic 2D Euler equations with additive noise, and their convergence in the infinite scale separation limit.

math.PR↗