Juhi Jang

Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System

The dynamics of an electron gas in a constant ion background can be decribed by the Vlasov-Poisson-Boltzmann system at the kinetic level, or by the compressible Euler-Poisson system at the fluid level. We prove that any solution of the Vlasov-Poisson-Boltzmann system near a smooth local Maxwellian with a small irrotational velocity converges global in time to the corresponding solution to the Euler-Poisson system, as the mean free path ε goes to zero. We use a recent L2 − L∞ framework in the Boltzmann theory to control the higher order remainder in the Hilbert expansion uniformly in ε and globally in time.

Instability theory of the Navier-Stokes-Poisson equations

The stability question of the Lane‐Emden stationary gaseous star configurations is an interesting problem arising in astrophysics. We establish both linear and nonlinear dynamical instability results for the Lane‐Emden solutions in the framework of the Navier‐Stokes‐Poisson system with adiabatic exponent 6 5 < < 4 3 .

Well and ill-posedness for compressible Euler equations with vacuum

An interesting problem arising in gas and fluid dynamics is to understand the behavior of vacuum states, namely, the behavior of the system in the presence of vacuum. In this paper, we review and expand some local-in-time well-posedness results for one-dimensional compressible Euler equations for polytropic gases with a range of behaviors at the interface including the physical vacuum case. We also present some sort of ill-posedness results in some other cases. Finally, we present some conjectures about the change of behavior at the interface. This paper is a mixture of rigorous and formal calculations.

Smooth global solutions for the two-dimensional Euler Poisson system

Abstract. The Euler–Poisson system is a fundamental two-fluid model to describe the dynamics of the plasma consisting of compressible electrons and a uniform ion background. By using the dispersive Klein–Gordon effect, Guo (1998) first constructed a global smooth irrotational solution in the three-dimensional case. It has been conjectured that same results should hold in the two-dimensional case. The main difficulty in 2D comes from the slow dispersion of the linear flow and certain nonlocal resonant obstructions in the nonlinearity. In this paper we develop a new method to overcome these difficulties and construct smooth global solutions for the 2D Euler–Poisson system.

The two-dimensional Euler-Poisson system with spherical symmetry

This article concerns the global-in-time existence of smooth solutions with small amplitude to two space dimensional Euler-Poisson system. The main difficulty lies in the slow time decay (1 + t)−1 of the linear system. Inspired by the work of Ozawa et al., [“Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions,” Math. Z. 222, 341–362 (1996)10.1007/BF02621870; “Remarks on the Klein-Gordon equation with quadratic nonlinearity in two space dimensions,” in Nonlinear Waves, Gakuto International Series: Mathematical Sciences and Applications Vol. 10 (Gakkotosho, Tokyo, 1997), pp. 383–392,] we show that such smooth solutions exist for radially symmetric flows.

High order asymptotic preserving nodal discontinuous Galerkin IMEX schemes for the BGK equation

In this paper, we develop high-order asymptotic preserving (AP) schemes for the BGK equation in a hyperbolic scaling, which leads to the macroscopic models such as the Euler and compressible Navier-Stokes equations in the asymptotic limit. Our approaches are based on the so-called micro-macro formulation of the kinetic equation which involves a natural decomposition of the problem to the equilibrium and the non-equilibrium parts. The proposed methods are formulated for the BGK equation with constant or spatially variant Knudsen number. The new ingredients for the proposed methods to achieve high order accuracy are the following: we introduce discontinuous Galerkin (DG) discretization of arbitrary order of accuracy with nodal Lagrangian basis functions in space; we employ a high order globally stiffly accurate implicit-explicit (IMEX) Runge-Kutta (RK) scheme as time discretization. Two versions of the schemes are proposed: Scheme I is a direct formulation based on the micro-macro decomposition of the BGK equation, while Scheme II, motivated by the asymptotic analysis for the continuous problem, utilizes certain properties of the projection operator. Compared with Scheme I, Scheme II not only has better computational efficiency (the computational cost is reduced by half roughly), but also allows the establishment of a formal asymptotic analysis. Specifically, it is demonstrated that when 0 < e ? 1 , Scheme II, up to O ( e 2 ) , becomes a local DG discretization with an explicit RK method for the macroscopic compressible Navier-Stokes equations, a method in a similar spirit to the ones in Bassi and Rebay (1997) 3, Cockburn and Shu (1998) 16. Numerical results are presented for a wide range of Knudsen number to illustrate the effectiveness and high order accuracy of the methods.

The Compressible Viscous Surface-Internal Wave Problem: Stability and Vanishing Surface Tension Limit

This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and the upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces. We establish a sharp nonlinear global-in-time stability criterion and give the explicit decay rates to the equilibrium. When the upper fluid is heavier than the lower fluid along the equilibrium interface, we characterize the set of surface tension values in which the equilibrium is nonlinearly stable. Remarkably, this set is non-empty, i.e., sufficiently large surface tension can prevent the onset of the Rayleigh-Taylor instability. When the lower fluid is heavier than the upper fluid, we show that the equilibrium is stable for all non-negative surface tensions and we establish the zero surface tension limit.

The Compressible Viscous Surface-Internal Wave Problem: Nonlinear Rayleigh–Taylor Instability

This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and t he upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces.We are concerned with the Rayleigh–Taylor instability when the upper fluid is heavier than the lower fluid along the equilibrium interface. When the surface tension at the free internal interface is below the critical value, we prove that the problem is nonlinear unstable.

Derivation of Ohm's Law from the Kinetic Equations

The goal of this article is to give a formal derivation of Ohms law of magnetohydrodynamics (MHD) starting from the Vlasov--Maxwell--Boltzmann system. The derivation is based on various physical scalings and the moment methods when the Knudsen number goes to zero. We also give a derivation of the so-called Hall effect as well as other limit models such as the Navier--Stokes--Maxwell system. Our results include both the compressible and incompressible MHD models.

Analysis of Asymptotic Preserving DG-IMEX Schemes for Linear Kinetic Transport Equations in a Diffusive Scaling

In this paper, some theoretical aspects will be addressed for the asymptotic preserving discontinuous Galerkin implicit-explicit (DG-IMEX) schemes recently proposed in [J. Jang, F. Li, J.-M. Qiu, and T. Xiong, High order asymptotic preserving DG-IMEX schemes for discrete-velocity kinetic equations in a diffusive scaling, http://arxiv.org/abs/1306.0227, 2013, submitted] for kinetic transport equations under a diffusive scaling. We will focus on the methods that are based on discontinuous Galerkin (DG) spatial discretizations with the