Gradient Decay in the Boltzmann theory of Non-isothermal boundary

We consider the Boltzmann equation in convex domain with non-isothermal boundary of diffuse reflection. For both unsteady/steady problems, we construct solutions belong to $W_x^{1,p}$ for any $p<3$. We prove that the unsteady solution converges to the steady solution in the same Sobolev space exponentially fast as $t\to \mathrm{\infty }$.