Vanishing diffusion limit and boundary layers for a nonlinear hyperbolic system with damping and diffusion

We consider an initial and boundary value problem for a nonlinear hyperbolic system with damping and diffusion. This system was derived from the Rayleigh–Benard equation. Based on a new observation of the structure of the system, two identities are found; then, the following results are proved by using the energy method. First, the well-posedness of the global large solution is established; then, the limit with a boundary layer as some diffusion coefficient tending to zero is justified. In addition, the $L^2$ convergence rate in terms of the diffusion coefficient is obtained together with the estimation of the thickness of the boundary layer.