Sharp Anisotropic Estimates for the Boltzmann Collision Operator and Its Entropy Production

Abstract. This article provides sharp constructive upper and lower bound estimates for the Boltzmann collision operator with the full range of physical non cut-off collision kernels (γ > −n and s ∈ (0, 1)) in the trilinear L(R) energy 〈Q(g, f), f〉. These new estimates prove that, for a very general class of g(v), the global diffusive behavior (on f) in the energy space is that of the geometric fractional derivative semi-norm identified in the linearized context in our earlier works [15, 16]. We further prove new global entropy production estimates with the same anisotropic semi-norm. This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non cut-off Boltzmann collision operator in the energy space L(R).