【作者】 魏金波；

【导师】 张显文；

【作者基本信息】 华中科技大学 ， 应用数学， 2005， 硕士

【摘要】 气体动力学是统计力学的重要组成部分,而统计力学的基本出发点就是对气体的微观状态以及人们对其微观状态的观测进行统计平均,并用统计的方法处理问题。它认为在任意给定的时刻一个气体分子的运动状态是不确定的,人们只能给出该分子在某一状态附近出现的概率。Boltzmann方程就是概率密度所满足的一类非线性微分积分方程,它刻画了相对稀疏气体的统计演化规律。 早在1972年,L.Arkeryd就利用紧性和单调性方法在一定条件下证明了空间齐次Boltzmann方程整体解的存在性和唯一性。随后,有许多人对该方程做了大量的研究,然而比较完善的结果是由S.Mischer和B.Wennberg近期给出的。而对空间非齐次的Boltzmann方程,在1978年,Kanie和Shinbrot通过合适的线性动力学方程,利用夹逼法得到了方程解的唯一性。随后,这种方法(称之为Kaniel-Shinbrot迭代方法)得到了广泛的应用。第一个整体存在性的证明是由Illner和Shinbrot给出的,他们所用的方法就是Kaniel-Shinbrot迭代方法,Toscani等利用它分别证明了t>0,初值接近局部Maxwell分布时,各种情况下Boltzmann方程整体解的存在唯一性。1988年,R.J.DiPerna和P.L.Lions考虑了具有Fokker-Planck型算子扰动时的空间非齐次Boltzmann方程,证明了该方程的一种弱解(renormalized solution)的整体存在性。 本文研究的是空间非齐次Boltzmann方程的L~1稳定性和永久型解的存在唯一性。事实上,C.Villani猜测:除Maxwell分布的永久型解外,Boltzmann方程没有其他的永久型解。这个问题被Bobylev和Cercignani在空间齐次的情况下讨论了,且证明是正确的,但我们利用一种新的迭代方法得到了空间非齐次的Boltzmann方程永久型解的存在唯一性定理,这说明C.Villani的猜测对于空间非齐次的情况是不成立的。另外,对于空间非齐次的Boltzmann方程经典解的L~1稳定性,Seung-Yeal Ha虽已做了一定的工作,但他只是针对硬球模型,本文利用Toscani等人所做的估计对硬位势,软位势等各种情况都做了统一的讨论。更多还原

【Abstract】 The kinetic theory of gases is an important part of statistical dynamics. However the basic springboard of statistical dynamics is the statistical average of microscopic state of gases and observation of microscopic state and to settle problems with statistical methods. It considers that the locomotion state of gases at arbitrary moment is uncertain. People only can give probability that the molecule occurs around certain state. The Boltzmann equation is an integro-differential equation that the probability density satisfies. It provides a mathematical model for the statistical evolution of the moderately rarefied gas.As early as 1972, L. Arkeryd proved the existence and uniqueness of the global solution for the spatially homogeneous Boltzmann equation under certain conditions with compactness and monotone methods[1]. Then many people did much work on the equation[2,3]. However, the perfect result was given by S. Mischer and B. Wennberg recently[4]. For the space-inhomogeneous boltzmann equation,in 1978,Kaniel and Shinbrot approximated the solution of the nonlinear Boltzmann equation from above and below by unique solutions of suitable kinetic equations[20]. The first globles exstence proof was given by Illner ang Shinbrot[19]. Toscni provided a global existence proof for initial values whose decay is polynomial[47,50,51]. All these results were essentially obtained by the application of fixed point theorems combined with the Kaniel and Shinbrot iteration scheme. In 1988, R. J. DiPerna and P. L. Lions considered the spatially non-homogeneous Boltzmann equation perturbed by the Fokker-Planck operator and proved the global existence of weak solution(renormalized solution)[5,14,15].In this paper, we study the L~1 stability of the classical solutions and the existense of the eternal solution for the Boltzmann equation with a small initial data. In fact, C. Villani conjectured[54,55] that except for Maxwell’s distributions, the nonlinear Boltzmann equation has no other type of positive eternal solutions with finite kinetic energy. This problem was first discussed by Bobylev and Cercignani in the spatially homogeneous case.[48,49] But it is shown in this paper that the Cauchy problem of the Boltzmann equation, with a cut-off soft potential and an initial datum close to a travelling Maxwellian, has a unique positive eternal solution. we use a new iterative scheme, which is a variant of the Kaniel-Shinbrot iterative method. This result gives a negative answer to the conjecture of Villani in the spatially inhomogeneous case.On the other hand, By means of theestimates given by Toscani et al , both hard potentials and soft potentials are discussed and hence the results obtained in’56’ about hard sphere model are included.更多还原