Quantitative pointwise estimates of the cooling process for inelastic Boltzmann equation

In this paper, we study the homogeneous inelastic Boltzmann equation for hard spheres. We first prove that the solution $f(t,v)$ is pointwisely bounded from above by Cf₀ t ³ and establish that the cooling time is infinite Tc = +∞ under the condition f₀ ∈ L¹₂ ∩ L∞ₛ for $s > 2 $. Away from the zero velocity, we further prove that f(t,v)≤ Cf₀, ε t for |v| ≥ ε at any time $t > 0 $ and ε>0. This time-growing pointwise upper-bound is natural in the cooling process, as we expect the density near $v = 0 $ to grow rapidly. As a consequence, via these results, we obtain Maxwellian upper-bounds of solutions for each time. Our upper-bounds hold for any constant normal restitution 0 < α ≤ 1 and are uniform in α.