INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES. Low Mach and Péclet number limit for a model of stellar tachocline and upper radiative zones

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1 INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES Low Mach and Péclet number limit for a model of stellar tachocline and upper radiative zones Donatella Donatelli Bernard Ducomet Marek Kobera Šárka Nečasová Preprint No PRAHA 216

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3 Low Mach and Péclet number limit for a model of stellar tachocline and upper radiative zones Donatella Donatelli 1, Bernard Ducomet 2, Marek Kobera 4 3, Šárka Nečasová 1 Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Universita degli Studi dell Aquila, 671 L Aquila, Italy donatella.donatelli@univaq.it 2 CEA, DAM, DIF, F Arpajon, France bernard.ducomet@cea.fr 3 Institute of Mathematics of the Academy of Sciences of the Czech Republic Žitná 25, Praha 1, Czech Republic matus@nath.cas.cz 4 Mathematical Institute of the Charles University at Prague Sokolovská 83, Praha 8, Czech Republic kobera@centrum.cz Abstract We study a hydrodynamical model describing the motion of internal stellar layers based on compressible Navier-Stokes-Fourier-Poisson system. We suppose that the medium is electrically charged, we include energy exchanges through radiative transfer and we assume that the system is rotating. We analyze the singular limit of this system when the Mach number, the Alfvén number, the Péclet number and the Froude number go to zero in a certain way and prove convergence to a 3D incompressible MHD system with a stationary linear transport equation for transport of radiation intensity. Finally, we show that the energy equation reduces to a steady equation for the temperature corrector. Key words: Navier-Stokes-Fourier-Poisson system, compressible magnetohydrodynamics, radiation transfer, rotation, stellar radiative zone, weak solution, elliptic-parabolic initial boundary value problem, vanishing Péclet number, vanishing Mach number, vanishing Alfvén number, classical physics, plasma. AMS classifications: 35Q35,76N15 1 Introduction Our motivation in this work is the rigorous analysis of the equations describing parts of stars called radiative zones which are one of the most basic structures constituting stars among cores, convection zones, photospheres and atmospheres. Our model can be also applied to tachoclines which are transition 1

4 layers between convection and radiative zones of stars. In this context it is conjectured that magnetic field of stars arises when poloidal orientation of magnetic fields changes to toroidal and that a dynamo effect is present in tachoclines 37]. Tachoclines are not homogeneous and stable structures and they move steadily. In their upper parts the Péclet numbers are high of the order 6, but in the vicinity of the radiative part they drop below 1. Their distinctive feature concerns rotation, naïvely speaking the convective zone behaves in this respect as a fluid and rotates differentially, whereas the radiative zone more like a solid and rotates as a rigid body. The origin of these rotational changes has preocuppied astrophysicists and astronomes, particularly in connection with helioseismological observations 3] Gravitational forces in these regions are high, however the fluid is no longer strongly stratified as show non-dimensional numbers associated to the solar tachocline. Namely the Froude number Fr measuring the strength of gravitational interactions see Section 3 below for precise definitions is Fr = U, where U is the referential speed of flow in SI units. The Mach number Ma measuring the compressibility is Ma = U, i. e. the fluid is almost incompressible for sufficiently slow motions and one has Fr 2 Ma Mach number is due to high temperatures when radiation dominates. Finally Péclet number Pe need not to be sufficiently small in the solar tachocline we assume Ma 2 = Pe, but thermal diffusivity in giant stars can be seven orders of magnitude larger than that of the Sun see 17], page 22. Notice in conclusion that our low stratification model can be applied to other compact stellar objects, as the fraction of F r and M a depends on the ratio of temperature, density and is inversely proportional to the square of characteristic length. Therefore white dwarfs are too cold to be described by low stratification models, but neutron stars, especially newly born are not. Validity of classical MHD may be restricted to their outer crusts though; in their superfluid cores a quantum description is inevitable. Let us complete this physical introduction by drawing the reader s attention to the fact that models in stellar physics are computationally time consuming. Rieutord 33] has estimated for example that modelling a single supergranule on the Sun would require having more than power of the Sun at our disposal! That is why Lignières 26] has initiated studies of models at small Péclet number as through the Boussinesq-Oberbeck approximation density variations with temperature enter through the buoyancy force only and moreover temperature can be expressed by the velocity field. In our previous work 5] we analyzed a thick disk model for the Mach number of order, whereas the Peclet number was of order 1. Instead as in 31], in the present one we consider a model where the Peclet number is of order ǫ 2 and the domain is general. The mathematical model we consider is the compressible heat conducting MHD system 7] describing the motion of a viscous plasma confined in, a 3D 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 cooling/heating see 7], 1], but neglect radiative accelerations. More precisely, the non-dimensional system of equations giving the evolution of the mass density = t, x, the velocity field u = ut, x, the divergence-free magnetic field B = Bx, t, and the radiative intensity I = Ix, t, ω, ν as functions of the time t, T, the spatial coordinate x = x 1, x 2, x 3 R 3, and for I the angular and frequency variables ω, ν S 2 R +, reads as follows t + div x u = in, T, 1.1 t u + div x u u + x p + 2 χ u = div x S + Ψ x χ x 2 + j B in, T, 1.2 2

5 t e + div x e u + div x q = S : x u pdiv x u + j E S E in, T, c ti + ω x I = S in, T, S t B + curlx B u + curl x λ curl x B = in, T. 1.5 Ψ = 4πG η + g in, T. 1.6 In the electromagnetic source terms, electric current j and electric field E are interrelated by Ohm s law and Ampère s law j = σ E + u B, ζ j = curl x B, where ζ > is the constant magnetic permeability and σ is the coefficient of electric conductivity. This is a simplified version of Ohm s law for plasmas as both the Hall effect and the ambipolar diffusion from density gradients and the electron inertia are neglected. Moreover in 1.5 λ = λϑ > is the magnetic diffusivity of the fluid. In 1.6 Ψ is the gravitational potential and the corresponding source term in 1.2 is the Newton force Ψ. G is the Newton constant and g is a given function, modelling an external gravitational effect. Supposing that is extended by outside and solving 1.6, we have Ψt, x = G Kx y η t, y + gy dy, where Kx = 1 x, and the parameter η may take the values or 1: for η = 1 selfgravitation is present and for η = gravitation only acts as an external field. 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 2 χ u appears in the system, together with the centrifugal force term x χ x 2 see 4]. We consider here the simplified model studied in 11] where radiation does not appear in the momentum equation see also 38]: only the source term S E is present, in the energy equation S E t, x = St, x, ω, ν d ω dν. S 2 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. 1.7 ϑ Furthermore, S is the Newtonian viscous stress tensor determined by S = µ x u + t x u 2 3 div x u I + η div x u I, 1.8 3

6 where the shear viscosity coefficient µ = µϑ > and the bulk viscosity coefficient η = ηϑ are effective functions of the temperature. Similarly, q is the heat flux given by Fourier s law with the heat conductivity coefficient κ = κϑ >. Finally, q = κ x ϑ, 1.9 S = S a,e + S s, 1.1 where S a,e = σ a Bν, ϑ I, S s = σ s Ĩ I In this formula Ĩ := 1 4π S I, ω d ω and Bν, ϑ = 2hν 3 c e 2 hν 2 kϑ 1 is the radiative equilibrium function where h and k are the Planck and Boltzmann constants, σ a = σ a ν, ϑ is the absorption coefficient and σ s = σ s ν, ϑ is the scattering coefficient. More restrictions on these structural properties of constitutive quantities will be imposed in Section 2 below. System is supplemented with the no-slip, thermal insulation, perfect conductor, no reflection, no radiative entropy flux boundary conditions: u =, q n =, B n =, E n =, 1.12 It, x, ν, ω = on Γ, q R nx = for x, 1.13 where n denotes the outer normal vector to, Γ := {x, ω S 2 : ω n x } and the radiative entropy flux q R will be defined in the next Section. Similarly we define Γ + := S 2 \ Γ. Let us mention that previous works have been achieved in the previous framework but, to our knowledge, not in the case of rotating fluid with radiation with an exception for 5]. Among them: Kukučka 23] studied the case when Mach and Alfvén number go to zero in the case of a bounded domain and Novotný and collaborators 31] investigated the problem in the case of strong stratification. Let us also mention the works of Trivisa et al. 25] and Wang et al.19], and related articles of Jiang et al. 21, 22, 2]. Our work differs from theirs in that we take a larger Froude number and add radiation and non-inertial effects. This 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 , and state the existence result for our model. In Section 3 we compute the formal asymptotics of the problem. Uniform bounds imposed on weak solutions by the data are derived in Section 4. The convergence theorem is proved in Section 5. Existence of a solution for the target system is briefly given in the Appendix. 2 Hypotheses and stability result As in 8] we consider a pressure law in the form p, ϑ = ϑ 5/2 P ϑ 3/2 + a 3 ϑ4, a >, 2.1 4

7 where P :,, is a given function with the following properties: P C 2,, P =, P Z > for all Z, 2.2 < 5 3 PZ P ZZ < c for all Z, Z 2.3 PZ lim Z Z = p 5/3 >. 2.4 According to Maxwell s equation 1.7, the specific internal energy e is and the associated specific entropy reads e, ϑ = 3 2 ϑϑ3/2 P + a ϑ4 ϑ 3/2, 2.5 s, ϑ = M ϑ 3/2 + 4a 3 ϑ 3, 2.6 with M Z = PZ P ZZ 2 Z 2 <. To ensure positivity of the total entropy production rate, as in 5], in this paper we explicitly introduce the entropy for the photon gas in the sequel. The transport coefficients µ, η, κ and λ are continuously differentiable and Lipschitz functions of the absolute temperature with the properties, c ϑ µϑ, µ ϑ < c 2, ηϑ c ϑ, 2.7 c ϑ r κϑ c ϑ r 2.8 c ϑ < λϑ c ϑ p 2.9 for any ϑ, for a 1 p < 17 6 and r = 3. Moreover we assume that σ a, σ s, B are continuous functions of ν, ϑ such that < σ a ν, ϑ c 1, σ s ν, ϑ, ϑ σ a ν, ϑ, ϑ σ s ν, ϑ c 1, 2.1 σ a ν, ϑbν, ϑ, ϑ {σ a ν, ϑbν, ϑ} c 2, 2.11 σ a ν, ϑ, σ s ν, ϑ, σ a ν, ϑbν, ϑ hν, h L 1, for all ν, ϑ, where c 1,2,3,4,5 are positive constants. Let us recall some definitions introduced in 8]. In the weak formulation of the Navier-Stokes-Fourier system the equation of continuity 1.1 is replaced by its weak renormalized version 6] represented by the family of integral identities T ] + b t ϕ + + b u x ϕ + b b div x u ϕ dxdt 5

8 satisfied for any ϕ Cc, T, and any b C,, b Cc includes the initial condition, =. Similarly, the momentum equation 1.2 is replaced by = T T = + b ϕ, dx 2.13,, where 2.13 implicitly u t ϕ + u u : x ϕ + p div x ϕ + 2 χ u ϕ dx dt 2.14 S : x ϕ x Ψ ϕ j B ϕ 1 2 x χ x 2 ϕ dx dt u ϕ, dx for any ϕ C c, T ; R3. As usual, for 2.14 to make sense, the field u must belong to a certain Sobolev space with respect to the spatial variable and we require that where 2.15 already includes the no-slip boundary condition The magnetic equation 1.5 is replaced by T u L 2, T; W 1,2 ; R 3, 2.15 B t ϕ B u + λcurl x B curlx ϕ to be satisfied for any vector field ϕ D, T ; R 3. Here, according to the boundary conditions, one has to take dx dt + B ϕ, dx =, 2.16 B L 2, div x B = in D, B n = Following Theorem 1.4 in 39], B belongs to the closure of all solenoidal functions from D with respect to the L 2 norm. Anticipating see 2.29 below we see that and we deduce from 2.16 that B L, T; L 2 ; R 3, curl x B L 2, T; L 2 ; R 3 div x Bt = in D, Bt n = for a.a. t, T. In particular, using Theorem 6.1 in 14], we conclude that B L 2, T; W 1,2 ; R 3, div x Bt =, B n = for a.a. t, T From 1.2 and 1.3 we find the energy conservation law 1 t 2 u 2 + e + 1 2ζ B 2 + div x 1 2 u 2 + e + p u + E B S u + q = x Ψ u x χ x 2 u S E

9 As the gravitational potential Ψ is determined by equation 1.6 considered on the whole space R 3, the density being extended to be zero outside we observe that x Ψ u dx = d 1 Ψ dx, dt 2 in the same stroke 1 x χ x 2 u dx = d 2 dt 1 2 χ x 2 dx. Denoting now by E R the radiative energy given by E R t, x = 1 It, x, ω, ν d ω dν, 2.2 c and integrating the radiative transfer equation 1.4, we get t E R dx + It, x, ω, ν ω n dν d ω ds x = S 2, ω n S 2 S E dx. so, by using boundary conditions, we can integrate 2.19, as follows, d 1 dt 2 u 2 + e + 1 2ζ B u 2 + e + p u + E B S u + q n ds = x Ψ u + 12 x χ x 2 u S E dx. d 1 dt 2 u 2 + e + 1 2ζ B 2 2 Ψ χ x 2 + E R dx + It, x, ω, ν ω n dν d ω ds x = 2.21 Γ + by 1.12 and Finally, dividing 1.3 by ϑ and using Maxwell s relation 1.7, we obtain the entropy equation q t s + div x s u + div x = ς, 2.22 ϑ where ς = 1 ϑ S : x u q xϑ + λ ϑ ζ curl xb 2 S E ϑ, 2.23 S : x u q xϑ ϑ + λ ζ curl x B 2 is the positive electromagnetic matter where the first term ς m := 1 ϑ entropy production. In order to identify the second term in 2.23, let us recall form 1] the formula for the entropy of a photon gas s R = 2k c 3 S 2 ν 2 n log n n + 1logn + 1] d ωdν,

10 where n = ni = c2 I 2hν is the occupation number. Defining the radiative entropy flux 3 q R = 2k c 2 ν 2 n logn n + 1logn + 1] ω d ωdν, 2.25 S 2 and using the radiative transfer equation, we get the equation t s R + div x q R = k 1 h S ν log n n + 1 S d ωdν =: ςr With the identity log nb nb+1 = hν kϑ with B = Bϑ, ν denoting Planck s function, and using the definition of S, the right-hand side of 2.26 rewrites ς R = S E ϑ k h k h S 2 1 ν 1 S ν 2 log log ni ni + 1 log ni ni + 1 log ] σ a B I d ωdν nb nb + 1 ] nĩ nĩ + 1 σ s Ĩ I d ωdν, where we used the hypothesis that the transport coefficients σ a,s do not depend on ω. So we obtain finally t s + s R + div x s u + q R q + div x = ς + ς R ϑ and equation 2.22 is replaced in the weak formulation by the inequality T q s + s R t ϕ + s u x ϕ + ϑ + qr x ϕ T s + s R 1 ϕ, dx ϑ T k h + S 2 1 ν log S 2 1 ν log ni ni + 1 log ni ni + 1 log nb nb + 1 ] ]σ s Ĩ I d ωdν dx dt 2.28 S : x u q xϑ + λ ϑ ζ curl xb 2 ϕ dx dt ] σ a B I d ωdν nĩ nĩ + 1 ϕ dx dt for any ϕ Cc, T, ϕ, where the sign of all the terms in the right hand side may be controlled. Since replacing equation 1.3 by inequality 2.28 would result in a formally under-determined problem, system 2.13, 2.14, 2.28 must be supplemented with the total energy balance 1 2 u 2 + e, ϑ + 1 2ζ B 2 2 Ψ χ x 2 + E R τ, dx 2.29 τ + Γ + It, x, ω, ν ω n dν d ω ds x dt 8

11 where E R 1 = u 2 + e ζ B 2 2 Ψ 1 2 χ 1 x 2 + E R is given by E R x = 1 c S 2 I, x, ω, ν d ω dν. The transport equation 1.4, can be extended to the whole physical space R 3 provided we set σ a x, ν, ϑ = I σ a ν, ϑ and σ s x, ν, ϑ = I σ s ν, ϑ, where I A is the characteristic function of a set A and take the initial distribution I x, ω, ν to be zero for x R 3 \. Accordingly, for any fixed ω S 2, equation 1.4 can be viewed as a linear transport equation defined in, T R 3, with a right-hand side S. With the above mentioned convention, extending u to be zero outside, we may therefore assume that both and I are defined on the whole physical space R 3. Definition 2.1 We say that, u, ϑ, B, I is a weak solution of problem iff, ϑ > for a.a. t, x, I a.a. in, T S 2,, L, T; L 5/3, ϑ L, T; L 4, u L 2, T; W 1,2 ; R 3, ϑ L 2, T; W 1,2, B L 2, T; W 1,2 ; R 3, B n = I L, T S 2,, I L, T; L 1 S 2,, and if, u, ϑ, B, I satisfy the integral identities 2.13, 2.14, 2.28, 2.16, 2.29, together with the transport equation 1.4. The stability result of 1] reads now Theorem 2.1 Let R 3 be a bounded C 2,α with α > domain. Assume that the thermodynamic functions p, e, s satisfy hypotheses , and that the transport coefficients µ, λ, κ, σ a, and σ s comply with Let {, u, ϑ, B, I } > be a family of weak solutions to problem in the sense of Definition 2.1 such that,, in L 5/3, 2.3 and Then 1 2 u 2 + e, ϑ + 1 2ζ B 2 2 Ψ 1 2 χ 1 x 2 + E R 1 u, 2 + e, + E, R + 1 2, 2ζ B, 2 2, χ 1 x 2 1, dx E, 2,Ψ s, ϑ + s R I ], dx s + s R, dx S, I, I, I, I,, ν hν for a certain h L 1,. in C weak, T]; L 5/3, dx,, dx

12 and u u weakly in L 2, T; W 1,2 ; R 3, ϑ ϑ weakly in L 2, T; W 1,2, B B weakly in L 2, T; W 1,2 ; R 3, B n = I I weakly-* in L, T S 2,, at least for suitable subsequences, where {, u, ϑ, B, I} is a weak solution of problem Formal scaling analysis In order to identify the appropriate limit regime we perform a general scaling, denoting by L ref, T ref, U ref, ρ ref, ϑ ref, p ref, e ref, µ ref, λ ref, κ ref, the reference hydrodynamical quantities length, time, velocity, density, temperature, pressure, energy, viscosity, conductivity, by I ref, ν ref, σ a,ref, σ s,ref, the reference radiative quantities radiative intensity, frequency, absorption and scattering coefficients, by χ ref the reference rotation velocity, and by ζ ref, B ref the reference electrodynamic quantities permeability and magnetic induction. We also assume the compatibility conditions p ref ρ ref e ref, ν ref = kϑ ref h, I ref = 2hν3 ref λ λ = ref L ref U ref and we denote by Sr := L ref U T ref U ref, Ma := ref, Re := U ref ρ ref L ref pref /ρ ref µ ref, Pe := U ref p ref L ref U ϑ ref κ ref, Fr := ref, C := c Gρref L 2 ref c 2, U ref, the Strouhal, Mach, Reynolds, Péclet, Froude and infrarelativistic dimensionless numbers corresponding to hydrodynamics, by Ro := by Al := U refρ 1/2 ref ζ1/2 ref B ref U ref χ ref L ref the Alfven number and by L := L ref σ a,ref, L s := σ s,ref σ a,ref, P := dimensionless numbers corresponding to radiation. Using these scalings and using carets to symbolize renormalized variables we get where S = I ref L ref Ŝ, Ŝ = Lˆσ a Bˆν, ˆϑ 1 Î + LL sˆσ s Î, ω d ω 4π Î. S 2 the Rossby number, 2k4 ϑ 4 ref h 3 c 3 ρ ref e ref, various Omitting the carets in the following, we get first the scaled equation for I, in the region, T, S 2 Sr 1 C ti + ω x I = s = Lσ a B I + LL s σ s I d ω I, 3.1 4π S 2 where we used the same notation B for the dimensionless Planck function Bν, ϑ = ν3 e ν ϑ 1. Denoting also by E R = I dν d ω, the renormalized radiative energy, by F R = I ωdν d ω, S 2 S 2 the renormalized radiative momentum, by s E = S 2 s dν d ω, the renormalized radiative energy source, by s R = S ν 2 n log n n + 1logn + 1] d ωdν, the renormalized radiative entropy with 2 1

13 n = ni = I ν, by q R = ν 2 n log n n + 1logn + 1] ω d ωdν, the renormalized radiative 3 S 2 entropy flux, and taking the first moment of 3.1 with respect to ω, we get first an equation for E R The continuity equation is now and the momentum equation reads Sr C te R + x F R = s E. 3.2 Sr t + div x u =, 3.3 Sr t u + div x u u + 1 Ma 2 xp, ϑ + Ro χ 2 u = 1 Re div xs + 1 Fr 2 Ψ + 1 2Ro 2 x χ x Al 2 j B. 3.4 The balance of internal energy rewrites Sr t e + 1 C ER + div x e u + F R + 1 Pe div x q = Ma2 Re S : x u pdiv x u + Sr Ma2 Al 2 j E, and we get the balance of matter fluid entropy with ς = 1 ϑ Sr t s + div x s u + 1 Pe div x Ma 2 and the balance of radiative entropy with ς R = PL Re S : x u 1 Pe +PLL s S 2 1 ν 1 S ν 2 log q x ϑ ϑ q = ς, 3.5 ϑ + Ma2 λ Al 2 ζ curl xb 2 S E ϑ, Sr C ts R + div x q R = ς R, 3.6 log ni ni + 1 log ni ni + 1 log The scaled equation for the electromagnetic field is nb nb + 1 ] nĩ nĩ + 1 ] σ a I B d ωdν σ s I Ĩ d ωdν + S E ϑ. Sr t B + curlx B u + curl x λ curl x B =. 3.7 The scaled equation for total energy gives finally the total energy balance Sr d Ma 2 u 2 + e + 1 dt 2 C ER + Ma2 1 2Al 2 ζ B 2 1 Ma 2 2 Fr 2 Ψ 1 Ma 2 χ x 2 2 Ro2 dx 11

14 + In the sequel we analyze the asymptotic regime defined by Γ + I ω n dγ + dν =. 3.8 Ma =, Al =, Fr = 1/2, C = 1, Pe = 2 where > is small and we put Sr = 1, Re = 1, Ro = 1, P = 1, L = L s = 1 in the previous system. Plugging this scaling into the previous system gives 1 t I + ω x I = σ a B I + σ s I d ω I, 3.9 4π S 2 t + div x u =, 3.1 t u + div x u u xp, ϑ + 2 χ u = div x S + 1 Ψ x χ x j B, 3.11 with t e + E R + div x e u + F R div x q = 2 S : x u pdiv x u + j E, 3.12 t s + s R + div x s u + q R + 1 q 2 div x ς, 3.13 ϑ + + ς = 1 ϑ 1 S ν 2 S 2 1 ν 2 S : x u q xϑ 2 ϑ + λ ζ curl xb 2 log log ni ni + 1 log ni ni + 1 log nb nb + 1 ] nĩ nĩ + 1 ] σ a I B d ωdν σ s I Ĩ d ωdν, and finally d 1 dt t B + curlx B u + curl x λ curl x B =, u 2 + e + E R ζ B Ψ χ x 2 To compute the limit system, we consider now the formal expansions Γ + ω ni dγ + dν = I,, u, ϑ, p, B = Ĭ,, ŭ, ϑ, p, B + I 1, 1, u 1, ϑ 1, p 1, B 1 + O We first observe from 3.11 that = const := and ϑ = const := ϑ, moreover x p 1 = x Ψ Let us fix the constants in the Neumann problem for perturbations of the temperature ϑ i dx = for any i dx

15 From 3.1 we derive the incompressibility condition div x ŭ =, 3.19 and t 1 + div x u 1 + 1ŭ =. 3.2 From 3.9 we get now two stationary linear transport equations for the two moments Ĭ and I 1 ω x Ĭ = σ a, B Ĭ + σ s, Ĭ Ĭ, 3.21 ω x I 1 = σ a, ϑ B ϑ 1 I 1 + ϑ σ a, B Ĭ ϑ 1 + ϑ σ s, Ĭ ϑ Ĭ 1 + σ s, Ĩ1 I 1, 3.22 where Ĩ := 1 4π I d ω, σ S 2 a, = σ a ν, ϑ, σ s, = σ s ν, ϑ and B = Bν, ϑ. The limit momentum equation reads t ŭ + div x ŭ ŭ + x Π + 2 χ ŭ = div x Sŭ + 1 ζ curl x B 1 B 1 + F, 3.23 where µ = µ ϑ is used in Sŭ, F = 1 x Ψ and Π is an effective pressure for which it holds x Π = 1 2 x χ x 2 + x Π 1 + p,, ϑ 1 x 1. Here we set ϑ 1 = which is consistent with the O 1 order of the internal energy equation 3.12 and the additional zero mean of ϑ ϑ requirement. The limit magnetic field B 1 solves t B1 + curl x B 1 ŭ + curl x λ curl x B1 =, 3.24 for λ = λ ϑ. At the lowest order O the energy equation 3.12 gives κ ϑ 2 = s E 3.25 where s E = σ S 2 a, Ĭ B d ω dν and κ = κϑ. At the order O we simplify the energy equation Observing that from 3.17 we have where D := t + ŭ x, and from 3.2 and after 3.22 and simplifying by 1.7 we end up with p, ϑd 1 + ŭ x Ψ =, 3.26 S E1 = t 1 + div x 1ŭ = α div x u 1 = D 1, S 2 σ a, I 1 d ω dν, κ ϑ σ a, I 1 d ω dν S 2

16 where α := ϑ θp, ϑ. Putting U = ŭ, Θ = ϑ 3, B = B1, =, ϑ = ϑ, µ = µ ϑ, σ a = σ a,, σ s = σ s,, B = B, D U = 1 2 ŭ + T ŭ, and G = σ S 2 a, Ĭ B d ω dν κ we observe that the solution of the equation 3.21 is up to the boundary condition Ĭ = B which in turn entails that the equation for ϑ 2 turns in into the Laplace homogeneous equation G = and therefore ϑ 2 = and we obtain the limit system in, T div x U =, 3.27 t U + divx U U + x Π = div x 2µ D U + 1 ζ curl x B B + F 3.28 t B + curlx B U + curl x λ curl x B =, 3.29 Θ = 1 ακ U x r 1 κ together with the Boussinesq relation 3.17 div x B =, 3.3 σ a S 2 I 1 d ω dν + ht 3.31 ω x I 1 = σ a I 1 + σ s Ĩ1 I 1, 3.32 x r = xψ p, ϑ, 3.33 where r := 1 and h is an undetermined function which allows satisfaction of We finally consider the boundary conditions U =, Θ n =, B n =, curl xb n = 3.34 for and I 1 x, ν, ω = for x, ω n 3.35 for 3.32, and the initial conditions U t= = U, B t= = B Moreover, we endow the system with the additional conditions div xb =, Θ dx = For this system we have the following existence result see the Appendix for a short proof 14

17 Theorem 3.1 Let R 3 be a bounded Lipschitz domain. For any T > the initial-bounday value problem has at least a weak solution U, Θ, B, I 1 such that 1. U L, T; H L 2, T; U, B L, T; V L 2, T; W, with H = { U L 2 ; R 3, div xu = in, U = }, U = H W 1,2 ; R 3, { } V = b L 2 ; R 3, div x b =, b n = and W = V W 1,2 ; R 3, Θ L, T; W 2,2 L 2, T; W q,2 for any q < 5 2, I 1 L, T S 2 R +, with ω x I 1 L p, T S 2 R +, for any p > 1 and any ω S 2. The remaining part of the paper is devoted to the proof of the convergence of the primitive system to the target system Global existence for the primitive system and uniform estimates For the system we prepare the initial data as follows, =, = + 1,, u, = u,, ϑ, = ϑ, = ϑ + 3 ϑ 3,, I,,, = I, = I + I 1,, B, = B, = B 1,, 4.1 where >, ϑ >, I > are spacetime constants and 1, dx = = ϑ3, dx for any >. As in 15], for any locally compact Hausdorff metric space X we denote by MX the set of signed Borel measures on X and by M + X the cone of non-negative elements of MX. From Theorem 2.1 we get immediately by combining the approximating schemes introduced in 8] and 7] the existence of a weak solution, u, ϑ, I, B to the radiative MHD system

18 Theorem 4.1 Let R 3 be a bounded C 2,α with α > domain. Assume that the thermodynamic functions p, e, s satisfy hypotheses , and that the transport coefficients µ, λ, κ, σ a, σ s and the equilibrium function B comply with Let the initial data,, u,, ϑ,, I,, B, be given by 4.1, where 1,, ϑ3,, I1,, B 1, are uniformly bounded measurable functions. Then for any > small enough in order to maintain positivity of, and ϑ,, there exists a weak solution, u, ϑ, I, B to the radiative Navier-Stokes system for t, x, ω, ν, T S 2 R +, supplemented with the boundary conditions and the initial conditions 4.1. More precisely we have T = T b t φ + u x φ dx dt β div x u φ dx dt, b, φ, dx, 4.2 for any β such that β L C,, b = b1 + 1 βz z 2 dz and any φ C c, T, T u t ϕ + u u : x ϕ + p 2 div x ϕ 2 χ u ϕ dx dt T = S : x ϕ 1 x Ψ ϕ 1 2 j B ϕ x χ x 2 ϕ dx dt, u, ϕ, dx, 4.3 for any ϕ C c, T ; R3 with p = p, ϑ, S = S u, ϑ, and j = 1 ζ curl x B, 2 2 u 2 + e + E R + 1 2ζ B Ψ χ x 2 dx dt T + ω n x I t, x, ω, ν dγ + dν dt Γ + 2 = 2, u, 2 +, e, + E, R + 1 2ζ B, 2 1 2,Ψ, 1 2 2, χ x 2 dx, 4.4 for a.a. t, T with e = e, ϑ, Ψ = Ψ, Ψ, = Ψ, and E R t, x = S I 2 t, x, ω, ν d ω dν T B t ϕ B u + λ curl x B curl x ϕ for any vector field ϕ D, T R 3, R 3, with λ = λϑ. dx dt + B, ϕ, dx =,

19 T where and s + s R T t ϕ + s u + q R q x ϕ dx dt + 2 x ϕ dx dt ϑ + ς m + ςr ; ϕ = s, + s R M;C],T, ϕ, dx, 4.6 ς R + ς m 1 2 S : x u q xϑ ϑ 2 + λ curl x 2 B, ϑ ζ 1 S ν 2 S 2 1 ν log log ni ni + 1 log nb nb + 1 ni ni + 1 log nĩ nĩ + 1 ] ] σ a B I d ωdν σ s Ĩ I d ωdν, for any ϕ Cc, T with ςm M +, T and ς R M +, T, and with σ a = σ a ν, ϑ, σ s = σ s ν, ϑ, B = Bν, ϑ, q = κϑ x ϑ, s = s, ϑ, s R = s R I, q R = qr I and Ĩ := 1 4π I S 2 t, x, ω, ν d ω, T = T + S 2 t ψ + ω x ψi d ω dν dx dt S 2 ] σ a B I + σ s Ĩ I ψ d ω dν dx dt, S 2 I, ψ, x, ω, ν d ω dν dx + for any ψ C c, T S2 R Uniform estimates T Γ + I ω n x ψ dγ + dν dt, 4.7 We recall from 15] the necessary definitions in the formalism of essential and residual sets see 11]. Given three numbers R +, ϑ R + and E R + we define Oess H the set of hydrodynamical essential values O H ess := {, ϑ R 2 : and O R ess the set of radiative essential values O R ess := {E R R : with O ess := Oess H OR ess, and their residual counterparts } 2 < < 2, ϑ 2 < ϑ < 2ϑ, 4.8 } E 2 < ER < 2E, 4.9 O H res := R + 2 \O H ess, O R res := R + \O R ess, O res := R + 3 \O ess

20 } Let {, u, ϑ, B, I > Theorem 4.1. We call M ess, T the set be a family of solutions of the scaled radiative Navier-Stokes system given in M ess = { t, x, T : t, x, ϑ t, x, E R t, x O ess}, and M res =, T \M ess the corresponding residual set. To any measurable function h we associate its decomposition into essential and residual parts h = h] ess + h] res, where h] ess = h I M ess and h] res = h I M res. Denoting by H ϑ the Helmholtz function for matter and for radiation H ϑ, ϑ = e ϑ s, H R ϑ I = ER ϑs R, and using 4.6 we rewrite 4.4 as 2 2 u 2 + H ϑ, ϑ + H R ϑ I + 1 2ζ B Ψ χ x 2 dx = 2 T + ω n x I t, x, ω, ν dγ dν dt + ϑ ς m + ς R ], T] Γ + 2, u, 2 +, e, + E R, + 1 2ζ B, 2 1 2,Ψ, 1 2 2, χ x 2 Observing that the total mass is a constant of motion M = dx = and using Hardy-Littlewood- Sobolev inequality, we get 2 Ψ dx G 2 CM2/3 4/3. By virtue of 2.1 and 2.5 we have L 4/3 also e, ϑ aϑ 4 + 3p 5/3 2, so we have the lower bound H ϑ, ϑ 12 ] Ψ dx c H ϑ, ϑ dx, for small and a c < 1 and we deduce finally the dissipation energy-entropy inequality 2 2 u 2 + H ϑ, ϑ H ϑ, ϑ H ϑ, ϑ + 1 2ζ B χ x 2 + H R ϑ I dx T + I t, x, ω, ν ω n x dγ dν dt + ϑ ς m + ] ςr, T] Γ + 2 C 2, u, 2 + H ϑ,, ϑ,, H ϑ, ϑ H ϑ, ϑ + 1 2ζ B, 2 + H R ϑ I, dx Now, according to Lemma 4.1 in 11] see 15] we have the following properties for material and radiative Helmholtz functions dx. 18

21 Lemma 4.1 Let > and ϑ > two given constants and let and H ϑ, ϑ = e ϑ s, H R ϑ I = ER ϑs R. Let O ess and O res be the sets of essential and residual values introduced in There exist positive constants C j = C j, ϑ for j = 1,, 4 and positive constants C j = C j E, ϑ for j = 5,, 8 such that 1. C ϑ ϑ 2 H ϑ, ϑ H ϑ, ϑ H ϑ, ϑ for all, ϑ O H ess, C ϑ ϑ 2, for all, ϑ O H res, for all, ϑ O H res, H ϑ, ϑ H ϑ, ϑ H ϑ, ϑ { } inf, ϑ O res H H ϑ, ϑ H ϑ, ϑ H ϑ, ϑ = C 3, 4.13 H ϑ, ϑ H ϑ, ϑ H ϑ, ϑ C 4 e, ϑ + s, ϑ, for all E O R ess, for all E O R res, for all E O R res. C 5 E R E 2 H R ϑ I C 6 E R E 2, 4.15 H R ϑ I inf H R ϑ Ĩ = C 7, 4.16 Ĩ Ores R H R ϑ I C 8 E R I + s R I 4.17 Using 4.11 and Lemma 4.1, we get the following energy estimates 19

22 Lemma 4.2 Suppose that the initial data satisfy, ] ess 2 L 2 C2, ϑ, ϑ] ess 2 L 2 C2, E R, E 2 L 2 C, B, 2 L 2 ;R 3 C2, and Then the following estimates hold, u, L 2 ;R 3 C. ess sup M res t C2, 4.18 t,t ess sup ] ess t 2 L 2 C2, 4.19 t,t ess sup ϑ ϑ] ess t 2 L 2 C2, 4.2 t,t ess sup E R E] ess t 2 L 2 C, 4.21 t,t ess sup e, ϑ ] res t L 1 C 2, 4.22 t,t ess sup s, ϑ ] res t L 1 C 2, 4.23 t,t ess sup E R I ] res t L 1 C, 4.24 t,t ess sup t,t s R I ] res t L 1 C ς m + ς R ], T] C 2, 4.26 ess sup t,t B t L 2 ;R 3 C, 4.27 ess sup u t L 2 ;R 3 C t,t ess sup t,t ] 5 3 res + ϑ ] 4 res t dx C 2, 4.29 u L 2,T;W 1,2 ;R 3 C, 4.3 ϑ ϑ 2 C, L 2,T;W 1, logϑ logϑ C, L2,T;W 1, B L 2,T;W 1,2 ;R 3 C

23 Proof: Estimate 4.18 follows from Bounds 4.19, 4.2 and 4.21 follow from 4.12 and Estimates 4.22 and 4.23 follow from Bounds 4.24 and 4.25 follow from Estimates 4.26, 4.27 and 4.28 follow from the dissipation energy-entropy inequality Bound 4.29 follows from 4.22 and 2.5 cf. a lower bound for e before From 4.26 we see that x u + T x u 2 3 div x u I C L2,T;L 2 ;R 3 3 From 2.7, 4.28 and 4.34 we get 4.3. Details can be found in 1] and 15]. From 4.26 we get ϑ L x 2 + log ϑ x 2,T;L 2 ;R 3 2 C, L 2,T;L 2 ;R 3 which, using Poincaré inequality, gives 4.31 and Finally by 2.9, 2.23 and 4.26 one gets curl xb C, L 2,T;L 2 ;R 3 and 4.33 follows by using Theorem 6.1 in 14]. Our goal in the next Section will be to prove that the incompressible system is the limit of the primitive system in the following sense Theorem 4.2 Let R 3 be a bounded domain of class C 2,ν. Assume that the thermodynamic functions p, e, s satisfy hypotheses with P C 1, C 2,, and that the transport coefficients µ, η, κ, λ, σ a, σ s and the equilibrium function B comply with Let, u, ϑ, B, I be a weak solution of the scaled system for t, x, ω, ν, T] S 2 R +, supplemented with the boundary conditions and initial conditions,, u,, ϑ,, B,, I, given by, = + 1,, u, = u,, ϑ, = ϑ + 3 ϑ 3,, I, = I + I 1,, B, = B 1,, where >, ϑ >, I > are constants in, T and 1, dx =, ϑ 3, dx =, I 1, dx =, Assume that Then B 1, 1, 1 weakly in L, u, U weakly in L ; R 3, ϑ 3, ϑ3 weakly in L, I 1, I1 weakly in L S 2 R +, B 1, B 1 weakly in L ; R 3, dx = for all >. ess sup t 5 L 3 C, 4.35 t,t 21

24 and up to subsequences u U weakly in L 2, T; W 1,2 ; R 3, 4.36 ϑ ϑ 3 =: ϑ 3 Θ weakly in L 4 3, T; W 1, I I = B weakly in L 2, T; L 2 S 2 R +, 4.38 and I I where U, Θ, B, I 1 solves the system Proof of Theorem 4.2 Let us first quote the following result of 11] see 15]. B = B 1 B weakly in L 2, T; W 1,2 ; R 3, 4.39 = I 1 I 1 weakly in L 2, T; L 2 S 2 R +, 4.4 Proposition 5.1 Let { } >, {ϑ } >, {I } > be three sequences of non-negative measurable functions such that ] 1 1 weakly in L, T; L 2, ess ] ϑ 1 ess ϑ1 weakly in L, T; L 2, ] I 1 ess I1 weakly in L, T; L 2, a.e. in S 2 R +, where Suppose that 1 =, ϑ 1 Let G, G R C 1 O ess be given functions. Then = ϑ ϑ, I 1 = I I. ess sup M res t C t,t G, ϑ ] ess G, ϑ G, ϑ 1 + G, ϑ ϑ ϑ 1, weakly in L, T; L 2, and if we denote G R I ] ess := G R I,, ω, ν ] ess = GR I I M ess, for a.a. ω, ν S2 R +, we have G R I ] ess GR I weakly in L, T; L 2, a.e. in S 2 R +. GI I I 1, 22

25 Moreover if G, G R C 2 O ess then G, ϑ ] ess G, ϑ G, ϑ 1] G, ϑ ess ϑ ϑ 1] ess L,T;L 1 C, and G R I ] ess GR I GI I I 1] ess L,T;L 1 C, for a.a. ω, ν S 2 R +. Clearly, this result provides us with the convergence properties , The convergence of radiative intensity 4.38 follows from 4.24, 4.21, and the linearity of 3.9, cf. the section 5.2 Radiative transfer equation. The equilibrium Planck function B does not satisfy the boundary condition however since it is isotropic; therefore has to be modified at the boundary. The last convergence 4.37 is postponed to Section 5.3. To conclude the proof of Theorem 4.2, let us prove that the limit quantities U, Θ, B, I 1 solve the target system As number of terms in the equations of our model are similar to those of the radiative Navier-Stokes- Fourier analyzed in 11] we focus on the new contributions only. 5.1 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, from Lemma 4.2 we know that T u t 2 dt C so passing to W 1,2 ;R 3 the limit after possible extraction of a subsequence, we deduce that u U, weakly in L 2, T; W 1,2 ; R In the same stroke, weakly in L, T; L 5/3 ; R 3. So we can pass to the limit in the weak continuity equation 4.2 which gives T U x φ dx dt = for all φ D, T, which rewrites div x U =, a.e. in, T, U =, provided is regular. For the momentum equation one knows that due to possible strong time oscillations of the gradient component of velocity, one has only u u U U weakly in L 2, T; L 3 29 ; R 3. However one can show by the analysis in 15] that one can pass to the limit in the convective term and obtain T T U U : xφdxdt U U : x φ dxdt. According to the hypotheses on the pressure law, the temperature ϑ is bounded in L, T; L 4 L 2, T; L 6, which together with the strong convergence of ϑ by 4.31 imply that S µϑ xu+ T x U weakly in L 34 23, T; L ; R 3. 23

26 So taking a divergence free test vector field φ in 4.3, we have = T T S : x φ u tφ + u u : xφ 2 χ u φ x Ψ φ 1 ζ curl x B dx dt B φ x χ x 2 φ dxdt, u, φ, dx. 5.3 Moreover, using 2.16 together with estimates 4.27, 4.33 and Aubin-Lions lemma we get B B weakly in L 2, T; W 1,2 ; R 3 and strongly in L 2, T; L q, R 3, curl xb B ζ 1 ζ curl xb B weakly in L 5 4, T ; R 3, for any 1 q < 6. Then passing to the limit and using , we get = T T U t φ + U U : x φ 2 χ U φ µϑ xu + T x U : xφ 1 x Ψ φ 1 ζ curl xb B φ dx dt dx dt U φ dx, provided that u, U weakly- in L ; R 3. Moreover as in 15], the formal relation between 1 and is recovered by multiplying the momentum equation by. One gets, using Proposition 5.1 and passing to the limit T which is the weak formulation of This rewrites as x p 1 x Ψ φ dx dt =, 5.5 p, ϑ x 1 + ϑ p, ϑ x ϑ 1 x Ψ =. 5.6 p, ϑ x 1 x Ψ =. 5.7 once we establish that ϑ 1 = ϑ 1 = ϑ 2 = in the section 5.3. That means we have got an explicit formula for 1 1 = Ψ + ht, 5.8 p, ϑ where h is an undetermined function. 24

27 5.2 Radiative transfer equation Using the L bound shown in 1] for I, based on the initial data bound 4.1, it is clear that I I weakly in L 2, T S 2 R +, and we have also by virtue of 4.31 ϑ ϑ weakly in L 2, T; W 1,2. By using the cut-off hypotheses 2.1, 2.12 and the same notation for any time-independent test function ψ Cc S 2 R +, we can pass to the limit in 4.7 and we get, ω x ψ I d ω dν dx + σ a ν, ϑ ] Bν, ϑ I + σs ν, ϑ Ĩ I ψ d ω dν dx S 2 S 2 = I ω n x ψ dγ dν, which is the weak formulation of the stationary problem Γ + ω x I = S, 5.9 with the boundary condition I = on Γ, 5.1 where S = σ a ν, ϑ Bν, ϑ I + σs ν, ϑ Ĩ I. The solution of is the function equal to Bν, ϑ = B in and on Γ. This solution is unique at least for a particular type of domains thanks to the linearity of 5.9. Substracting from 4.7 and dividing by gives T t ψ + ω x ψ I I T ] S S d ω dν dx dt + ψ d ω dν dx dt, S 2 S 2 = I, I S 2 ψ, x, ω, ν d ω dν dx + T Γ + I I ω n x ψ dγ dν dt, for any ψ C c, T S2 R +, with S S := SI SI. From Proposition 5.1, we get S S S 1 := ϑ σ a Bν, ϑϑ 1 ϑ σ a ν, ϑϑ 1 I σ a ν, ϑi 1 + ϑ σ s ν, ϑϑ 1 Ĩ + σ s ν, ϑĩ1 ϑ σ s ν, ϑϑ 1 I σ s ν, ϑi 1 = σ a ν, ϑi 1 + σ s ν, ϑ Ĩ1 I 1 weakly-* in L, T; L 2 S 2 R + with I 1 := I 1. Passing to the limit we find the limit equation based on the assumption I 1, I1 weakly-* in L S 2 R + ω x ψ I 1 d ω dν dx + S 1 ψ d ω dν dx, = I 1 ω n x ψ dγ dν, 5.11 S 2 S 2 using the same notation for any time-independent test function ψ C c S2 R + which is the weak formulation of the stationary problem ω x I 1 = S 1, 5.12 with the boundary condition Γ + I 1 = on Γ

28 5.3 Entropy balance First of all we analyze the weak limit of 4.6, then we substract it with weak limits denoted by bars from 4.6 and divide by as in the last section. We follow the ideas of 16] and 24]. The most obvious convergence in 4.6 is in the entropy production rate measures. By virtue of 4.26 it holds ς m + ς R ; ϕ as +, 5.14 M;C],T and 1 ς m + ς R ; ϕ as M;C],T We split the heat flux term into residual and essential parts as follows: T q 2 x ϕdxdt ϑ = T κϑ ] res x ϑ T ϑ ] res 2 x ϕdxdt + The first term on the rhs vanishes. The argument is as follows: Firstly, from 4.26, 2.23 and 2.8 we get an exact estimate From 4.31 we know that T ϑ ϑ L 2,T;W 1,2 ϑ x ϑ ϑ 2 2 c, thus κϑ ] ess ϑ ] ess x ϑ 2 x ϕdxdt dxdt c ϑ ϑ in L 2, T; W 1, strongly. On the residual set we now apply the Sobolev embedding and interpolate 5.18 with the information in Similarly we get ϑ 1 in L 2, T; W 1,2 as well. This leads to the convergence ϑ ] res ϑ] res = in L , T; L 3, 5.19 meaning that the first integral in 5.16 converges. With the intention that its limit is zero we apply 2.8 and split the integral into two parts. The second part, namely, T ϑ ] 3 2 res ϑ ϑ 3 ϑ 2 ϑ ] res x 2 xϕdxdt 5.2 converges to zero as with the rate 2 by the Poincaré inequality ϑ 3 2 ϑ 3 2 c L2 ϑ x ϑ L2,T;L 2,T;L 2 C by 5.17, 4.18 and The first part, namely, T 1] res ϑ 1 ϑ x 26 2 xϕdxdt 5.22

29 converges to zero as with the rate by Cauchy-Schwarz inequality, 4.32 and The second term on the rhs of 5.16 converges by virtue of 5.17 and 5.18, at least for a subsequence, to T κϑϑ 1 x ϑ 2 x ϕdxdt For the convergence of the initial entropies in 4.6 we use Proposition 5.1 and we get { ] s, ] ess s, ϑ s R + }, ess sr I ϕ, dx In particular s, ϑ 1 φ, dx s, ] ess s, ϑ +, 5.25 s R, ] ess sr I +, 5.26 weakly in L, T; L 2. Residual parts of the initial conditions disappear thanks to the L weak star convergences of the initial data in Theorem 4.2 for sufficiently small. For the convergence in advective part of the entropy balance 4.6 we use 5.18 and the fact that in L, T; L 5 3. This allows to make the limit of entropy to a constant for a subsequence s, ϑ s, ϑ + a. e. in, T The convergence of entropy of a photon gas follows from Proposition 5.1 as s R ] ess sr, 5.29 s R sr 5.3 weakly in L, T; L 2 as + according to 4.25 and 4.18 again. The convergence of the next term containing s u is again split into two terms, first one on the residual, second one on the essential set. For the second one we use again Proposition 5.1, the first one T s, ϑ ] res u x ϕdxdt 5.31 converges to just in L 1, T as + because of the estimates 4.18, 4.23, While the convergence of the equilibrial radiative entropy flux can be readily improved, e. g. to the space L 12 11, T because of the Gibbs relation between specific entropy and energy, cf. 2.5 and 2.6, the integral with the material entropy flux part does not seem to have a right regularity to be 27

30 meaningful. However, we can use usual cut-off functions T K z := mink, z, choose K large enough, e. g. K = 1 6 and split the integral to two parts T s, ϑ ] res u x ϕ dxdt = T T s, ϑ ] res T 1 6 u x ϕ dxdt+ ] s, ϑ ] res u T 1 u 6 x ϕ dxdt. The first part converges to by 4.23, the second one is of order O by Sobolev embedding, estimate 4.18 and Markov-Chebyshev inequality. The limiting part of this estimate is the first part, where the need to improve the regularity the material part of the entropy flux faces the problem that we have not got generally a better estimate than Previous works 31] 24] 16] rely on the closedness of the equation of state to the ideal gas law so that s is estimated essentially by log, ϑ 3 and log ϑ, the last one being the most restrictive, leading to the convergence in 5.31 in L 2, T; L Without such an assumption we would estimate the entropy by 2 u which is not tractable in view of Nevertheless, in our case of low stratification we do not need to identify the limit of the entropy flux on the essential set since it vanishes after an integration by parts. After 5.18 and 5.27 T s, ϑ ] ess u x ϕdxdt T s, ϑ U x ϕdxdt = The last term contains the nonequlibirial radiative entropy flux q R. Let us recall q R = q R I = ν 2 {n log n n + 1logn + 1} ω d ω dν, S 2 with n = ni = I ν 3. We claim J := T q R x ϕdxdt T q R x ϕdxdt =: J 5.33 because of the convergence on the essential set M ess that follows from 5.41 and on the residual set M res we use Collecting now all the aforementioned convergences in this section we readily get the weak formulation of With 3.18 we see that ϑ 2 and search for x Θ = x ϑ 3 := w lim L 4 3,T, + x ϑ ϑ 3. Let us recall that r = 3 that is needed for the proof of existence and try to relax it for the current proof of convergence + and realize that we can extract from 4.26 the bound T ϑ r 2 x ϑ ϑ 2 4 dxdt < c 5.34 with a constant c independent of. Therefore for r 2 xϑ ϑ 4 is bounded in L 3, T and Θ 3 exists. 28

31 We substract equation 4.6 from its limit and divide by T { s s t ϕ + u x ϕ + sr sr t ϕ + qr qr x ϕ } dx dt T κϑ ϑ + x ϑ 3 xϕ dx dt + 1 ς m + ς R ; ϕ M;C],T] = { s, s, + sr, } sr ϕ, dx We claim that all the terms in 5.35 are uniformly bounded, especially κϑ ϑ ϑ x ϑ 3 κϑ ϑ xϑ 3, 5.36 weakly in L 6r+16 6r+15, T ; R 3 which gives for r = 3 the summability with the exponent of To show this we restrict ourselves to the residual set M res, since on the essential set M ess the boundedness is easy. For the set A := {t, x : x ϑ t, x < 1} we use the estimates 4.18, 4.29 with Hölder s inequality and r 3, 5] K := ϑ ] r 1 res ϑ x dxdt 3 ϑ r 1 dxdt T 3 ϑ ] res r 1 L M res A 3 M res ess sup I M res t 4 t,t L 5 r C 3 r = C r 3 2 c with c independent of. In the opposite case the complement of this set in M res we estimate as follows K 1 := ϑ ] r 1 res ϑ x M 3 dxdt = ϑ ] r res x ϑ 1 2 ϑ 3 4 r 3 2 x ϑ 3 2 res \A M 3 res \A }{{} dxdt T ϑ ] r res ϑ 3 4 r 3 2 L 4 3 M res \A 5.38 x ϑ 3 2 dxdt < c 5.39 provided ϑ ] r res is uniformly bounded in L 4, T, that is ϑ ] r+2 res is uniformly bounded in the space L 1, T. However we know that ϑ ] res is bounded in L, T; L 4 L r, T; L 3r as in 15]. By interpolation we get ϑ ] res is uniformly bounded in L r+8 3, T and that is why K 1 converges; moreover when we reiterate the same argument with a s power of its integrand, we obtain the bound s 6r+16 6r+15 for Hölder s inequality. Similarly to 15], using Proposition 5.1 and energy estimates, we see that s s s, ϑ 1 + ϑ s, ϑϑ 1 = 1 s, ϑ 29 3

32 weakly- in L, T; L 2 ; R 3, and remind 5.15, 5.29, 4.25, 4.18 and Moreover the advective part weakly converges according to Proposition 5.1 again s s u s, ϑ 1 + ϑ s, ϑϑ 1 U = s, ϑ 1 U, weakly in L 2, T; L 3/2 ; R 3. This allows to pass to the limit in all terms of 5.35 except the nonequilibrial radiative entropy flux term T q R q R x ϕ dx dt. 5.4 Let us compute the limit of qr qr. Applying once more Proposition 5.1 with G R I = nilog ni ni + 1lognI + 1 and integrating on S 2 R +, we find q R q R S 2 1 ν weakly- in L, T; L 2 ; R 3 on the essential set M ess and as log ni + 1 log ω I 1 d ω dν, ni ] ni+1 = ν, we have got ni ϑ q R qr 1 F R I 1, ϑ with the radiative momentum F R I 1 = ω I 1 d ω dν. So S 2 T q R q R T R div xf I 1 x ϕdxdt ϑ ϕ dx dt 5.41 by the Proposition 5.1, 4.25 and 4.18 on M res. As we have from 5.12 R div xf = ϑ σ a ν, ϑ ] Bν, ϑ I ϑ 1 + σ a ν, ϑ ϑ Bν, ϑϑ 1 I 1 d ω dν, the limit contribution in 5.35 becomes S 2 T σ a ν, ϑi 1 t, x, ω, ν S 2 ϑ ϕ d ω dν dx dt. Gathering all of these terms, we find the limit equation for entropy T s, ϑ 1 t φ + U x φ dx dt + κ T x Θ x ϕ dx dt ϑ 1 T σ a ν, ϑ I 1 t, x, ω, νϕd ω dν dx dt = s, ϑ 1 ϕ, dx. ϑ S 2 Using 5.7 we easily verify that we finally obtained the thermal equation 3.31 once we take the Maxwell relation ϑ p = s into account. 3

33 5.4 Maxwell equation From 5.2 and 5.4 we get and B u B U weakly in L q, T; L q, R 3 for q 1, 5, 3 λϑ curl x B λcurl x B weakly in L 34 6p+17, T, L 34 6p+17, R Then it is easy to pass to the limit in 4.5, realizing that This last step ends the proof of Theorem 4.2. A Appendix: Proof of Theorem 3.1 6p+17 > 1 for 1 p < Consider now the linearly coupled problem for the remaining equations div xu =, tu + U x U + x Π µ U + 1 B 2 ζ x 1 2 ζ B x B = F, A.1 A.2 tb + U x B + B x U λ B =, A.3 Θ = U β 1 κ div x B =, A.4 σ a S 2 I 1 d ω dν + h A.5 ω x I 1 + σ a I 1 σ s Ĩ1 I 1 =, A.6 where β L 3, together with the boundary conditions U =, Θ n =, B n =, curl xb n = for A.1-A.5 and I 1 x, ν, ω = for x, ω n for A.6, and the initial conditions A.7 A.8 U t= = U, B t= = B. A.9 We first consider the solution U, B, I 1 of the radiative-mhd problem div x U =, t U + U x U + x Π µ U = 1 ζ curl x B B + F, A.1 A.11 31

34 t B + U x B + B x U λ B =, div x B =, A.12 A.13 ω x I 1 + σ a I 1 σ s Ĩ1 I 1 =, A.14 with and U =, B n =, curl x B n =, U t= = U, B t= = B I 1 x, ν, ω = for x, ω n. The MHD part has a weak solution U L 2, T; U, B L 2, T; W thanks to an extension of the Leray-Hopf Theorem see 36]. Moreover the stationary radiative equation A.14 also has a weak solution I 1 L 2, T S 2 R + according to Theorem 1 and Proposition 2 of 2]. Then we consider the solution Θ of the stationary diffusion equation with Θ = U β 1 κ Θ n = σ a S 2 I 1 d ω dν + h A.15 subject to Θ dx = for all times. It admits a weak solution Θ L, T; W 2,2 L 2, T; W q,2 q < 5 2, thanks to classical elliptic regularity theory and due to regularity of the rhs due to 18]. Since the radiative-mhd problem does not depend on the temperature perturbation Θ the proof is complete. Acknowlegments: Š. N. was supported by the Czech Agency of the Czech Republic No S and by RVO Bernard Ducomet is partially supported by the ANR project INFAMIE ANR-15-CE4-11. References 1] R. Balian: From microphysics to macrophysics. Methods and applications of statistical physics Vol. II. Springer Verlag, Berlin, Heidelberg, New York, ] C. Bardos, F. Golse, B. Perthame, R. Sentis: The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation. J. Funct. Anal ] A.-M. Broomhall, P. Chatterjee, R. Howe, A. A. Norton, M. J. Thompson: The Sun s interior structure and dynamics, and the solar cycle. ArXiv: v2 astro-ph.sr] 25 Nov ] A.R. Choudhuri: The physics of fluids and plasmas, an introduction for astrophysicists. Cambridge University Press, ] D. Donatelli, B. Ducomet, Submitted Š. Nečasová: Low Mach number limit for a model of accretion disk, 32

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