Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a Maxwellian gas

Results obtained in recent years in the theory of the nonlinear Boltzmann equation for Maxwellian molecules are reviewed. The general theory of spatially homogeneous relaxation based on Fourier transformation with respect to the velocity is presented. The behavior of the distribution function $f({\mathbf v},t)$ is studied in the limit $|{\mathbf v}|\rightarrow\infty$ (the formation of the MaxwelIian tails) and $t\rightarrow\infty$ (relaxation rate). An analytic transformation relating the nonlinear and linearized equations is constructed. It is shown that the nonlinear equation has a countable set of invariants, families of particular solutions of special form are constructed, and an analogy with equations of Korteweg–de Vries type is noted.