Pierangelo Marcati

Università degli Studi dell'Aquila | Università dell'Aquila · Department of Information Engineering, Computer Science and Mathematics

In this paper we consider the multi-dimensional Quantum Hydrodynamics (QHD) system, by adopting an intrinsically hydrodynamic approach. The present paper continues the analysis initiated in Antonelli et al. (Commun Math Phys 383:2113–2161, 2021) where the one dimensional case was investigated. Here we extend the analysis to the multi-dimensional pr...

On April 6, 2009, a strong earthquake (6.1 Mw) struck the city of L’Aquila, which was severely damaged as well as many neighboring towns. After this event, a digital model of the region affected by the earthquake was built and a large amount of data was collected and made available. This allowed us to obtain a very detailed dataset that accurately...

The Cauchy-Problem for 2D and 3D nonlinear Schr\"odinger equations with non-vanishing conditions at infinity is investigated. Local well-posedness in the energy space for energy-subcritical nonlinearities merely satisfying Kato-type assumptions is proven. Our result provides the analogue of the well-established local $H^1$-theory for solutions vani...

Plants growth is a complex and delicate balance among different factors involving environmental and physiological conditions. In this context, we propose a mechanistic model that considers the main internal processes of plant growth and reproduces a wide range of plant behaviors observed experimentally. In particular, we describe the model plant of...

In this paper, we analyse how the combination of fault zone shape and material properties affects the propagation of seismic waves in a two-dimensional domain. We focus on SH wave propagation through several faults with different thicknesses and bending radii, but the theory is easily generalized to the three-dimensional case. We show how the densi...

In this paper we simulate the earthquake that hit the city of L’Aquila on the 6th of April 2009 using SPEED (SPectral Elements in Elastodynamics with Discontinuous Galerkin), an open-source code able to simulate the propagation of seismic waves in complex three-dimensional (3D) domains. Our model includes an accurate 3D reconstruction of the Quater...

In this paper, we study the linear stability properties of perturbations around the homogeneous Couette flow for a 2D isentropic compressible fluid in the domain $$\mathbb {T}\times \mathbb {R}$$ T × R . In the inviscid case there is a generic Lyapunov type instability for the density and the irrotational component of the velocity field. More preci...

We prove the existence of global in time, finite energy, weak solutions to a quantum magneto--hydrodynamic system (QMHD), modeling a charged quantum fluid interacting with it self-generated electromagnetic field, with the additional presence of a pure-power, quantum statistical pressure which comes from the electron degeneracy, due to Heisenberg's...

In this paper we consider a genuinely hydrodynamic approach for the one dimensional quantum hydrodynamics system. In the recent years, the global existence of weak solutions with large data has been obtained in Antonelli and Marcati (Commun Math Phys 287(2):657–686, 2009; Arch Ration Mech Anal 203:499–527, 2012), in several space dimensions, by usi...

We construct solutions to the randomly-forced Navier–Stokes–Poisson system in periodic three-dimensional domains or in the whole three-dimensional Euclidean space. These solutions are weak in the sense of PDEs and also weak in the sense of probability. As such, they satisfy the system in the sense of distributions and the underlying probability spa...

In this paper we present a new tool which can be used to simplify and speed up the reconstruction of real Earth surfaces, cake-layered domains and planar fault sources for numerical simulations. The tool makes use of the CUBIT-Python interface in order to directly 'communicate' and to allow for maximum portability across different operating systems...

In this paper, we study the linear stability properties of perturbations around the homogeneous Couette flow for a 2D isentropic compressible fluid in the domain $\mathbb{T}\times \mathbb{R}$. In the inviscid case there is a generic Lyapunov type instability for the density and the irrotational component of the velocity field. More precisely, we pr...

Hydrodynamic systems for quantum fluids are systems for compressible fluid flows for which quantum effects are macroscopically relevant. We discuss how the presence of the dispersive tensor describing the quantum effects alters the acoustic dispersion at the example of the Quantum Hydrodynamic system (QHD). For the QHD system the dispersion relatio...

The aim of this paper is to study solutions of one dimensional compressible Euler system with dissipation–dispersion terms, where the dispersive term is originated by the quantum effects described through the Bohm potential, as customary in quantum hydrodynamic models. We shall investigate numerically the sensitivity of the profiles with respect to...

In recent years, there has been a rise in interest in the development of self-growing robotics inspired by the moving-by-growing paradigm of plants. In particular, climbing plants capitalize on their slender structures to successfully negotiate unstructured environments while employing a combination of two classes of growth-driven movements: tropic...

In this paper we analyze a 2D free-boundary viscoelastic fluid model of Oldroyd-B type at infinite Weissenberg number. Our main goal is to show the existence of the so-called splash singularities, namely points where the boundary remains smooth but self-intersects. The combination of existence and stability results allows us to construct a special...

This paper aims to propose a novel approach to model the dynamics of objects that move within the soil, e.g. plants roots. One can assume that external forces are significant only at the tip of the roots, where the plant's growth is actuated. We formulate an optimal control problem that minimises the energy spent by a growing root subject to physic...

We construct solutions to the randomly-forced Navier--Stokes--Poisson system in periodic three-dimensional domains or in the whole three-dimensional Euclidean space. These solutions are weak in the sense of PDEs and also weak in the sense of probability. As such, they satisfy the system in the sense of distributions and the underlying probability s...

In recent years there has been a rise in interest in the development of self-growing robotics inspired by the moving-by-growing paradigm of plants. In particular, climbing plants capitalize on their slender structures to successfully negotiate unstructured environments, while employing a combination of two classes of growth-driven movements: tropic...

In this paper, we investigate linear stability properties of the 2D isentropic compressible Euler equations linearized around a shear flow given by a monotone profile, close to the Couette flow, with constant density, in the domain $\mathbb{T}\times \mathbb{R}$. We begin by directly investigating the Couette shear flow, where we characterize the li...

In this paper we study a 2D free-boundary Oldroyd-B model which describes the evolution of a viscoelastic fluid. We prove the existence of splash singularities, namely points where the free-boundary remains smooth but self-intersects. This paper extends the previous results obtained for the infinite Weissenberg number by the authors in Di Iorio et...

This paper aims to propose a novel approach to model the dynamics of objects that move within the soil, e.g. plants roots. Since it can be assumed that external forces are significant only at the tip of the roots, where the plant's growth is actuated, we have formulated an optimal control problem that minimises the energy spent by a growing root su...

This paper aims to propose a novel approach to model the dynamics of objects that move within the soil, e.g. plants roots. Since it can be assumed that external forces are significant only at the tip of the roots, where the plant's growth is actuated, we have formulated an optimal control problem that minimises the energy spent by a growing root su...

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 relevan...

In this paper we consider the one dimensional quantum hydrodynamics (QHD) system, with a genuine hydrodynamic approach. The global existence of weak solutions with large data has been obtained in [2, 3], in several space dimensions, by using the connection between the hydrodynamic variables and the Schr\"odinger wave function. One of the main purpo...

In this paper we study the Cauchy problem associated to the Maxwell-Schrodinger 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 electro-magnetic potential. We show the global well-posedne...

In this paper we study existence and stability of shock profiles for a 1-D compressible Euler system in the context of quantum hydrodynamic models. The dispersive term is originated by the quantum effects described through the Bohm potential; moreover we introduce a (linear) viscosity to analyze its interplay with the former while proving existence...

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 anal...

This article is devoted to the analysis of two models which describe different processes in the cerebrospinal fluid dynamics: the cerebrospinal flow in the ventricles of the brain and the reabsorption of the fluid. We investigate the local existence and uniqueness of solutions for the systems of equations that characterize both models by means of a...

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 (see Section 4). 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 th...

This paper is concerned with an existence and stability result on the nonlinear derivative Schrödinger equation in 1-D, which is originated by the study of the stability of nontrivial steady states in Quantum Hydrodynamics. The problem is equivalent to a compressible Euler fluid system with a very specific Korteweg–Kirchhoff stress K(ρ)=[Formula pr...

In this paper we investigate the low Mach number limit for the quantum Navier-Stokes system considered in the three-dimensional space. For general ill-prepared initial data of finite energy, we prove strong convergence of finite energy weak solutions towards weak solutions of incompressible Navier Stokes equations. Our approach relies on a careful...

In this paper we study existence and stability of shock profiles for a 1-D compressible Euler system in the context of Quantum Hydrodynamic models. The dispersive term is originated by the quantum effects described through the Bohm potential; moreover we introduce a (linear) viscosity to analyze its interplay with the former while proving existence...

Plants are far from being passive organisms being able to exhibit complex behaviours in response to environmental stimuli. How these stimuli are combined, triggered and managed is still an open and complex issue in biology. Mathematical models have helped in understanding some of the pieces in the complexity of intra-plant communication, but a larg...

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 probl...

We discuss some results on the Maxwell–Schrödinger system with a nonlinear power-like potential. We prove 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. Then we apply these results to the analysis of finite energy weak solutions for Quantum Magnetohydrodynami...

In this paper we analyze a 2D free-boundary viscoelastic fluid model of Oldroyd-B type at infinite Weissenberg number. Our main goal is to show the existence of the so-called splash singularities, namely points where the boundary remains smooth but self-intersects. The combination of existence and stability results allows us to construct a special...

In this paper we study a 2D Oldroyd free-boundary model which describes the evolution of a viscoelastic fluid. We prove existence of splash singularities, namely points where the boundary remains smooth but self-intersects. This paper extends the previous results obtained for infinite Weissenberg number to the case of any finite Weissenberg number....

Numerical computations in viscoelasticity show the failure of many numerical schemes when the Weissenberg number is beyond a critical value Keunings (J Non-Newtonian Fluid Mech 20:209–226, 1986, [6]). The existence of singularities in the continuum model could be the way to explain instability appearing in numerical simulations. We consider here a...

In this paper consider a 2-D free boundary Oldroyd-B model at infinite Weissenberg number, under the assumption that the Piola-Kirchoff tensor, entering in the description of the extra-stress tensor, is given by a quadratic, convex energy functional. Our main goal is to investigate the existence of splash type singularities, namely points of self-i...

The main purpose of this manuscript is to analyze an intracranial 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...

We present the Wigner-Lohe model for quantum synchronization which can be derived from the Schr\"{o}dinger-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.

Motivated by some models arising in quantum plasma dynamics, in this paper we study the Maxwell-Schr\"odinger 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 en...

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 Schr\"odinger equations coupled by nonlinear self-interacting...

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 perf...

This paper describes the reachable set and resolves an optimal control problem for the scalar conservation laws with discontinuous flux. We give a necessary and sufficient criteria for the reachable set. A new backward resolution has been described to obtain the reachable set. Regarding the optimal control problem, we first prove the existence of a...

The paper reports results on the rigorous analysis of the quasineutral limit for a hydrodynamical models for a hydrodynamical models of viscous plasmas with capillarity effects represented by the Navier Stokes Poisson system in 3-D. A common feature of this kind of limits in the ill prepared data framework is the high plasma oscillations, namely th...

In this paper we prove the global existence of large amplitude finite energy solutions for a system describing Quantum Fluids with nonlinear nonlocal interaction terms. The system may also (but not necessarily) include dissipation terms which do not provide any help to get the global existence. The method is based on the polar factorization of the...

We derive rigorously a set of boundary conditions for heterogenous devices using a description via the quantum hydrodynamic system provided by the Madelung transformations. In particular, we show that the generalized enthalpy should be constant at the interface between classical and quantum domains. This condition provides a set of boundary conditi...

In the setting of general initial data and whole space we perform a rigorous analysis of the quasineutral limit for a hydrodynamical model of a viscous plasma with capillarity tensor represented by the Navier Stokes Korteweg Poisson system. We shall provide a detailed mathematical description of the convergence process by analyzing the dispersion o...

We consider a general Euler-Korteweg-Poisson system in $R^3$, supplemented with the space periodic boundary conditions, where the quantum hydrodynamics equations and the classical fluid dynamics equations with capillarity are recovered as particular examples. We show that the system admits infinitely many global-in-time weak solutions for any suffi...

This paper is a first attempt to describe the quasineutral limit for a Navier-Stokes-Poisson system where the thermal effects are taken into consideration. In the framework of weak solutions and ill-prepared data, we show that as λ → 0 the velocity field u λ strongly converges towards an incompressible velocity vector field u, the density fluctuati...

a b s t r a c t We study a hybrid model linking the quantum hydrodynamics equation with classical hydrodynamics deriving the transmission conditions between the two PDE systems modeling the quantum and the classical dynamics. These conditions are derived under the assumption of constant scaled temperature and assuming a current jump across the inte...

We describe the main difficulties that arise in performing the quasineutral limit for the Navier Stokes Poisson system.

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...

We perform a rigorous analysis of the quasineutral limit for a hydrodynamical model of a viscous plasma represented by the Navier–Stokes–Poisson system in three dimensions. We show that as λ → 0 the velocity field u λ strongly converges towards an incompressible velocity vector field u and the density fluctuation ρ λ −1 weakly converges to zero. In...

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....

This paper is concerned with the low Mach number limit for the compressible Navier-Stokes equations in an exterior domain. We present here an approach based on Strichartz estimate defined on a non trapping exterior domain and we will be able to show the compactness and strong convergence of the velocity vector field.

We perform a rigorous analysis of the quasineutral limit for a hydrodynamical model of a viscous plasma represented by the Navier Stokes Poisson system in $3-D$. We show that as $\lambda\to 0$ the velocity field $u^{\lambda}$ strongly converges towards an incompressible velocity vector field $u$ and the density fluctuation $\rho^{\lambda}-1$ weakly...

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 [24], have been extensively used in Physics to investigate Supefluidity and Superconductivity phenomena [10], [19] and more r...

In this paper we first prove short time existence of a classical solution for the problem which describes the evolution by Gaussian curvature of a strictly convex hypersurface in . Then we give a proof of the existence of a viscosity solution for this problem in such a way as to define a generalized motion existing for each time.

In this paper we study the Leray weak solutions of the incompressible Navier Stokes equation in an exterior domain.We describe, in particular, an hyperbolic version of the so called artificial compressibility method investigated by J.L.Lions and Temam. The convergence of these type of approximation show in general a lack of strong convergence due t...

We present a generalization of the div-curl lemma to a Banach space framework which is not included in the almost existing generalizations. An example is shown where this generalization is needed.

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 m...

In this paper we investigate a quasineutral type limit for the Navier–Stokes–Poisson system. We prove that the projection of the approximating velocity fields on the divergence-free vector field is relatively compact and converges to a Leray weak solution of the incompressible Navier–Stokes equation. By exploiting the wave equation structure of the...

Without Abstract

The ambient space for the system of balance laws, introduced in Chapter I, will be visualized here as space-time, and the central notion of hyperbolicity in the time direction will be motivated and defined. Companions to the flux, considered in Section 1.5, will now be realized as entropy-entropy flux pairs. Numerous examples will be presented of h...

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...

This paper is devoted to study the asymptotic behaviors of the solutions to a model of hyperbolic balance laws with damping on the quarter plane (x,t) Î \mathbbR+ \mathbbR+ .(x,t) \in \mathbb{R}_ + \times \mathbb{R}_ + . We show the optimal convergence rates of the solutions to their corresponding nonlinear diffusion waves, which are the solutions...

We study the asymptotic behavior of a compressible isentropic flow through a porous medium when the initial mass is finite. The model system is the compressible Euler equation with frictional damping. As t→∞, the density is conjectured to obey the well-known porous medium equation and the momentum is expected to be formulated by Darcy’s law. In thi...

This paper is devoted to the study of the existence and the time-asymptotic of multi-dimensional quantum hydrodynamic equations for the electron particle density, the current density and the electrostatic potential in spatial periodic domain. The equations are formally analogous to classical hydrodynamics but differ in the momentum equation, which...

This paper deals with the asymptotic stability of the null solution of a semilinear partial differential equation. The La Salle Invariance Principle has been used to obtain the stability results. The first result is given under quite general hypotheses assuming only the precompactness of the orbits and the local existence. In the second part, under...

We first obtain the Lp–Lq estimates of solutions to the Cauchy problem for one-dimensional damped wave equationVtt−Vxx+Vt=0,(V,Vt)|t=0=(V0,V1)(x),(x,t)∈R×R+,corresponding to that for the parabolic equationφt−φxx=0φ|t=0=(V0+V1)(x).The estimates are shown by (∗)(V−φ)(·,t)−e−t/2V0(·+t)+V0(·−t)2Lp⩽Ct−121q−1p−1||V0,V1||Lq,t⩾1,etc. for 1⩽q⩽p⩽∞. To show (...

We study the initial value problem for a hyperbolic–elliptic coupled system with arbitrary large discontinuous initial data. We prove existence and uniqueness for that model by means of L1-contraction and comparison properties. Moreover, after suitable scalings, we study both the hyperbolic–parabolic and the hyperbolic–hyperbolic relaxation limits...

. The combined quasineutral and relaxation time limit for a bipolar hydrodynamic model is considered. The resulting limit problem is a nonlinear diffusion equation describing a neutral fluid. We make use of various entropy functions and the related entropy productions in order to obtain strong enough uniform bounds. The necessary strong convergence...

We investigate the singular limit for the solutions to the compressible gas dynamics equations with damping term, after a parabolic scaling, in the one-dimensional isentropic case. In particular, we study the convergence in Sobolev norms towards diffusive prophiles, in case of well-prepared initial data and small perturbations of them. The results...

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 para...

In this paper we are interested in the study of the singular limits for the following hyperbolic nonlinear system of partial differential equation of the form $$ {W_{t}} + A\left( {x,W,D} \right)W = F\left( W \right), $$ (1.1) where W =W(x, t) takes values in ℝN and denotes the density vector of some physical quantities over the space variable, x ∈...

We study the Cauchy problem for the system of one dimensional compressible adiabatic flow through porous media and the related diffusive problem. We introduce a new approach which combines the usual energy methods with special L-1-estimates and use the weighted Sobolev norms to prove the global existence and large time behavior for the solutions of...

The combined relaxation and vanishing Debye length limit for the hydrodynamic model for semiconductors is considered in both the unipolar and the bipolar case. The resulting limit problems are non-linear drift driven hyperbolic equations. We make use of non-standard entropy functions and the related entropy productions in order to obtain uniform es...

We study the relaxation of multi-D semilinear hyperbolic systems to parabolic systems. The singular limits are studied combining Tartar’s and Gérard’s generalized compensated compactness and by using the properties of the pseudodifferential symmetrizer of the system.

A small Debye length limit combined with a relaxation limit in the hydrodynamic model for semiconductors is analysed. The limit problem is identified. Entropy methods andL ∞type estimates originating from the existence proof are used in order to proof the result.

We consider a model of hyperbolic balance laws with damping on the quarter plane (x,t)∈ℝ + ×ℝ + . By means of a suitable shift function, which will play a key role to overcome the difficulty of large boundary perturbations, we show that the IBVP solutions converge time-asymptotically to the shifted nonlinear diffusion wave solutions of the Cauchy p...

We consider the Cauchy problem for the system of 1D compressible Euler equations with damping and the related diffusive problem. By combining the usual weighted energy methods with special L 1 -estimate, we prove the global existence and large time behavior for the smooth solutions and the time-asymptotically equivalence with convergence rates for...

In this paper we study the limiting behavior of nonhomogeneous hyperbolic systems of balance laws when the relaxed equilibria are described by means of systems of parabolic type. In particular we obtain a complete theory for the 2×2 systems of genuinely nonlinear hyperbolic balance laws in 1≳D with a strong dissipative term. A different method, whi...

In this paper we study the relaxation of semilinear hyperbolic systems to parabolic system. The singular limits are studied using Gérard's generalized compensated compactness.

We are concerned with the study of the relaxation limit of the 3-D hydrodynamic model for semiconductors. We prove the convergence of the weak solutions to the Euler-Poisson system toward the solutions to the drift-diffusion system, as the relaxation time tends to zero.

In this paper we consider the asymptotic stability of the solutions to the nonlinear damped wave equation in 2-D of space. In particular we deal with initial data which are small perturbation (in Sobolev norms) of a self-similar plane diffusive profile which solve a related parabolic equation. The results are achieved by using the classical energy...