Quantitative pointwise estimates of the cooling process for inelastic Boltzmann equation

2024/06/21 by An, Gayoung, Jang, Jin Woo, Lee, Donghyun
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2406.15077

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_f0⟨ t ⟩³ and establish that the cooling time is infinite T_c = +∞ under the condition f₀ ∈ L¹₂ ∩ L^∞_s for s > 2. Away from zero velocity, we further prove that f(t,v)≤ C_{f0, |v|} ⟨ t ⟩ for v ≠ 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, α∈ (0,1]. This upper bound becomes constant when α= 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.

Related

• Exponential trend to equilibrium for the inelastic Boltzmann equationdriven by a particle bath
• Inelastic Boltzmann Equation Driven by a Particle Thermal Bath
• Fluid dynamic limit of Boltzmann equation for granular hard--spheres in a nearly elastic regime
• Propagation of L¹ and L^∞ Maxwellian weighted bounds for derivatives of solutions to the homogeneous elastic Boltzmann Equation
• Long time validity of the linearized Boltzmann equation for hard spheres: a proof without billiard theory