Boltzmann方程及其动力学模型
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摘要
稀薄气体的动力学方程是数学物理的重要研究分枝之一.该方向涉及自然科学的许多领域,应用十分广泛,例如:非平衡态统计力学,天体物理,等离子体,半导体,航空技术和核反应堆理论等.本学位论文以经典Boltzmann方程,(Gauss-)BGK模型以及描述耗散气体的Boltzmann型方程等为研究对象,以尚待研究的问题为核心,建立一系列有关定性数学结果.
在第一章,我们简要的介绍了所要研究的几类数学物理方程;就本学位论文所涉及的问题分析了它们的研究现状和存在的问题.最后,我们给出本文所得到的一些主要结果,它们将在以后各章节中得到证明.
第二章研究一类刻划散体介质流气体的Boltzmann型方程,这是一个全新的研究方向.问题的物理背景是:气体中任何两分子在每次相互作用后保持动量守恒,但能量有一定的损失;并且气体被置于一个大热库中.在这种情况下气体的微观状态满足一个Fokker-Planck-Boltzmann型方程.我们在拟Maxwell分子模型近似情况下研究其空间均匀解的光滑性和大时间渐近行为.首先,在初始分布是一个整体速度为零温度有限的概率密度这一最弱的假设下,证明了解的光滑性并给出了其Sobolev范数在零时刻的渐近估计.其次,利用这一正则性结果和插值方法证明了当l→∞时解指数收敛于平衡态并给出了收敛指数的估计.这些结果改进了M.Bisi,J.A.Carrillo和G.Toscani在|24|中所得到的结论.
第三章以研究经典Boltzmann方程空间均匀的概率解为核心,我们采用了Grad弱角截断的硬位势.首先,我们利用Boltzmann碰撞算子在连续函数空间上定义了一个连续线性泛函,由此将碰撞算子的定义域拓展到概率测度空间.在这种情况下,给出了空间均匀Boltzmann方程在概率测度空间中的等价表述.其次,借鉴于[44]等中的方法给出了概率解的一些先验估计,在初始分布分别属于概率测度空间P_2(R~3)和P_4(R~3)的情况下证明了守恒解的存在性和唯一性.最后,利用Mischler和Wennberg在|37|中推广的Povzner不等式和Desvillettes的方法|42|,建立了高阶矩的生成定理.这些结果将[33,36,37,39,40,41,42,44]中部分已知经典结论推广到概率解;同时也改进了[38,56,57,58]中有关概率解的结论.
在第四章,我们研究一般经典Boltzmann方程指数型衰减的能量无限解,我们采用了Grad弱角截断的软位势.首先,我们建立了碰撞算子在一类指数衰减(能量可以无限)可测函数类中的一致性估计,结合Kaniel-Shinbrot方法[59]证明了小振幅初始值问题解的存在唯一性以及稳定性.其次,我们在一般情况下建立了这种能量无限解在该函数类中的稳定性(但是未获得存在性)以及解的大时间渐近行为.这些结果改进了Mischler和Perthame在[70]中证明的结论.我们注意到在[74]中建立了碰撞算子在连续函数空间上的一种极其重要的表示并在连续函数类中建立了类似的结果,这里我们采用了Kaniel-Shinbrot方法.
第五章在R~N中研究气体动力学理论中极其重要的BGK模型.首先,我们证明了Perthame在其重要工作[12]中所构造的整体分布解具有高阶矩的传播性质.其次,我们在碰撞频率v(?)ρ~μ(l,x)(μ≥1)的一般情况下,在一类加权L~∞空间中证明了BGK模型解的唯一性,同时我们还给出了一个局部存在性结果,它们部分的改进了[87,88]中已有结论.
第六章在R~N中研究BGK模型的L~p解.首先,我们将[87]中有关流体动力学量的L~∞估计推广到L~p估计,由此建立局部Maxwell分布的加权L~p估计,它是对BGK碰撞模型的一种有效的估计方法.其次,在以上估计的基础之上建立了方程的一致性L~p估计,然后利用速度平均紧引理和矩引理证明了逼近解的强收敛性,由此获得了L~p解的存在性;同时还给出了解的守恒性等一系列性质.最后,证明了上述解对某些L~p矩的传播性.其中,解的存在性是一个不同于Perthame存在定理[12]的结果.
第七章研究BGK模型的一个修正情形,即椭圆统计模型(ellipsoidal statisticalmodel),又称为Gauss-BGK模型.它是一类极其复杂的动力学模型,关于其解的研究到现在为止还没有任何结果.本章将研究它的一个特殊情形,即空间均匀解.我们给出了空间均匀解的结构和表示,在此基础之上给出了解的Maxwell分布型下界估计.其次,证明了解随时间增大而趋于平衡态并给出了收敛指数的精确值,建立了相应的熵定理.
Kinetic equations arising from rarefied gas dynamics are important research objectsin mathematical physics. This topic involves many fields of natural science and has extensive applications, for example: nonequilibrium statistical physics, astrophysics,plasma physics, semiconductor, aeronautical engineering, nuclear reactor and so on. The present thesis is devoted to studying the classical Boltzmann equation, the (Gauss-)BGK model and an equation of Boltzmann type arising from the theory of dissipative gases, we mainly focus on some matters remaining to be settled in this field, and establish a series of qualitative mathematical results.
In chapter 1, we briefly introduce several equations of mathematical physics which are the main objects in this thesis, then the current situation and problems to be solved relating to these equations are analysed, finally we give the main results obtained in this thesis, which will be discussed and proved in the following chapters.Chapter 2 is devoted to studying a Boltzmann type equation describing the evolutionof a dissipative gas consisting of granular media, this is a new direction in kinetic theory. Concerning the physical background of this problem, we assume that when two molecules in the gas collide, the conservation of momentum holds but a definite part of kinetic energy is lost. Furthermore, we also assume that the gas is put in a thermal bath. Under those assumptions, the microscopic state of the gas is governedby a partial differential equation of Fokker-Planck-Boltzmann type. We discuss smoothness and long time behavior of its spatially homogeneous solutions in the case of quasi-Maxwell molecules approximation. First, under the very weak assumption that the initial distribution is a probability density having null bulk velocity and finite temperature, we prove the smoothness of its solutions and establish the asymptotic estimates of its Sobolev's norms near t = 0. Then, combining this regularity result and interpolation method we show that the solution converges exponentially towards a unique equilibrium as t→∞, furthermore, an estimate of the exponent governing the exponential convergence is given. These results sharpen some theorems obtained by M. Bisi, J. A. Carrillo and G. Toscani[24].
In chapter 3, we study the spatially homogeneous probability solutions of the classical Boltzmann equation for hard potentials with Grad's weakly angular cut off assumption. Firstly, we can define a continuous linear functional on a continuous function space by the Boltzmann collision operator, as a consequence, we can extendthe domain of the collision operator to the space consisting of some probability measures. Having this result in mind, we give an equivalent formulation of the spatiallyhomogeneous Boltzmann equation in the space of probability measures. Secondly,by using the method introduced in [44], we establish some a priori estimates for probability solutions, and prove the existence and uniqueness of conversation probabilitysolutions for initial distributions respectively in P_2(R~3) and P_4(R~3). Finally, Using extended Povzner's inequality given by Mischler and Wennberg [37] and inspiredby Desvillettes method [42], we prove the production of higher moments for probability solutions. Those results extend some of the classical theorems obtained in [33, 36, 37, 39, 40, 41, 42, 44] to the probability solutions; meanwhile, we also improve some results on probability solutions obtained in [38, 56, 57, 58].
In chapter 4, for the general Boltzmann equation for soft potentials with Grad's weakly angular cut off, we discuss a class of its distributional solutions having infiniteenergy and decaying exponentially at infinity along its characteristics. Firstly, we establish an a priori estimate of the collision operator for a class of measurable functionsdecaying exponentially at infinity (maybe having infinite energy). Combining this result and Kaniel-Shinbrot method [59], we obtain the existence, uniqueness and stability of solutions with small amplitude. Then, we prove, for general initial data, the stability of solutions in this function class (but without existence result) and establishtheir long time behavior. These results generalize theorems obtained by Mischler and Perthame [70] for Maxwell molecules to soft potentials. It should be mentioned here that in [74] an important representation of the collision operator for continuous functions was given and similar results for mild continuous solutions were established, here we use Kaniel-Shinbrot method.
In chapter 5, we study the BGK model in R~N, which is extremely important in kinetic theory. Firstly, we prove that the global distributional solution, which was constructed by Perthame in his major work [12], propagates higher moments. Then, supposing that the collision frequency satisfies v =ρ~μ(t,x)(μ≥1), we prove the uniqueness of solutions to the BGK model in a weighted L~∞space and give a local existence theorem, which partially improves some known results obtained in [87, 88].
Chapter 6 is devoted to studying the L~p solutions to the BGK model in R~N. Firstly, we extend the method used to prove the L~∞estimates for hydrodynamic quantities in [87] and obtain some desired L~p estimates. On the basis of these estimates, we can establish weighted L~p estimates for local Maxwellians which is very efficient for us to proceed. Secondly, with these estimates, we obtain a uniform L~p estimate for solutions to the BGK model, then we employ velocity averaging lemma and moment lemma to prove the strong convergence of approximating solutions and as a consequence we obtain an L~p solution; further, we also get a series of properties for this kind of solutions, for example conservation of mass, momentum and energy. Finally, we prove that the solutions constructed above propagate some of the L~p moments. We note that the existence result obtain in this chapter is different from Perthame's theorem [12].
In chapter 7, we discuss a corrected version of the BGK model, i. e., the ellipsoidal statistical model, also called Gauss-BGK model. It is an extremely complicated kineticmodel, as far as we know, there is no mathematical results about its solvability. Here we discuss a special case: spatially homogeneous solutions. Firstly, we give a representation of its spatially homogeneous solution, and analyse the structure of this representation. Consequently, we obtain a Maxwellian lower bound for the solution. Secondly, we prove the solution converges exponentially (with an explicit exponent) towardsits equilibrium as time goes to infinity. An entropy theorem is also established.
在第一章,我们简要的介绍了所要研究的几类数学物理方程;就本学位论文所涉及的问题分析了它们的研究现状和存在的问题.最后,我们给出本文所得到的一些主要结果,它们将在以后各章节中得到证明.
第二章研究一类刻划散体介质流气体的Boltzmann型方程,这是一个全新的研究方向.问题的物理背景是:气体中任何两分子在每次相互作用后保持动量守恒,但能量有一定的损失;并且气体被置于一个大热库中.在这种情况下气体的微观状态满足一个Fokker-Planck-Boltzmann型方程.我们在拟Maxwell分子模型近似情况下研究其空间均匀解的光滑性和大时间渐近行为.首先,在初始分布是一个整体速度为零温度有限的概率密度这一最弱的假设下,证明了解的光滑性并给出了其Sobolev范数在零时刻的渐近估计.其次,利用这一正则性结果和插值方法证明了当l→∞时解指数收敛于平衡态并给出了收敛指数的估计.这些结果改进了M.Bisi,J.A.Carrillo和G.Toscani在|24|中所得到的结论.
第三章以研究经典Boltzmann方程空间均匀的概率解为核心,我们采用了Grad弱角截断的硬位势.首先,我们利用Boltzmann碰撞算子在连续函数空间上定义了一个连续线性泛函,由此将碰撞算子的定义域拓展到概率测度空间.在这种情况下,给出了空间均匀Boltzmann方程在概率测度空间中的等价表述.其次,借鉴于[44]等中的方法给出了概率解的一些先验估计,在初始分布分别属于概率测度空间P_2(R~3)和P_4(R~3)的情况下证明了守恒解的存在性和唯一性.最后,利用Mischler和Wennberg在|37|中推广的Povzner不等式和Desvillettes的方法|42|,建立了高阶矩的生成定理.这些结果将[33,36,37,39,40,41,42,44]中部分已知经典结论推广到概率解;同时也改进了[38,56,57,58]中有关概率解的结论.
在第四章,我们研究一般经典Boltzmann方程指数型衰减的能量无限解,我们采用了Grad弱角截断的软位势.首先,我们建立了碰撞算子在一类指数衰减(能量可以无限)可测函数类中的一致性估计,结合Kaniel-Shinbrot方法[59]证明了小振幅初始值问题解的存在唯一性以及稳定性.其次,我们在一般情况下建立了这种能量无限解在该函数类中的稳定性(但是未获得存在性)以及解的大时间渐近行为.这些结果改进了Mischler和Perthame在[70]中证明的结论.我们注意到在[74]中建立了碰撞算子在连续函数空间上的一种极其重要的表示并在连续函数类中建立了类似的结果,这里我们采用了Kaniel-Shinbrot方法.
第五章在R~N中研究气体动力学理论中极其重要的BGK模型.首先,我们证明了Perthame在其重要工作[12]中所构造的整体分布解具有高阶矩的传播性质.其次,我们在碰撞频率v(?)ρ~μ(l,x)(μ≥1)的一般情况下,在一类加权L~∞空间中证明了BGK模型解的唯一性,同时我们还给出了一个局部存在性结果,它们部分的改进了[87,88]中已有结论.
第六章在R~N中研究BGK模型的L~p解.首先,我们将[87]中有关流体动力学量的L~∞估计推广到L~p估计,由此建立局部Maxwell分布的加权L~p估计,它是对BGK碰撞模型的一种有效的估计方法.其次,在以上估计的基础之上建立了方程的一致性L~p估计,然后利用速度平均紧引理和矩引理证明了逼近解的强收敛性,由此获得了L~p解的存在性;同时还给出了解的守恒性等一系列性质.最后,证明了上述解对某些L~p矩的传播性.其中,解的存在性是一个不同于Perthame存在定理[12]的结果.
第七章研究BGK模型的一个修正情形,即椭圆统计模型(ellipsoidal statisticalmodel),又称为Gauss-BGK模型.它是一类极其复杂的动力学模型,关于其解的研究到现在为止还没有任何结果.本章将研究它的一个特殊情形,即空间均匀解.我们给出了空间均匀解的结构和表示,在此基础之上给出了解的Maxwell分布型下界估计.其次,证明了解随时间增大而趋于平衡态并给出了收敛指数的精确值,建立了相应的熵定理.
Kinetic equations arising from rarefied gas dynamics are important research objectsin mathematical physics. This topic involves many fields of natural science and has extensive applications, for example: nonequilibrium statistical physics, astrophysics,plasma physics, semiconductor, aeronautical engineering, nuclear reactor and so on. The present thesis is devoted to studying the classical Boltzmann equation, the (Gauss-)BGK model and an equation of Boltzmann type arising from the theory of dissipative gases, we mainly focus on some matters remaining to be settled in this field, and establish a series of qualitative mathematical results.
In chapter 1, we briefly introduce several equations of mathematical physics which are the main objects in this thesis, then the current situation and problems to be solved relating to these equations are analysed, finally we give the main results obtained in this thesis, which will be discussed and proved in the following chapters.Chapter 2 is devoted to studying a Boltzmann type equation describing the evolutionof a dissipative gas consisting of granular media, this is a new direction in kinetic theory. Concerning the physical background of this problem, we assume that when two molecules in the gas collide, the conservation of momentum holds but a definite part of kinetic energy is lost. Furthermore, we also assume that the gas is put in a thermal bath. Under those assumptions, the microscopic state of the gas is governedby a partial differential equation of Fokker-Planck-Boltzmann type. We discuss smoothness and long time behavior of its spatially homogeneous solutions in the case of quasi-Maxwell molecules approximation. First, under the very weak assumption that the initial distribution is a probability density having null bulk velocity and finite temperature, we prove the smoothness of its solutions and establish the asymptotic estimates of its Sobolev's norms near t = 0. Then, combining this regularity result and interpolation method we show that the solution converges exponentially towards a unique equilibrium as t→∞, furthermore, an estimate of the exponent governing the exponential convergence is given. These results sharpen some theorems obtained by M. Bisi, J. A. Carrillo and G. Toscani[24].
In chapter 3, we study the spatially homogeneous probability solutions of the classical Boltzmann equation for hard potentials with Grad's weakly angular cut off assumption. Firstly, we can define a continuous linear functional on a continuous function space by the Boltzmann collision operator, as a consequence, we can extendthe domain of the collision operator to the space consisting of some probability measures. Having this result in mind, we give an equivalent formulation of the spatiallyhomogeneous Boltzmann equation in the space of probability measures. Secondly,by using the method introduced in [44], we establish some a priori estimates for probability solutions, and prove the existence and uniqueness of conversation probabilitysolutions for initial distributions respectively in P_2(R~3) and P_4(R~3). Finally, Using extended Povzner's inequality given by Mischler and Wennberg [37] and inspiredby Desvillettes method [42], we prove the production of higher moments for probability solutions. Those results extend some of the classical theorems obtained in [33, 36, 37, 39, 40, 41, 42, 44] to the probability solutions; meanwhile, we also improve some results on probability solutions obtained in [38, 56, 57, 58].
In chapter 4, for the general Boltzmann equation for soft potentials with Grad's weakly angular cut off, we discuss a class of its distributional solutions having infiniteenergy and decaying exponentially at infinity along its characteristics. Firstly, we establish an a priori estimate of the collision operator for a class of measurable functionsdecaying exponentially at infinity (maybe having infinite energy). Combining this result and Kaniel-Shinbrot method [59], we obtain the existence, uniqueness and stability of solutions with small amplitude. Then, we prove, for general initial data, the stability of solutions in this function class (but without existence result) and establishtheir long time behavior. These results generalize theorems obtained by Mischler and Perthame [70] for Maxwell molecules to soft potentials. It should be mentioned here that in [74] an important representation of the collision operator for continuous functions was given and similar results for mild continuous solutions were established, here we use Kaniel-Shinbrot method.
In chapter 5, we study the BGK model in R~N, which is extremely important in kinetic theory. Firstly, we prove that the global distributional solution, which was constructed by Perthame in his major work [12], propagates higher moments. Then, supposing that the collision frequency satisfies v =ρ~μ(t,x)(μ≥1), we prove the uniqueness of solutions to the BGK model in a weighted L~∞space and give a local existence theorem, which partially improves some known results obtained in [87, 88].
Chapter 6 is devoted to studying the L~p solutions to the BGK model in R~N. Firstly, we extend the method used to prove the L~∞estimates for hydrodynamic quantities in [87] and obtain some desired L~p estimates. On the basis of these estimates, we can establish weighted L~p estimates for local Maxwellians which is very efficient for us to proceed. Secondly, with these estimates, we obtain a uniform L~p estimate for solutions to the BGK model, then we employ velocity averaging lemma and moment lemma to prove the strong convergence of approximating solutions and as a consequence we obtain an L~p solution; further, we also get a series of properties for this kind of solutions, for example conservation of mass, momentum and energy. Finally, we prove that the solutions constructed above propagate some of the L~p moments. We note that the existence result obtain in this chapter is different from Perthame's theorem [12].
In chapter 7, we discuss a corrected version of the BGK model, i. e., the ellipsoidal statistical model, also called Gauss-BGK model. It is an extremely complicated kineticmodel, as far as we know, there is no mathematical results about its solvability. Here we discuss a special case: spatially homogeneous solutions. Firstly, we give a representation of its spatially homogeneous solution, and analyse the structure of this representation. Consequently, we obtain a Maxwellian lower bound for the solution. Secondly, we prove the solution converges exponentially (with an explicit exponent) towardsits equilibrium as time goes to infinity. An entropy theorem is also established.
引文
[1] Maxwell J C. Illustrations of the dynamical theory of gases I. Philosophical Magazine,1860, 19: 19-32.
[2] Maxwell J C. Illustrations of the dynamical theory of gases Ⅱ. Philosophical Magazine,1860, 20: 21-32.
[3] Maxwell J C. Illustrations of the dynamical theory of gases Ⅲ. Philosophical Magazine,1860, 20: 33-37.
[4] Maxwell J. C. On the dynamical theory of gases. Phil.Trans.Roy.Soc.London,1867, 157: 49-88.
[5] Boltzmann L.Weitere studien (?)ber das w(?)rmegleichgenicht unfer gasmolek(?)ler.Stizungsberichte der Akademie der Wissenschaften Wien, 1872, 66: 275-370.
[6] Boltzmann L.(?)ber dasw(?)rmegleichgewicht vongasen auf welche (?)ussere ka(?)fte wirken.Stizungsberichte der Akademie der Wissenschaften Wien, 1875, 72: 427-457.
[7] Boltzmann L.(?)ber die aufstellung und integration der gleichungen, welche die molekularbewegungen ingasenbestimmen. Stizungsberichte der Akademie der Wissenschaften Wien, 1876, 74: 503-552.
[8] Cercignani C, Illner R, Pulvirenti M. "The Mathematical Theory of Dilute Gases."New York: Springer, 1994.
[9] Cercignani C. "The Boltzmann Equation and Its Applications." New York: Springer,1988.
[10] Bhatnagar P L, Gross E P, Krook M. A model for collision processes in gases. Phys.Rev., 1954, 94: 511-514.
[11] Cercignani C. "Mathematical Methods in Kinetic Theory." New York: Plenum Press,1990.
[12] Perthame B. Global existence to the BGK model of Boltzmann equation. J. Differential Equations, 1989, 82: 191-205.
[13] Holway L H. Kinetic theory of shock structure using an ellipsoidal distribution function. in: edited by J. H. de Leeuw. "Rarefied Gas Dynamics," Vol. I. New York: Academic Press, 1966. pp. 193-215.
[14] Andries P, Le Tallee P, Perlat J-P, Perthame B. The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B-Fluids, 2000, 19: 813-830.
[15] Cercignani C. Recent developments in the mechanics of granular materials. In Fisica matematica e ingeneria delle struttre:rapporti e compatibilit(?), G. Ferrarese, Ed.Pitagora Ed., Bologna, 1995, pp.119-132.
[16] Bellomo N, Esteban M, Lachowicz M. Nonliear kinetic equations with dissipative collisions. Appl. Math. Lett.,1995, 8: 47-52.
[17] Benedetto D, Caglioti E, Carrillo J A, Pulvirenti M. A non-Maxwellian steady distribution for one-dimensional granular media. J. Stat. Phys.,1998, 91: 979-990.
[18] Bobylev A V, Carrillo J A, Gamba I M. On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Stat. Phys., 2000, 98: 743-773.
[19] Carrillo J A, Cercignani C, Gamba I M. Steady states of a Boltzmann equation for driven granular media. Phys. Rev. E ,2000, 62: 7700-7707.
[20] Cercignani C.Shear flow of a granular material. J. Stat. Phys.,2001, 102:1407-1415.
[21] Cercignani C, Illner R, Stoica C. On diffusive equilibria in generalized kinetic theory.J. Stat. Phys.,2001, 105: 337-352.
[22] Bobylev A V, Cercignani C. Moment equations for a granular material in a themal bath. J. Stat. Phys., 2002, 106: 547-567.
[23] Gamba M, Panferov V, Villani C. On the Boltzmann equation for diffusively excited granular media. Comm. Math. Phys.,2004, 246: 503-541.
[24] Bisi M, Carrillo J A, Toscani G. Contractive metrics for a Boltzmann equation for granular gases: diffusive equilibria. J. Stat. Phys., 2005, 118: 301-331.
[25] Hilbert D. Begrundung der kinetischen gastheorie. Math. Ann.,1916/1917,72:562-577.
[26] Chapman S. The kinetic theory of simple and composite gases: viscosity, thermal conduction and diffusion. Proc.Royal Soc. London, 1916/1917, A93: 1-20.
[27] Enskog D. "Kinetische Theorie der Vorg(?)nge in m(?)ssig verd(?)nnten Gasen, I. Allgemeiner Teil." Uppsala: Almqvist & Wiksell, 1917.
[28] Carleman T. Sur la th(?)orie de l'(?)quation int(?)gro-differentielle de Boltzmann.Ada Math., 1933, 60: 91-146.
[29] Carleman T. "Probl(?)es Mathematiques das la th(?)ory Cin(?)tique des gaz." Uppsala:Almqvist and Wiksell, 1957.
[30] Morgenstern D. General existence and uniqueness proof for spatially homogenous solutions of the Maxwell-Boltzmann equation in the case of Maxwellian molecules. Proc.Nat. Acad. Sci.U.S.A.,1954, 40: 719-721.
[31] Morgenstern D. Analytical studies related to the Maxwell-Boltzman equation. J. Rational Mech. Anal, 1955, 4: 533-555.
[32] Povzner A J. About the Boltzmann equation in kinetic gas theory. Mat. Shorn., 1962,58:65-86.
[33] Arkeryd L. On the Boltzmann equation. Arch. Rational Mech. Anal.,1972, 45: 1-34.
[34] Arkeryd L. Intermolecular forces of infinite range and the Boltzmann equation. Arch.Rational Mech. Anal,1981,77: 11-21.
[35] Arkeryd L. Stability in L~1 for the spatially homogeneous Boltzmann equation. Arch.Rational Mech. Anal.,1988, 103: 151-167.
[36] Lu X. Conservation of energy, entropy identity, and local stability for the spatially homogeneous Boltzmann equation.J. Stat. Phys.,1999, 96: 765-796.
[37] Mischler S, Wennberg B. On the spatially homogeneous Boltzmann equation. Ann.Inst. Henri Poincar(?), Analyse non lin(?)aire, 1999, 16: 467-501.
[38] Toscani G, Villani C. Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Stat. Phys.,1999, 94 :619-637.
[39] Wennberg B. On moments and uniqueness for solutions to the space homogenous Boltzmann equation. Transport Theory Stat. Phys., 1994, 24: 533-539.
[40] Elmroth T. Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range. Arch. Rational Mech. Anal., 1983, 82: 1-12.
[41] Elmroth T. On the H-function and convergence towards equilibrium for a space-homogeneous molecular density. SIAM J. Appl. Math., 1984, 44: 150-159.
[42] Desvillettes L. Some applications of the method of moments for the homogeneous Boltzmann and Kac equations. Arch. Rational Mech. Anal, 1993, 123: 387-404.
[43] Desvillettes L. Boltzmann's kernel and the spatially homogeneous Boltzmann equation.Rivita di Matematica dell'Universita di Parma,2001, 6: 1-22.
[44] Lu X, Wennberg B. Solutions with increasing energy for the spatially homogeneous Boltzmann equation. Nonlinear Anal. Real World Appl., 2002, 3: 243-258.
[45] Wennberg B. Entropy dissipation and moment production for the Boltzmann equation.J. Stat. Phys.,1997, 86: 1053-1066.
[46] Carlen E A, Gabetta E, Toscani G. Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Comm.Math. Phys. , 1999, 305: 521-546.
[47] Gabetta E, Toscani G, Wennberg B. Metrics for probability distributions and trend to equilibrium for solutions of the Boltzmann equation. J. Stat. Phys., 1995, 81:901-934.
[48] Pulvirenti A, Wennberg B. Lower bounds for the solutions to the Kac and the Boltzmann equation. Transport Theory Stat. Phys.,1996, 25: 437-446.
[49] Pulvirenti A, Wennberg B. A Maxwellian lower bound for solutions to the Boltzmann equation. Comm. Math. Phys., 1997, 183: 145-160.
[50] Gustafsson T. L~p-estimates for the nonlinear spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal., 1986, 92: 23-57.
[51] Gustafsson T. Global L~p-properties for the spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal., 1988, 103: 1-38.
[52] Arkeryd L. L~∞ estimates for the space-homogeneous Boltzmann equation. J. Stat.Phys., 1983, 31: 347-361.
[53] Wennberg B. Stability and exponential convergence in L~p for the spatially homogeneous Boltzmann equation.Nonlinear Anal. TMA,1993, 20: 935-964.
[54] Mouhot C, Villani C. Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Rational Mech. Anal.,2004, 173: 169-212.
[55] Villani C. A review of mathematical topics in collisional kinetic theory. in: edited by S. Friedlander and D. Serre. "Handbook of Mathematical Fluid Dynamics." Vol. Ⅰ.Amsterdam: North-Holland,2002. pp71-305.
[56] Tanaka H. Probabilistic treatment of the Boltzmann equation of Maxwellian molecules.Z. Wahrsch.Verw. Geb.,1978, 46: 67-105.
[57] Fournier N, M(?)l(?)ard S. A Markov process associated with a Boltzmann equation without cutoff and for non-Maxwell molecules. J. Stat. Phys.,2001, 104: 359-385.
[58] Desvillettes L, Graham C, Meleard S. Probabilistic interpretation and numerical approximation of a Kac equation without cutoff. Stochastic Process. Appl., 1999, 84:115-135.
[59] Kaniel S, Shinbrot M. The Boltzmann equation I: uniqueness and local existence.Comm. Math. Phys., 1978, 58: 65-84.
[60] Illner R, Shinbrot M. The Boltzmann equation: global existence for a rare gas in an infinite vacuum. Comm. Math. Phys., 1984, 95: 217-226.
[61] Bellomo N, Toscani G. On the Cauchy problem for the nonlinear Boltzmann equation:global existence, uniqueness and asymptotic behaviour. J. Math. Phys., 1985, 26:334-338.
[62] Hamdache K. Existence in the large and asymptotic behaviour for the Boltzmann equation. Japan. J. Appl. Math.,1985, 2: 1-15.
[63] Toscani G. Global solution of the initial value problem for the Boltzmann equation near a local Maxwellian. Arch. Rational Mech. Anal., 1988, 102: 231-241.
[64] Toscani G. H-theorem and asymptotic trend of the solution for a rarefied gas in the vacuum. Arch. Rational. Mech. Anal., 1987, 100: 1-12.
[65] Toscani G. On the Boltzmann equation in unbouned domains. Arch. Rational. Mech.nal, 1986, 95: 37-49.
[66] Toscani G, Bellomo N. Global existence, uniqueness and stability of the nonlinear Boltzmann equation with general gas-particle interaction potential. Rend. Circ. Mat.Palermo(2)Suppl., 1985, 8: 419-433.
[67] Polewczak J. Classical solution of the nonlinear Boltzmann equation in all R~3: Asymptotic behavior of solutions. J. Stat. Phys., 1988, 50: 611-632.
[68] Palczewski A, Toscani G. Global solution of the Boltzmann equation for rigid spheres and initial data close to a local Maxwellian. J. Math. Phys., 1989, 30: 2445-2450. [69] Bellomo N, Palczewski A, Toscani G. "Mathematical Topics in Nonlinear Kinetic Theory." Singapore: World Scientific, 1988.
[70] Mischler S, Perthame B. Boltzmann equation with infinite energy: renormanized solutions and distributional solutions for small initial data and initial data close to a Maxwellian. SIAM J. Math. Anal., 1997, 28: 1015-1027.
[71] Wei J, Zhang X. Eternal solutions of the Boltzmann equation near travelling Maxwellians. J. Math. Anal. Appl., 2006, 314: 219-232.
[72] Wei J, Zhang X. Eternal solutions of the Boltzmann equation. Acta Math. Sci. Ser.A,2007, 27: 240-247(in Chinese).
[73] Duan R J, Yang T, Zhu C J. Global existence to the Boltzmann equation with external forces in infinite vacuum. J. Math. Phys., 2005, 46: 053307.
[74] Lu X. Spatial decay solutions of the Boltzmann equation: converse properties of long times limiting brhavior. SIAM J. Math. Anal., 1999, 30: 1151-1174.
[75] Ukai S. On the existence of global solutions of mixed problems for the nonlinear Boltzmann equation. Proc. Japan Acad. Sci.,1974, 50: 179-184.
[76] Ukai S. Solutions of the Boltzmann equation. in: edited by T. Nishida. "Patterns and Waves-Qualitative Analysis of Differential Equations". Amsterdam: North-Holland,1986, pp37-96.
[77] Yang T, Zhao H J. A new energy method for the Boltzmann equation. J. Math. Phys.,2006, 47: 053301.
[78] Diperna R J, Lions P L. On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math., 1989, 130: 321-366.
[79] Diperna R J, Lions P L. Global solutions of Boltzmann's equation and the entropy inequality. Arch. Rational Mech. Anal., 1991, 114: 47-55.
[80] Arkeryd L, Maslova N. On diffuse reflection at the boundary for the Boltzmann equation and related equations. J. Stat.Phys.,1994, 77: 1051-1077.
[81] Desvillettes L. Convergence to equilibrium in large time for Boltzmann and B.G.K.equation. Arch. Rational Mech. Anal., 1990, 110: 73-91.
[82] Hamdache K. Initial-boundary value problems for the Boltzmann equation: global existence of weak solutions. Arch. Rational Mech. Anal., 1992, 119: 309-353.
[83] Lions P L. Compactness in Boltzmann's equation via Fourier integral operators and applications I. J. Math. Kyoto Univ., 1994, 34: 391-427.
[84] Lions P L. Compactness in Boltzmann's equation via Fourier integral operators and applications Ⅱ. J. Math. Kyoto Univ., 1994, 34: 429-461.
[85] Lions P L. Compactness in Boltzmann's equation via Fourier integral operators and applications Ⅲ. J. Math. Kyoto Univ., 1994, 34: 539-584.
[86] Lions P L. Regularity and compactness for Boltzmann collision operators without angular cut-off. C. R. Acad. Sci. Paris, 1998, 326: 37-41.
[87] Perthame B, Pulvirenti M. Weighted L~∞ bounds and uniqueness for the Boltzmann BGK model. Arch. Rational Mech. Anal., 1993, 125: 289-295.
[88] Mischler S. Uniqueness for the BGK-equation in R~N and rate of convergence for a semi-discrete scheme. Differential Integral Equations, 1996, 9: 1119-1138.
[89] Bellouquid A. Global existence and large-times behavior for BGK model for a gas with non constant cross section. Transport Theory Stat. Phys., 2003, 32: 157-184.
[90] Barros-Neto J. "An Introduction to the Theory of Distributions." New York: Marcel Dekker, 1973.
[91] Adams R A. "Sobolev Spaces." New York: Academic Press, 1975.
[92] Zhang X. Regularity and long time behavior of the Boltzmann equation for granular gases. J. Math. Anal. Appl., 2006, 324: 650-662.
[93] Truesdell C, Muncaster R G. "Fundamentals of Maxwell's Kinetic Theory of a Simple Monoatomic Gas." New York: Academic Press, 1980.
[94] Zhang X, Zhang X. Probability approaches to spatially homogeneous Boltzmann equation. Stochastic Anal. Appl., 2007, 25: 1129-1150.
[95] Kallenberg O. "Foundations of Modern Probability." Berlin: Springer, 1997.
[96] Mukherjea A, Pothoven K. "Real and Fucntional Analysis." New York: Plenum Press,1984.
[97] Zhang X, Zhang X. Supports of measure solutions for spatially homogeneous Boltzmann equations. J. Stat. Phys., 2006, 124: 485-495.
[98] Greenberg W, Polewczak J. A global existence theorem for the nonlinear BGK equation.J. Stat. Phys., 1989, 55: 1313-1321.
[99] Polewczak J, Greenberg W. Some remarks about continuity properties of local Maxwellians and an existence theorem for the BGK model of the Boltzmann equation. J. Stat. Phys., 1983, 33: 307-316.
[100] Zhang X, Hu S. Moment propagation of the BGK equation. Math. Appl.,2006, 19:857-862.
[101] Zhang X, Hu S. The Cauchy problem of the BGK equation with collision frequency depending on macroscopic density. submitted.
[102] Zhang X, Hu S. L~p solutions to the Cauchy problem of the BGK equation. J. Math.Phys.,2007, (11)48: 113304.
[103] Perthame B. Mathematical tools for kinetic equations. Bull. Amer. Math. Soc., 2004,41: 205-244.
[104] Horst E. On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation. I General theory. Math.Meth.Appl.Sci.,1981, 3:229-248.
[105] Horst E. On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation. Ⅱ Special cases. Math. Meth. Appl. Sci., 1982, 4: 19-32.
[106] Horst E, Hunze R. Weak solutions of the initial value problem for the unmodified non-linear Vlasov equation. Math. Meth. Appl. Sci., 1984, 6: 262-279.
[107] Illner R, Neunzert H. An existence theorem for the unmodified Vlasov equation. Math.Meth. Appl. Sci.,1979,1: 530-554.
[108] Carrillo J A, Soler J. On the initial value problem for the Vlasov-Poisson-Fokker-Planck system with initial data in L~p spaces. Math. Meth. Appl. Sci., 1995, 18:825-839.
[109] Castella F. Propagation of space moments in the Vlasov-Poisson equation and further results. Ann. Inst. Henri Poincar(?), Analyse non lin(?)aire, 1999, 16:503-533.
[110] Dunford N, Schwartz J T. "Linear Operators, I. General Theory." New York: Inter-science, 1958.
[111] Golse F, Perthame B, Sentis R. Un r(?)sultat pour les (?)quations de transport et application au calcul de la limite de la valeur propre principale d'un op(?)rateur de transport.C. R. Acad. Sci. Paris, 1985, 301: 341-344.
[112] Golse F, Lions P L, Perthame B, Sentis R. Regularity of the moments of the solution of a transport equation. J. Funct. Anal., 1988, 76: 110-125.
[113] Diperna R J, Lions P L. Global weak solutions of Vlasov-Maxwell system. C. R. Acad. Sci. Paris, 1988, 307: 655-658.
[114] Diperna R J, Lions P L. Global weak solutions of Vlasov-Maxwell system. Comm.Pure Appl. Math., 1989, 42: 729-757.
[115] Bardos C, Golse F, Perthame B, Sentis R. The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximatiom. J. Funct. Anal., 1988,77: 434-460.
[116] Yosida K. "Functional Analysis." Sixth edition. New York: Springer, 1980.
[117] Zhang X, Hu S, Zhang C. On the spatially homogeneous ES model of the Boltzmann equation. Math. Appl., 2006, 19: 426-432.
[118] Lu X. A result on uniqueness of mild solutions of Boltzmann equation. Transport Theory Stat. Phys., 1997, 26: 209-220.
[2] Maxwell J C. Illustrations of the dynamical theory of gases Ⅱ. Philosophical Magazine,1860, 20: 21-32.
[3] Maxwell J C. Illustrations of the dynamical theory of gases Ⅲ. Philosophical Magazine,1860, 20: 33-37.
[4] Maxwell J. C. On the dynamical theory of gases. Phil.Trans.Roy.Soc.London,1867, 157: 49-88.
[5] Boltzmann L.Weitere studien (?)ber das w(?)rmegleichgenicht unfer gasmolek(?)ler.Stizungsberichte der Akademie der Wissenschaften Wien, 1872, 66: 275-370.
[6] Boltzmann L.(?)ber dasw(?)rmegleichgewicht vongasen auf welche (?)ussere ka(?)fte wirken.Stizungsberichte der Akademie der Wissenschaften Wien, 1875, 72: 427-457.
[7] Boltzmann L.(?)ber die aufstellung und integration der gleichungen, welche die molekularbewegungen ingasenbestimmen. Stizungsberichte der Akademie der Wissenschaften Wien, 1876, 74: 503-552.
[8] Cercignani C, Illner R, Pulvirenti M. "The Mathematical Theory of Dilute Gases."New York: Springer, 1994.
[9] Cercignani C. "The Boltzmann Equation and Its Applications." New York: Springer,1988.
[10] Bhatnagar P L, Gross E P, Krook M. A model for collision processes in gases. Phys.Rev., 1954, 94: 511-514.
[11] Cercignani C. "Mathematical Methods in Kinetic Theory." New York: Plenum Press,1990.
[12] Perthame B. Global existence to the BGK model of Boltzmann equation. J. Differential Equations, 1989, 82: 191-205.
[13] Holway L H. Kinetic theory of shock structure using an ellipsoidal distribution function. in: edited by J. H. de Leeuw. "Rarefied Gas Dynamics," Vol. I. New York: Academic Press, 1966. pp. 193-215.
[14] Andries P, Le Tallee P, Perlat J-P, Perthame B. The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B-Fluids, 2000, 19: 813-830.
[15] Cercignani C. Recent developments in the mechanics of granular materials. In Fisica matematica e ingeneria delle struttre:rapporti e compatibilit(?), G. Ferrarese, Ed.Pitagora Ed., Bologna, 1995, pp.119-132.
[16] Bellomo N, Esteban M, Lachowicz M. Nonliear kinetic equations with dissipative collisions. Appl. Math. Lett.,1995, 8: 47-52.
[17] Benedetto D, Caglioti E, Carrillo J A, Pulvirenti M. A non-Maxwellian steady distribution for one-dimensional granular media. J. Stat. Phys.,1998, 91: 979-990.
[18] Bobylev A V, Carrillo J A, Gamba I M. On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Stat. Phys., 2000, 98: 743-773.
[19] Carrillo J A, Cercignani C, Gamba I M. Steady states of a Boltzmann equation for driven granular media. Phys. Rev. E ,2000, 62: 7700-7707.
[20] Cercignani C.Shear flow of a granular material. J. Stat. Phys.,2001, 102:1407-1415.
[21] Cercignani C, Illner R, Stoica C. On diffusive equilibria in generalized kinetic theory.J. Stat. Phys.,2001, 105: 337-352.
[22] Bobylev A V, Cercignani C. Moment equations for a granular material in a themal bath. J. Stat. Phys., 2002, 106: 547-567.
[23] Gamba M, Panferov V, Villani C. On the Boltzmann equation for diffusively excited granular media. Comm. Math. Phys.,2004, 246: 503-541.
[24] Bisi M, Carrillo J A, Toscani G. Contractive metrics for a Boltzmann equation for granular gases: diffusive equilibria. J. Stat. Phys., 2005, 118: 301-331.
[25] Hilbert D. Begrundung der kinetischen gastheorie. Math. Ann.,1916/1917,72:562-577.
[26] Chapman S. The kinetic theory of simple and composite gases: viscosity, thermal conduction and diffusion. Proc.Royal Soc. London, 1916/1917, A93: 1-20.
[27] Enskog D. "Kinetische Theorie der Vorg(?)nge in m(?)ssig verd(?)nnten Gasen, I. Allgemeiner Teil." Uppsala: Almqvist & Wiksell, 1917.
[28] Carleman T. Sur la th(?)orie de l'(?)quation int(?)gro-differentielle de Boltzmann.Ada Math., 1933, 60: 91-146.
[29] Carleman T. "Probl(?)es Mathematiques das la th(?)ory Cin(?)tique des gaz." Uppsala:Almqvist and Wiksell, 1957.
[30] Morgenstern D. General existence and uniqueness proof for spatially homogenous solutions of the Maxwell-Boltzmann equation in the case of Maxwellian molecules. Proc.Nat. Acad. Sci.U.S.A.,1954, 40: 719-721.
[31] Morgenstern D. Analytical studies related to the Maxwell-Boltzman equation. J. Rational Mech. Anal, 1955, 4: 533-555.
[32] Povzner A J. About the Boltzmann equation in kinetic gas theory. Mat. Shorn., 1962,58:65-86.
[33] Arkeryd L. On the Boltzmann equation. Arch. Rational Mech. Anal.,1972, 45: 1-34.
[34] Arkeryd L. Intermolecular forces of infinite range and the Boltzmann equation. Arch.Rational Mech. Anal,1981,77: 11-21.
[35] Arkeryd L. Stability in L~1 for the spatially homogeneous Boltzmann equation. Arch.Rational Mech. Anal.,1988, 103: 151-167.
[36] Lu X. Conservation of energy, entropy identity, and local stability for the spatially homogeneous Boltzmann equation.J. Stat. Phys.,1999, 96: 765-796.
[37] Mischler S, Wennberg B. On the spatially homogeneous Boltzmann equation. Ann.Inst. Henri Poincar(?), Analyse non lin(?)aire, 1999, 16: 467-501.
[38] Toscani G, Villani C. Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Stat. Phys.,1999, 94 :619-637.
[39] Wennberg B. On moments and uniqueness for solutions to the space homogenous Boltzmann equation. Transport Theory Stat. Phys., 1994, 24: 533-539.
[40] Elmroth T. Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range. Arch. Rational Mech. Anal., 1983, 82: 1-12.
[41] Elmroth T. On the H-function and convergence towards equilibrium for a space-homogeneous molecular density. SIAM J. Appl. Math., 1984, 44: 150-159.
[42] Desvillettes L. Some applications of the method of moments for the homogeneous Boltzmann and Kac equations. Arch. Rational Mech. Anal, 1993, 123: 387-404.
[43] Desvillettes L. Boltzmann's kernel and the spatially homogeneous Boltzmann equation.Rivita di Matematica dell'Universita di Parma,2001, 6: 1-22.
[44] Lu X, Wennberg B. Solutions with increasing energy for the spatially homogeneous Boltzmann equation. Nonlinear Anal. Real World Appl., 2002, 3: 243-258.
[45] Wennberg B. Entropy dissipation and moment production for the Boltzmann equation.J. Stat. Phys.,1997, 86: 1053-1066.
[46] Carlen E A, Gabetta E, Toscani G. Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Comm.Math. Phys. , 1999, 305: 521-546.
[47] Gabetta E, Toscani G, Wennberg B. Metrics for probability distributions and trend to equilibrium for solutions of the Boltzmann equation. J. Stat. Phys., 1995, 81:901-934.
[48] Pulvirenti A, Wennberg B. Lower bounds for the solutions to the Kac and the Boltzmann equation. Transport Theory Stat. Phys.,1996, 25: 437-446.
[49] Pulvirenti A, Wennberg B. A Maxwellian lower bound for solutions to the Boltzmann equation. Comm. Math. Phys., 1997, 183: 145-160.
[50] Gustafsson T. L~p-estimates for the nonlinear spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal., 1986, 92: 23-57.
[51] Gustafsson T. Global L~p-properties for the spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal., 1988, 103: 1-38.
[52] Arkeryd L. L~∞ estimates for the space-homogeneous Boltzmann equation. J. Stat.Phys., 1983, 31: 347-361.
[53] Wennberg B. Stability and exponential convergence in L~p for the spatially homogeneous Boltzmann equation.Nonlinear Anal. TMA,1993, 20: 935-964.
[54] Mouhot C, Villani C. Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Rational Mech. Anal.,2004, 173: 169-212.
[55] Villani C. A review of mathematical topics in collisional kinetic theory. in: edited by S. Friedlander and D. Serre. "Handbook of Mathematical Fluid Dynamics." Vol. Ⅰ.Amsterdam: North-Holland,2002. pp71-305.
[56] Tanaka H. Probabilistic treatment of the Boltzmann equation of Maxwellian molecules.Z. Wahrsch.Verw. Geb.,1978, 46: 67-105.
[57] Fournier N, M(?)l(?)ard S. A Markov process associated with a Boltzmann equation without cutoff and for non-Maxwell molecules. J. Stat. Phys.,2001, 104: 359-385.
[58] Desvillettes L, Graham C, Meleard S. Probabilistic interpretation and numerical approximation of a Kac equation without cutoff. Stochastic Process. Appl., 1999, 84:115-135.
[59] Kaniel S, Shinbrot M. The Boltzmann equation I: uniqueness and local existence.Comm. Math. Phys., 1978, 58: 65-84.
[60] Illner R, Shinbrot M. The Boltzmann equation: global existence for a rare gas in an infinite vacuum. Comm. Math. Phys., 1984, 95: 217-226.
[61] Bellomo N, Toscani G. On the Cauchy problem for the nonlinear Boltzmann equation:global existence, uniqueness and asymptotic behaviour. J. Math. Phys., 1985, 26:334-338.
[62] Hamdache K. Existence in the large and asymptotic behaviour for the Boltzmann equation. Japan. J. Appl. Math.,1985, 2: 1-15.
[63] Toscani G. Global solution of the initial value problem for the Boltzmann equation near a local Maxwellian. Arch. Rational Mech. Anal., 1988, 102: 231-241.
[64] Toscani G. H-theorem and asymptotic trend of the solution for a rarefied gas in the vacuum. Arch. Rational. Mech. Anal., 1987, 100: 1-12.
[65] Toscani G. On the Boltzmann equation in unbouned domains. Arch. Rational. Mech.nal, 1986, 95: 37-49.
[66] Toscani G, Bellomo N. Global existence, uniqueness and stability of the nonlinear Boltzmann equation with general gas-particle interaction potential. Rend. Circ. Mat.Palermo(2)Suppl., 1985, 8: 419-433.
[67] Polewczak J. Classical solution of the nonlinear Boltzmann equation in all R~3: Asymptotic behavior of solutions. J. Stat. Phys., 1988, 50: 611-632.
[68] Palczewski A, Toscani G. Global solution of the Boltzmann equation for rigid spheres and initial data close to a local Maxwellian. J. Math. Phys., 1989, 30: 2445-2450. [69] Bellomo N, Palczewski A, Toscani G. "Mathematical Topics in Nonlinear Kinetic Theory." Singapore: World Scientific, 1988.
[70] Mischler S, Perthame B. Boltzmann equation with infinite energy: renormanized solutions and distributional solutions for small initial data and initial data close to a Maxwellian. SIAM J. Math. Anal., 1997, 28: 1015-1027.
[71] Wei J, Zhang X. Eternal solutions of the Boltzmann equation near travelling Maxwellians. J. Math. Anal. Appl., 2006, 314: 219-232.
[72] Wei J, Zhang X. Eternal solutions of the Boltzmann equation. Acta Math. Sci. Ser.A,2007, 27: 240-247(in Chinese).
[73] Duan R J, Yang T, Zhu C J. Global existence to the Boltzmann equation with external forces in infinite vacuum. J. Math. Phys., 2005, 46: 053307.
[74] Lu X. Spatial decay solutions of the Boltzmann equation: converse properties of long times limiting brhavior. SIAM J. Math. Anal., 1999, 30: 1151-1174.
[75] Ukai S. On the existence of global solutions of mixed problems for the nonlinear Boltzmann equation. Proc. Japan Acad. Sci.,1974, 50: 179-184.
[76] Ukai S. Solutions of the Boltzmann equation. in: edited by T. Nishida. "Patterns and Waves-Qualitative Analysis of Differential Equations". Amsterdam: North-Holland,1986, pp37-96.
[77] Yang T, Zhao H J. A new energy method for the Boltzmann equation. J. Math. Phys.,2006, 47: 053301.
[78] Diperna R J, Lions P L. On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math., 1989, 130: 321-366.
[79] Diperna R J, Lions P L. Global solutions of Boltzmann's equation and the entropy inequality. Arch. Rational Mech. Anal., 1991, 114: 47-55.
[80] Arkeryd L, Maslova N. On diffuse reflection at the boundary for the Boltzmann equation and related equations. J. Stat.Phys.,1994, 77: 1051-1077.
[81] Desvillettes L. Convergence to equilibrium in large time for Boltzmann and B.G.K.equation. Arch. Rational Mech. Anal., 1990, 110: 73-91.
[82] Hamdache K. Initial-boundary value problems for the Boltzmann equation: global existence of weak solutions. Arch. Rational Mech. Anal., 1992, 119: 309-353.
[83] Lions P L. Compactness in Boltzmann's equation via Fourier integral operators and applications I. J. Math. Kyoto Univ., 1994, 34: 391-427.
[84] Lions P L. Compactness in Boltzmann's equation via Fourier integral operators and applications Ⅱ. J. Math. Kyoto Univ., 1994, 34: 429-461.
[85] Lions P L. Compactness in Boltzmann's equation via Fourier integral operators and applications Ⅲ. J. Math. Kyoto Univ., 1994, 34: 539-584.
[86] Lions P L. Regularity and compactness for Boltzmann collision operators without angular cut-off. C. R. Acad. Sci. Paris, 1998, 326: 37-41.
[87] Perthame B, Pulvirenti M. Weighted L~∞ bounds and uniqueness for the Boltzmann BGK model. Arch. Rational Mech. Anal., 1993, 125: 289-295.
[88] Mischler S. Uniqueness for the BGK-equation in R~N and rate of convergence for a semi-discrete scheme. Differential Integral Equations, 1996, 9: 1119-1138.
[89] Bellouquid A. Global existence and large-times behavior for BGK model for a gas with non constant cross section. Transport Theory Stat. Phys., 2003, 32: 157-184.
[90] Barros-Neto J. "An Introduction to the Theory of Distributions." New York: Marcel Dekker, 1973.
[91] Adams R A. "Sobolev Spaces." New York: Academic Press, 1975.
[92] Zhang X. Regularity and long time behavior of the Boltzmann equation for granular gases. J. Math. Anal. Appl., 2006, 324: 650-662.
[93] Truesdell C, Muncaster R G. "Fundamentals of Maxwell's Kinetic Theory of a Simple Monoatomic Gas." New York: Academic Press, 1980.
[94] Zhang X, Zhang X. Probability approaches to spatially homogeneous Boltzmann equation. Stochastic Anal. Appl., 2007, 25: 1129-1150.
[95] Kallenberg O. "Foundations of Modern Probability." Berlin: Springer, 1997.
[96] Mukherjea A, Pothoven K. "Real and Fucntional Analysis." New York: Plenum Press,1984.
[97] Zhang X, Zhang X. Supports of measure solutions for spatially homogeneous Boltzmann equations. J. Stat. Phys., 2006, 124: 485-495.
[98] Greenberg W, Polewczak J. A global existence theorem for the nonlinear BGK equation.J. Stat. Phys., 1989, 55: 1313-1321.
[99] Polewczak J, Greenberg W. Some remarks about continuity properties of local Maxwellians and an existence theorem for the BGK model of the Boltzmann equation. J. Stat. Phys., 1983, 33: 307-316.
[100] Zhang X, Hu S. Moment propagation of the BGK equation. Math. Appl.,2006, 19:857-862.
[101] Zhang X, Hu S. The Cauchy problem of the BGK equation with collision frequency depending on macroscopic density. submitted.
[102] Zhang X, Hu S. L~p solutions to the Cauchy problem of the BGK equation. J. Math.Phys.,2007, (11)48: 113304.
[103] Perthame B. Mathematical tools for kinetic equations. Bull. Amer. Math. Soc., 2004,41: 205-244.
[104] Horst E. On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation. I General theory. Math.Meth.Appl.Sci.,1981, 3:229-248.
[105] Horst E. On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation. Ⅱ Special cases. Math. Meth. Appl. Sci., 1982, 4: 19-32.
[106] Horst E, Hunze R. Weak solutions of the initial value problem for the unmodified non-linear Vlasov equation. Math. Meth. Appl. Sci., 1984, 6: 262-279.
[107] Illner R, Neunzert H. An existence theorem for the unmodified Vlasov equation. Math.Meth. Appl. Sci.,1979,1: 530-554.
[108] Carrillo J A, Soler J. On the initial value problem for the Vlasov-Poisson-Fokker-Planck system with initial data in L~p spaces. Math. Meth. Appl. Sci., 1995, 18:825-839.
[109] Castella F. Propagation of space moments in the Vlasov-Poisson equation and further results. Ann. Inst. Henri Poincar(?), Analyse non lin(?)aire, 1999, 16:503-533.
[110] Dunford N, Schwartz J T. "Linear Operators, I. General Theory." New York: Inter-science, 1958.
[111] Golse F, Perthame B, Sentis R. Un r(?)sultat pour les (?)quations de transport et application au calcul de la limite de la valeur propre principale d'un op(?)rateur de transport.C. R. Acad. Sci. Paris, 1985, 301: 341-344.
[112] Golse F, Lions P L, Perthame B, Sentis R. Regularity of the moments of the solution of a transport equation. J. Funct. Anal., 1988, 76: 110-125.
[113] Diperna R J, Lions P L. Global weak solutions of Vlasov-Maxwell system. C. R. Acad. Sci. Paris, 1988, 307: 655-658.
[114] Diperna R J, Lions P L. Global weak solutions of Vlasov-Maxwell system. Comm.Pure Appl. Math., 1989, 42: 729-757.
[115] Bardos C, Golse F, Perthame B, Sentis R. The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximatiom. J. Funct. Anal., 1988,77: 434-460.
[116] Yosida K. "Functional Analysis." Sixth edition. New York: Springer, 1980.
[117] Zhang X, Hu S, Zhang C. On the spatially homogeneous ES model of the Boltzmann equation. Math. Appl., 2006, 19: 426-432.
[118] Lu X. A result on uniqueness of mild solutions of Boltzmann equation. Transport Theory Stat. Phys., 1997, 26: 209-220.
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