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The Kompaneets equation governs dynamics of the photon energy spectrum in certain high temperature (or low density) plasmas. We prove several results concerning the long time convergence of solutions to Bose--Einstein equilibria and the failure of photon conservation. In particular, we show the total photon number can decrease with time via an outflux of photons at the zero-energy boundary. The ensuing accumulation of photons at zero energy is analogous to Bose-Einstein condensation. We provide two conditions that guarantee that photon loss occurs, and show that once loss is initiated then it persists forever. We prove that as t → ∞, solutions necessarily converge to equilibrium and we characterize the limit in terms of the total photon loss. Additionally, we provide a few results concerning the behavior of the solution near the zero-energy boundary, an Oleinik inequality, a comparison principle, and show that the solution operator is a contraction in L¹. None of these results impose a boundary condition at the zero-energy boundary. [ABSTRACT FROM AUTHOR]

The Kompaneets equation describes a field of photons exchanging energy by Compton scattering with the free electrons of a homogeneous, isotropic, nonrelativistic, thermal plasma. This paper strives to advance our understanding of how this equation captures the phenomenon of Bose--Einstein condensation through the study of a model equation. For this model we prove existence and uniqueness theorems for global weak solutions. In some cases a Bose--Einstein condensate will form in finite time, and we show that it will continue to gain p hotons forever afterward. Moreover we show that every solution approaches a stationary solution for large time. Key tools include a universal super solution, a one-sided Oleinik type inequality, and an L¹ contraction. [ABSTRACT FROM AUTHOR]

We establish a low Mach number limit for classical solutions over the whole space of a compressible fluid dynamic system that includes dispersive corrections to the Navier-Stokes equations. The limiting system is similar to a ghost effect system [Y. Sone, Kinetic Theory and Fluid Dynamics, Model. Simul. Sci. Eng. Technol., Birkhäuser, Boston, 2002]. Our analysis builds upon the framework developed by Métivier and Schochet [Arch. Ration. Mech. Anal., 158 (2001), pp. 61-90] and Alazard [Arch. Ration. Mech. Anal., 180 (2006), pp. 1-73] for nondispersive systems. The strategy involves establishing a priori estimates for the slow motion as well as a priori estimates for the fast motion. The desired convergence is obtained by establishing the local decay of the energy of the fast motion. [ABSTRACT FROM AUTHOR]

A common method for constructing a function from a finite set of moments is to solve a constrained minimization problem. The idea is to find, among all functions with the given moments, that function which minimizes a physically motivated, strictly convex functional. In the kinetic theory of gases, this functional is the kinetic entropy; the given moments are macroscopic densities; and the solution to the constrained minimization problem is used to formally derive a closed system of partial differential equations which describe how the macroscopic densities evolve in time. Moment equations are useful because they simplify the kinetic, phase-space description of a gas, and with entropy-based closures, they retain many of the fundamental properties of kinetic transport. Unfortunately, in many situations, macroscopic densities can take on values for which the constrained minimization problem does not have a solution. Essentially, this is because the moments are not continuous functionals with respect to the L1 topology. In this paper, we give a geometric description of these so-called degenerate densities in the most general possible setting. Our key tool is the complementary slackness condition that is derived from a dual formulation of a minimization problem with relaxed constraints. We show that the set of degenerate densities is a union of convex cones and, under reasonable assumptions, that this set is small in both a topological and a measure-theoretic sense. This result is important for further assessment and implementation of entropy-based moment closures. [ABSTRACT FROM AUTHOR]

In highly diffusive regimes, the transfer equation with anisotropic boundary conditions has an asymptotic behavior as the mean free path ∈ tends to zero that is governed by a diffusion equation and boundary conditions obtained through a matched asymptotic boundary layer analysis. A numerical scheme for solving this problem has an ∈-1 contribution to the truncation error that generally gives rise to a nonuniform consistency with the transfer equation for small ∈, thus degrad- ing its performance in diffusive regimes. In this paper we show that whenever the discrete-ordinate method has the correct diffusion limit, both in the interior and at the boundaries, its solutions converge to the solution of the transport equation uniformly in ∈. Our proof of the convergence is based on an asymptotic diffusion expansion and requires error estimates on a matched boundary layer approximation to the solution of the discrete-ordinate method. [ABSTRACT FROM AUTHOR]

The moment closure method of Levermore applied to the Boltzmann equation for rarefied gas dynamics leads to a hierarchy of symmetric hyperbolic systems of partial differential equations. The Euler system is the first member of this hierarchy of closures. In this paper we investigate the next member, the 10 moment Gaussian closure. We first reduce the collision term to an integral which may be explicitly evaluated for the special case of Maxwell molecular interaction. The resulting collision term for this case is shown to be equivalent to the term obtained by replacing the Boltzmann collision operator with the Bhatnagar, Gross, and Krook (BGK) approximation. We then analyze the Gaussian system applied to the canonical flow problem of a stationary planar shock. An analytic shock profile for the Gaussian closure is derived and compared with the numerical solutions of the Boltzmann and Navier-Stokes equations. The results show reasonable agreement for weak shocks and close agreement between the downstream Gaussian and Navier-Stokes profiles. The results also suggest what may be expected from higher moment closure systems. In particular, the presence of discontinuities in the solution are seen not to prohibit the development of significant profiles. [ABSTRACT FROM AUTHOR]

The compressible Euler equations possess a family of generalized entropy densities of the form pf(σ), where ρ is the mass density, σ is the specific entropy, and f is an arbitrary function. Entropy inequalities associated with convex entropy densities characterize physically admissible shocks. For polytropic gases, Harten has determined which pf(σ) are strictly convex. In this paper we extend this determination to gases with an arbitrary equation of state. Moreover, we show that at every state where the sound speed is positive (i.e., where the Euler equations are hyperbolic) there exist pf(σ) that are strictly convex, thereby establishing the converse of the general fact that the existence of a strictly convex entropy density implies hyperbolicity. [ABSTRACT FROM AUTHOR]