This paper is devoted to studying the initial-boundary value problem for the radiative full Euler equations, which are a fundamental system in the radiative hydrodynamics with many practical applications in astrophysical and nuclear phenomena, with the slip boundary condition on an impermeable wall. Different from our recent paper named “Asymptotic stability of rarefaction wave with slip boundary condition for radiative Euler flow”, in this paper we study the initial-boundary value problem with the Neumann boundary condition instead of the Dirichlet boundary on the temperature. Based on the Neumann boundary condition on the temperature, we obtain that the pressure also satisfies the Neumann boundary condition. This observation allows us to establish the local existence and a priori estimates more easily than the case of the Dirichlet boundary condition which is studied in the mentioned paper. Since for the impermeable problem, there are quite a few results available for the Navier-Stokes equations and the radiative Euler equations, it will contribute a lot to our systematical study on the asymptotic behaviors of the rarefaction wave with the radiative effect and different boundary conditions such as the inflow/outflow problem and the impermeable boundary problem in our series papers.

We study the hyperbolic version of the Prandtl system derived from the hyperbolic Navier-Stokes system with no-slip boundary condition. Compared to the classical Prandtl system, the quasi-linear terms in the hyperbolic Prandtl equation leads to an additional instability mechanism. To overcome the loss of derivatives in all directions in the quasi-linear term, we introduce a new auxiliary function for the well-posedness of the system in an anisotropic Gevrey space which is Gevrey class 3/2 in the tangential variable and is analytic in the normal variable.

We consider the non-cutoff Vlasov-Poisson-Boltzmann (VPB) system of two species with soft potential in the whole space $\mathbb{R}^3$ when an initial data is near Maxwellian. Continuing the work Deng [Comm. Math. Phys. 387 (2021)] for hard potential case, we prove the global regularity of the Cauchy problem to VPB system for the case of soft potential in the whole space for the whole range $0<s<1.$ This completes the smoothing effect of the Vlasov-Poisson-Boltzmann system, which shows that any classical solutions are smooth with respect to $(t,x,v)$ for any positive time $t>0.$ The proof is based on the time-weighted energy method building upon the pseudo-differential calculus.