Gradient Decay in the Boltzmann Theory of Non-isothermal Boundary

We consider the Boltzmann equation in a convex domain with a non-isothermal boundary of diffuse reflection. For both unsteady/steady problems, we construct solutions belonging 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 quickly as \(t \rightarrow \infty \).