arXiv subjects

Marco Bagnara

Publications and source records attributed to Marco Bagnara.

7 recordsLinked to original sources

arXiv2026

Regularity thresholds for anomalous dissipation and related phenomena in passive scalars

We prove the absence of anomalous dissipation for passive scalars driven by some random autonomous divergence-free vector fields in $\mathbb T^d$. In dimension $d=2$ we just need continuity almost surely and a mild nondegeneracy condition on the randomness. In dimension $d\geq 3$ we assume a special geometric structure and almost sure Hölder regularity with a Hölder exponent bigger than $\frac{1}{8}$. No regularity is assumed on the passive scalar except for boundedness in the initial data. The proof relies on dimension-theoretic arguments, as opposed to commutator estimates. A consequence of these results is that the same assumptions prevent (almost surely) many other expected properties of turbulent flows, such as anomalous regularization, the Yaglom-Obukhov-Corrsin law, and Richardson diffusion.

math.AP↗

arXiv2026

Refined uniqueness results for 2D Euler and gSQG with rough Kraichnan noise

We prove strong well-posedness results for the stochastic 2D Euler equations in vorticity form and generalized SQG equations, with $L^p$ initial data and driven by a spatially rough, incompressible transport noise of Kraichnan type. Previous works addressed this problem with noise of spatial regularity $α\in (0,1/2)$, in a setting where a rougher noise yields a stronger regularization. We remove this limitation by allowing any $α\in (0,1)$, covering the same range of parameters for which anomalous regularization effects are known to occur in passive scalars. In particular, this covers the physically relevant case $α=2/3$, associated with the Richardson-Kolmogorov scaling of energy cascade.

math.PR↗

arXiv2025

No blow-up by nonlinear Itô noise for the Euler equations

By employing a suitable multiplicative Itô noise with radial structure and with more than linear growth, we show the existence of a unique, global-in-time, strong solution for the stochastic Euler equations in two and three dimensions. More generally, we consider a class of stochastic partial differential equations (SPDEs) with a superlinear growth drift and suitable nonlinear, multiplicative Itô noise, with the stochastic Euler equations as a special case within this class. We prove that the addition of such a noise effectively prevents blow-ups in the solution of these SPDEs.

math.PR↗

arXiv2025

Regularization by rough Kraichnan noise for the generalised SQG equations

We consider the generalised Surface Quasi-Geostrophic (gSQG) equations in $\mathbb R^2$ with parameter $β\in (0,1)$, an active scalar model interpolating between SQG ($β=1$) and the 2D Euler equations ($β=0$) in vorticity form. Existence of weak $(L^1\cap L^p)$-valued solutions in the deterministic setting is known, but their uniqueness is open. We show that the addition of a rough Stratonovich transport noise of Kraichnan type regularizes the PDE, providing strong existence and pathwise uniqueness of solutions for initial data $θ_0\in L^1\cap L^p$, for suitable values $p\in[2,\infty]$ related to the regularity degree $α$ of the noise and the singularity degree $β$ of the velocity field; in particular, we can cover any $β\in (0,1)$ for suitable $α$ and $p$ and we can reach a suitable ("critical") threshold. The result also holds in the presence of external forcing $f\in L^1_t (L^1\cap L^p)$ and solutions are shown to depend continuously on the data of the problem; furthermore, they are well approximated by vanishing viscosity and regular approximations. With similar techniques, we also show well-posedness for two-dimensional linear transport equation with random drift, with the same noise.

math.PR↗

arXiv2024

Anomalous Regularization in Kazantsev-Kraichnan Model

This work investigates a passive vector field which is transported and stretched by a divergence-free Gaussian velocity field, delta-correlated in time and poorly correlated in space (spatially nonsmooth). Although the advection of a scalar field (Kraichnan's passive scalar model) is known to enjoy regularizing properties, the potentially competing stretching term in vector advection may induce singularity formation. We establish that the regularization effect is actually retained in certain regimes. While this is true in any dimension $d\ge 3$, it notably implies a regularization result for linearized 3D Euler equations with stochastic modeling of turbulent velocities, and for the induction equation in magnetohydrodynamic turbulence.

math.PR↗

arXiv2024

A suitable nonlinear Stratonovich noise prevents blow-up in the Euler equations and other SPDEs

We perturb the 3D Euler equations by a particular non-linear Stratonovich noise. We show the existence and uniqueness of a global-in-time (i.e. no blow-up) smooth solution. The result is a corollary of a more general theorem valid in an abstract framework, where the addition of such noise prevents the blow-up possibly induced by a drift with super-linear growth. The result is new with Stratonovich noise.

math.PR↗

arXiv2022

On the Hausdorff Measure of $\R^n$ with the Euclidean Topology

In this paper we answer a question raised by David H. Fremlin about the Hausdorff measure of $\mathbb{R}^2$ with respect to a distance inducing the Euclidean topology. In particular we prove that the Hausdorff $n$-dimensional measure of $\mathbb{R}^n$ is never $0$ when considering a distance inducing the Euclidean topology. Finally, we show via counterexamples that the previous result does not hold in general if we remove the assumption on the topology.

math.MG↗