arXiv subjects
Publications and source records attributed to Donatella Donatelli.
We consider the compressible Navier-Stokes system describing the motion of a viscous fluid confined to a straight layer $Ω_δ=(0,δ)\times\mathbb{R}^2$. We show that the weak solutions in the 3D domain converge strongly to the solution of the 2D incompressible Navier-Stokes equations (Euler equations) when the Mach number $ε$ tends to zero as well as $δ\rightarrow 0$ (and the viscosity goes to zero).
In this paper we consider a fluid dynamic model which describes the atmospheric flow and we perform an asymptotic analysis for different time and length scales. In particular we will focus on the two following cases: when the Mach number dominates on the Rossby number yielding an incompressible regime or when the high rotating (low Rossby number) and incompressibility regime act on the same scale ending up with a geostrophic balance. The limit analysis is performed in the ill prepared data setting and therefore we will develop a rigorous and detailed analysis of the local decay and dispersive behavior of the related acoustic waves. Finally we point out, that the set of equations analyzed in the paper may also fit in the artificial compressibility approximation methods.
We investigate the global existence of weak solutions to a free boundary problem governing the evolution of finitely extensible bead-spring chains in dilute polymers. We construct weak solutions of the two-phase model by performing the asymptotic limit as the adiabatic exponent $γ$ goes to $\infty$ for a macroscopic model which arises from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids. In this context the polymeric molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. This class of models involves the unsteady, compressible, isentropic, isothermal Navier-Stokes system in a bounded domain $Ω$ in $\mathbb{R}^d,$ $d=2, 3$ coupled with a Fokker-Planck-Smoluchowski-type diffusion equation (cf. Barrett and Süli [4], [5], [9]). The convergence of these solutions, up to a subsequence, to the free-boundary problem is established using weak convergence methods, compactness arguments which rely on the monotonicity properties of certain quantities in the spirit of [19].
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.
The Navier-Stokes-Maxwell-Stefan system describes the dynamics of an incompressible gaseous mixture in isothermal condition. In this paper we set up an artificial compressibility type approximation. In particular we focus on the existence of solution for the approximated system and the convergence to the incompressible case. The existence of the approximating system is proved by means of semidiscretization in time and by estimating the fractional time derivative.
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.
We investigate the stability and global existence of weak solutions to a free boundary problem governing the evolution of polymeric fluids. We construct weak solutions of the two-phase model by performing the asymptotic limit of a macroscopic model governing the suspensions of rod-like molecules (known as Doi-Model) in compressible fluids as the adiabatic exponent $γ$ goes to $\infty.$ The convergence of these solutions, up to a subsequence, to the free-boundary problem is established using techniques in the spirit of Lions and Masmoudi \cite{LionsMasmoudi-1999}.
We investigate the dynamics of a nonlinear system modeling tumor growth with drug application. The tumor is viewed as a mixture consisting of proliferating, quiescent and dead cells as well as a nutrient in the presence of a drug. The system is given by a multi-phase flow model: the densities of the different cells are governed by a set of transport equations, the density of the nutrient and the density of the drug are governed by rather general diffusion equations, while the velocity of the tumor is given by Darcy's equation. The domain occupied by the tumor in this setting is a growing continuum $Ω$ with boundary $\partial Ω$ both of which evolve in time. Global-in-time weak solutions are obtained using an approach based on the vanishing viscosity of the Brinkman's regularization. Both the solutions and the domain are rather general, no symmetry assumption is required and the result holds for large initial data.
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.
We investigate the dynamics of a nonlinear model for tumor growth within a cellular medium. In this setting the "tumor" is viewed as a multiphase flow consisting of cancerous cells in either proliferating phase or quiescent phase and a collection of cells accounting for the "waste" and/or dead cells in the presence of a nutrient. Here, the tumor is thought of as a growing continuum $Ω$ with boundary $\partial Ω$ both of which evolve in time. The key characteristic of the present model is that the total density of cancerous cells is allowed to vary, which is often the case within cellular media. We refer the reader to the articles \cite{Enault-2010}, \cite{LiLowengrub-2013} where compressible type tumor growth models are investigated. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion, viscosity and pressure in the weak formulation, as well as convergence and compactness arguments in the spirit of Lions \cite{Lions-1998} (see also \cite{Feireisl-book, DT-MixedModel-2013}).
We investigate the dynamics of a nonlinear system modeling tumor growth with drug application. The tumor is viewed as a mixture consisting of proliferating, quiescent and dead cells as well as a nutrient in the presence of a drug. The system is given by a multi-phase flow model: the densities of the different cells are governed by a set of transport equations, the density of the nutrient and the density of the drug are governed by rather general diffusion equations, while the velocity of the tumor is given by Brinkman's equation. The domain occupied by the tumor in this setting is a growing continuum $Ω$ with boundary $\partial Ω$ both of which evolve in time. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion and viscosity in the weak formulation. Both the solutions and the domain are rather general, no symmetry assumption is required and the result holds for large initial data. This article is part of a research program whose aim is the investigation of the effect of drug application in tumor growth.
In this paper we propose an existence and uniqueness theory for the solutions of a system of non-linear hyperbolic conservation laws, structured in age and maturity variables, representing a tissue environment. In particular we are interested in the investigation of the role of stem cells in its homeostasis. The main result presented in this paper is the consistence of the stability results arising from the analisys of the model we designed with the experimental observations on which several branches of medicine are currently attempting to trade on their research activity.
In this paper we analyze a method of to approximation for the weak solutions of the incompressible magnetohydrodynamic equations (MHD) in unbounded domains. In particular we describe an hyperbolic version of the so called artificial compressibility method adapted to the MHD system. By exploiting the wave equation structure of the approximating system we achieve the convergence of the approximating sequences by means of dispersive estimate of Strichartz type. We prove that the soleinoidal component of the approximating velocity and magnetic fields is relatively compact and converges strongly to a weak solution of the MHD equation.
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.
We consider the zero-electron-mass limit for the Navier-Stokes-Poisson system in unbounded spatial domains. Assuming smallness of the viscosity coefficient and ill-prepared initial data, we show that the asymptotic limit is represented by the incompressible Navier-Stokes system, with a Brinkman damping, in the case when viscosity is proportional to the electron-mass, and by the incompressible Euler system provided the viscosity is dominated by the electron mass. The proof is based on the RAGE theorem and dispersive estimates for acoustic waves, and on the concept of suitable weak solutions for the compressible Navier-Stokes system.
This paper deals with the approximation of the weak solutions of the incompressible Navier Stokes Fourier system. In particular it extends the artificial compressibility method for the Leray weak solutions of the Navier Stokes equation, used by Temam, in the case of a bounded domain and later in the case of the whole space. By exploiting the wave equation structure of the pressure of the approximating system the convergence of the approximating sequences is achieved by means of dispersive estimate of Strichartz type. It will be proved that the projection of the approximating velocity fields on the divergence free vectors is relatively compact and converges to a weak solution of the incompressible Navier Stokes Fourier system.
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.
This paper deals with the rigorous study of the diffusive stress relaxation in the multidimensional system arising in the mathematical modeling of viscoelastic materials. The control of an appropriate high order energy shall lead to the proof of that limit in Sobolev space. It is shown also as the same result can be obtained in terms of relative modulate energies.