Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials

Details Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials

2006-06-14

arxiv.org/abs/math/0606328v1

Mathematics

Analysis of PDEs

12 pages

Scientific paper

We consider the spatially homogeneous Boltzmann equation for regularized soft potentials and Grad's angular cutoff. We prove that uniform (in time) bounds in $L^1 ((1 + |v|^s)dv)$ and $H^k$ norms, $s, k \ge 0$ hold for its solution. The proof is based on the mixture of estimates of polynomial growth in time of those norms together with the quantitative results of relaxation to equilibrium in $L^1$ obtained by the so-called "entropy-entropy production" method in the context of dissipative systems with slowly growing a priori bounds (see reference [14]).

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