Donatella Donatelli

A quasineutral type limit for the Navier?Stokes?Poisson system with large data

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 density fluctuation we achieve the convergence of the approximating sequences by means of a dispersive estimate of the Strichartz type.

Convergence of singular limits for multi-D semilinear hyperbolic systems to parabolic systems

In this paper we investigate the diffusive zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form: W t (x,t) + 1 eA(x,D)W(x,t) = 1 e 2 B(x, W(x,t)) + 1 e D(W(x,t)) + E(W(x,t)). We analyze the singular convergence, as e ↓ 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 e, by assuming the existence of a symmetrizer having the so-called block structure and by assuming dissipativity conditions on B. (iii) We assume a Kawashima type condition and perform the convergence analysis by using generalizations of compensated compactness due to Tartar and Gerard. Finally, we include examples that show how to use our theory to approximate any quasilinear parabolic systems, satisfying the Petrowski parabolicity condition, or general reaction diffusion systems, including Chemotaxis and Brusselator type systems.

Well/Ill Posedness for the Euler-Korteweg-Poisson System and Related Problems

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 sufficiently smooth initial data including the case of a vanishing initial density - the vacuum zones. Moreover, there is a vast family of initial data, for which the Cauchy problem possesses infinitely many dissipative weak solutions, i.e. the weak solutions satisfying the energy inequality. Finally, we establish the weak-strong uniqueness property in a class of solutions without vacuum. In this paper we show that, even in presence of a dispersive tensor, we have the same phenomena found by De Lellis and Székelyhidi.

Analysis of Oscillations and Defect Measures for the Quasineutral Limit in Plasma Physics

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 uλ strongly converges towards an incompressible velocity vector field u and the density fluctuation ρλ−1 weakly converges to zero. In general, the limit velocity field cannot be expected to satisfy the incompressible Navier–Stokes equation; indeed, the presence of high frequency oscillations strongly affects the quadratic nonlinearities and we have to take care of self-interacting wave packets. We provide a detailed mathematical description of the convergence process by using microlocal defect measures and by developing an explicit correctors analysis. Moreover, we were able to identify an explicit pseudo-parabolic PDE satisfied by the leading correctors terms. Our results include all the previous results in the literature; in particular, we show that the formal limit holds rigorously in the case of well prepared data.

A DISPERSIVE APPROACH TO THE ARTIFICIAL COMPRESSIBILITY APPROXIMATIONS OF THE NAVIER–STOKES EQUATIONS IN 3D

In this paper we study how to approximate the Leray weak solutions of the incompressible Navier–Stokes equations. 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 estimates 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.

Quasi-neutral limit, dispersion, and oscillations for korteweg-type fluids

In the setting of general initial data and the whole space we perform a rigorous analysis of the quasi-neutral limit for a hydrodynamical model of a viscous plasma with capillarity tensor represented by the Navier--Stokes--Poisson--Korteweg system. We shall provide a detailed mathematical description of the convergence process by analyzing the dispersion of the fast oscillating acoustic waves. However the standard acoustic wave analysis is not sufficient to control the high frequency oscillations in the electric field but it is necessary to estimates the dispersive properties induced by the capillarity terms. Therefore by using these additional estimates we will be able to control, via compensated compactness, the quadratic nonlinearity of the stiff electric force field. In conclusion, opposite to the zero capillarity case [D. Donatelli and P. Marcati, Arch. Ration. Mech. Anal., 206 (2012), pp. 159--188] where persistent space localized time high frequency oscillations need to be taken into account, we sh...

Scale analysis of a hydrodynamic model of plasma

We examine a hydrodynamic model of the motion of ions in plasma in the regime of small Debye length, a small ratio of the ion/electron temperature, and high Reynolds number. We analyze the associated singular limit and identify the limit problem — the incompressible Euler system. The result leans on careful analysis of the oscillatory component of the solutions by means of Fourier analysis.

On the Vanishing Electron-Mass Limit in Plasma Hydrodynamics in Unbounded Media

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.

ON THE ARTIFICIAL COMPRESSIBILITY METHOD FOR THE NAVIER-STOKES-FOURIER 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.

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 toward 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.