arXiv subjects

Pierangelo Marcati

Publications and source records attributed to Pierangelo Marcati.

33 records · Page 2Linked to original sources

arXiv2019

Dispersive shocks and spectral analysis for linearized Quantum Hydrodynamics

In this paper we perform the analysis of spectral properties of the linearized system around constant states and dispersive shock for a 1-D compressible Euler system with dissipation--dispersion terms. The dispersive term is originated by the quantum effects described through the Bohm potential, as customary in Quantum Hydrodynamic models. The analysis performed in this paper includes the computation of the linearized operator and the spectral stability through the Evans function method.

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arXiv2018

A comparison of two mathematical models of the cerebrospinal fluid dynamics

In this paper we provide the numerical simulations of two cerebrospinal fluid dynamics models by comparing our results with the real data available in literature. The models describe different processes in the cerebrospinal fluid dynamics: the cerebrospinal flow in the ventricles of the brain and the reabsorption of the fluid. In the appendix we show in detail the mathematical analysis of both models and we identify the set of initial conditions for which the solutions of the systems of equations do not exhibit blow up. We investigate step by step the accuracy of these theoretical outcomes with respect to the real cerebrospinal physiology and dynamics.

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arXiv2018

Non-relativistic limit analysis of the Chandrasekhar-Thorne relativistic Euler equations with physical vacuum

Our results provide a first step to make rigorous the formal analysis in terms of $\frac{1}{c^2}$ proposed by Chandrasekhar \cite{Chandra65b}, \cite{Chandra65a}, motivated by the methods of Einstein, Infeld and Hoffmann, see Thorne \cite{Thorne1}. We consider the non-relativistic limit for the local smooth solutions to the free boundary value problem of the cylindrically symmetric relativistic Euler equations, when the mass energy density includes the vacuum states at the free boundary. For large enough (rescaled) speed of light $c$ and suitably small time $T,$ we obtain uniform, with respect to $c,$ \lq\lq a priori\rq\rq estimates for the local smooth solutions. Moreover, the smooth solutions of the cylindrically symmetric relativistic Euler equations converge to the solutions of the classical compressible Euler equation, at the rate of order $\frac{1}{c^2}$.

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arXiv2017

Analysis of solutions for a cerebrospinal fluid model

The main purpose of this manuscript is to analyze an intracranic fluid model from a mathematical point of view. By means of an iterative process we are able to prove the existence and uniqueness of a local solution and the existence and uniqueness of a global solution under some restriction conditions on the initial data. Moreover the last part of the paper is devoted to present numerical simulations for the analyzed cerebrospinal model. In particular, in order to assess the reliability of the stated theoretical results, we carry out the numerical simulations in two different cases: first, we fix initial data which satisfy the conditions for the global existence of solution then, we choose initial data that violate them.

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arXiv2017

The Wigner-Lohe model for quantum synchronization and its emergent dynamics

We present the Wigner-Lohe model for quantum synchronization which can be derived from the Schrödinger-Lohe model using the Wigner formalism. For identical one-body potentials, we provide a priori sufficient framework leading the complete synchronization, in which $L^2$-distances between all wave functions tend to zero asymptotically.

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arXiv2017

Nonlinear Maxwell-Schroedinger system and Quantum Magneto-Hydrodynamics in 3D

Motivated by some models arising in quantum plasma dynamics, in this paper we study the Maxwell-Schrödinger system with a power-type nonlinearity. We show the local well-posedness in $H^2(\mathbb{R}^3)\times H^{3/2}(\mathbb{R}^3)$ and the global existence of finite energy weak solutions, these results are then applied to the analysis of finite energy weak solutions for Quantum Magnetohydrodynamic systems.

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arXiv2017

A model of synchronization over quantum networks

We investigate a non-Abelian generalization of the Kuramoto model proposed by Lohe and given by $N$ quantum oscillators ("nodes") connected by a quantum network where the wavefunction at each node is distributed over quantum channels to all other connected nodes. It leads to a system of Schrödinger equations coupled by nonlinear self-interacting potentials given by their correlations. We give a complete picture of synchronization results, given on the relative size of the natural frequency and the coupling constant, for two non-identical oscillators and show complete phase synchronization for arbitrary $N>2$ identical oscillators. Our results are mainly based on the analysis of the ODE system satisfied by the correlations and on the introduction of a quantum order parameter, which is analogous to the one defined by Kuramoto in the classical model. As a consequence of the previous results, we obtain the synchronization of the probability and the current densities defined via the Madelung transformations.

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arXiv2016

Low Mach number limit for the Quantum-Hydrodynamics system

In this paper we deal with the low Mach number limit for the system of quantum-hydrodynamics, far from the vortex nucleation regime. More precisely, in the framework of a periodic domain and ill-prepared initial data we prove strong convergence of the solutions towards regular solutions of the incompressible Euler system. In particular we will perform a detailed analysis of the time oscillations and of the relative entropy functional related to the system.

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arXiv2016

Exact and optimal controllability for scalar conservation laws with discontinuous flux

This paper deals with an optimal control problem and describes the reachable set for the scalar 1-D conservation laws with discontinuous flux. Regarding the optimal control problem we first prove the existence of a minimizer and then we prescribe an algorithm to compute it. The same method also applies to compute the initial data control. The proof relies on the explicit formula for the conservation laws with the discontinuous flux and finer properties of the characteristics.

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arXiv2012

Incompressible type limit analysis of a hydrodynamic model for charge-carrier transport

This paper is concerned with the rigorous analysis of the zero electron mass limit of the full Navier-Stokes-Poisson. This system has been introduced in the literature by Anile and Pennisi (see [5]) in order to describe a hydrodynamic model for charge-carrier transport in semiconductor devices. The purpose of this paper is to prove rigorously zero electron mass limit in the framework of general ill prepared initial data. In this situation the velocity field and the electronic fields develop fast oscillations in time. The main idea we will use in this paper is a combination of formal asymptotic expansion and rigorous uniform estimates on the error terms. Finally we prove the strong convergence of the full Navier Stokes Poisson system towards the incompressible Navier Stokes equations.

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arXiv2011

Convergence of Ginzburg-Landau Approximations for a Liquid Crystal Flow in 2D

In this paper we prove the convergence for all time for a Ginzburg- Landau type approximation of a simplified Ericksen-Leslie model in two dimension. Moreover, we are able to show that the singular set consists in at most finitely many singular points and we give a characterizations of the singularities.

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arXiv2010

The Quantum Hydrodynamics system in two space dimensions

In this paper we study global existence of weak solutions for the Quantum Hydrodynamics System in 2-D in the space of energy. We do not require any additional regularity and/or smallness assumptions on the initial data. Our approach replaces the WKB formalism with a polar decomposition theory which is not limited by the presence of vacuum regions. In this way we set up a self consistent theory, based only on particle density and current density, which does not need to define velocity fields in the nodal regions. The mathematical techniques we use in this paper are based on uniform (with respect to the approximating parameter) Strichartz estimates and the local smoothing property.

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arXiv2008

On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics

In this paper we consider the global existence of weak solutions to a class of Quantum Hydrodynamics (QHD) systems with initial data, arbitrarily large in the energy norm. These type of models, initially proposed by Madelung, have been extensively used in Physics to investigate Supefluidity and Superconductivity phenomena and more recently in the modeling of semiconductor devices . Our approach is based on various tools, namely the wave functions polar decomposition, the construction of approximate solution via a fractional steps method, which iterates a Schrödinger Madelung picture with a suitable wave function updating mechanism. Therefore several \emph{a priori} bounds of energy, dispersive and local smoothing type allow us to prove the compactness of the approximating sequences. No uniqueness result is provided.

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arXiv2006

A dispersive approach to the artificial compressibility approximations of the Navier Stokes equations in 3-D

In this paper we study how to approximate the Leray weak solutions of the incompressible Navier Stokes equation. In particular we describe an hyperbolic version of the so called artificial compressibility method investigated by J.L.Lions and Temam. By exploiting the wave equation structure of the pressure of the approximating system we achieve the convergence of the approximating sequences by means of dispersive estimate of Strichartz type. We prove that the projection of the approximating velocity fields on the divergence free vectors is relatively compact and converges to a Leray weak solution of the incompressible Navier Stokes equation.

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arXiv2002

Convergence of Singular Limits for Multi-D Semilinear Hyperbolic Systems to Parabolic Systems

In this paper we investigate the zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form: W_{t}(x,t) + (1/epsilon) A(x,D) W(x,t) = (1/epsilon^2) B(x,W(x,t)) + (1/epsilon) D(W(x,t)) + E(W(x,t)). We analyse the singular convergence, as epsilon tends to 0, in the case which leads to a limit system of parabolic type. The analysis is carried out by using the following steps: (i) We single out algebraic ``structure conditions'' on the full system, motivated by formal asymptotics, by some examples of discrete velocity models in kinetic theories. (ii) We deduce ``energy estimates'', uniformly in epsilon, by assuming the existence of a symmetrizer having the so called block structure and by assuming ``dissipativity conditions'' on B. (iii) We perform the convergence analysis by using generalizations of Compensated Compactness due to Tartar and Gerard. Finally we include examples which show how to use our theory to approximate prescribed general quasilinear parabolic systems, satisfying Petrowski parabolicity condition, or general reaction diffusion systems.

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