Global Solutions to the Initial Value Problem for the Nonlinear Boltzmann Equation

Abstract

The nonlinear Boltzmann equation in the rarefied gas dynamics is investigated for the gas molecules with the cut-off hard potential in the sense of Grad. The solution to the initial value problem is proved to exist uniquely in the large in time and to have the decay order of $(1+t{)}^{–3/4}$ as $t\to +\infty$ for the small initial data in the space $H_{3,3}\cap L^1(x;L^2(v))$. The decay order is improved to $(1+t{)}^{–5/4}$ by the additional assumptions on the initial data.