On nonnegativity of solutions of the Boltzmann equation

Nonnegativity of solutions of the Boltzmann equation with nonnegative initi al data is the most necessary physical property. This paper considers colli sion kernels with angular cutoff. We prove that for collision kernels of no n-soft potentials, if f is a mild solution of the Boltzmann equation on R-3 x R-3 x [0, t(infinity)) and satisfies the following condition ess sup (x is an element of R3)integral (R3) \f(x, v, t)\ (1 + \v \ (2))(5/ 2) dv is an element of L-loc(2)[0, t(infinity)) (which includes the case of x-priodic solutions), then f must be nonnegativ e on R-3 x R-3 x [0, t(infinity)), where 0 < t(infinity) less than or equal to infinity. In particular, for Maxwell molecules model, if f is an x-preo dic solution and satisfies ess sup (x is an element of R3)integral (R3) \f(x, v, t)\ dv is an element of L-loc(2)[0, t(infinity)), then f is nonnegative on R-3 x R-3 x [0, t(infinity)). Under similar condit ions, the same results about nonnegativity of solutions hold also true for collision kernels of soft potentials. As an application, we show that these conditions are easily satisfied by the solutions f obtained before which a re close to the global Maxwellian M (v) in the form f(x, v, t) = (v) + [M(v )](1/2) F(x, v, t) (i.e., F are small functions), and therefore f are actua lly nonnegative.