Searcharxiv⌕ Search

arXiv subjects

Nicola De Nitti

Publications and source records attributed to Nicola De Nitti.

At least 19 recordsLinked to original sources
arXiv2026

Superharmonicity of the fractional ground state

We prove that the positive ground state $u$ of the fractional Laplacian $(-Δ)^s$, $s\in (0,1)$, on an arbitrary open set $Ω$ satisfies $-Δu>λ_s(Ω)^{1/s}u$ in $Ω$. In one dimension, this yields strong concavity and settles a conjecture of Bañuelos, Kulczycki, and Méndez-Hernández. We establish a hierarchy of pointwise inequalities comparing different powers of the Laplacian. In balls, these inequalities give a quantitative Hessian estimate for $\log u$, and hence log-concavity in every dimension. By contrast, we show that log-concavity fails in general convex domains by constructing counterexamples in sufficiently thin ellipsoids. We also extend the superharmonicity principle to ground states of $ψ(-Δ)$ for complete Bernstein functions $ψ$.

math.AP↗
arXiv2026

A Liouville theorem for the two-dimensional stationary hypodissipative Navier--Stokes system

We study the two-dimensional stationary incompressible Navier--Stokes equations on $\mathbb R^2$ with fractional dissipation $(-Δ)^s$. In the full range $s \in (0,1)$, we prove that every smooth solution satisfying the natural energy condition $u\in\dot{\mathrm H}^s(\mathbb R^2;\mathbb R^2)$ has $u\equiv0$ and constant pressure. This is a fractional counterpart of the planar finite-Dirichlet theorem of Gilbarg and Weinberger at $s=1$. The proof uses different arguments in three ranges. For $0<s<\frac13$, we combine an $\mathrm L^2$-estimate derived from the equation with a stream-function truncation argument. For $\frac13\leq s\leq\frac23$, we use a localized energy estimate whose boundary terms are supported on expanding annuli. For $\frac23<s<1$, we establish regularity and decay via a Lorentz-space bootstrap and then apply the maximum principle to the vorticity. We also treat the stationary damped Euler system at $s=0$ by combining the Bernoulli identity with a cut-off argument under an annular growth condition that includes $u\in \mathrm L^r(\mathbb R^2)$ for every $1\le r\le2$.

math.AP↗
arXiv2026

Asymptotic stability of Landau solutions to the MHD system and energy decay

We consider the three-dimensional incompressible MHD system. Any weak solution satisfying a strong energy inequality is $L^2$-asymptotically stable around a Landau solution. Under an additional integrability assumption on the initial perturbation, we also obtain an explicit algebraic decay rate for the $L^2$-norm of the velocity and magnetic perturbations.

math.AP↗
arXiv2026

Stability and instability of a one-dimensional MHD model

We consider a one-dimensional magnetohydrodynamics model introduced by Dai \textit{et al.}~(2023), in a parameter regime where, in the absence of a magnetic field, the system reduces to the De Gregorio model for the Euler equations. We analyze stability and instability near the first excited state on the torus, thus generalizing the recent results obtained by Guo and Jiu~(2025) for the De Gregorio model. Specifically, we establish global well-posedness of the linearized system, local well-posedness for the nonlinear system, and demonstrate both linear and nonlinear instability for a broad class of initial data in the weighted Sobolev space introduced by Lai \textit{et al.}~(2020). We identify the principal linearized operator, which is structurally equivalent to that of the De Gregorio model, as the primary mechanism of instability. Moreover, we prove global well-posedness and stability of both linear and nonlinear systems for initial data in a particular subspace of the aforementioned weighted Sobolev space.

math.AP↗
arXiv2025

A Volterra equation approach to the local limit of nonlocal traffic models

We consider a class of nonlocal conservation laws modeling traffic flow, given by $ \partial_t u_\varepsilon + \partial_x(V(u_\varepsilon \ast γ_\varepsilon)\, u_\varepsilon) = 0 $ with $ γ_\varepsilon(\cdot) := \varepsilon^{-1}γ(\cdot/\varepsilon) $ for a suitable convex convolution kernel $γ$. Since the work of Colombo et al. (Arch. Ration. Mech. Anal., 2023), thanks to uniform $ \mathrm{L}^\infty $- and TV-estimates, it is known that $ w_\varepsilon := u_\varepsilon \ast γ_\varepsilon $ converges to the entropy solution of the local scalar conservation law $ \partial_t u + \partial_x(V(u)\, u) = 0 $ as $\varepsilon \searrow 0$. However, the convergence of $ \{u_\varepsilon\}_{\varepsilon > 0} $ itself has not been fully addressed so far. In this direction, a known result applies specifically to the case of an exponential kernel, where the identity $ \varepsilon \partial_x w_\varepsilon = w_\varepsilon - u_\varepsilon $ is fundamental. In this work, we address this gap in the literature and prove that $ \{u_\varepsilon\}_{\varepsilon > 0} $ converges to the same limit $u$ under the mild additional assumption that the initial datum belongs to $\mathrm L^1(\mathbb{R})$. Our analysis exploits, through a Fourier approach, the stability properties of the more general Volterra-type equation $\partial_xw_\varepsilon=γ'_\varepsilon\ast u_\varepsilon-γ_\varepsilon(0)u_\varepsilon$, thereby deducing the convergence of $u_\varepsilon$ from that of $w_\varepsilon$.

math.AP↗
arXiv2025

Singular limit for a class of nonlocal conservation laws via compensated compactness

We consider a class of nonlocal conservation laws modeling traffic flows, given by $ \partial_t u_\varepsilon + \partial_x(V(u_\varepsilon \ast γ_\varepsilon) u_\varepsilon) = 0$, with a rescaled convolution kernel $γ_\varepsilon(\cdot) := \varepsilon^{-1}γ(\cdot/\varepsilon)$. We establish the strong $\mathrm L^1_{\mathrm{loc}}$-convergence of weak solutions $u_\varepsilon$ toward the entropy-admissible solution of the corresponding local conservation law as the kernel $γ_\varepsilon$ concentrates to a Dirac delta distribution when $\varepsilon \searrow 0$. In contrast to previous literature, we obtain compactness of the family $\{u_\varepsilon \ast γ_\varepsilon\}_{\varepsilon>0}$ without relying on total variation bounds or Ole\uınik-type estimates. Instead, we establish $\mathrm L^2$-type bounds on its entropy production and use the theory of compensated compactness, assuming that the initial datum merely belongs to $\mathrm L^1\cap \mathrm L^\infty$. Our results are twofold. First, we establish the nonlocal-to-local limit for the piecewise constant kernel $γ(\cdot) := {1}_{[-1,0]}(\cdot)$ combined with the affine velocity function from Greenshields' traffic model. Second, we prove the limit for strictly monotone kernels along with decreasing velocity functions. These results settle a long-standing open problem concerning the nonlocal-to-local convergence for non-convex kernels.

math.AP↗
arXiv2025

Asymptotically compatible entropy-consistent discretization for a class of nonlocal conservation laws

We consider a class of nonlocal conservation laws modeling traffic flows, given by $ \partial_t ρ_\varepsilon + \partial_x(V(ρ_\varepsilon \ast γ_\varepsilon) ρ_\varepsilon) = 0 $ with a suitable convex kernel $ γ_\varepsilon $, and its Godunov-type numerical discretization. We prove that, as the nonlocal parameter $ \varepsilon $ and mesh size $ h $ tend to zero simultaneously, the discrete approximation $ W_{\varepsilon,h} $ of $ W_\varepsilon := ρ_\varepsilon \ast γ_\varepsilon $ converges to the entropy solution of the (local) scalar conservation law $ \partial_t ρ+ \partial_x(V(ρ) ρ) = 0 $, with an explicit convergence rate estimate of order $ \varepsilon+h+\sqrt{\varepsilon\, t}+\sqrt{h\,t} $. In particular, with an exponential kernel, we establish the same convergence result for the discrete approximation $ ρ_{\varepsilon,h} $ of $ ρ_\varepsilon $, along with an $ \mathrm{L}^1 $-contraction property for $ W_\varepsilon $. The key ingredients in proving these results are uniform $ \mathrm{L}^\infty $- and $\mathrm{TV}$-estimates that ensure compactness of approximate solutions, and discrete entropy inequalities that ensure the entropy admissibility of the limit solution.

math.NA↗
arXiv2025

Controllability of a One-Dimensional Dynamic Debonding Model

We investigate a one-dimensional dynamic debonding model, introduced by Freund (1990), in which the wave equation is coupled with a Griffith criterion governing the propagation of the fracture. In particular, we study the boundary controllability of the system to a prescribed target state. Our main results provide precise characterizations of the reachable target states, in both \( C^{ 0, 1 } \) and \( C^1 \) regularity settings, and construct exact controls toward these target states.

math.AP↗
arXiv2025

Scaling limits for fractional polyharmonic Gaussian fields

This work is concerned with fractional Gaussian fields, i.e. Gaussian fields whose covariance operator is given by the inverse fractional Laplacian $(-Δ)^{-s}$ (where, in particular, we include the case $s >1$). We define a lattice discretization of these fields and show that their scaling limits -- with respect to the optimal Besov space topology (up to an endpoint case) -- are the original continuous fields. As a byproduct, in dimension $d<2s$, we prove the convergence in distribution of the maximum of the fields. A key tool in the proof is a sharp error estimate for the natural finite difference scheme for $(-Δ)^s$ under minimal regularity assumptions, which is also of independent interest.

math.PR↗
arXiv2025

Existence and finite speed of propagation of solutions for a multi-dimensional fractional thin-film equation

In this paper, we discuss existence and finite speed of propagation for the solutions to an initial-boundary value problem for a family of fractional thin-film equations in a bounded domain in $\mathbb{R}^d$. The nonlocal operator we consider is the spectral fractional Laplacian with Neumann boundary conditions. In the case of a ``strong slippage'' regime with ``complete wetting'' interfacial conditions, we prove local entropy estimates that entail finite speed of propagation of the support and a lower bound for the waiting time phenomenon.

math.AP↗
arXiv2025

Optimal transport of measures via autonomous vector fields

We study the problem of transporting one probability measure to another via an autonomous velocity field. We rely on tools from the theory of optimal transport. In one space-dimension, we solve a linear homogeneous functional equation to construct a suitable autonomous vector field that realizes the (unique) monotone transport map as the time-$1$ map of its flow. Generically, this vector field can be chosen to be Lipschitz continuous. We then use Sudakov's disintegration approach to deal with the multi-dimensional case by reducing it to a family of one-dimensional problems.

math.OC↗
arXiv2024

Regularity aspects of Leray-Hopf solutions to the 2D Inhomogeneous Navier-Stokes system and applications to weak-strong uniqueness

We characterize the Leray--Hopf solutions of the 2D inhomogeneous Navier--Stokes system that become strong for positive times. This characterization relies on the strong energy inequality and the regularity properties of the pressure. As an application, we establish a weak-strong uniqueness result and provide a unified framework for several recent advances in the field.

math.AP↗
arXiv2024

Feedback stabilization for entropy solutions of a 2x2 hyperbolic system of conservation laws at a junction

We consider the p-system in Eulerian coordinates on a star-shaped network. Under suitable transmission conditions at the junction and dissipative boundary conditions in the exterior vertices, we show that the entropy solutions of the system are exponentially stabilizable. Our proof extends the strategy by Coron et al. (2017) and is based on a front-tracking algorithm used to construct approximate piecewise constant solutions whose BV norms are controlled through a suitable exponentially-weighted Glimm-type Lyapunov functional.

math.AP↗
arXiv2024

Non-degeneracy, stability and symmetry for the fractional Caffarelli-Kohn-Nirenberg inequality

The fractional Caffarelli-Kohn-Nirenberg inequality states that $$ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(u(x)-u(y))^2}{|x|^α|x-y|^{n+2s} |y|^α} \mathrm{d} x \, \mathrm{d} y \geq Λ_{n, s, p, α,β} \|u |x|^{-β}\|_{L^p}^2, $$ for $0<s<\min\{1, n/2\}$, $2<p<2^*_s$, and $α,β\in\mathbb R$ so that $β-α= s - n\big(\frac12 - \frac1p\big)$ and $-2s < α< \frac{n-2s}{2}$. Continuing the program started in Ao et al. (2022), we establish the non-degeneracy and sharp quantitative stability of minimizers for $α\ge 0$. Furthermore, we show that minimizers remain symmetric when $α<0$ for $p$ very close to $2$. Our results fit into the more ambitious goal of understanding the symmetry region of the minimizers of the fractional Caffarelli-Kohn-Nirenberg inequality. We develop a general framework to deal with fractional inequalities in $\mathbb R^n$, striving to provide statements with a minimal set of assumptions. Along the way, we discover a Hardy-type inequality for a general class of radial weights that might be of independent interest.

math.AP↗
arXiv2023

Exponential convergence to steady-states for trajectories of a damped dynamical system modelling adhesive strings

We study the global well-posedness and asymptotic behavior for a semilinear damped wave equation with Neumann boundary conditions, modelling a one-dimensional linearly elastic body interacting with a rigid substrate through an adhesive material. The key feature of of the problem is that the interplay between the nonlinear force and the boundary conditions allows for a continuous set of equilibrium points. We prove an exponential rate of convergence for the solution towards a (uniquely determined) equilibrium point.

math.AP↗
arXiv2023

Sharp bounds on enstrophy growth for viscous scalar conservation laws

We prove sharp bounds on the enstrophy growth in viscous scalar conservation laws. The upper bound is, up to a prefactor, the enstrophy created by the steepest viscous shock admissible by the $L^\infty$ and total variation bounds and viscosity. This answers a conjecture by D. Ayala and B. Protas (\emph{Physica D}, 2011), based on numerical evidence, for the viscous Burgers equation.

math.AP↗
arXiv2023

Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion

We study the quantitative stability of critical points of the fractional Sobolev inequality. We show that, for a non-negative function $u \in \dot H^s(\mathbb R^N)$ whose energy satisfies $$\tfrac{1}{2} S^\frac{N}{2s}_{N,s} \le \|u\|_{\dot H^s(\mathbb R^N)} \le \tfrac{3}{2}S_{N,s}^\frac{N}{2s},$$ where $S_{N,s}$ is the optimal Sobolev constant, the bound $$ \|u -U[z,λ]\|_{\dot{H}^s(\mathbb R^N)} \lesssim \|(-Δ)^s u - u^{2^*_s-1}\|_{\dot{H}^{-s}(\mathbb R^N)}, $$ holds for a suitable fractional Talenti bubble $U[z,λ]$. {For functions $u$ which are close to Talenti bubbles, we give the sharp asymptotic value of the implied constant in this inequality.} As an application {of this}, we derive an explicit polynomial extinction rate for positive solutions to a fractional fast diffusion equation.

math.AP↗