Existence Of The Non-homogeneous Boltzmann Equation Without Angular Cutoff

The Boltzmann equation is an integro-differential equation. It provides a mathematical model for the statistical evolution of the moderately rarefied gas.In photon transport theory,elsctron flow in semiconductors theory and population model theory fields etc., the Boltzmann equation has theoric and applied value.As early as 1972, L. Arkeryd[13] proved the existence and uniqueness of the global solution for the spatially homogeneous Boltzmann equation under certain conditions with compactness and monotone methodshl. Then many people did much work on the equa-tionf[l,2]. However the perfect result was given by S. Mischer and B. Wennberg recently [15].For the non—homogeneous 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 equtions,this methods was called Kaniel—Shinbrot iteration scheme.Using the same method,Toscani provided a unique global existence proof with initial values close to travelling Maxwellians for hard potential ,soft potential and hard sphere,etc.in[31],Jinbo Wei and Xianweng Zhang showed a unique positive eternal solution for the inhomogeneous Boltzmann equation with cut—off soft potential.In this paper,we recall existence solutions and external solution of the non—homogeneous Boltzmann equation with cut—off assumption. At last, enlightened by the method given by R.Alexandre[39] we discuss that there is a local non—homogeneous solution with sigular cross—section.