arXiv subjects

Paolo Antonelli

Publications and source records attributed to Paolo Antonelli.

At least 19 recordsLinked to original sources

arXiv2026

Instability of gray solitons in a Gross-Pitaevskii model with a moving impurity

The effect of a moving impurity in a dilute Bose-Einstein condensate is investigated by means of the one-dimensional Gross-Pitaevskii model (GP) with non-zero boundary conditions at infinity. The impurity is modeled as a localized external potential, that travels at constant speed $v \in \mathbf{R}$. In a co-moving reference frame, we study the existence and stability of time-independent solutions. The latter are of physical relevance, being associated with the superfluid behavior of the condensate. For every non-zero velocity $v$ in the subsonic regime, we show the existence of a family of time-independent solutions which bifurcates from a (displaced) gray soliton $\phi_{0,v}(x-s_0)$, with $s_0 \in \mathbf{R}$, of the GP equation. The position $s_0$ is determined as an extremal point of an effective potential explicitly defined. Moreover, we study the spectral stability of these states. For small values of the potential strength, we show that the families originating from the maxima of the effective potential are spectrally unstable. For this last result, we employ an Evans function approach. Finally, we formally apply the instability result to the case of a repulsive delta potential.

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arXiv2026

On the Gross-Pitaevskii model with a moving impurity: Cauchy problem and superfluidity criterion

We study the one-dimensional Gross-Pitaevskii equation with a traveling delta potential and non-zero conditions at infinity. This model describes the effect of a moving impurity in a quantum fluid. Firstly, we show that the associated Cauchy problem is globally well-posed in the energy space. This requires the definition of a conserved energy, which involves the notion of renormalized momentum. Secondly, we study the existence and stability of stationary states in a co-moving reference frame. It is known that there exists an impurity-dependent critical velocity above which no stationary state exists. For velocities below the critical one, two different stationary states appear. We show the orbital stability of the one with higher minimal density.

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arXiv2026

Initial layer analysis of relaxation-time limit of the collisional QHD

We study the structure of the initial layer arising in the relaxation-time limit of the collisional quantum hydrodynamic (QHD) system. When the initial data are not well prepared, a fast transient regime appears near the initial time, which prevents the uniform-in-time convergence of the momentum density to its limiting value. Using the method of matched asymptotic expansions, we derive a systematic asymptotic description of the solution with respect to the relaxation-time parameter $\tau$. In particular, we identify the fast time scale $t/\tau^{2}$ governing the initial layer and explicitly construct the corresponding inner expansion for the momentum density together with the outer expansion describing the slow dynamics. The leading-order outer dynamics are shown to coincide with the quantum drift-diffusion equation. The asymptotic expansion is rigorously justified by establishing uniform in $\tau$ energy estimates for the remainder terms under suitable regularity assumptions on the solutions. As a consequence, we prove the strong convergence of the momentum density in $L^\infty$ in time after subtracting the leading initial-layer correction. The analysis further shows that the convergence rate of order $\tau$ is optimal for general initial data and explains the improved rate in the well-prepared case.

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arXiv2026

Stability of dispersive boundary layers for scalar conservation laws in one space dimension

We study the zero-dispersion limit for a class of Korteweg--de Vries (KdV)-type initial-boundary value problems on the half-line, with Dirichlet boundary conditions assigned at $x=0$. We focus on the outflow regime, where the solution of the limiting scalar conservation law does not attain the boundary condition imposed on the dispersive problem. We construct a boundary layer profile, depending on the fast variable, which is uniquely determined, through the associated stationary third-order boundary layer equation, by the mismatch between the boundary conditions, and by the exponential decay at infinity in the fast variable. Our main result shows that, under suitable regularity and compatibility assumptions on the data, the dispersive solution is well approximated by a WKB expansion given by the sum of the smooth solution of the conservation law and the boundary layer profile. In particular, we establish stability of the boundary layer profile by proving quantitative estimates for the remainder term in a weighted energy norm, and show that it converges to $0$ in $H^1$, uniformly in time and up to the lifespan of the smooth solution of the conservation law. The proof is based on the analysis of a linearized energy functional and does not rely on complete integrability or inverse scattering techniques. It applies to general fluxes and requires no smallness assumption on the amplitude of the boundary layer. To the best of our knowledge, this is the first stability result for boundary layers of KdV-type equation on the half line.

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arXiv2025

The zero capillarity limit for the Euler-Korteweg system with no-flux boundary conditions

In this article, we study the small dispersion limit of the Euler-Korteweg system in a domain with a smooth boundary and no-flux boundary conditions. We exploit a relative energy approach to study the convergence of finite energy weak solutions towards strong solutions to the compressible Euler system. Given the boundary conditions under consideration, our approach requires a correction for the limiting particle density, due to the appearance of a boundary layer. Unlike conditional result on the vanishing viscosity limit, our analysis does not require additional conditions on the lack of anomalous concentration of capillary energy. This is due to the fact that the boundary layer appearing in our context is weaker than the one formed in the vanishing viscosity limit. We believe this approach can be adapted to study similar singular limits involving non-trivial boundary conditions.

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arXiv2025

Global weak solutions of the Navier-Stokes-Korteweg Equations in one dimension

We prove the global existence of weak solutions of the one-dimensional Navier-Stokes-Korteweg (NSK) equations when the viscosity and the capillarity coefficients are power functions of the density, which may be zero on a set with positive measure. The proofs are based on a truncation argument combined with the Energy estimate and BD Entropy. Notably, we do not require any upper bound on the exponent of the power of the viscosity coefficient. In particular, we are able to consider very degenerate viscosity coefficient and to substantially improve previous results.

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arXiv2025

An intrinsically hydrodynamic approach to multidimensional QHD systems

In this paper we consider the multi-dimensional Quantum Hydrodynamics (QHD) system, by adopting an intrinsically hydrodynamic approach. The present work continues the analysis initiated in [6] where the one dimensional case was studied. Here we extend the analysis to the multi-dimensional problem, in particular by considering two physically relevant classes of solutions. First of all we consider two-dimensional initial data endowed with point vortices; by assuming the continuity of the mass density and a quantization rule for the vorticity we are able to study the Cauchy problem and provide global finite energy weak solutions. The same result can be obtained also by considering spherically symmetric initial data in the multi-dimensional setting. For rough solutions with finite energy, we are able to provide suitable dispersive estimates, which also apply to a more general class of Euler-Korteweg equations. Moreover we are also able to show the sequential stability of weak solutions with positive density. Analogously to the one dimensional case this is achieved through the a priori bounds given by a new functional first introduced in [6].

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arXiv2024

Lyapunov stability and exponential phase-locking of Schrödinger-Lohe quantum oscillators

We study the well known Schrödinger-Lohe model for quantum synchronization with non-identical natural frequencies. The main results are related to the characterization and convergence to phase-locked states for this quantum system. The results of this article are four-fold. Via a characterization of the fixed points of the system of correlations, we uncover a direct correspondence to the fixed points of the classical Kuramoto model. Depending on the coupling strength, $κ$, relative to natural frequencies, $Ω_j$, a Lyapunov function is revealed which drives the system to the phase-locked state exponentially fast. Explicit bounds on the asymptotic configurations are granted via a parametric analysis. Finally, linear stability (instability) of the fixed points is provided via an eigenvalue perturbation argument. Although the Lyapunov and linear stability are related, their arguments and results are of a different nature. The Lyapunov stability provides a specific value $κ(Ω_j)$, where for $κ>κ(Ω_j)$, there exists a set of initial data, quantitatively defined, such that the system relaxes to the fixed point at a quantitatively defined exponential rate. While the linear stability is given by analysis of the Jacobian of the fixed points for values of $κ$ large, considering $\varepsilon=\frac{1}κ$ as the perturbation parameter. Under certain assumptions this provides stability for a wider range of $κ$ than that which is given in the Lyapunov argument.

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arXiv2024

The time-relaxation limit for weak solutions to the quantum hydrodynamics system

This paper analyzes weak solutions of the quantum hydrodynamics (QHD) system with a collisional term posed on the one-dimensional torus. The main goal of our analysis is to rigorously prove the time-relaxation limit towards solutions to the quantum drift-diffusion (QDD) equation. \newline The existence of global in time, finite energy weak solutions can be proved by straightforwardly exploiting the polar factorization and wave function lifting tools previously developed by the authors. However, the sole energy bounds are not sufficient to show compactness and then pass to the limit. \newline For this reason, we consider a class of more regular weak solutions (termed GCP solutions), determined by the finiteness of a functional involving the chemical potential associated with the system. For solutions in this class and bounded away from vacuum, we prove the time-relaxation limit and provide an explicit convergence rate. \newline Our analysis exploits compactness tools and does not require the existence (and smoothness) of solutions to the limiting equations or the well-preparedness of the initial data. \newline As a by-product of our analysis, we also establish the existence of global in time $H^2$ solutions to a nonlinear Schr\"odinger-Langevin equation and construct solutions to the QDD equation as strong limits of GCP solutions to the QHD system.

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arXiv2024

Existence and asymptotic behavior for $L^2$-norm preserving nonlinear heat equations

We consider a nonlinear parabolic equation with a nonlocal term, which preserves the $L^2$-norm of the solution. We study the local and global well posedness on a bounded domain, as well as the whole Euclidean space, in $H^1$. Then we study the asymptotic behavior of solutions. In general, we obtain weak convergence in H^1 to a stationary state. For a ball, we prove strong asymptotic convergence to the ground state when the initial condition is positive.

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arXiv2023

Stability of Cnoidal Waves for the Damped Nonlinear Schrödinger Equation

We consider the cubic nonlinear Schrödinger (NLS) equation with a linear damping on the one dimensional torus and we investigate the stability of some solitary wave profiles within the dissipative dynamics. The undamped cubic NLS equation is well known to admit a family of periodic waves given by Jacobi elliptic functions of cnoidal type. We show that the family of cnoidal waves is orbitally stable. More precisely, by considering a sufficiently small perturbation of a given cnoidal wave at initial time, the evolution will always remain close (up to symmetries of the equation) to the cnoidal wave whose mass is modulated according to the dissipative dynamics. This result extends the concept of orbital stability to this non-Hamiltonian evolution. Since cnoidal waves are not exact solutions to the damped NLS, the perturbation is forced away from the family of solitary wave profiles. In order to control this secular growth of the error, we find a first order approximation of the solitary wave that takes into account the dissipative term. Then we use a suitable, exponentially decreasing Lyapunov functional that controls the $H^1$-norm of the perturbation around the approximated solitons.

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arXiv2023

On the formation of singularities for the slightly supercritical NLS equation with nonlinear damping

We consider the focusing, mass-supercritical NLS equation augmented with a nonlinear damping term. We provide sufficient conditions on the nonlinearity exponents and damping coefficients for finite-time blow-up. In particular, singularities are formed for focusing and dissipative nonlinearities of the same power, provided that the damping coefficient is sufficiently small. Our result thus rigorously proves the non-regularizing effect of nonlinear damping in the mass-supercritical case, which was suggested by previous numerical and formal results. We show that, under our assumption, the damping term may be controlled in such a way that the self-similar blow-up structure for the focusing NLS is approximately retained even within the dissipative evolution. The nonlinear damping contributes as a forcing term in the equation for the perturbation around the self-similar profile, that may produce a growth over finite time intervals. We estimate the error terms through a modulation analysis and a careful control of the time evolution of total momentum and energy functionals.

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arXiv2023

Existence and Large Time Behavior for a Dissipative Variant of the Rotational NLS Equation

We study a dissipative variant of the Gross-Pitaevskii equation with rotation. The model contains a nonlocal, nonlinear term that forces the conservation of $L^2$-norm of solutions. We are motivated by several physical experiments and numerical simulations studying the formation of vortices in Bose-Einstein condensates. We show local and global well-posedness of this model and investigate the asymptotic behavior of its solutions. In the linear case, the solution asymptotically tends to the eigenspace associated with the smallest eigenvalue in the decomposition of the initial datum. In the nonlinear case, we obtain weak convergence to a stationary state. Moreover, for initial energies in a specific range, we prove strong asymptotic stability of ground state solutions.

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arXiv2023

Schrödinger-Lohe type models of quantum synchronization with nonidentical oscillators

We study the asymptotic emergent dynamics of two models that can be thought of as extensions of the well known Schrödinger-Lohe model for quantum synchronization. More precisely, the interaction strength between different oscillators is determined by intrinsic parameters, following Cucker-Smale communication protocol. Unlike the original Schrödinger-Lohe system, where the interaction strength was assumed to be uniform, in the cases under our consideration the total mass of each quantum oscillator is allowed to vary in time. A striking consequence of this property is that these extended models yield configurations exhibiting phase, but not space, synchronization. The results are mainly based on the analysis of the ODE systems arising from the correlations, control over the well known Cucker-Smale dynamics, and the dynamics satisfied by the quantum order parameter.

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arXiv2023

Finite energy well-posedness for nonlinear Schr\"odinger equations with non-vanishing conditions at infinity

Relevant physical phenomena are described by nonlinear Schr\"odinger equations with non-vanishing conditions at infinity. This paper investigates the respective 2D and 3D Cauchy problems. Local well-posedness in the energy space for energy-subcritical nonlinearities, merely satisfying Kato-type assumptions, is proven, providing the analogue of the well-established local $H^1$-theory for solutions vanishing at infinity. The critical nonlinearity will be simply a byproduct of our analysis and the existing literature. Under an assumption that prevents the onset of a Benjamin-Feir type instability, global well-posedness in the energy space is proven for a) non-negative Hamiltonians, b) sign-indefinite Hamiltonians under additional assumptions on the zeros of the nonlinearity, c) generic nonlinearities and small initial data. The cases b) and c) only concern the 3D case

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arXiv2022

Existence and Stability of almost finite energy weak solutions to the Quantum Euler-Maxwell system

We prove the existence of global in time, finite energy, weak solutions to a quantum magnetohydrodynamic system (QMHD) with large data, modeling a charged quantum fluid interacting with a self-generated electromagnetic field. The analysis of QMHD relies upon the use of Madelung transformations. The rigorous derivation requires non-trivial smoothing estimates, which are obtained by assuming slightly higher regularity for the electromagnetic potential. These assumptions are motivated by the nonlinear dependence of the hydrodynamic system in terms of the underlying wave function dynamics, which is supercritical with respect to the bare energy bounds. Due to quantum effects on the dispersive properties of QMHD, our approach requires neither smallness nor high regularity, unlike a large amount of existing literature for Euler-Maxwell's classical system. For quantum MHD system the irrotationality and the presence of a highly nonlinear quantum stress tensor induce much stronger dispersive properties, as a byproduct of a close relationship with the classical Maxwell-Schrödinger system. Therefore the core argument is shifted to the analysis of the nonlinearities related to the formulation of the hydrodynamic variables through the Madelung transformations. The analysis carried out in section 4 shows that it is necessary to go through non-trivial smoothing estimates and these require us to assume regularity conditions, just above the energy norms, for the initial data of the Maxwellian electromagnetic potential. In the same regime of regularity, with the help of suitable local smoothing estimates, we also prove stability of both the hydrodynamic variables and the Lorentz force associated with the electromagnetic field. (Abbreviated version, see full abstract in the paper).

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arXiv2021

Global well-posedness for the non-linear Maxwell-Schrödinger system

In this paper we study the Cauchy problem associated to the Maxwell-Schrödinger system with a defocusing pure-power non-linearity. This system has many applications in physics, for instance in the description of a charged non-relativistic quantum plasma, interacting with its self-generated electromagnetic potential. One consequence of our analysis is to demonstrate that the Lorentz force associated with the electromagnetic field is well-defined for solutions slightly more regular than the finite energy class. This aspect is of fundamental importance since all the related physical models require the observability of electromagnetic effects. The well-posedness of the Lorentz force still seems to be a major open problem in the class of solutions which are only finite energy. We show the global well-posedness at high regularity for the cubic and sub-cubic case, and we provide polynomial bounds for the growth of the Sobolev norm of the solutions, for a certain range of non-linearities. An important role is played by appropriate a priori dispersive estimates, obtained by means of Koch-Tzvetkov type bounds for the non-homogeneous Schrödinger equation, which overcome the lack of Strichartz estimates for the magnetic Schrödinger flow. Because of the power-type non-linearity, the propagation of higher regularity, globally in time, cannot be achieved via a bootstrap argument as done in [44]. Our approach then exploits the analysis of a modified energy functional, combined with the a priori bounds coming from the dispersive estimates obtained previously.

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arXiv2021

On the compactness of weak solutions to the Navier-Stokes-Korteweg equations for capillary fluids

In this paper we consider the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effects in three space dimensions. We prove compactness of finite energy weak solutions for large initial data. In contrast with previous results regarding this system, vacuum regions are allowed in the definition of weak solutions and no additional damping terms are considered. The compactness is obtained by introducing suitable truncations of the velocity field and the mass density at different scales and use only the a priori bounds obtained by the energy and the BD entropy.

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