On Smooth Solutions to the Thermostated Boltzmann Equation with Deformation

Abstract

This paper concerns a kinetic model of the thermostated Boltzmann equation with a linear deformation force described by a constant matrix. The collision kernel under consideration includes both the Maxwell molecule and general hard potentials with angular cutoff. We construct the smooth steady solutions via a perturbation approach when the deformation strength is sufficiently small. The steady solution is a spatially homogeneous non Maxwellian state and may have the polynomial tail at large velocities. Moreover, we also establish the long time asymptotics toward steady states for the Cauchy problem on the corresponding spatially inhomogeneous equation in torus, which in turn gives the non-negativity of steady solutions.

Time-Velocity Decay of Solutions to the Non-Cutoff Boltzmann Equation in the Whole Space

“…In fact, one of two exponential weights can be reduced to be only polynomial, cf. [19][20][21]. Furthermore, the velocity-pointwise estimates on the Boltzmann collision term weighted by the polynomial velocity functions were first considered by Arkeryd et al [6,Proposition 3.1] and the technique has led to many applications [26,27] for the fluid dynamic limit problem on the Boltzmann equation.…”

“…Motivated by [9] as well as [19], the second and third authors of this paper together with Liu [18] first proved the well-posedness of polynomial tail solutions for the Boltzmann equation with cut-off assumption in the whole space. We should also emphasize that the enlarged functional spaces are useful when we consider the kinetic shear flow problem such that one can control the polynomial growth caused by the shear force, see [19][20][21] as mentioned before as well as [23] and many references therein.…”