TOPICAL PROBLEMS OF FLUID MECHANICS 53 ON THE PROBLEM OF SINGULAR LIMITS IN A MODEL OF RADIATIVE FLOW

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1 TOPICAL PROBLEMS OF FLUID MECHANICS 53 ON THE PROBLEM OF SINGULAR LIMITS IN A MODEL OF RADIATIVE FLOW Bernard Ducomet, Šárka Nečasová 2 CEA, DAM, DIF, F-9297 Arpajon, France 2 Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 5 67 Praha, Czech Republic Abstract We consider a semi-relativistic model of radiative viscous compressible Navier-Stokes-Fourier system coupled to the radiative transfer equation extending the classical model introduced in 8 and we study diffusion limits in the case of well-prepared initial data and Dirichlet boundary condition for the velocity field. Keywords: Radiation hydrodynamics, Navier-Stokes-Fourier system, weak solution, diffusion limits, well-prepared initial data Introduction In recent works singular limits low Mach number limit and diffusion limits for a simplified model of radiation hydrodynamics introduced by Teleaga, Seaïd, Gasser, Klar and Struckmeier in 23 have been presented. A more realistic model was studied in 8 however this more complete model suffers from a non manifestly positive production rate of total entropy, preventing ones from studying these singular limits. Our idea in the present paper is to introduce in the complete model of 8 a perturbed Planck s function and a suitable relativistic velocity cut off this is the meaning we give to semi-relativistic model allowing to recover this crucial positivity property for the production rate of total entropy. As the perturbation will be small going formally to zero as c, one can expect to obtain the correct limit regimes. The motion of the fluid is still described by standard non-relativistic fluid mechanics giving the evolution of the mass density ϱ = ϱt, x, the velocity field u = ut, x, and the temperature ϑ = ϑt, x as functions of the time t and the spatial coordinate x R 3. The effect of radiation is still incorporated in the radiative intensity I = It, x, ω, ν, depending on the direction ω S 2, where S 2 R 3 denotes the unit sphere, and the frequency ν, but we take into account their relativistic corrections. The evolution of I is described by a transport equation with a source term and the fluid-radiation coupling is expressed through radiative sources in the momentum and energy equations. More precisely, the system of equations to be studied reads as follows: t ϱ + div x ϱ u = in, T, t ϱ u + div x ϱ u u + x pϱ, ϑ = div x S S F in, T, 2 t 2 ϱ u 2 + ϱeϱ, ϑ + div x 2 ϱ u 2 + ϱeϱ, ϑ + p u + q S u = S E in, T, 3 c ti + ω x I = S in, T, S 2. 4 The symbol p = pϱ, ϑ denotes the thermodynamic pressure and e = eϱ, ϑ is the specific internal energy, related through Maxwell s equation e ϱ = ϱ 2 pϱ, ϑ ϑ p. 5 ϑ In 2 S is the viscous stress tensor given by S = µ x u + t x u 2 3 div x u + η div x u I, where the viscosity coefficients µ = µϑ > and η = ηϑ are effective functions of the temperature.

2 54 Prague, February -3, 25 Similarly in 3 q is the heat flux given by Fourier s law q = κ x ϑ, with the heat conductivity coefficient κ = κϑ >. We suppose that the radiative source S is given by S = σ a Bν, ω, u, ϑ It, x, ν, ω + σ s It, x, ν, ω d ω It, x, ν, ω =: S a,e + S s. 6 4π S 2 In the right-hand side the first term is the emission-absorption contribution where σ a > is the absorption coefficient and B is a perturbation of the equilibrium Planck s function given by Bν, ω, u, ϑ = 2h ν 3 c 2 hν kϑ e α ω u c, 7 where h is the Planck s constant, k is the Boltzmann s constant and αϑ is a smooth function, to be determined below. One observes that for u c << one recovers the standard equilibrium Planck s function Bν, ϑ = 2h ν 3 c. 2 e kϑ hν Note that the idea of this kind of perturbation is not new and has been extensively used in recent works on radiative transfer 4,5,7,6, for exemple in the M Levermore model 6,7. The second term in S is the scattering contribution where σ s > is the scattering coefficient and in the right-hand sides of 2 and 3 appear the coupling sources. S F t, x = c S 2 ωs d ω dν, S E t, x = S 2 S d ω dν. 8 We first suppose that the transport coefficients are smooth functions satisfying σ a ϑ, u = χ u σ a ϑ and σ s ϑ and that both depend neither on angular variable - 4 isotropy of radiation, nor on frequency the so called grey hypothesis. The function χ appearing in the emission-absorption coefficient is a C cut-off satisfying { if s c, χs = if s c + β, for an arbitrary β >. The role of this cut-off is to deal with the singularity of B and its meaning is the following: in the over-relativistic regime u c where special relativity would be violated, we decide to decouple matter and radiation. Of course this is an arbitrary choice but only a meaningless region with respect to physics is concerned recall that in the relativistic setting 5, /2 Lorentz factors of the type u2 c become singular for u = c. Finally system - 4 is 2 supplemented with the boundary conditions: u =, q n =, 9 It, x, ν, ω Γ =, Γ {{x, ω} S 2, ω n }, where n denotes the outer normal vector to, and initial conditions ϱt, x, ut, x, ϑt, x, It, x, ω, ν t= = ϱ x, u x, ϑ x, I x, ω, ν, for any x, ω S 2,ν R +. The relativistic version of system - has been introduced by Pomraning 2 and Mihalas and Weibel-Mihalas 2 and investigated more recently in astrophysics and laser applications in the inviscid case by Lowrie, Morel and Hittinger 8 and Buet and Desprès 5, with a special attention to asymptotic regimes, and this last paper was a deep source of inspiration for the present work. Let us mention that a simplified version of the system non relativistic non conducting fluid at rest has been investigated by Golse and Perthame in 5 where global existence was proved under very mild hypotheses transport coefficients may be singular. A global existence result was

3 TOPICAL PROBLEMS OF FLUID MECHANICS 55 also proved in 8 for the simplified model without relativistic corrections, under some cut-off hypotheses on transport coefficients. Hypotheses: We consider the pressure in the form pϱ, ϑ = ϑ 5/2 P ϱ ϑ 3/2 + a 3 ϑ4, a >, 2 where P :,, is a given function with the following properties: P C,, P =, P Z >, for all Z, 3 < 5 3 P Z P ZZ Z lim Z < c for all Z, 4 P Z Z 5/3 = p >. 5 After Maxwell s equation 5, the specific internal energy e is eϱ, ϑ = 3 ϑ 5/2 ϱ P + a ϑ4 2 ϱ ϑ 3/2 ϱ, 6 and the associated specific entropy reads sϱ, ϑ = M ϱ ϑ 3/2 + 4a 3 ϑ 3 ϱ with M Z = P Z P ZZ Z 2 <. 7 The transport coefficients µ, η, and κ are continuously differentiable functions of the absolute temperature such that < c + ϑ µϑ, µ ϑ < c 2, ηϑ c + ϑ, 8 < c + ϑ 3 κϑ c 2 + ϑ 3 9 for any ϑ. Moreover we assume that σ a and σ s are smooth functions such that σ a ϑ, u, σ s ϑ c, σ a ϑ, ubν, ω, u, ϑ c 2, 2 σ a ϑ, ubν, ω, u, ϑ hν, h L,, 2 where c,2,3 are positive constants. Relations 2-2 represent cut-off hypotheses neglecting the effect of radiation at large temperature and ultra relativistic velocities see 22 for physical motivations. 2 Diffusion limits Diffusion limits consist in supposing that one of the transport coefficient is small while the other is large. These regimes have been introduced by Lowrie, Morel et Hittinger 8 and also considered by Buet and Desprès 5. In order to identify the appropriate limit regimes we perform two different scalings. The first one corresponds to the equilibrium diffusion regime defined in 5 by leading to the primitive system Ma = Sr = P e = Re = P =, C = ε, L s = ε 2 and L = ε, ε t I + ω x I = ε σ a B I + εσ s 4π S 2 I d ω I, 22 t ϱ + div x ϱ u =, 23

4 56 Prague, February -3, 25 with t ϱ u + εf R + div x ϱ u u + P R + x p div x S =. 24 t 2 ϱ u 2 + ϱe + E R F + div x 2 ϱ u 2 + ϱe + p u + R + q S u =, 25 ε q t ϱs + εs R + div x ϱ us + q R + div x ϑ ϑ nb + ε +ε S ν 2 S 2 ν ni ni + ni ni + d ϱe + E R dx + dt ε nb + nĩ nĩ + where E = 2 u 2 + e, the entropy of a photon gas is defined as n = ni = as c2 I 2hα 3 ν 3 nb nb+ = hν kϑ s R = 2k c 3 S : x u q xϑ ϑ σ a I B d ωdν, σ s I Ĩ d ωdν 26 Γ + ω n I dγ + dν =, 27 S 2 ν 2 n n n + n + d ωdν, is the occupation number, the radiative entropy flux is defined as q R = 2k c 2 α ω u c and the radiative momentum F R t, x = ν 2 n n n + n + ω d ωdν, S 2, where B is the Planck s function. The radiative energy E R is defined E R t, x = It, x, ω, ν d ω dν, 28 c S 2 S 2 ωit, x, ω, ν d ω dν. 29 The second one is the non-equilibrium diffusion regime also defined in 5 by Ma = Sr = P e = Re = P =, C = ε, L = ε 2 and L s = ε. One checks that equations and 27 still hold in this scaling. The new transport equation is ε t I + ω x I = εσ a B I + ε σ s I d ω I, 3 4π S 2 and the new entropy inequality is q t ϱs + εs R + div x ϱ us + q R + div x ϑ ϑ +ε + ε S ν 2 S 2 ν ni ni + ni ni + nb nb + nĩ nĩ + Target system in the equilibrium-diffusion regime S : x u q xϑ ϑ σ a I B d ωdν σ s I Ĩ d ωdν. 3 t ϱ + div x ϱ u =, 32

5 TOPICAL PROBLEMS OF FLUID MECHANICS 57 t ϱ u + div x ϱ u u + x p = div x S, 33 t 2 ϱ u 2 + ϱe + div x 2 ϱ u 2 + ϱe + p u + q S u =, 34 q t ϱs + div x ϱs u + div x = ϑ ϑ S : x u q xϑ, 35 ϑ I = Bν, ϑ, 36 where pϱ, ϑ = pϱ, ϑ + a 3 ϑ4, eϱ, ϑ = eϱ, ϑ + a ϱ ϑ4, kϑ = κϑ + 4a 3σ a ϑ 3, q = kϑ x ϑ and ϱs = ϱs aϑ3. We also get boundary conditions and initial conditions u =, ϑ n =, 37 ϱx, t, ux, t, ϑx, t t= = ϱ x, u x, ϑ x, 38 for any x with the following compatibility conditions u x =, ϑ n =. 39 As expected, this system corresponds to a viscous compressible heat-conductive fluid at local thermodynamical equilibrium with radiation, equilibrium being achieved between matter and radiation with radiative intensity I = Bν, ϑ, corresponding to the black-body radiation at temperature ϑ with radiative energy E R ϑ = aϑ 4. From the classical results of Matsumura and Nishida 9 it can be derived the existence results see 2. We adapt from 3 the necessary definitions to the formalism of essential and residual sets. Given three numbers ϱ R +, ϑ R + and E R + we define Oess H the set of hydrodynamical essential values O H ess := { ϱ, ϑ R 2 : and O R ess the set of radiative essential values O R ess := { E R R : with O ess := O H ess O R ess, and their residual counterparts } ϱ 2 < ϱ < 2ϱ, ϑ 2 < ϑ < 2ϑ, 4 } E 2 < ER < 2E, 4 O H res := R + 2 \O H ess, O R res := R + \O R ess, O res := R + 3 \O ess. 42 Theorem 2. Let R 3 be a bounded domain of class C 2,ν. Assume that the thermodynamic functions p, e, s satisfy hypotheses 2-7 with P C, C 2,, and that the transport coefficients µ, η, κ, σ a, σ s and the equilibrium function B comply with 8-2. Let ϱ ε, u ε, ϑ ε, I ε be a weak solution to the scaled radiative Navier-Stokes system for t, x, ω, ν, T S 2 R +, supplemented with boundary conditions 9 - and initial conditions ϱ,ε, u,ε, ϑ,ε, I,ε such that ϱ ε, = ϱ + εϱ,ε, u ε, = u,ε, ϑ ε, = ϑ + εϑ,ε, where ϱ, u, ϑ H 3 are smooth functions such that ϱ, ϑ belong to the set Oess H where ϱ >, ϑ >, are two constants and ϱ,ε dx =, ϑ,ε dx =. Suppose also that u,ε u strongly in L ; R 3, ϱ,ε ϱ strongly in L 2, ϑ,ε ϑ strongly in L 2.

6 58 Prague, February -3, 25 Then up to subsequences ϱ ε ϱ strongly in L, T ; L 5 3, uε u strongly in L 2, T ; W,2 ; R 3, ϑ ε ϑ strongly in L, T ; L 4, I ε Bν, θ strongly in L, T S 2,, where ϱ, u, ϑ is the smooth solution of the equilibrium decoupled system on, T, with initial data ϱ, u, ϑ. The target system in the non-equilibrium diffusion regime We obtain a compressible Navier-Stokes-Fourier system with sources coupled to a diffusion equation for N. t ϱ + div x ϱ u =, 43 t ϱ u + div x ϱ u u + x p = div x S, 44 t 2 ϱ u 2 + ϱe + div x 2 ϱ u 2 + ϱe + p u + q S u =, 45 q t ϱs + div x ϱs u + div x = ϑ ϑ t N 3 div x S : x u q xϑ ϑ σ s ϑ xn + x N u σ aϑ aϑ 4 N, 46 3 ϑ ϑ = σ a ϑ aϑ 4 N, 47 where p = p + 3 N, e = e + N ϱ and q = κ xϑ + 3σ s x N with boundary conditions initial conditions u =, ϑ n =, N =, 48 ϱx, t, ux, t, ϑx, t, Nx, t t= = ϱ x, u x, ϑ x, N x, 49 for any x, with N x = I x, ν, ω d ω dν and the compatibility conditions S 2 u 2 =, ϑ n =, N =. 5 It will be useful as in 5 to define the non-equilibrium temperature θ r by N = aθ 4 r. 5 In anay with previous works on asymptotic analysis of radiative transfer equation see 2, 3 we call the Navier-Stokes-Rosseland system. As in the equilibrium case, we have a global existence result for solutions of this problem for small data for more details see 2. Theorem 2.2 Let R 3 be a bounded domain of class C 2,ν. Assume that the thermodynamic functions p, e, s satisfy hypotheses 2-7 with P C, C 2,, and that the transport coefficients µ, λ, κ, σ a, σ s and the equilibrium function B comply with 8-2. Let ϱ ε, u ε, ϑ ε, I ε be a weak solution to the system , 27, 3, 3 for t, x, ω, ν, T S 2 R +, supplemented with the boundary conditions 9 - and the initial conditions ϱ,ε, u,ε, ϑ,ε, I,ε such that ϱ ε, = ϱ + εϱ,ε, u ε, = u,ε, ϑ ε, = ϑ + εϑ,ε, I ε, = I + εi,ε, where the functions ϱ, u, ϑ and x I x, ω, ν belong to H 3 and are such that ϱ, ϑ, E R I belong to the set O ess. Suppose also that u,ε u strongly in L ; R 3, ϱ,ε ϱ strongly in L 2, ϑ,ε ϑ strongly in L 2, I,ε I strongly in L, T,.

7 TOPICAL PROBLEMS OF FLUID MECHANICS 59 Then up to subsequences ϱ ε ϱ strongly in L, T ; L 5 3, uε u strongly in L 2, T ; W,2 ; R 3, ϑ ε ϑ strongly in L, T ; L 4, N ε N strongly in L, T, where N ε = I S 2 ε d ω dν and ϱ, u, ϑ, N is the smooth solution of the Navier-Stokes-Rosseland system on, T with initial data ϱ, u, ϑ, N. Proofs of Theorems 2., 2.2 are based on the theory of singular limits 3 and the relative entropy inequality 4. We just give the sketch of the proof. We introduce a relative entropy inequality satisfied by any weak solution ϱ, u, ϑ, I of the radiative Navier-Stokes -Fourier system. Let us consider a set {r, Θ, U} of arbitrary smooth functions such that r and Θ are bounded below away from zero and U =. We call ballistic free energy the thermodynamical potential given by H Θ ϱ, ϑ = ϱeϱ, ϑ Θϱsϱ, ϑ, and radiative ballistic free energy the potential H R Θ I = E R I Θs R I. The relative entropy is then defined by Eϱ, ϑ r, Θ := H Θ ϱ, ϑ ρ H Θ r, Θϱ r H Θ r, Θ. Then the relative entropy inequality of the radiative Navier-Stokes-Fourier system is the following τ 2 ϱ ε u ε U 2 + E ϱ ε, ϑ ε r, Θ + εh R I ε τ, dx + ω n x I ε t, x, ω, ν dγ dν dt Γ + τ Θ + S ε : x u ε q ε x ϑ ε dx dt τ + τ + S 2 2 Θ ν ϑ ε ϑ ε ni ε ni ε + nb ε nb ε + j σ a ε B ε I ε d ω dν dx dt Θ ni ε S ν ni 2 ε + nĩε j σ s ε Ĩε I ε d ω dν dx dt, nĩε + ϱ,ε u,ε U, 2 + E ϱ,ε, ϑ,ε r,, Θ, + εh R I,ε dx τ + ϱ ε u ε U xu U uε τ τ τ For more details see 2. τ + p ε div xu Sε : xu ϱ ε s ε sr, Θ t Θ q ε ϑ ε x Θ dx dt + τ τ dx dt + ϱ ε s ε sr, Θ U uε x Θ dx dt U uε ϱ ε t U + U x U τ τ dx dt dx dt τ dx dt εs R ε t Θ + q R ε x Θ dx dt ϱ ε s ε sr, Θ U x Θ dx dt ϱ ε t pr, Θ ϱ ε r r u ε x pr, Θ dx dt εf ε R tu + P R ε : xu dx dt. 52 Acknowlegment: Šárka Nečasová acknowledges the support of the GAČR Czech Science Foundation project P S in the framework of RVO: Part of work has been done during her visit of CEA and she would like to thank for wonderfull hospitality of Prof. Ducomet and his colleagues.

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