John M. Hong

A Bound on the Total Variation of the Conserved Quantities for Solutions of a General Resonant Nonlinear Balance Law

We introduce a new potential interaction functional and use it to define a new Glimm-type functional that bounds the total variation of the conserved quantities at time t>0 by the total variation at time t=0+ in Glimm approximate solutions of a general resonant nonlinear balance law.

A version of the Glimm method based on generalized Riemann problems

We introduce a generalization of Glimms random choice method, which provides us with an approximation of entropy solutions to quasilinear hyperbolic system of balance laws. The flux-function and the source term of the equations may depend on the unknown as well as on the time and space variables. The method is based on local approximate solutions of the generalized Riemann problem, which form building blocks in our scheme and allow us to take into account naturally the effects of the flux and source terms. To establish the nonlinear stability of these approximations, we investigate nonlinear interactions between generalized wave patterns. This analysis leads us to a global existence result for quasilinear hyperbolic systems with source-term, and applies, for instance, to the compressible Euler equations in general geometries and to hyperbolic systems posed on a Lorentzian manifold.

An extension of Glimm's method to the gas dynamical model of transonic flows

We extend Glimms method for conservation laws to a larger class of sources so that it can be applied to the compressible Euler equations in transonic flow. Our approximate solutions to the Cauchy problem are constructed by combining the Glimm scheme with the splitting algorithm. We study the nonlinear interaction between wave patterns and the perturbations, and establish the stability of our scheme. Therefore, a more general Glimm-type argument is developed.

The generalized Buckley-Leverett and the regularized Buckley-Leverett equations

This paper studies solutions of the generalized Buckley-Leverett (BL) equation with variable porosity and solutions of the regularized BL equation with the Burgers-Benjamin-Bona-Mahony-type regularization. We construct global in time weak solutions to the Cauchy problem and to the initial- and boundary-value problem for the generalized BL equation and global classical solutions for the regularized BL equation. Solutions of the regularized BL equation are shown to converge to the corresponding solution of the generalized BL equation when the coefficient γ of the BBM term and the coefficient ν of the Burgers term obey γ = O(ν2).

A new solution representation for the BBM equation in a quarter plane and the eventual periodicity

The initial-boundary-value problem for the Benjamin–Bona–Mahony (BBM) equation is studied in this paper. The goal is to understand the periodic behaviour (termed as eventual periodicity) of its solutions corresponding to periodic boundary condition or periodic forcing. Towards this end, we derive a new formula representing solutions of this initial- and boundary-value problem by inverting the operator ∂t +α∂x − γ∂ xxt defined in the space–time quarter plane. The eventual periodicity of the linearized BBM equation with periodic boundary data and forcing term is established by combining this new representation formula and the method of stationary phase. The eventual periodicity of the full BBM equation is obtained under a suitable assumption imposed on its solution.

Global Transonic Solutions of Planetary Atmospheres in a Hydrodynamic Region---Hydrodynamic Escape Problem Due to Gravity and Heat

The hydrodynamic escape problem (HEP), which is characterized by a free boundary value problem of Euler equation with gravity and heat, is crucial for investigating the evolution of planetary atmospheres. In this paper, the global existence of transonic solutions to the HEP is established using the generalized Glimm method. The new version of Riemann and boundary-Riemann solvers, are provided as building blocks of the generalized Glimm method by inventing the contraction matrices for the homogeneous Riemann (or boundary-Riemann) solutions. The extended Glimm-Goodman wave interaction estimates are investigated for obtaining a stable scheme and positive gas velocity, which matches the physical observation. The limit of approximation solutions serves as an entropy solution of bounded variations. Moreover, the range of the hydrodynamical region is also obtained.

Characterization of the Transonic Stationary Solutions of the Hydrodynamic Escape Problem

This paper presents the characterization of the transonic stationary solutions for the hydrodynamic escape problem (HEP), which is an important issue in the study of the evolution of planetary atmospheres. The transonic stationary solutions of the HEP involve effects of gravity, heat, and conduction, which are based in reality in this study and the sonic points are singular points in the time independent model. The characterization is established by the geometric singular perturbation method on the adiabatic wind solution of the HEP. The existence and nonexistence of the adiabatic wind solution with or without heat effect has been explored. The smooth transonic stationary solution under the effect of conduction is verified by analyzing the adiabatic wind solution. The singularity at the sonic points can be moved to the thermal critical points under the effect of conduction. By introducing the artificial viscosity, the dynamics at the thermal critical points and sonic points is studied in detail. Such a sm...

Approximation of generalized Riemann solutions to compressible Euler-Poisson equations of isothermal flows in spherically symmetric space-times

In this paper, we consider the compressible Euler-Poisson system in spherically symmetric space-times. This system, which describes the conservation of mass and momentum of physical quantity with attracting gravitational potential, can be written as a

Invariant solutions of the equation of motion for a sphere composed of Blatz-Ko material

3\times 3

An extension of Glimm's method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem

mixed-system of partial differential systems or a