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 concern the local well-posedness of the moving boundary problem of the corresponding Navier-Stokes-Fourier-Poisson system and construct a prior estimates of strong and classical solutions. Our results explore the vacuum behaviour of density and temperature near the free boundary for the equilibrium and capture such degeneracy in the evolutionary problem.
It shall be emphasised that the physical vacuum causes big challenges in the study of the evolutionary problem of . That is to study the following Euler-Poisson system Indeed, the physical vacuum boundary implies that the sound speed @math is only @math -H "older continuous rather than Lipschitz continuous across the boundary. As pointed out in @cite_30 , this singularity makes the standard hyperbolic method fail. It is only recently that the local well-posedness theory is studied by Coutand, Lindblad and Shkoller @cite_13 @cite_32 @cite_23 , Jang and Masmoudi @cite_19 @cite_9 , Gu and Lei @cite_3 @cite_15 , Luo, Xin and Zeng @cite_5 in the setting of one spatial dimension, three spatial dimension and spherical symmetry with or without self-gravitation. See @cite_24 for a viscous flow. We refer to @cite_30 @cite_10 @cite_26 @cite_8 for other discussions on the physical vacuum problem and the references therein for other vacuum problems.
@cite_30 @cite_26 @cite_8 @cite_10 @cite_9 @cite_32 @cite_3 @cite_24 @cite_19 @cite_23 @cite_5 @cite_15 @cite_13
2011819936 2163734752 2064683888 2130710172
For the physical vacuum free boundary problem with the sound speed being C1 2-Holder continuous near vacuum boundaries of the one-dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the porous media equation with the same total mass when the initial datum is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in the supremum norm, and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher-order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher-order nonlinear energy estimates, and elliptic estimates.© 2016 Wiley Periodicals, Inc. This paper concerns the well-posedness theory of the motion of a physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in (, Commun Math Phys 296:559–587, 2010; Coutand and Shkoller, Arch Ration Mech Anal 206:515–616, 2012; Jang and Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, 2008) by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary. In this paper we establish the global existence of smooth solutions to vacuum free boundary problems of the one-dimensional compressible isentropic Navier–Stokes equations for which the smoothness extends all the way to the boundaries. The results obtained in this work include the physical vacuum for which the sound speed is C1 2-Holder continuous near the vacuum boundaries when 1 < γ < 3. The novelty of this result is its global-in-time regularity which is in contrast to the previous main results of global weak solutions in the literature. Moreover, in previous studies of the one-dimensional free boundary problems of compressible Navier–Stokes equations, the Lagrangian mass coordinates method has often been used, but in the present work the particle path (flow trajectory) method is adopted, which has the advantage that the particle paths and, in particular, the free boundaries can be traced. This paper considers a trapped characteristic initial value problem for the spherically symmetric Einstein-Maxwell-scalar field equations. For an open set of initial data whose closure contains in particular Reissner-Nordstrdata, the future boundary of the maximal domain of development is found to be a light-like surface along which the curvature blows up, and yet the metric can be continuously extended beyond it. This result is related to the strong cosmic censorship conjecture of Roger Penrose. The principle of determinism in classical physics is expressed mathemat- ically by the uniqueness of solutions to the initial value problem for certain equations of evolution. Indeed, in the context of the Einstein equations of general relativity, where the unknown is the very structure of space and time, uniqueness is equivalent on a fundamental level to the validity of this principle. The question of uniqueness may thus be termed the issue of the predictability of the equation. The present paper explores the issue of predictability in general relativity. Since the work of Leray, it has been known that for the Einstein equations, contrary to common experience, uniqueness for the Cauchy problem in the large does not generally hold even within the class of smooth solutions. In other words, uniqueness may fail without any loss in regularity; such failure is thus a global phenomenon. The central question is whether this violation of predictability may occur in solutions representing actual physical processes. Physical phenomena and concepts related to the general theory of relativity, namely gravitational collapse, black holes, angular momentum, etc., must cer- tainly come into play in the study of this problem. Unfortunately, the math- ematical analysis of this exciting problem is very difficult, at present beyond reach for the vacuum Einstein equations in the physical dimension. Conse-