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 L²_ (Ḃ₂, ₁^1/2̇₂, ₁^3/2). In addition, under the condition that the low-frequency part of initial perturbation is bounded in L²_ (Ḃ₂, ^₀) with -3/2₀₀, and the microscopic part decays at an enhanced rate of O (t^- (-₀) /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.