On Smooth Solutions to the Thermostated Boltzmann Equation with Deformation

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