Optimal decay of the Boltzmann equation

Abstract

The Boltzmann equation is a typical example of partially dissipative equations, where the linearized collision operator is positive definite with respect to the microscopic part and the dissipation of the hydrodynamic part is discovered from the coupling structure between the transport operator and the linearized collision operator. Guo and Wang (Comm. Partial Differential Equations, 37, 2012) developed a general energy method for proving the optimal time decay rates of the solution to such type of equations in the whole space; however, the decay rate of the highest order spatial derivatives of the solution is not optimal. In this paper, by incorporating the high-low frequency decomposition in the energy estimates, both linearly and nonlinearly, we prove the optimal decay rates of any high order spatial derivatives of the low frequency part of the solution to the Boltzmann equation and the almost exponential decay rate of the high frequency part, which imply in particular the optimal decay rate of the highest order spatial derivatives of the solution. Moreover, the velocity-weighted assumption of the initial data required in Guo and Wang (Comm. Partial Differential Equations, 37, 2012) is removed by capturing the time-weighted dissipation estimates via the time-weighted energy method. The method can be applied to the compressible Navier–Stokes equations and many partially dissipative equations in kinetic theory and fluid dynamics.