Quantitative pointwise estimates of the cooling process for inelastic Boltzmann equation

TL;DR

This paper provides pointwise upper bounds for solutions to the inelastic Boltzmann equation, showing the cooling process persists infinitely and establishing bounds that depend on initial conditions and restitution coefficient.

Contribution

It introduces new time-dependent pointwise upper bounds for the inelastic Boltzmann equation solutions, including bounds near zero velocity and dependence on the restitution coefficient.

Findings

Solution bounded by $C_{f_0} extless t extgreater^3$ over time

Cooling time is infinite under certain initial conditions

Upper bounds depend on the restitution coefficient and match elastic case when $eta=1$

Abstract

In this paper, we study the homogeneous inelastic Boltzmann equation for hard spheres. We first prove that the solution $f(t,v)$ is bounded pointwise from above by $C_{f_0}\langle t \rangle^3$ and establish that the cooling time is infinite $T_c = +\infty$ under the condition $f_0 \in L^1_2 \cap L^{\infty}_{s}$ for $s > 2$. Away from zero velocity, we further prove that $f(t,v)\leq C_{f_0, |v|} \langle t \rangle$ for $v \neq 0$ at any time $t > 0$. This time-dependent pointwise upper bound is natural in the cooling process, as we expect the density near $v = 0$ to grow rapidly. We also establish an upper bound that depends on the coefficient of normal restitution constant, $\alpha \in (0,1]$. This upper bound becomes constant when $\alpha = 1$, restoring the known upper bound for elastic collisions [8]. Consequently, through these results, we obtain Maxwellian upper bounds on the solutions at each time.