Non-Isothermal Plane Couette Flow for the Boltzmann Equation in Three Dimensions
We study the Boltzmann equation for a non-isothermal plane Couette flow in the three-dimensional channel $\mathbb T_x\times(-1,1)_y\times\mathbb T_z$ for Maxwell molecules. For sufficiently small shear rate and wall temperature perturbations, we construct a unique steady solution near the global Maxwellian. For the corresponding unsteady boundary value problem, we establish the global existence and uniqueness of solutions arising from small perturbations of the steady state, together with their exponential convergence to the steady state as $t\to\infty$. A major difficulty in both the steady and unsteady problems is to close the outgoing mass fluxes generated by diffuse reflection. To overcome this difficulty, we develop a new boundary flux transmission framework which, in the slab geometry, yields steady and time-dependent wall flux estimates through the boundary-to-boundary transmission between the two walls. We further extend this framework to damped transport equations with diffuse reflection in general bounded $C^1$ domains, even when the damping coefficient degenerates at large velocities.