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

### Takaaki Nishida

Kyoto University, Japan### Kazuo Imai

Kyoto University, Japan

## 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* → + ∞ for the small initial data in the space _H_3,3 ∩ _L_1(*x*; _L_2(*v*)). The decay order is improved to (1 + *t*)–5/4 by the additional assumptions on the initial data.