L～1 Stability And Eternal Solutions Of The Boltzmann Equation

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.