The Boltzmann equation without angular cutoff in the whole space: I,Global existence for soft potential

It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and a possible gain of weight in the velocity variable. By defining and analyzing a non-isotropic norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross-sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces.     

The Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cut-off.This equation is partially elliptic in the velocity direction and degenerates in the spatial variable.We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in the Gevrey class at positive time with the Gevrey index depending on the angular singularity.Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of the linearized Boltzmann operator.     

Continual Distribution for the Bryan–Pidduck Equation

Ukrainian Mathematical Journal - For a nonlinear kinetic Boltzmann equation that describes the model of rough spheres, we construct its approximate solution in the form of a continual distribution...     

Self-similar solutions of a nonlinear friction equation in higher dimensions

We study the class of self-similar solutions of certain multi-dimensional kinetic models of granular flows, which have been recently introduced in connection with the quasi elastic limit of a model Boltzmann equation with dissipative collisions and variable coefficient of restitution. The importance of these solutions in connection with the cooling of the dissipative gas is subsequently discussed.

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Sunto Si studia la classe delle soluzioni di similarità di alcune equazioni cinetiche per flussi granulari in più dimensioni. Queste equazioni sono state introdotte di recente in connessione con il limite quasi elastico di un’ equazione di Boltzmann per collisioni dissipative con coefficiente di restituzione variabile. Nella seconda parte del lavoro si discute l’importanza di tali soluzioni nello studio del raffreddamento del gas dissipativo.

     

Optimal time decay of the Boltzmann equation with frictional force

In this paper, we use the combination of energy method and Fourier analysis to obtain the optimal time decay of the Boltzmann equation with frictional force towards equilibrium. Precisely speaking, we decompose the equation into macroscopic and microscopic partitions and perform the energy estimation. Then, we construct a special solution operator to a linearized equation without source term and use Fourier analysis to obtain the optimal decay rate to this solution operator. Finally, combining the decay rate with the energy estimation for nonlinear terms, the optimal decay rate to the Boltzmann equation with frictional force is established.     

Branching of Solutions of the Abstract Kinetic Equation

In this paper, bifurcation of solutions of a special nonlinear operator equation used in mathematical physics is considered. In the case of an equation for which the Fréchet derivative of the associated operator is a locally perturbed Fredholm operator, sufficient conditions for branching of solutions are studied. The methodology of application of the formalism developed in the paper is demonstrated by the example of the Boltzmann equation.     

n阶变系数线性差分方程的解

本文利用变数算符￣［２］以及给出变数算符和移动算符的乘积关系，并定义变系数移动算符幂级数间的乘积且证明其在Ｍｉｋｕｉｕｓｋｉ收敛意义下是正确的；另外，把一般的ｎ阶变系数线性差分方程转化为一个恰当的算符方程组，从而获得一般ｎ阶变系数线性差分方程的解。     

Approximate biflow solutions of the kinetic Bryan–Pidduck equation

Some explicit approximate solutions of the non‐linear Bryan–Pidduck equation (that is the Boltzmann equation for the model of rough spheres) are proposed. They have a form of spatially non‐homogeneous linear combination of two global Maxwellians with zero mass angular velocities but arbitrary mass linear velocities. The low‐temperature asymptotics of the uniform‐integral and the pure integral errors between the sides of this equation are found. Sufficient conditions of the infinitesimality of these errors are received, which are based on some requirements on coefficient functions and parameters of the distribution. Copyright © 2000 John Wiley & Sons, Ltd.     

A Fixed Point Operator for a Nonlinear Boundary Value Problem

We study a semilinear second order equation with a nonlinear boundary condition for the axial deformation of a nonlinear elastic beam in the presence of friction. Under appropriate conditions we define a fixed point operator in order to obtain solutions for this equation.     

Spectrum structure and decay rate estimates on the Landau equation with Coulomb potential

In this paper we first deduce the estimates on the linearized Landau operator with Coulomb potential and then analyze its spectrum structure by using semigroup theory and linear operator perturbation theory. Based on these estimates, we give the precise time decay rate estimates on the semigroup generated by the linearized Landau operator so that the optimal time decay rates of the nonlinear Landau equation follow. In addition, we present a similar result for the non-angular cutoff Boltzmann equation with soft potentials.

     

Application of a direct method for solving the Boltzmann equation in supersonic flow computation

The Boltzmann kinetic equation is used to numerically study the evolution of separated flows over a backward-facing step at low Knudsen numbers. The Boltzmann equation is solved by applying an explicit–implicit scheme. To improve the efficiency of the solution algorithm, it is parallelized with the help of MPI. The solution obtained with the kinetic equation is compared with those based on continuous medium equations. It is shown that the kinetic approach makes it possible to reproduce the distributions of surface pressure, friction coefficient, and heat transfer, as well as to obtain a flow structure close to experimental data.     

A new fractional finite volume method for solving the fractional diffusion equation

The inherent heterogeneities of many geophysical systems often gives rise to fast and slow pathways to water and chemical movement. One approach to model solute transport through such media is by fractional diffusion equations with a space–time dependent variable coefficient. In this paper, a two-sided space fractional diffusion model with a space–time dependent variable coefficient and a nonlinear source term subject to zero Dirichlet boundary conditions is considered.Some finite volume methods to solve a fractional differential equation with a constant dispersion coefficient have been proposed. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann–Liouville fractional derivatives at control volume faces in terms of function values at the nodes. However, these finite volume methods have not been extended to two-dimensional and three-dimensional problems in a natural manner. In this paper, a new weighted fractional finite volume method with a nonlocal operator (using nodal basis functions) for solving this two-sided space fractional diffusion equation is proposed. Some numerical results for the Crank–Nicholson fractional finite volume method are given to show the stability, consistency and convergence of our computational approach. This novel simulation technique provides excellent tools for practical problems even when a complex transition zone is involved. This technique can be extend to two-dimensional and three-dimensional problems with complex regions.     

空间分数阶电报方程的格子Boltzmann方法

应用格子Boltzmann方法（LBM）对Riemann Liouville空间分数阶电报方程进行了数值模拟研究.首先,将分数阶算子中的积分项进行离散化处理,并进行了收敛阶分析.然后,构建了带修正函数项的一维三速度(D1Q3)的LBM演化模型.利用Chapman Enskog多尺度技术和Taylor展开技术,推导出各平衡态分布函数和修正函数的具体表达式,准确地从所建的演化模型恢复出宏观方程.最后,数值计算结果表明该模型是稳定、有效的.     

Small mass asymptotic for the motion with vanishing friction

We consider the small mass asymptotic (Smoluchowski–Kramers approximation) for the Langevin equation with a variable friction coefficient. The friction coefficient is assumed to be vanishing within certain region. We introduce a regularization for this problem and study the limiting motion for the 1-dimensional case and a multidimensional model problem. The limiting motion is a Markov process on a projected space. We specify the generator and the boundary condition of this limiting Markov process and prove the convergence.     

Boltzmann collision operator for the infinite range potential: A limit problem

The conventional Boltzmann collision operator for the infinite range inverse power law model was derived by Maxwell by adopting a collision kernel which is a limit of that for the finite range model by ignoring the glancing angles. Since the interpretation of collision operator for the infinite range potential through limit process to the one with finite range potential is natural in regard to the derivation of the Boltzmann equation. It is the purpose of this paper to clarify the physical meaning of the conventional collision operator for the infinite range inverse power law model through the study of the limiting process of the collision operator as the cutoff radius tends to infinity. We first estimate the extent in which the glancing angles can be ignored in the limiting process. Furthermore we prove that taking limit to collision operator with finite range potential directly will lead to the conventional one with algebraic convergence rate.     

Lattice Boltzmann model for two-dimensional generalized sine-Gordon equation

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical model based on lattice Boltmann method to obtain the numerical solutions of two-dimensional generalized sine-Gordon equation, including damped and undamped sine-Gordon equation. By choosing properly the conservation condition between the macroscopic quantity $u_t$ and the distribution functions and applying the Chapman-Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained. The numerical results of the first three examples agree well with the analytic solutions, which indicates the lattice Boltzmann model is satisfactory and efficient. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given. Numerical experiments also show that the present scheme has a good long-time numerical behavior for the generalized sine-Gordon equation. Moreover, the model can also be applied to other two-dimensional nonlinear wave equations, such as nonlinear hyperbolic telegraph equation and Klein-Gordon equation.     

A study of integrability and symmetry for the (p + 1)th Boltzmann equation via Painlevé analysis and Lie‐group method

In this paper,we applied the Painlevé property test on Krook‐Wu model of the nonlinear Boltzmann equation (p = 1). As a result, by using Bäcklund transformation, we obtained three solutions two of them were known earlier, while the third one is new and more general, we have also two reductions one of them is Abel's equation. Also, Lie‐group method is applied to the (p + 1)th Boltzmann equation. The complete Lie algebra of infinitesimal symmetries is established. Three nonequivalent sub‐algebraic of the complete Lie algebra are used to investigate similarity solutions and similarity reductions in the form of nonlinear ordinary equations for (p + 1)th Boltzmann equation; we obtained two general solutions for (p + 1)th Boltzmann equation and new solutions for Krook‐Wu model of Boltzmann equation (p = 1). Copyright © 2014 John Wiley & Sons, Ltd.     

An exponentially convergent algorithm for nonlinear differential equations in Banach spaces

An exponentially convergent approximation to the solution of a nonlinear first order differential equation with an operator coefficient in Banach space is proposed. The algorithm is based on an equivalent Volterra integral equation including the operator exponential generated by the operator coefficient. The operator exponential is represented by a Dunford-Cauchy integral along a hyperbola enveloping the spectrum of the operator coefficient, and then the integrals involved are approximated using the Chebyshev interpolation and an appropriate Sinc quadrature. Numerical examples are given which confirm theoretical results.

     

Propagation of and Maxwellian weighted bounds for derivatives of solutions to the homogeneous elastic Boltzmann equation

We consider the n-dimensional space homogeneous Boltzmann equation for elastic collisions for variable hard potentials with Grad (angular) cutoff. We prove sharp moment inequalities, the propagation of L¹-Maxwellian weighted estimates, and consequently, the propagation L^∞-Maxwellian weighted estimates to all derivatives of the initial value problem associated to the afore mentioned equation. More specifically, we extend to all derivatives of the initial value problem associated to this class of Boltzmann equations corresponding sharp moment (Povzner) inequalities and time propagation of L¹-Maxwellian weighted estimates as originally developed Bobylev [A.V. Bobylev, Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems, J. Statist. Phys. 88 (1997) 1183–1214] in the case of hard spheres in 3 dimensions. To achieve this goal we implement the program presented in Bobylev–Gamba–Panferov [A.V. Bobylev, I.M. Gamba, V. Panferov, Moment inequalities and high-energy tails for Boltzmann equation with inelastic interactions, J. Statist. Phys. 116 (5–6) (2004) 1651–1682], which includes a full analysis of the moments by means of sharp moment inequalities and the control of L¹-exponential bounds, in the case of stationary states for different inelastic Boltzmann related problems with ‘heating’ sources where high energy tail decay rates depend on the inelasticity coefficient and the type of ‘heating’ source. More recently, this work was extended to variable hard potentials with angular cutoff by Gamba–Panferov–Villani [I.M. Gamba, V. Panferov, C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, ARMA (2008), in press] in the elastic case collision case where the L¹-Maxwellian weighted norm was shown to propagate if initial states have such property. In addition, we also extend to all derivatives the propagation of L^∞-Maxwellian weighted estimates, proven in [I.M. Gamba, V. Panferov, C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, ARMA (2008), in press], to solutions of the initial value problem to the Boltzmann equations for elastic collisions for variable hard potentials with Grad (angular) cutoff.     

On the Langevin equation with variable friction

We study two asymptotic problems for the Langevin equation with variable friction coefficient. The first is the small mass asymptotic behavior, known as the Smoluchowski–Kramers approximation, of the Langevin equation with strictly positive variable friction. The second result is about the limiting behavior of the solution when the friction vanishes in regions of the domain. Previous works on this subject considered one dimensional settings with the conclusions based on explicit computations.