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