J. Math. Fluid Mech. 20 (2018), 697–719
2018 Springer International Publishing AG,
part of Springer Nature
Journal of Mathematical
Derivation of the Navier–Stokes–Poisson System with Radiation for an Accretion Disk
S´arka Neˇcasov´a, Milan Pokorn´y and M. Angeles Rodr´ıguez–Bellido
Communicated by H. Beirao da Veiga
Abstract. We study the 3-D compressible barotropic radiation ﬂuid dynamics system describing the motion of the compress-
ible rotating viscous ﬂuid with gravitation and radiation conﬁned to a straight layer Ω
= ω×(0,), where ω is a 2-D domain.
We show that weak solutions in the 3-D domain converge to the strong solution of—the rotating 2-D Navier–Stokes–Poisson
system with radiation in ω as → 0 for all times less than the maximal life time of the strong solution of the 2-D system
when the Froude number is small (Fr = O(
)),—the rotating pure 2-D Navier–Stokes system with radiation in ω as → 0
when Fr = O(1).
Keywords. Navier–Stokes–Poisson system, Radiation, Rotation, Froude number, Accretion disk, Weak solution, Thin domain,
Our aim in this work is the rigorous derivation of the equations describing objects called “accretion disks”
which are quasi planar structures observed in various places in the universe.
From a naive point of view, if a massive object attracts matter distributed around it through the
Newtonian gravitation in presence of a high angular momentum, this matter is not accreted isotropically
around the central object but forms a thin disk around it. As the three main ingredients claimed by astro-
physicists for explaining the existence of such objects are gravitation, angular momentum and viscosity
(see [23,24,26] for detailed presentations), a reasonable framework for their study seems to be a viscous
self-gravitating rotating ﬂuid system of equations.
These disks are indeed three-dimensional but their size in the “third” dimension is usually very small,
therefore they are often modeled as two-dimensional structures. Our goal in this paper is to derive
rigorously the ﬂuid equations of the disk from the equations set in a “thin” cylinder of thickness by
passing to the limit → 0
and applying recent techniques of dimensional reduction introduced and
applied in various situations by Bella, Feireisl, Maltese, Novotn´y and Vod´ak (see [2,19,29,30]).
The mathematical model which we consider is the compressible barotropic Navier–Stokes–Poisson
system with radiation ([9–11]) describing the motion of a viscous radiating ﬂuid conﬁned in a bounded
straight layer Ω
= ω × (0,), where ω ⊂ R
has smooth boundary. Moreover, as we suppose a global
rotation of the system, some new terms appear due to the change of frame.
Concerning gravitation a modelization diﬃculty appears as we consider the restriction to Ω
solution of the Poisson equation in R
: when the thickness of the cylinder tends to zero, a simple argument
shows that the gravitational potential given by the Poisson equation in the whole space goes to zero. So if
we want to recover the presence of gravitation at the limit, and then keep track of the physical situation,
we will have to impose some scaling conditions. In fact as the limit problem will not depend on x
ﬂow is stratiﬁed and we expect that the scaling involves naturally the Froude number; see also .
More precisely, the system of equations giving the evolution of the mass density = (t, x) and the
velocity ﬁeld u = u(t, x)=(u
), as functions of the time t ∈ (0,T) and the spatial coordinate
) ∈ Ω
, reads as follows: