Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation

This paper studies Lp-estimates for solutions of the nonlinear, spatially homogeneous Boltzmann equation. The molecular forces considered include inverse kth-power forces with k > 5 and angular cut-off.The main conclusions are the following. Let f be the unique solution of the Boltzmann equation with f(v,t)(1 + ¦v2¦)(s1+ β/p)/2 ∈ L1, when the initial value f0 satisfies f0(v) ≧ 0, f0(v) (1 + ¦v¦2)(s1+ β/p)/2 ∈ L1, for some s1 ≧ 2 + β/p′, and f0(v) (1 + ¦v¦2)s/2 ∈ Lp. If s ≧ 2/p and 1 < p < ∞, then f(v, t)(1 + ¦v¦2)(s ∧ s1)/2 ∈ Lp, t > 0. If s >2 and 3/(1+ β) < p < ∞, thenf(v,t) (1 + ¦v¦2)(s∧(s1+ 3/p′))/2 ∈ Lp, t > 0. If s >2 + 2C0/C1 and 3/(l + β) < p < ∞, then f(v,t)(1 + ¦v¦2)s/2 ∈ Lp, t > 0. Here 1/p + 1/p′ = 1, x ∧ y = min (x, y), and C0, C1, 0 < β ≦ 1, are positive constants related to the molecular forces under consideration; β = (k − 5)/ (k − 1) for kth-power forces.Some weaker conclusions follow when 1 < p ≦ 3/ (1 + β).In the proofs some previously known L∞-estimates are extended. The results for Lp, 1 < p < ∞, are based on these L∞-estimates coupled with nonlinear interpolation.