Pointwise behavior of the linearized Boltzmann equation on a torus

We study the pointwise behavior of the linearized Boltzmann equation on a torus for nonsmooth initial perturbations. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as part of the long-wave expansion in the spectrum of the Fourier modes for the space variable, and the time decay rate of the fluid-like waves depends on the size of the domain. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which are carried by the transport equations and have exponential time decay rate. The mixture lemma plays an important role in constructing the kinetic-like waves, and we supply a new proof of this lemma without using the explicit solution of the damped transport equations (compare with Liu and Yu's proof [H. W. Kuo, T. P. Liu, and S. E. Noh, Bull. Inst. Math. Acad. Sin. (N.S.), 5 (2010), pp. 1-10; T.-P. Liu and S.-H. Yu, Comm. Pure Appl. Math., 57 (2004), pp. 1543-1608]).