Global existence of solutions for the relativistic Boltzmann equation with arbitrarily large initial data on a Bianchi Type I space-time

We prove, for the relativistic Boltzmann equation on a Bianchi Type I space-time, a global existence and uniqueness theorem, for arbitrarily large initial data.     

On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics

Solutions are analyzed of the linearized relativistic Boltzmann equation for initial data fromL ²(r, p) in long-time and/or small-mean-free-path limits. In both limits solutions of this equation converge to approximate ones constructed with solutions of the set of differential equations called the equations of relativistic hydrodynamics.     

A planar perturbation of the Bianchi type I metric

The standard procedure for finding analytic perturbations in General Relativity suffers from the drawback that it is cumbersome to use beyond linear order perturbations. Following up on our previous work, we continue to use an alternate method of finding perturbations. We find a plane symmetric perturbation of the cosmological Bianchi type I metric. The perturbation corresponds to a fluid with heat flow moving perpendicularly to a singular plane in a region which can be made to be either overdense or underdense relative to the background spacetime. The fluid satisfies both the Strong and Dominant energy conditions everywhere except the region close to the singularity.     

Stability of the Boltzmann equation with potential forces on torus

In this paper, we are concerned with the stability of solutions to the Cauchy problem of the Boltzmann equation with potential forces on torus. It is shown that the natural steady state with the symmetry of origin is asymptotically stable in the Sobolev space with exponential rate in time for any initially smooth, periodic, origin symmetric small perturbation, which preserves the same total mass, momentum and mechanical energy. For the non-symmetric steady state, it is also shown that it is stable in L¹-norm for any initial data with the finite total mass, mechanical energy and entropy.     

New exact solutions to the Eddington-inspired-Born-Infeld equations for a relativistic perfect fluid in a Bianchi type I spacetime

We consider a Bianchi type I physical metric g, an auxiliary metric q and a density matter ρ in Eddington-inspired-Born-Infeld theory. We first derive a system of second order nonlinear ordinary differential equations. Then, by a suitable change of variables, we arrive at a system of first order nonlinear ordinary differential equations. Using both the solution-tube concept for the first order nonlinear ordinary differential equations and the nonlinear analysis tools such as the Arzelá–Ascoli theorem, we prove an existence result for the nonlinear system obtained. The resolution of this last system allows us to obtain new exact solutions for the model considered. Finally, by studying the asymptotic behaviour of the exact solutions obtained, we conclude that this solution is the counterpart of the Friedman–Lemaître–Robertson–Walker spacetime in Eddington-inspired-Born-Infeld theory.     

Global solutions of the Boltzmann equation on a lattice

The nonlinear Boltzmann equation with a discretized spatial variable is studied in a Banach space of absolutely integrable functions of the velocity variables. Conservation laws and positivity are utilized to extend weak local solutions to a global solution. This is shown to be a strong solution by analytic semigroup techniques.Supported by National Science Foundation Grant ENG-7515882.     

Trend to equilibrium of weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions

Recently R. Illner and the author proved that, under a physically realistic truncation on the collision kernel, the Boltzmann equation in the one-dimensional slab [0, 1] with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense. Here it is proved that when the Maxwellians associated with the boundary conditions atx=0 andx=1 are the same MaxwellianM _w , then the solution is uniformly bounded and tends toM _w fort.     

Two half-space problems based on a synthetic-kernel model of the linearized Boltzmann equation

An analytical discrete-ordinates method is used to solve two basic half-space problems based on a new synthetic-kernel model of the linearized Boltzmann equation. In particular, Kramers’ problem and the half-space problem of thermal creep, both basic to the general area of rarefied-gas dynamics, are defined by model equations that are solved (essentially) analytically in terms of a modern version of the discrete-ordinates method. The developed algorithms are implemented to yield numerical results for the slip coefficients and the velocity and heat-flow profiles that compare well with solutions derived from much more computationally intensive techniques.     

Pseudospin symmetry of the Dirac equation for a MSbius square plus Mie type potential with a Coulomb-like tensor interaction via SUSYQM

We investigate the approximate solution of the Dirac equation for a combination of Mobius square and Mie type potentials under the pseudospin symmetry limit by using supersymmetry quantum mechanics. We obtain the bound-state energy equation and the corresponding spinor wave functions in an approximate analytical manner. We comment on the system via various useful figures and tables.     

A new hierarchy system on the basis of a “Master” Boltzmann equation for microscopic density

It is shown that Boltzmann's equation written in terms of microscopic density (namely the unaveraged Boltzmann function) has a wider range of validity as well as finer resolvability for fluctuations than the conventional Boltzmann equation governing Boltzmann's function. In fact the new Boltzmann equation for ideal gases has implications as a microscopically exact continuity equation like Klimontovich's equation for plasmas, and can be derived without invoking any statistical concepts, e.g., distribution functions, or molecular chaos. The Boltzmann equation in the older formalism is obtained by averaging this equation only under a restricted condition of the molecular chaos. The new Boltzmann equation is seen to contain information comparable with Liouville's equation, and serves as a master kinetic equation. A new hierarchy system is formulated in a certain parallelism to the BBGKY hierarchy. They are shown to yield an identical one-particle equation. The difference between the two hierarchy systems first appears in the two-particle equation. The difference is twofold. First, the present formalism includes thermal fluctuations that are missing in the BBGKY formalism. Second, the former allows us to formulate multi-time correlations as well, whereas the latter is restricted to simultaneous correlation. These two features are favorably utilized in deriving the Landau-Lifshitz fluctuation law in a most straightforward manner. Also, equations describing the nonequilibrium interaction between thermal and fluid-dynamical fluctuations are derived.     

Effect of viscosity on stability and accuracy of the two-component lattice Boltzmann method with a multiple-relaxation-time collision operator investigated by the acoustic attenuation model

A two-component lattice Boltzmann method (LBM) with a multiple-relaxation-time (MRT) collision operator is presented to improve the numerical stability of the single relaxation time (SRT) model. The macroscopic and the momentum conservation equations can be retrieved through the Chapman—Enskog (C-E) expansion analysis. The equilibrium moment with the diffusion term is calculated, a diffusion phenomenon is simulated by utilizing the developed model, and the numerical stability is verified. Furthermore, the binary mixture channel model is designed to simulate the sound attenuation phenomenon, and the obtained simulation results are found to be consistent with the analytical solutions. The sound attenuation model is used to study the numerical stability and calculation accuracy of the LBM model. The simulation results show the stability and accuracy of the MRT model and the SRT model under different viscosity conditions. Finally, we study the influence of the error between the macroscopic equation of the MRT model and the standard incompressible Navier—Stokes equation on the calculation accuracy of the model to demonstrate the general applicability of the conclusions drawn by the sound attenuation model in the present study.     

Additional symmetries and string equations of the noncommutative B and C type KP hierarchies

In this paper, we construct the noncommutative B and C type KP hierarchies using pseudo-differential operators and reducing conditions. Further a series of additional flows of the noncommutative B and C type KP hierarchies will be defined and the additional symmetries constitute the B and C type infinite dimensional Lie algebra W_1+∞. In addition, the generating function of the additional symmetries can also be proved to have a nice form in terms of wave functions. Further, the string equations of the noncommutative B and C type KP hierarchies are derived.     

Chetaev型非完整约束相对运动动力学系统Nielsen方程的Mei对称性和Mei守恒量

研究Chetaev型非完整约束相对运动动力学系统Nielsen方程的Mei对称性和Mei守恒量.对Chetaev型非完整约束相对运动力学系统Nielsen方程的运动微分方程、Mei对称性定义和判据进行具体的研究,得到了Mei对称性直接导致的Mei守恒量的表达式.最后举例说明结果的应用.     

Chetaev型约束的相对运动动力学系统Nielsen方程的Noether对称性与Noether守恒量

研究Chetaev型约束的相对运动动力学系统Nielsen方程的Noether对称性与Noether守恒量. 对Chetaev型约束的相对运动力学系统Nielsen方程的运动微分方程、Noether对称性定义和判据进行具 体的研究, 得到了Noether对称性直接导致的Noether守恒量的表达式. 最后举例说明结果的应用.     

Chetaev型约束的相对运动动力学系统Nielsen方程的Noether对称性与Noether守恒量

研究Chetaev型约束的相对运动动力学系统Nielsen方程的Noether对称性与Noether守恒量. 对Chetaev型约束的相对运动力学系统Nielsen方程的运动微分方程、Noether对称性定义和判据进行具 体的研究, 得到了Noether对称性直接导致的Noether守恒量的表达式. 最后举例说明结果的应用.