Scaling Solutions of Inelastic Boltzmann Equations with Over-Populated High Energy Tails

This paper deals with solutions of the nonlinear Boltzmann equation for spatially uniform freely cooling inelastic Maxwell models for large times and for large velocities, and the nonuniform convergence to these limits. We demonstrate how the velocity distribution approaches in the scaling limit to a similarity solution with a power law tail for general classes of initial conditions and derive a transcendental equation from which the exponents in the tails can be calculated. Moreover on the basis of the available analytic and numerical results for inelastic hard spheres and inelastic Maxwell models we formulate a conjecture on the approach of the velocity distribution function to a scaling form.     

An Exact Solution to the Two-Particle Boltzmann Equation System for Maxwell Gases

An exact solution to the two-particle Boltzmann equation system for Maxwell gases is obtained with use of Bobylev approach.The relationship between the exact solution and the self-similar solution of the boltzmann equation is also given.     

Exact Eternal Solutions of the Boltzmann Equation

We construct two families of self-similar solutions of the Boltzmann equation in an explicit form. They turn out to be eternal and positive. They do not possess finite energy. Asymptotic properties of the solutions are also studied.     

Nonlinear Transport in Inelastic Maxwell Mixtures Under Simple Shear Flow

The Boltzmann equation for inelastic Maxwell models is used to analyze nonlinear transport in a granular binary mixture in the steady simple shear flow. Two different transport processes are studied. First, the rheological properties (shear and normal stresses) are obtained by solving exactly the velocity moment equations. Second, the diffusion tensor of impurities immersed in a sheared inelastic Maxwell gas is explicitly determined from a perturbation solution through first order in the concentration gradient. The corresponding reference state of this expansion corresponds to the solution derived in the (pure) shear flow problem. All these transport coefficients are given in terms of the restitution coefficients and the parameters of the mixture (ratios of masses, concentration, and sizes). The results are compared with those obtained analytically for inelastic hard spheres in the first Sonine approximation and by means of Monte Carlo simulations. The comparison between the results obtained for both interaction models shows a good agreement over a wide range values of the parameter space.     

Transport Coefficients for Inelastic Maxwell Mixtures

The Boltzmann equation for inelastic Maxwell models (IMM) is used to determine the Navier–Stokes transport coefficients of a granular binary mixture in d-dimensions. The Chapman–Enskog method is applied to solve the Boltzmann equation for states near the (local) homogeneous cooling state. The mass, heat, and momentum fluxes are obtained to first order in the spatial gradients of the hydrodynamic fields, and the corresponding transport coefficients are identified. There are seven relevant transport coefficients: the mutual diffusion, the pressure diffusion, the thermal diffusion, the shear viscosity, the Dufour coefficient, the pressure energy coefficient, and the thermal conductivity. All these coefficients are exactly obtained in terms of the coefficients of restitution and the ratios of mass, concentration, and particle sizes. The results are compared with known transport coefficients of inelastic hard spheres (IHS) obtained analytically in the leading Sonine approximation and by means of Monte Carlo simulations. The comparison shows a reasonably good agreement between both interaction models for not too strong dissipation, especially in the case of the transport coefficients associated with the mass flux     

Exact solutions to the time-dependent Lorentz gas Boltzmann Equation: The approach to hydrodynamics

New exact solutions to the time-dependent Lorentz gas Boltzmann equation are presented for two classes of nonequilibrium initial value problems: thedecay of localized disturbances and theresponse to applied electric fields. These exact results are used to gain some insight into the crossover of the nonequilibrium state from the early-timekinetic regime to the late-timehydrodynamic regime.     

An Algebraic Method for Constructing Exact Solutions to Difference-Differential Equations

In this paper, we present a method to solve difference differential equation(s). As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation(s).     

Estimating the Solutions of the Boltzmann Equation

We discuss some possible estimates of the solutions of the Boltzmann equation, which might permit a progress in the theory of existence of weak solutions.     

A Maxwell Field Equations Class of Exact Solutions on Matter-Dominated Spatially Closed Friedman-Robertson-Walker Universe

Working in a spatially closed Friedman-Robertson-Walker universe with cosmological dust, we investigate a particular source-free Maxwell field generated by a rotationally-symmetric potential A with one of the components in the direction of and the other along . Using the (, , )-Euler coordinates on S ³ and a compact timelike coordinate f, we obtain a class of parametric solutions that allows us to write down the essential components of the Maxwell tensor as well as the induction and the electric field intensity pointing out, besides the non-propagating fundamental electric field, a burst of electromagnetic radiation.     

Self-Similar Solutions of the Boltzmann Equation and Their Applications

We consider a class of solutions of the Boltzmann equation with infinite energy. Using the Fourier-transformed Boltzmann equation, we prove the existence of a wide class of solutions of this kind. They fall into subclasses, labelled by a parameter a, and are shown to be asymptotic (in a very precise sense) to the self-similar one with the same value of a (and the same mass). Specializing to the case of a Maxwell-isotropic cross section, we give evidence to the effect that the only self-similar closed form solutions are the BKW mode and the two solutions recently found by the authors. All the self-similar solutions discussed in this paper are eternal, i.e., they exist for –<t<, which shows that a recent conjecture cannot be extended to solutions with infinite energy. Eternal solutions with finite moments of all orders, and different from a Maxwellian, are also studied. It is shown that these solutions cannot be positive. Moreover all such solutions (partly negative) must be asymptotically (for large negative times) close to the exact eternal solution of BKW type.     

Self-Similar Solutions of the Boltzmann Equation for Non-Maxwell Molecules

We show that the method previously used by the authors to obtain self-similar, eternal solutions of the space-homogeneous Boltzmann equation for Maxwell molecules yields different results when extended to other power-law potentials (including hard spheres). In particular, self-similar solutions cease to exist for a positive time for hard potentials. In the case of soft potentials, the solutions exist for all potive times, but are not eternal.     

Exact solutions of the Boltzmann equation in the VHP model with removal interaction

The linear and nonlinear Boltzmann equation for very hard particles (VHP) is considered in the case when the collision between two particles may lead not only to elastic scattering, but also to a removal event with the disappearance of the molecules. The extended transport equation is solved for arbitrary initial distributions. The computations are carried out explicitly for a special class of initial distributions and for various removal rates. The results are demonstrated graphically. Finally, source terms fulfilling physically reasonable conditions are introduced into the VHP model, and the time-dependent particle number is calculated.     

Supports of Measure Solutions for Spatially Homogeneous Boltzmann Equations

We prove that the support of the unique measure solution for the spatially homoge-neous Boltzmann equation in R3 is the whole space, if the initial distribution is not a Dirac measure and has 4-order moment. More precisely, we obtain the lower bound of exponential type for the probability of any small ball in ℝ³ relative to the measure solution.     

New Exact Travelling Wave Solutions for Generalized Zakharov-Kuzentsov Equations Using General Projective Riccati Equation Method

Applying the generalized method, which is a direct and unified algebraic method for constructing multiple travelling wave solutions of nonlinear partial differential equations (PDEs), and implementing in a computer algebraic system, we consider the generalized Zakharov-Kuzentsov equation with nonlinear terms of any order. As a result, we can not only successfully recover the previously known travelling wave solutions found by existing various tanh methods and other sophisticated methods, but also obtain some new formal solutions. The solutions obtained include kink-shaped solitons, bell-shaped solitons, singular solitons, and periodic solutions.     

Exact solution of the Boltzmann equation in the homogeneous color conductivity problem

An exact solution of the Boltzmann equation for a binary mixture of colored Maxwell molecules is found. The solution corresponds to a nonequilibrium homogeneous steady state created by a nonconservative external force. Explicit expressions for the moments of the distribution function are obtained. By using information theory, an approximate velocity distribution function is constructed, which is exact in the limits of small and large field strengths. Comparison is made between the exact energy flux and the one obtained from the information theory distribution.     

Lie Point Symmetries and Exact Solutions of Couple KdV Equations

The simple Lie point symmetry reduction procedure is used to obtain infinitely many symmetries to a new integrable system of coupled KdV equations. Using some symmetry subalgebra of the equations, five types of the significant similarity reductions are obtained by virtue of the Lie group approach, and obtain abundant solutions of the coupled KdV equations, such as the solitary wave solution, exponential solution, rational solution, polynomial solution, etc.     

Symbolic Computations and Exact and Explicit Solutions of Some Nonlinear Evolution Equations in Mathematical Physics

With the aid of symbolic computation system Mathematica, several explicit solutions for Fisher＇s equation and CKdV equation are constructed by utilizing an auxiliary equation method, the so called G′/G-expansion method, where the new and more general forms of solutions are also constructed. When the parameters are taken as special values, the previously known solutions are recovered.     

Invariant Sets and Exact Solutions to Higher-Dimensional Wave Equations

The invariant sets and exact solutions of the （1 ＋ 2）-dimensional wave equations are discussed. It is shown that there exist a class of solutions to the equations which belong to the invariant set E0 = {u ： ux = vxF（u）,uy = vyF（u） }. This approach is also developed to solve （1 ＋ N）-dimensional wave equations.     

Global Weak Solutions of the Boltzmann Equation

A new definition of the concept of weak solution of the nonlinear Boltzmann equation is introduced. It is proved that, without any truncation on the collision kernel, the Boltzmann equation in the one-dimensional case has a global weak solution in this sense. Global conservation of energy follows.     

Solutions of the Boltzmann equation for Maxwell interactions and singular angle-dependent cross sections

We consider the spatially homogeneous and isotropic Boltzmann distribution function in the case of nonisotropic, binary cross sections inversely proportional to the relative speed of the colliding particles. Further, we allow the angle dependence of the differential cross section() to be singular in the forward direction ( 0). We assume (), d < which includes the case of a Maxwellian interaction. We explicitly show how to construct the solutions of the Boltzmann equation, study their properties, and obtain for a class of solutions sufficient conditions for their existence at any positive time value. We extend the formalism to the more general case of arbitrary dimensionality. We observe an effect noticed previously by Krook, Wu, and Tjon in other models of the Boltzmann equations-namely, for special initial distributions, we find solutions which exhibit an excess of higher energy particles at later time.