Searcharxiv⌕ Search

arXiv subjects

Renato Lucà

Publications and source records attributed to Renato Lucà.

At least 19 recordsLinked to original sources
arXiv2026

Magnetic relaxation for the MHD equations via the stable manifold method

We prove that given any sufficiently small and regular solution $B$ of the stationary Euler equations there exists an infinite dimensional family of solutions $(u,b)$ of the non-resistive magnetohydrodynamics equations (MHD) that relax to $(0, B)$. More precisely, $(u,b) \to (0, B)$ exponentially fast as $t \to +\infty$. This family may be viewed as lying in the stable manifold of the non-resistive MHD equations around the equilibrium state $(0, B)$. The problem whether it actually coincides with the stable manifold remains open. As a byproduct of our result, we provide a large class of global regular solutions of the non-resistive MHD equations. Another consequence is that any sufficiently small and regular solution of the stationary Euler equation is (non-trivially) topologically accessible via MHD from a large class of magnetic fields according to the definition of Moffatt and, in this scenario, the topology of the magnetic lines is (entirely) preserved in the limit $t \to + \infty$.

math.AP↗
arXiv2026

Pointwise convergence of the Klein-Gordon flow

We consider the PDEs version of the Carleson problem in the context of the cubic nonlinear Klein-Gordon equation. This means that we aim to establish the lowest regularity class for which one has almost everywhere pointwise convergence of the solutions to the initial data, as $t \to 0$. We prove sharp results for initial data in Sobolev spaces and for their randomized counterparts.

math.AP↗
arXiv2026

On the topology of the magnetic lines of large solutions to the Magnetohydrodynamic equations in $\mathbb{R}^3$

The purpose of this article is twofold: first, we introduce a new class of global strong solutions to the magnetohydrodynamic system in $\mathbb{R}^3$ with initial data $(u_0,b_0)$ of arbitrarily large size in any critical space. To do so, we impose a smallness condition on the difference $u_0-b_0$. Then we use this result to prove magnetic reconnection for a suitable class of (large) solutions. With this, we mean a change of topology of the integral lines of the magnetic field $b$ under the evolution. The proof relies on counting the number of hyperbolic critical points of the solutions, and this instance is structurally stable.

math.AP↗
arXiv2025

On low regularity well-posedness of the binormal flow

We focus on a class of solutions of the binormal flow, model of the evolution of vortex filaments, that generate several corner singularities in finite time. This phenomenon has been studied earlier in the regular case, which in this context is in terms of the summability of the angles of the corners generated. Our goal here is to investigate the lower regularity case, using further the Hasimoto approach that allows to use the 1D cubic nonlinear Schrödinger to study the binormal flow. We first obtain a deterministic result by proving an existence result for general binormal flow solutions at low regularity. Then we obtain improved results on the above class of solutions by a suitable randomization of the curvature and torsion of the vortex filament. To do so, we prove a scattering result for a quasi-invariance measure associated with a suitable 1D cubic nonlinear Schrödinger equation that we consider of independent interest. An interesting feature of this result is that we are able to identify a limit measure, which is usually not possible when working on quasi-invariant Gaussian measures for Hamiltonian PDEs on bounded domains.

math.AP↗
arXiv2025

Accelerated and fast magnetic reconnection through enhanced resistive dissipation for MHD equations

We consider the phenomenon of magnetic reconnection, namely a change in the topology of magnetic lines, for sufficiently regular solutions of the three-dimensional periodic magnetohydrodynamic (MHD) equations. We provide examples where magnetic reconnection occurs on time scales shorter than the resistive one, due to enhanced dissipation emerging from advective effects. This is the first analytical result where the advection term plays an active role in the reconnection process. A key aspect of our approach is a new estimate for enhanced diffusion of high Sobolev norms, which is of independent interest beyond its application to the MHD equations.

math.AP↗
arXiv2024

Localization of Beltrami fields: global smooth solutions and vortex reconnection for the Navier-Stokes equations

We introduce a class of divergence-free vector fields on $\mathbb{R}^3$ obtained after a suitable localization of Beltrami fields. First, we use them as initial data to construct unique global smooth solutions of the three dimensional Navier-Stokes equations. The relevant fact here is that these initial data can be chosen to be large in any critical space for the Navier-Stokes problem, however they satisfy the nonlinear smallness assumption introduced in [10]. As a further application of the method, we use these vector fields to provide analytical example of vortex-reconnection for the three-dimensional Navier-Stokes equations on $\mathbb{R}^3$. To do so, we exploit the ideas developed in [14] but differently from this latter we cannot rely on the non-trivial homotopy of the three-dimensional torus. To overcome this obstacle we use a different topological invariant, i.e. the number of hyperbolic critical points of the vector field.

math.AP↗
arXiv2023

Blow-up for the 1D cubic NLS

We consider the 1D cubic NLS on $\mathbb R$ and prove a blow-up result for functions that are of borderline regularity, i.e. $H^s$ for any $s<-\frac 12$ for the Sobolev scale and $\mathcal F L^\infty$ for the Fourier-Lebesgue scale. This is done by identifying at this regularity a certain functional framework from which solutions exit in finite time. This functional framework allows, after using a pseudo-conformal transformation, to reduce the problem to a large-time study of a periodic Schrödinger equation with non-autonomous cubic nonlinearity. The blow-up result corresponds to an asymptotic completeness result for the new equation. We prove it using Bourgain's method and exploiting the oscillatory nature of the coefficients involved in the time-evolution of the Fourier modes. Finally, as an application we exhibit singular solutions of the binormal flow. More precisely, we give conditions on the curvature and the torsion of an initial smooth curve such that the constructed solutions generate several singularities in finite time.

math.AP↗
arXiv2022

Convergence over fractals for the periodic Schrödinger equation

We consider a fractal refinement of Carleson's problem for pointwise convergence of solutions to the periodic Schrödinger equation to their initial datum. For $α\in (0,d]$ and \[ s < \frac{d}{2(d+1)} (d + 1 - α), \] we find a function in $H^s(\mathbb{T}^d)$ whose corresponding solution diverges in the limit $t \to 0$ on a set with strictly positive $α$-Hausdorff measure. We conjecture this regularity threshold to be optimal. We also prove that \[ s > \frac{d}{2(d+2)}\left( d+2-α\right) \] is sufficient for the solution corresponding to every datum in $H^s(\mathbb T^d)$ to converge to such datum $α$-almost everywhere.

math.AP↗
arXiv2022

Magnetic Reconnection in Magnetohydrodynamics

We provide examples of periodic solutions (in both 2 and 3 dimension) of the Magnetohydrodynamics equations such that the topology of the magnetic lines changes during the evolution. This phenomenon, known as magnetic reconnection, is relevant for physicists, in particular in the study of highly conducting plasmas. Although numerical and experimental evidences exist, analytical examples of magnetic reconnection were not known.

math.AP↗
arXiv2022

Transport of Gaussian measures with exponential cut-off for Hamiltonian PDEs

We show that introducing an exponential cut-off on a suitable Sobolev norm facilitates the proof of quasi-invariance of Gaussian measures with respect to Hamiltonian PDE flows and allows us to establish the exact Jacobi formula for the density. We exploit this idea in two different contexts, namely the periodic fractional Benjamin-Bona-Mahony (BBM) equation with dispersion~$γ>1$ and the periodic one dimensional quintic defocussing nonlinear Schrödinger equation (NLS). For the BBM equation we study the transport of the cut-off Gaussian measures on fractional Sobolev spaces, while for the NLS equation we study the measures based on the modified energies introduced by Planchon-Visciglia and the third author. Moreover for the BBM equation we also show almost sure global well-posedness for data in~$C^\a(\T)$ for arbitrarily small~$\a>0$ and invariance of the Gaussian measure associated with the $H^{γ/2}(\T)$ norm.

math.AP↗
arXiv2021

Convergence over fractals for the Schrödinger equation

We consider a fractal refinement of the Carleson problem for the Schrödinger equation, that is to identify the minimal regularity needed by the solutions to converge pointwise to their initial data almost everywhere with respect to the $α$-Hausdorff measure ($α$-a.e.). We extend to the fractal setting ($α< n$) a recent counterexample of Bourgain \cite{Bourgain2016}, which is sharp in the Lebesque measure setting ($α= n$). In doing so we recover the necessary condition from \cite{zbMATH07036806} for pointwise convergence~$α$-a.e. and we extend it to the range $n/2<α\leq (3n+1)/4$.

math.AP↗
arXiv2020

Quasi-invariance of low regularity Gaussian measures under the gauge map of the periodic derivative NLS

The periodic DNLS gauge is an anticipative map with singular generator which revealed crucial in the study of the periodic derivative NLS. We prove quasi-invariance of the Gaussian measure on $L^2(\T)$ with covariance $[1+(-\D)^{s}]^{-1}$ under these transformations for any $s>\frac12$. This extends previous achievements by Nahmod, Ray-Bellet, Sheffield and Staffilani (2011) and Genovese, Lucà and Valeri (2018), who proved the result for integer values of the regularity parameter $s$.

math.PR↗
arXiv2019

Local Central Limit Theorem for a Random Walk Perturbed in One Point

We consider a symmetric random walk on the $ν$-dimensional lattice, whose exit probability from the origin is modified by an antisymmetric perturbation and prove the local central limit theorem for this process. A short-range correction to diffusive behaviour appears in any dimension along with a long-range correction in the one-dimensional case.

math.PR↗
arXiv2019

Polynomial mixing under a certain stationary Euler flow

We study the mixing properties of a scalar $ρ$ advected by a certain incompressible velocity field $u$ on the two dimensional unit ball, which is a stationary radial solution of the Euler equation. The scalar $ρ$ solves the continuity equation with velocity field $u$ and we can measure the degree of mixedness of~$ρ$ with two different scales commonly used in this setting, namely the geometric and the functional mixing scale. We develop a physical space approach well adapted for the quantitative analysis of the decay in time of the geometric mixing scale, which turns out to be polynomial for a large class of initial data. This extends previous results for the functional mixing scale, based on the explicit expression for the solution in Fourier variable, results that are also partially recovered by our approach.

math.AP↗
arXiv2018

Invariant measures for the periodic derivative nonlinear Schrödinger equation

We construct invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation (DNLS) for small data in $L^2$ and we show these measures to be absolutely continuous with respect to the Gaussian measure. The key ingredient of the proof is the analysis of the gauge group of transformations associated to DNLS. As an intermediate step for our main result, we prove quasi-invariance with respect to the gauge maps of Gaussian measures on $L^2$.

math-ph↗
arXiv2018

A limiting absorption principle for the Helmholtz equation with variable coefficients

We prove a limiting absorption principle for a generalized Helmholtz equation on an exterior domain with Dirichlet boundary conditions \begin{equation*} (L+λ)v=f, \qquad λ\in \mathbb{R} \end{equation*} under a Sommerfeld radiation condition at infinity. The operator $L$ is a second order elliptic operator with variable coefficients, the principal part is a small, long range perturbation of $-Δ$, while lower order terms can be singular and large. The main tool is a sharp uniform resolvent estimate, which has independent applications to the problem of embedded eigenvalues and to smoothing estimates for dispersive equations.

math.AP↗
arXiv2017

Stability properties of the regular set for the Navier--Stokes equation

We investigate the size of the regular set for small perturbations of some classes of strong large solutions to the Navier--Stokes equation. We consider perturbations of the data which are small in suitable weighted $L^{2}$ spaces but can be arbitrarily large in any translation invariant critical Banach space. We give similar results in the small data setting.

math.AP↗