Quantitative Pointwise Estimates of the Cooling Process for Inelastic Boltzmann Equation

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 \( f(t,v)\le C_{f_0, |v|} \langle t \rangle \) for \(v \ne 0\) 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.