The Boltzmann equation and nonlinear microlocal analysis

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W e considered the Cauchy problem for the Boltzmann equation without angular cutoff approximation. To this Cauchy problem, we showed the structure of solutions, such as, the existence of solutions, the smoothing effect of solutions, the convergence of the equilibrium state, the uniqueness and the non-negativityof solutions, under the physically reasonable assumptions of the collision cross sections. The key ingredient of this study was to give precise lower and upper estimates forthe Boltzmann collision integral operator which contains the singularity with respect to the angular variable between collision particles, by means of the microlocal analysis developed in the theory of linear partial differential equations.