LOW MACH NUMBER LIMIT FOR A MODEL OF ACCRETION DISK

. We study an hydrodynamical model describing the motion of thick astrophysical disks relying on compressible Navier-Stokes-Fourier-Poisson sys- tem. We also suppose that the medium is electrically charged and we include energy exchanges through radiative transfer. Supposing that the system is ro- tating, we study the singular limit of the system when the Mach number, the Alfven number and Froude number go to zero and we prove convergence to a 3D incompressible MHD system with radiation with two stationary linear transport equations for transport of radiation intensity.


1.
Introduction. Our motivation in this work is the study of the equations describing objects called "accretion disk" 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 Newtonian gravitation in presence of angular momentum, the matter is not accreted isotropically around the central object but forms a disk around it. As the three main ingredients claimed by astrophysicists for explaining the existence of such objects are: gravitation, angular momentum and viscosity (see [24] [28] [29] for detailed presentations), a reasonable framework for their study seems to be a viscous self-gravitating rotating fluid.
In previous works we derived thin disks models [10] [11] corresponding to limit domains Ω ε = ω × (0, ε) for ε → 0. In the present one we consider a thick model where is no more small and is replaced by 1 in the sequel.
The mathematical model we consider is basically the compressible heat conducting MHD system [6] describing the motion of a viscous charged fluid confined to the thick disk Ω = ω × (0, 1), where ω ∈ R 2 is a 2-D domain, moreover as we suppose a global rotation of the system, some new terms appear due to the change of frame and we also suppose that the fluid exchanges energy with radiation through radiative transfers (see [6] [9]).
More precisely, the non-dimensional system of equations giving the evolution of the mass density = (t, x), the velocity field u = u(t, x), the (divergence free) magnetic field B = B(x, t), and the radiative intensity I = I(x, t, ω,ν) as functions of the time t ∈ (0, T ), the spatial coordinate x = (x 1 , x 2 , x 3 ) ∈ Ω ⊂ R 3 , and (for I) the angular and frequency variables ( ω,ν) ∈ S 2 × R + , reads as follows ∂ t ( e) + div x ( e u) + div x q = S : In the electromagnetic source terms, electric current j and electric field E are interrelated by Ohm's law j = σ( E + u × B), and Ampère's law ζ j = curl x B, where ζ > 0 is the (constant) magnetic permeability.
In (6) Ψ is the gravitational potential and the corresponding source term in (2) is the Newton force ∇Ψ. G is the Newton constant and g is a given function, modeling an external gravitational effect. Supposing that is extended by 0 outside Ω we have Ψ(t, x) = G Ω K(x − y)(η (t, y) + g(y)) dy, where K(x) = 1 |x| , and the parameter η may take the values 0 or 1: for η = 1 self-gravitation is present and for η = 0 gravitation only acts as an external field (some astrophysicists consider self-gravitation of accretion disks as small compared to the external attraction by a given massive central object modeled by g [29]) . We also assume that the system is globally rotating at uniform velocity χ around the vertical direction e 3 and we note χ = χ e 3 . Then Coriolis acceleration term χ × u appears in the system, together with the centrifugal force term ∇ x | χ × x| 2 (see [3]).
In (5) λ = λ(ϑ) > 0 is the magnetic diffusivity of the fluid. Observe that we consider here the simplified model studied in [12] where radiation does not appear in the momentum equation. Only the source S E appears in the energy equation The symbol p = p( , ϑ) denotes the thermodynamic pressure and e = e( , ϑ) is the specific internal energy, interrelated through Maxwell's relation ∂e ∂ = 1 2 p( , ϑ) − ϑ ∂p ∂ϑ .
System (1 -6) is supplemented with the boundary conditions: where n denotes the outer normal vector to ∂Ω.
Let us mention that there are already existing works in this field but not in the case of rotating fluid with radiation. We can mention some of existing works. First one was done by Kukučka [19] when Mach and Alfven number go to zero in the case of bounded domain. In [27] Novotný and his collaborators investigated the problem in the case of strong stratification. See also work of Trivisa et al. [21], work of Wang et al. [15], or works of Jiang et al. [17,18,16].

Remark 1.
• Let us mention that we consider a simplified model without radiative momentum as explained in our reference [12] quoted in the text. In fact one can observe that the radiative momentum is of order c −1 with respect to radiative energy S E , which is very small (as c is the velocity of light) and consequently can be neglected in the model. • The realistic physical domain of our thick disk is indeed a free boundary region. As we consider a well-formed disk we can suppose that the domain is fixed and the no slip boundary condition reflects an heuristic simulation of the exterior vacuum.
The paper is organized as follows.
In Section 2, we list the principal hypotheses imposed on constitutive relations, introduce the concept of weak solution to problem (1 -13), and state the existence result for our model. In Section 3 we compute the formal asymptotics of the problem characterized by an infra relativistic matter velocity C >> 1, a low Mach number Ma << 1 a sound speed small with respect to the velocity of light (accretion disk regime) Ma · C >> 1, a small Froude number Fr << 1 and a small Alfven number Al << 1.
Uniform bounds imposed on weak solutions by the data are derived in Section 4. The convergence Theorem is proved in Section 5. We conclude the paper with an Appendix. In the Appendix A we perform a slightly different scaling than those one of Section 3, while existence of a solution for the target system is briefly given in the Appendix B.

2.
Hypotheses and stability result. We consider the pressure in the form where P : [0, ∞) → [0, ∞) is a given function with the following properties: After Maxwell's equation (7), the specific internal energy e is and the associated specific entropy reads with A new feature of the present paper (see below) will be the explicit introduction of the entropy for the photon gas. The transport coefficients µ, η, κ and λ are continuously differentiable functions of the absolute temperature such that for any ϑ ≥ 0. Moreover we assume that σ a , σ s , B are continuous functions of ν, ϑ such that (24) for all ν ≥ 0, ϑ ≥ 0, where c 1,2,3 are positive constants.

Remark 2.
We consider a hot gas and the θ 4 contribution is the classical Stefan-Boltzmann term. However we wish also to keep track of non equilibrium phenomena described by the radiative transfer equation. This model has also been described in our previous work [6].
Let us recall some definitions introduced in [7].
• In the weak formulation of the Navier-Stokes-Fourier system the equation of continuity (1) is replaced by its (weak) renormalized version [5] represented by the family of integral identities (25) implicitly includes the initial condition (0, ·) = 0 .
• The magnetic equation (5) is replaced by to be satisfied for any vector field ϕ ∈ D([0, T ) × R 3 ).
Here, according the boundary conditions, one has to take Following Theorem 1.4 in [32], B 0 belongs to the closure of all solenoidal functions from D(Ω) with respect to the L 2 −norm.
In particular, using Theorem 6.1 in [13], we conclude • From (2) and (3) we have the energy conservation law Let us rearrange the right hand side. As the gravitational potential Ψ is determined by equation (6) considered on the whole space R 3 , the density being extended to be zero outside Ω we observe that In the same stroke Ω ∇ x | χ × x| 2 · u dx = d dt Ω | χ × x| 2 dx. Denoting now by E R the radiative energy given by Using boundary conditions, we deduce the identity • Finally, dividing (3) by ϑ and using Maxwell's relation (7), we obtain the entropy equation where where the first term ς m := 1 ϑ S : ∇ x u − q·∇xϑ ϑ + λ ζ |curl x B| 2 is the (positive) electromagnetic matter entropy production.
In order to identify the second term in (35), let us recall [1] the formula for the entropy of a photon gas where n = n(I) = c 2 I 2hν 3 is the occupation number. Defining the radiative entropy flux and using the radiative transfer equation, we get the equation Checking the identity log n(B) n(B)+1 = hν kϑ with B = B(ϑ, ν) the Planck's function, and using the definition of S, the right-hand side of (38) rewrites where we used the hypothesis that the transport coefficients σ a,s do not depend on ω. So we obtain finally and equation (34) is replaced in the weak formulation by the inequality where the sign of all the terms in the right hand side may be controlled.
• Since replacing equation (3) by inequality (40) would result in a formally under-determined problem, system (25), (26), (40) must be supplemented with the total energy balance where E R 0 is given by Concerning the transport equation (4), it can be extended to the whole physical space R 3 provided we set σ a (x, ν, ϑ) = 1 Ω σ a (ν, ϑ) and σ s (x, ν, ϑ) = 1 Ω σ s (ν, ϑ) and take the initial distribution I 0 (x, ω, ν) to be zero for x ∈ R 3 \ Ω. Accordingly, for any fixed ω ∈ S 2 , equation (4) can be viewed as a linear transport equation defined in (0, T ) × R 3 , with a right-hand side S. With the above mentioned convention, extending u to be zero outside Ω, we may therefore assume that both and I are defined on the whole physical space R 3 .
3. Formal scaling analysis. In order to identify the appropriate limit regime we perform a general scaling, denoting by L ref , We also assume the compatibility conditions Using these scalings and using carets to symbolize renormalized variables we get Omitting the carets in the following, we get first the scaled equation for I, in the where we used the same notation B for the dimensionless Planck function ] ω d ωdν, the renormalized radiative entropy flux, and taking the first moment of (44) with respect to ω, we get first an equation The continuity equation is now and the momentum equation The balance of internal energy rewrites and we get the balance of matter (fluid) entropy and the balance of radiative entropy The scaled equation for the electromagnetic field is The scaled equation for total energy gives finally the total energy balance In the sequel we analyze a simple asymptotic regime characterized by an infra relativistic matter velocity C >> 1, a low Mach number Ma << 1 a sound speed small with respect to the velocity of light (accretion disk regime) Ma · C >> 1, a small Froude number Fr << 1 and a small Alfven number Al << 1. So, given a small number ε > 0, we assume that the regime under study is defined by and we put Sr = 1, Pe = 1, Re = 1, Ro = 1, P = 1, L = 1, L s = 1, in the previous system. Plugging this scaling into the previous system gives with and finally In order to compute the limit system, we consider now the formal expansions • We first observe from (54) and using the arguments of [14] that we can choose From (53) we derive the incompressibility condition and (52) we get now two stationary linear transport equations for the two moments I 0 and I 1 whereĨ : where µ 0 = µ(ϑ 0 ) and Π is an effective pressure. In the right-hand side, the force term is We compute the first term by using (60) which • The limit magnetic field B 1 solves • At lowest order the energy equation gives Observing that from (60) we have where D := ∂ t + u 0 · ∇ x , and from (60) and we obtain the limit system in (0, together with the Boussinesq relation (60) where the (linear) sources F and G are given by (69) and (70). We finally consider the boundary conditions for (73)-(77) and I 0 (x, ν, ω) = 0 for x ∈ ∂Ω, ω · n ≤ 0 (82) I 1 (x, ν, ω) = 0 for x ∈ ∂Ω, ω · n ≤ 0 (83) for (78) and (79), and the initial conditions For this system we have the following existence result (see the Appendix for a short proof) For any T > 0 the initial-boundary value problem (73) -(84) has at least a weak solution ( U , Θ, B, I 0 , I 1 ) such that 1.
In the following we introduce the convergence result from the primitive system (1)-(13) to the incompressible limit (73)-(84).
Given three numbers ∈ R + , ϑ ∈ R + and E ∈ R + we define O H ess the set of hydrodynamical essential values and O R ess the set of radiative essential values O R ess := E R ∈ R : Let { ε , u ε , ϑ ε , I ε } ε>0 ) a family of solutions of the scaled radiative Navier-Stokes-Fourier system given in Theorem 4.1. We call M ε ess ⊂ (0, T ) × Ω the set  (88) as Observing that the total mass is a constant of motion M = Ω ε dx = |Ω| and using Hardy-Littlewood-Sobolev inequality, we get ε . After (14) and (18) we have also ε e( ε , ϑ ε ) ≥ aϑ 4 ε + 3p∞ 2 5/3 ε , so we have the lower bound for ε small and a c(ε) < 1 and we deduce finally the energy-entropy inequality Now, after Lemma 4.1 in [12] (see [14]) we have the following properties for matter and radiative Helmholtz functions There exist positive constants C j = C j ( , ϑ) for j = 1, · · ·, 8 such that 1.
for all E ∈ O R res , 6.
Clearly, this result provides us with the convergence properties (119-124).
To conclude the proof of Theorem 4.4, let us prove that the limit quantities ( U , Θ, B, I 0 , I 1 ) solve the target system (73)-(79).
As number of terms in the equations of our model are similar to those of the radiative Navier-Stokes-Fourier analyzed in [12] we only focus on the new contributions.

Continuity and momentum equations.
For the continuity equation, one expects that in the low Mach number limit, it reduces to the incompressibility constraint. In fact after Lemma 4.3 we know that dt ≤ C so passing to the limit after possible extraction of a subsequence, we deduce that In the same stroke ε → , weakly in L ∞ (0, T ; L 5/3 (Ω; R 3 )). So we can pass to the limit in the weak continuity equation (86) which gives For the momentum equation one knows that due to possible strong time oscillations of the gradient component of velocity, one has only ε u ε ⊗ u ε → U ⊗ U weakly in L 2 (0, T ; L 30 29 (Ω; R 3 )). However one can show after the analysis in [14] that one can pass to the limit in the convective term and obtain Moreover after the hypotheses on the pressure law, the temperature ϑ ε is bounded in L ∞ ((0, T ); L 4 (Ω)) ∩ L 2 (0, T ; L 6 (Ω)), which implies that S ε → µ(ϑ)(∇ x U + ∇ t x U ) weakly in L q (0, T ; L q (Ω; R 3 )) for a q > 1.

DONATELLA DONATELLI, BERNARD DUCOMET ANDŠÁRKA NEČASOVÁ
Now from (91) Passing to the limit we find the limit equation using the same notation for any time-independent test function ψ ∈ C ∞ c (Ω × S 2 × R + ) which is the weak formulation of the stationary problem with the boundary condition I 1 = 0 on Γ + .
Let us compute the limit of with n ε = n(I ε ) = Iε ν 3 . Applying once more Proposition 1 with G R (I) = n(I) log n(I)−(n(I)+1) log(n(I) + 1) and integrating on S 2 × R + , we find and as n(I)+1 with the radiative momentum F R (I (1) ) = ∞ 0 As we have, from (134) div the limit contribution in the right-hand side becomes Gathering all of these terms, we find the limit equation for entropy and Then it is easy to pass to the limit in (89). Appendix.
Appendix A. A different scaling. Here, for the worth of completeness we show how the asymptotic regime characterized by a small amount of radiation in the flow and a sound speed small with respect to the velocity of light (accretion disk regime), can be defined through a different scaling than the one performed in Section 3, namely for any ε > 0 we set and by Sr = 1, P e = 1, Re = 1, Ro = 1, Al = 1, L = L s = 1 in the previous system. Plugging this scaling into the system (44)-(51) we get ∂ t e + ε 2 E R + div x e u + F R + div x q = ε 2 S : ∇ x u − pdiv x u + ε 2 j · E (139) with and finally where Γ + = {(x, ω) ∈ ∂Ω × S 2 : ω · n x > 0}. Note that the main differences compared with the scaled system (52)-(58) are in the momentum equation (138) and in the boundary on radiation intensity term in the total energy identity (142). The limit analysis follows the same line of arguments of the previous section so we mention here only the points were we have the main differences.

Remark 3.
Concerning regularity of a domain Ω let us mention that an existence of weak solution of the barotropic case for more general domain was studied by Kukučka in the case of Lipschitz domain. It can be generalized to John domain. Important is that we need to apply the Bogovski operator, Korn inequality for more details,see [4].
Then it is easy to pass to the limit in (89) Appendix B. Proof of Theorem 3.1.
So we can follow verbatim the scheme of proof of Proposition 12.6 in [25] to conclude.