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 $W^{1,p}_x$ for any $p<3$. We prove that the unsteady solution converges to the steady solution in the same Sobolev space exponentially fast as $t \rightarrow \infty$.