In this work, we study the stability of the expanding configurations for radiation gaseous stars. Such expanding configurations exist for the isentropic model and the thermodynamic model. In particular, we focus on the effect of the monatomic gas viscosity tensor on the expanding configurations. With respect to small perturbation, it is shown the self-similarly expanding homogeneous solutions of the isentropic model is unstable, while the linearly expanding homogeneous solutions of the isentropic model and the thermodynamic model are stable. This is an extensive study of the result in arXiv: 1605.08083 by Hadzic and Jang.

This work is devoted to establishing the local-in-time well-posedness of strong solutions to the three-dimensional compressible primitive equations of atmospheric dynamics. It is shown that strong solutions exist, unique, and depend continuously on the initial data, for a short time in two cases: with gravity but without vacuum, and with vacuum but without gravity. We also introduce the free boundary problem for the compressible primitive equations.

We introduce a model concerning radiational gaseous stars and establish the existence theory of stationary solutions to the free boundary problem of hydrostatic equations describing the radiative equilibrium. We also establish in the spherically symmetric case the local well-posedness of the moving boundary problem of the corresponding Navier--Stokes--Fourier--Poisson system and construct a priori estimates of strong and classical solutions. Our results explore the vacuum behavior of density and temperature near the free boundary for the equilibrium and capture such degeneracy in the evolutionary problem.

We give a partial converse to [8, Theorem 1.3] (as a resolution of [2, Problem 8.4] for the quasiconformal Q-composition) for Q0<α<2−1(Rn≥2), and yet demonstrate that if f:R2→R2 is a homeomorphism then the boundedness of u↦u∘f on Q2−1<α<1(R2)⊂BMO(R2) yields the quasiconformality of f.

For the free surface problem of the highly subsonic heat-conducting inviscid flow in 2-D and 3-D, a priori estimates for geometric quantities of free surfaces, such as the second fundamental form and the injectivity radius of the normal exponential map, and the Sobolev norms of fluid variables are proved by investigating the coupling of the boundary geometry and the interior solutions. An interesting feature for the free surface problem studied in this paper is the loss of one more derivative than the problem of incompressible Euler equations for which a geometric approach was introduced by Christodoulou and Lindblad in [11]. Due to the loss of one more derivative and loss of symmetry of equations, the geometric approach in [11] needs to be substantially developed by exploring the interaction of large variation of temperature, heat-conduction, non-zero divergence of the fluid velocity and the evolution of free surfaces.