Self-Similar Asymptotics for the Boltzmann Equation with Inelastic and Elastic Interactions

We consider some questions related to the self-similar asymptotics in the kinetic theory of both elastic and inelastic particles. In the second case we have in mind granular materials, when the model of hard spheres with inelastic collisions is replaced by a Maxwell model, characterized by a collision frequency independent of the relative speed of the colliding particles. We first discuss how to define the n-dimensional (n = 1,2,...) inelastic Maxwell model and its connection with the more basic Boltzmann equation for inelastic hard spheres. Then we consider both elastic and inelastic Maxwell models from a unified viewpoint. We prove the existence of (positive in the inelastic case) self-similar solutions with finite energy and investigate their role in large time asymptotics. It is proved that a recent conjecture by Ernst and Brito devoted to high energy tails for inelastic Maxwell particles is true for a certain class of initial data which includes Maxwellians. We also prove that the self-similar asymptotics for high energies is typical for some classes of solutions of the classical (elastic) Boltzmann equation for Maxwell molecules. New classes of (not necessarily positive) finite-energy eternal solutions of this equation are also studied.     

The LSW Model for Domain Coarsening: Asymptotic Behavior for Conserved Total Mass

In the classical theory of domain coarsening the particles of the coarsening phase evolve by diffusional mass transfer with a mean field. We study the long-time behavior of measure-valued solutions with compact support to this model coupled with the constraint of conserved total mass, including mean-field mass. Unlike the case of conserved volume fraction, this system has no precisely self-similar solutions, and sufficiently low supersaturation can lead to the finite-time extinction of all particles. We find a new explicit family of asymptotically self-similar solutions, and in case that the largest particle size is unbounded we establish results similar to the volume-conserved case. These include necessary criteria for asymptotic self-similarity, and sensitive dependence of long-time behavior on the distribution of largest particles in the system.     

Proof of an Asymptotic Property of Self-Similar Solutions of the Boltzmann Equation for Granular Materials

We consider a question related to the kinetic theory of granular materials. The model of hard spheres with inelastic collisions is replaced by a Maxwell model, characterized by a collision frequency independent of the relative speed of colliding particles. Our main result is that, in the space-homogeneous case, a self-similar asymptotics holds, as conjectured by Ernst–Brito. The proof holds for any initial distribution function with a finite moment of some order greater than two.     

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.     

From Electromagnetism to Relativistic Quantum Mechanics

We study the relationship between Maxwell and Dirac equations for a class of solutions of Maxwell equations that can represent purely electromagnetic particles.     

Non-Self-Similar Behavior in the LSW Theory of Ostwald Ripening

The classical Lifshitz–Slyozov–Wagner theory of domain coarsening predicts asymptotically self-similar behavior for the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field. Here we consider the long-time behavior of measure-valued solutions for systems in which particle size is uniformly bounded, i.e., for initial measures of compact support. We prove that the long-time behavior of the size distribution depends sensitively on the initial distribution of the largest particles in the system. Convergence to the classically predicted smooth similarity solution is impossible if the initial distribution function is comparable to any finite power of distance to the end of the support. We give a necessary criterion for convergence to other self-similar solutions, and conditional stability theorems for some such solutions. For a dense set of initial data, convergence to any self-similar solution is impossible.     

自相似Euler方程的数值方法

Euler方程某些问题的解具有自相似特点，可以使用更准确的方法求解.提出了两种数值方法，分别称为自相似和准自相似方法，新方法可以使用现有守恒律方程的数值格式，无须设计特殊方法.对一维激波管问题、二维Riemann问题、激波反射以及激波折射问题进行了数值计算.对自相似Euler方程，一维计算结果显示数值解基本等同于精确解，二维结果也比现有文献计算的结果有更高的分辨率.对准自相似Euler方程，新方法可以求解不具有自相似性但接近自相似的问题，并在计算时间足够长时可以取得自相似Euler方程的效果.数值求解自相似Euler方程对自相似问题的研究，高分辨率、高精度格式的设计乃至Euler方程的精确解都有重要启示.      

Nonlinear dynamics of magnetohydrodynamic flows of a heavy fluid in the shallow water approximation

The system of the magnetohydrodynamic equations for a heavy fluid has been analyzed in the shallow water approximation. All discontinuous self-similar solutions and all continuous centered self-similar solutions have been found. It has been shown that magnetogravity compression waves are broken with the formation of a magnetogravity shock wave. The initial decay discontinuity problem for the magnetohydrodynamic equations has been solved in the explicit form in the shallow water approximation. The existence of five different configurations implementing the solution of the decay of an arbitrary discontinuity has been demonstrated. The conditions necessary and sufficient for the implementation of each configuration have been found.     

Solution to the nonlinear boltzmann equation for maxwell models for nonisotropic initial conditions

The known solution to the spatially homogeneous nonlinear Boltzmann equation for Maxwell models in a series of Laguerre polynomials is extended to include nonisotropic initial conditions. Existence proofs for a class of solutions are supplied. The equations for the generalized (nonisotropic Laguerre) moments are derived in explicit form for two- and three-dimensional models. Further it is shown that the ordinary moments satisfy the same set of equations as the (Hermite) polynomial moments.     

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.     

Nonlinear dynamics of magnetohydrodynamic flows of a heavy fluid on slope in the shallow water approximation

Magnetohydrodynamic equations for a heavy fluid over an arbitrary surface are studied in the shallow water approximation. While solutions to the shallow water equations for a neutral fluid are well known, shallow water magnetohydrodynamic (SMHD) equations over a nonflat boundary have an additional dependence on the magnetic field, and the number of equations in the magnetic case exceeds that in the neutral case. As a consequence, the number of Riemann invariants defining SMHD equations is also greater. The classical simple wave solutions do not exist for hyperbolic SMHD equations over an arbitrary surface due to the appearance of a source term. In this paper, we suggest a more general definition of simple wave solutions that reduce to the classical ones in the case of zero source term. We show that simple wave solutions exist only for underlying surfaces that are slopes of constant inclination. All self-similar discontinuous and continuous solutions are found. Exact explicit solutions of the initial discontinuity decay problem over a slope are found. It is shown that the initial discontinuity decay solution is represented by one of four possible wave configurations. For each configuration, the necessary and sufficient conditions for its realization are found. The change of dependent and independent variables transforming the initial equations over a slope to those over a flat plane is found.     

Particle-like solutions of the Einstein–Dirac–Maxwell equations

We consider the coupled Einstein–Dirac–Maxwell equations for a static, spherically symmetric system of two fermions in a singlet spinor state. Soliton-like solutions are constructed numerically. The stability and the properties of the ground state solutions are discussed for different values of the electromagnetic coupling constant. We find solutions even when the electromagnetic coupling is so strong that the total interaction is repulsive in the Newtonian limit. Our solutions are regular and well-behaved; this shows that the combined electromagnetic and gravitational self-interaction of the Dirac particles is finite.     

From Multi-Site to On-Site Transfer Matrix Models for Self-Similar Chains

We show how transfer matrix models on chains that are self-similar (renormalizable) with respect to a substitution rule can be transformed from multi-site models in which transfer matrices depend on the nature of a finite number of neighboring sites, to on-site models in which transfer matrices depend on the nature of one site only. We present sufficient conditions and show that these conditions are satisfied in the case of quasiperiodic chains of two symbols that are renormalizable with respect to an invertible substitution rule. We illustrate the application of our results to tight-binding Schrödinger equations modeling the electronic behavior of self-similar chains of atoms and to models describing the transmission of light through self-similarly stacked multilayers.     

Nonlinear Maxwell Equations in Inhomogeneous Media

We study the basic properties of the Maxwell equations for nonlinear inhomogeneous media. Assuming the classical nonlinear optics representation for the nonlinear polarization as a power series, we show that the solution exists and is unique in an appropriate space if the excitation current is not too large. The solution to the nonlinear Maxwell equations is represented as a power series in terms of the solution of the corresponding linear Maxwell equations. This representation holds at least for the time period inversely proportional to the appropriate norm of the solution to the linear Maxwell equation. We derive recursive formulas for the terms of the power series for the solution including an explicit formula for the first significant term attributed to the nonlinearity.     

Nonlinear dynamics of flows of a heavy compressible gas in the shallow water approximation

The system of the equations of motion for a compressible gas in the gravitational field over a smooth underlying surface has been analyzed in the shallow water approximation. All continuous centered self-similar solutions and all discontinuous self-similar solutions have been obtained. The problem of the decay of an arbitrary discontinuity for the equations of motion of the compressible gas has been solved in the explicit form in the shallow water approximation. The existence of four different configurations implementing the solution of the problem of the decay of an arbitrary discontinuity has been demonstrated. The conditions necessary and sufficient for the implementation of each configuration have been determined.     

Solution of the Einstein-Maxwell equations for a static distribution of massive charged particles

Exact solutions of the general relativistic field equations of Einstein and Maxwell have been found for a general static distribution of massive charged particles. As in the Newtonian case, the particles must have unit charge to mass ratioe ²/m ²=1. The active gravitational mass of the system of particles is precisely the sum of individual masses of the constituent particles.     

Stress wave propagation in maxwell fluid contained in rigid spherical shells

For stress wave propagation in a rigid spherical shell containing Maxwell fluid subjected to translational and rotational acceleration, the solutions to the governing equations are obtained by employing a finite difference technique, when the input acceleration is a unit step function. The solutions can be extended to accelerations which are general functions of time with the proper discretization of the input acceleration curve. The radial and temporal distribution of the stress waves in both cases are presented. The solutions are also specialized for the case of purely viscous fluids. The applicability of this model for brain injury simulation is briefly discussed.     

Asymptotic Properties of the Inelastic Kac Model

We discuss the asymptotic behavior of certain models of dissipative systems obtained from a suitable modification of Kac caricature of a Maxwellian gas. It is shown that global equilibria different from concentration are possible if the energy is not finite. These equilibria are distributed like stable laws, and attract initial densities which belong to the normal domain of attraction. If the initial density is assumed of finite energy, with higher moments bounded, it is shown that the solution converges for large-time to a profile with power law tails. These tails are heavily dependent of the collision rule.