The Boltzmann equation in the homogeneous critical regularity framework

We construct a unique global solution to the Cauchy problem of the 3D Boltzmann equation for initial data around the Maxwellian in the spatially critical homogeneous Besov space L²ξ(Ḃ2,11/2̇2,13/2). In addition, under the condition that the low-frequency part of initial perturbation is bounded in L²ξ(Ḃ2,∞^σ₀) with -3/2≤σ₀<1/2, it is shown that the solution converges to its equilibrium in large times with the optimal rate of O(t^-(σ-σ₀)/2) in L²ξ(Ḃ2,1σ) with some σ>σ₀, and the microscopic part decays at an enhanced rate of O(t^-(σ-σ₀)/2-1/2). In contrast to [19], the usual L² estimates are not necessary in our approach, which provides a new understanding of hypocoercivity theory for the Boltzmann equation allowing to construct the Lyapunov functional with different dissipation rates at low and high frequencies. Furthermore, a time-weighted Lyapunov energy argument can be developed to deduce the optimal time-decay estimates.