Diffusion limits in a model of radiative flow

Size: px

Start display at page:

Download "Diffusion limits in a model of radiative flow"

Augustine Parker
5 years ago
Views:

1 Institute of Mathematics of the Academy of Sciences of the Czech Republic Workshop on PDE s and Biomedical Applications

2 Basic principle of mathematical modeling Johann von Neumann [ ] In mathematics you don t understand things. You just get used to them.

3 General questions Compressible vs. incompressible Is air compressible? Is it important? Is the physical space bounded or unbounded?

4 Viscous vs. inviscid What is turbulence? Do extremely viscous fluids exhibit turbulent behavior?

5 Gaseous stars in astrophysics The effect of coupling between the macroscopic description of the fluid and the statistical character of the motion of the massless photons.

6 Eulerian description of motion Physical space time t [, ) position x Ω R 3 Leonhard Paul Euler [ ]

7 Thermostatic variable mass density = (t,x) Motion macroscopic velocity u = u(t, x) d X(t, x) dt ( = u t, X(t, x) ), X(, x) = x

8 Momentum balance t ( u) + div x ( u u) + x p( ) = div x S + f Newton s rheological law ( S = µ x u + t x u 2 ) 3 div x ui + ηdiv x ui Mass conservation t + div x ( u) =

9 A mathematical model of radiative flow t + div x ( u) = in (,T) Ω, (1.1) t ( u) + div x ( u u) + x p(,ϑ) = div x S+ S F in (,T) Ω, ( )) ( ( ) ) (1.2) 1 1 t ( 2 u 2 + e(,ϑ) + div x 2 u 2 + e(,ϑ) u (1.3) ) +div x (p u + q S u = S E in (,T) Ω, 1 c ti + ω x I = S in (,T) Ω (, ) S 2. (1.4)

10 the mass density = (t,x), the velocity field u = u(t,x), the temperature ϑ = ϑ(t, x) the radiative intensity I = I(t, x, ω, ν), depending on the direction ω S 2, S 2 R 3 the unit sphere, the frequency ν.

11 p = p(,ϑ) the thermodynamic pressure e = e(,ϑ) is the specific internal energy Maxwell s equation e = 1 2 ( p(,ϑ) ϑ p ). (1.5) ϑ S is the viscous stress tensor ( S = µ x u + t x u 2 ) 3 div x u + η div x u I, (1.6) the shear viscosity coefficient µ = µ(ϑ) > the bulk viscosity coefficient η = η(ϑ) are effective functions of the temperature

12 q is the heat flux given by Fourier s law q = κ x ϑ, (1.7) the heat conductivity coefficient κ = κ(ϑ) >. S = S a,e + S s, (1.8) ) S a,e = σ a (B(ν,ϑ) I ), S s = σ s (Ĩ I. (1.9) S E = S 2 S(,ν, ω) dν d ω, S F = 1 c the absorption coefficient σ a = σ s (ν,ϑ), the scattering coefficient σ s = σ s (ν,ϑ) S 2 ωs(, ν, ω) dν d ω, (1.1)

13 Ĩ := 1 I(, ω) d ω 4π S 2 B(ν,ϑ) = 2hν 3 c 2 ( e hν kϑ 1 ) 1 the radiative equilibrium function h and k are the Planck and Boltzmann constants,

14 the boundary conditions u Ω =, q n Ω =, (1.11) I(t,x,ν, ω) = for x Ω, ω n, (1.12) n the outer normal vector to Ω.

15 Pomraning Mihalas and Weibel-Mihalas in the framework of special relativity. astrophysics, laser applications (in the relativistic and inviscid case) by Lowrie, Morel and Hittinger, Buet and Després with a special attention to asymptotic regimes Dubroca and Feugeas, Lin, Lin, Coulombel and Goudon a simplified version of the system (non conducting fluid at rest) - investigated by Golse and Perthame, where global existence was proved by means of the theory of nonlinear semi-groups under very general hypotheses.

16 Global weak solutions barotropic case P. L. Lions (98) p( ) = γ, γ 9/5 generalization to a larger class of exponents γ > 3/2 E. Feireisl, A. Novotný and H. Petzeltová Full system - the Navier - Stokes - Fourier system global existence of weak solution E. Feireisl Singular limits for Navier type of boundary conditions, Concept of weak- strong uniqueness E. Feireisl, A. Novotný

17 Hypotheses Pressure p(,ϑ) = ϑ 5/2 P ( ϑ 3/2 ) + a 3 ϑ4, a >, (2.1) P : [, ) [, ) P C 1 [, ), P() =, P (Z) > for all Z, (2.2) < 5 3 P(Z) P (Z)Z < c for all Z, Z (2.3) P(Z) lim Z Z 5/3 = p >. (2.4) a 3 ϑ4 - equilibrium radiation pressure.

18 the specific internal energy e is e(,ϑ) = 3 2 ϑ ( ϑ 3/2 ) the associated specific entropy reads s(,ϑ) = M P ( ϑ 3/2 ) + a ϑ4, (2.5) ( ) ϑ 3/2 + 4a ϑ 3 3, (2.6) with M (Z) = P(Z) P (Z)Z Z 2 <.

19 < c 1 (1 + ϑ) µ(ϑ), µ (ϑ) < c 2, η(ϑ) c(1 + ϑ), (2.7) for any ϑ. < c 1 (1 + ϑ 3 ) κ(ϑ) c 2 (1 + ϑ 3 ) (2.8) σ a (ν,ϑ),σ s (ν,ϑ), ϑ σ a (ν,ϑ), ϑ σ s (ν,ϑ) c 1, (2.9) σ a (ν,ϑ)b(ν,ϑ), ϑ {σ a (ν,ϑ)b(ν,ϑ)} c 2, (2.1) σ a (ν,ϑ),σ s (ν,ϑ),σ a (ν,ϑ)b(ν,ϑ) h(ν), h L 1 (, ). (2.11) for all ν, ϑ, where c 1,2,3 are positive constants. Relations (2.9) - (2.11) represent cut-off hypotheses neglecting the effect of radiation at large frequencies ν

20 Weak formulation: (weak) renormalized version represented by the family of integral identities T T + Ω Ω (b( ) t ϕ + b( ) u x ϕ) dx dt (2.12) (( ) ) b( ) b ( ) div x uϕ = b( )ϕ(, ) dx Ω dxdt satisfied for any ϕ Cc ([,T) Ω), and any b C [, ), b Cc [, ), where (2.12) implicitly includes the initial condition (, ) =.

21 The momentum equation (1.2) is replaced by T Ω T = ( u t ϕ + u u : x ϕ + pdiv x ϕ) dx dt (2.13) S : x ϕ + S F ϕ dx dt Ω for any ϕ C c ([,T) Ω; R 3 ). Ω ( u) ϕ(, ) dx u L 2 (,T;W 1,2 (Ω; R 3 )), (2.14) where (2.14) already includes the no-slip boundary condition (1.11).

22 the entropy equation is replaced by the inequality T Ω ( s t ϕ + u x ϕ + qϑ x ϕ) dx dt (2.15) T Ω T ( 1 ϑ Ω Ω ( s) ϕ(, ) dx S : x u q xϑ ϑ ) ϕ dx dt 1 ( u ϑ S ) F S E ϕ dx dt for any ϕ C c ([,T) Ω), ϕ. where not the sign of all the terms in the right hand side may be controlled.

23 the total energy balance ( ) 1 2 u 2 + e(,ϑ) + E R (τ, ) dx (2.16) Ω τ + = Ω S 2, ω n Ω where E R is given by ω ni(t,x, ω,ν) dν d ω ds x dt ( ) 1 ( u) 2 + ( e) + E R, dx, 2 E R (t,x) = 1 c S 2 and E R, = S 2 I (, ω,ν) d ω dν. I(t,x, ω,ν) d ω dν. (2.17)

24 Definition 2.1 We say that, u,ϑ,i is a weak solution of problem ( ) if, ϑ > for a.a. (t,x) Ω, I a.a. in (,T) Ω S 2 (, ), L (,T;L 5/3 (Ω)), ϑ L (,T;L 4 (Ω)), u L 2 (,T;W 1,2 (Ω; R 3 )), ϑ L 2 (,T;W 1,2 (Ω)), I L ((,T) Ω S 2 (, )), I(t, ) L (,T;L 1 (Ω S 2 (, )), and if, u, ϑ, I satisfy the integral identities (2.12), (2.13), (2.15), (2.16), together with the transport equation (1.4).

25 Theorem (B. Ducomet, E.Feireisl,Š. Nečasová) Let Ω R3 be a bounded Lipschitz 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 { ε, u ε,ϑ ε,i ε } ε> be a family of weak solutions to problem ( ) in the sense of Definition 2.1 such that ε(, ) ε, in L 5/3 (Ω), (2.18) ( ) 1 ε Ω 2 ε u 2 + εe( ε,ϑ ε ) + E R,ε (, ) dx (2.19) ( ) 1 ( u),ε 2 + ( e),ε + E R,,ε dx E, 2,ε Ω

26 Then and ε in C weak ([,T];L 5/3 (Ω)), u ε u weakly in L 2 (,T;W 1,2 (Ω; R 3 )), ϑ ε ϑ weakly in L 2 (,T;W 1,2 (Ω)), I ε I weakly-(*) in L ((,T) Ω S 2 (, )), at least for suitable subsequences, where {, u,ϑ,i } is a weak solution of problem ( ).

27 Asymptotic analysis Scaling Scaled equations Mass conservation X X X char Momentum balance [Sr] t + div x ( u) = [Sr] t ( u) + div x ( u u) + [Ro] ω 1 [ ] 1 u + Ma 2 x p( ) [ ] 1 = ( x u + λ x div x u) + (external forces) Re

28 Characteristic numbers - Strouhal number Čeněk Strouhal [ ] Strouhal number [Sr] = length char time char velocity char Scaling by means of Strouhal number is used in the study of the long-time behavior of the fluid system, where the characteristic time is large

29 Mach number Ernst Mach [ ] Mach number [Ma] = velocity char pressurechar /density char Mach number is the ratio of the characteristic speed to the speed of sound in the fluid. Low Mach number limit, where, formally, the speed of sound is becoming infinite, characterizes incompressibility

30 Reynolds number Osborne Reynolds [ ] Reynolds number [Re] = density charvelocity char length char viscosity char High Reynolds number is attributed to turbulent flows, where the viscosity of the fluid is negligible

31 Rossby number Carl Gustav Rossby [ ] Rossby number [Ro] = velocity char ω char length char Rossby number characterizes the speed of rotation of the fluid

32 1.Simplified model S F = Derivation of the entropy inequality The internal energy equation t ( e) + div x ( e u) + div x q = S : x u pdiv x u S E. (2.2) the entropy equation ( ) q t ( s) + div x ( s u) + div x = ς, (2.21) ϑ where ς m := 1 ϑ production. ( ς = 1 ϑ ( S : x u q xϑ ϑ ) S E ϑ, (2.22) S : x u q xϑ ϑ ) is the (positive) matter entropy

33 The formula for the entropy of a photon gas s R = 2k c 3 ν 2 [n log n (n + 1)log(n + 1)] d ωdν, (2.23) S 2 n = n(i) = c2 I is the occupation number. 2hν 3 Defining the radiative entropy flux q R = 2k c 2 t s R + div x q R = k h S 2 ν 2 [n log n (n + 1)log(n + 1)] ω d ωdν, (2.24) 1 S 2 ν log n n + 1 S d ωdν =: ςr. (2.25)

34 ( t s + s R) ( + div x s u + q R) ( ) q + div x = ς + ς R. (2.26) ϑ The right-hand side of (2.65) rewrites S E ϑ k h k h 1 S 2 ν S 2 1 ν [ [ log log ς R =: n(i) n(i) + 1 log n(i) n(i) + 1 log ] n(b) σ a (B I) d ωdν n(b) + 1 ] n(ĩ) σ s (Ĩ I) d ωdν, n(ĩ) + 1

35 Rescaling We denote by Sr := L ref T ref U ref, Ma := U ref ρref p ref, Re := U ref ρ ref L ref µ ref, Pe := U ref p ref L ref ϑ ref κ ref, C := c U ref, the Strouhal, Mach, Reynolds, Péclet (dimensionless) and infrarelativistic numbers corresponding to hydrodynamics, and by L := L ref σ a,ref, L s := σ s,ref σ a,ref, P := 2k4 B ϑ4 ref h 3 c 3 ρ ref e ref, various dimensionless numbers corresponding to radiation.

36 ( Sr 1 C ti + ω x I = s = Lσ a (B I) + LL s σ s 4π S 2 I d ω I ), (2.27)

37 The continuity equation is now and the momentum equation Sr t + div x ( u) =, (2.28) Sr t ( u)+div x ( u u)+ 1 Ma 2 xp(,ϑ) 1 Re div xt =. (2.29)

38 The balance of matter (fluid) entropy Sr t ( s) + div x ( s u + P q R) + 1 Pe div x ( ) q = ς, (2.3) ϑ with ς = 1 ϑ ( Ma 2 Re S : x u 1 Pe ) q x ϑ S E ϑ ϑ,

39 The balance of radiative entropy with ς R = L +LL s 1 S 2 ν S 2 1 ν [ Sr C ts R + div x q R = ς R, (2.31) [ log log n(i) n(i) + 1 log n(i) n(i) + 1 log n(b) n(b) + 1 ] n(ĩ) n(ĩ) + 1 ] σ a (I B) d ωdν σ s (I Ĩ) d ωdν+s E ϑ.

40 The scaled equation for total energy gives finally the total energy balance ( d Ma 2 Sr u 2 + e + 1 ) dt Ω 2 C ER dx+ (2.32) +P ω ni dγ + dν =. Γ +

41 Diffusion limit equilibrium diffusion regime P = O(ε)- a small amount of radiation is present C = O(ε 1 ) -the flow is strongly under-relativistic Ma = Sr = Pe = Re = 1, P = ε, C = ε 1, L s = ε 2 and L = ε 1, ε t I + ω x I = 1 ( ) 1 ε σ a (B I) + εσ s I d ω I, (2.33) 4π S 2 t + div x ( u) =, (2.34) t ( u) + div x ( u u) + x p(,ϑ) div x T =. (2.35)

42 ( ) q t ( s + εs R )+div x ( us + q R )+ div x 1 ( S : x u q ) xϑ ϑ ϑ ϑ + 1 ε + ε 1 S 2 ν S 2 1 ν [ log [ log n(i) n(i) + 1 log ( ) d 1 dt Ω 2 u 2 + e + E R ] n(b) σ a (I B) d ωdν n(b) + 1 ] n(ĩ) n(ĩ) + 1 σ s (I Ĩ) d ωdν, (2.36) dx + 1 ω n I dγ + dν =. ε Γ + (2.37) n(i) n(i) + 1 log

43 the non-equilibrium diffusion regime Ma = Sr = Pe = Re = 1, P = ε, C = ε 1, L = ε 2 and L s = ε 1. ε t I + ω x I = εσ a (B I) + 1 ( ) 1 ε σ s I d ω I, (2.38) 4π S 2 t + div x ( u) =, (2.39) t ( u) + div x ( u u) + x p(,ϑ) div x T =. (2.4)

44 ( ) q t ( s + εs R )+div x ( us + q R )+div x 1 ( S : x u q ) xϑ ϑ ϑ ϑ +ε + 1 ε 1 S 2 ν S 2 1 ν [ log [ log n(i) n(i) + 1 log n(i) n(i) + 1 log ] n(b) σ a (I B) d ωdν n(b) + 1 ] n(ĩ) n(ĩ) + 1 σ s (I Ĩ) d ωdν. (2.41)

45 The limit system equilibrium system t + div x ( u) =, (2.42) t ( u) + div x ( u u) = div x T(,ϑ), (2.43) t ( E(,ϑ)) + div x ( e(,ϑ) u) + div x (K(,ϑ) x ϑ) = S(,ϑ) : x u p(,ϑ)div x u, (2.44) I = B(ν,ϑ). (2.45) u Ω =, ϑ n Ω =, (2.46) ( (x,t), u(x,t), ϑ(x,t)) t= = ( (x), u (x), ϑ (x) ), (2.47) with the following compatibility conditions u (x) Ω =, ϑ n Ω =. (2.48)

46 E(,ϑ) = e(,ϑ) + B(ϑ) 1, and K(ϑ) = κ(ϑ) 3σ a(ϑ) ϑb(ϑ).

47 existence of a global solution for the small data - Matsumura and Nishida Remark: When one considers the formal nonconducting at rest situation where κ = and u = and in the no-scattering case (σ s ), one obtains the simplified system introduced by Bardos, Golse and Perthame for which they proved global existence and diffusion limit (called Rosseland approximation ) under assumptions much more general than ours.

48 The non-equilibrium diffusion regime t + div x ( u) =, (2.49) t ( u) + div x ( u u) = div x T(,ϑ), (2.5) t ( e(,ϑ)) + div x ( e(,ϑ) u) + div x (κ(ϑ) x ϑ) = S(,ϑ) : x u p(,ϑ)div x u σ a (ϑ)[b(ϑ) N], (2.51) t N 1 ( ) 1 3 div x σ s (ϑ) xn = σ a (ϑ)(b(ϑ) N). (2.52) u Ω =, ϑ n Ω =, (2.53) N := I dν

49 N Ω =. (2.54) ( (x,t), u(x,t), ϑ(x,t),n(x,t)) t= = ( (x), u (x), ϑ (x),n (x) ), (2.55) and the compatibility conditions u 2 Ω =, ϑ n Ω =, N Ω =. (2.56) N (x) = S I (x,ν, ω) d ω dν. 2

50 Strong solution of the limit system for small data

51 Given three numbers R +, ϑ R + and E R + we define O H ess the set of hydrodynamical essential values O H ess := {(,ϑ) R 2 : } 2 < < 2, ϑ 2 < ϑ < 2ϑ, (2.57) O R ess the set of radiative essential values O R ess := {E R R : } E 2 < ER < 2E, (2.58) with O ess := Oess H OR ess, and their residual counterparts O H res := (R + ) 2 \O H ess, O R res := R + \O R ess, O res := (R + ) 3 \O ess. (2.59)

52 Theorem Let Ω 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, λ, κ, σ a, σ s and the equilibrium function B comply with ( ), B C 1. Let ( ε, u ε,ϑ ε,i ε ) be a weak solution to the scaled radiative Navier-Stokes system for (t,x, ω,ν) [,T] Ω S 2 R +, supplemented with the boundary conditions ( ) and the initial conditions (,ε, u,ε,ϑ,ε,i,ε )

53 such that ε(, ) = + ε (1),ε, u ε(, ) = u,ε, ϑ ε (, ) = ϑ + εϑ (1),ε, where (, u,ϑ ) H 3 (Ω) are smooth functions such that (,ϑ ) belong to the set Oess H defined in (2.59) where >, >, are two constants and (1) Ω,ε dx =, Ω ϑ(1),ε dx =.

54 Suppose also that u,ε u strongly in L (Ω; R 3 ), (1),ε (1) strongly in L 2 (Ω), ϑ (1),ε ϑ(1) strongly in L 2 (Ω), I (1),ε I(1) strongly in L ((,T) Ω (, )). Then up to subsequences ε strongly in L (,T;L 5 3 (Ω)), u ε u strongly in L 2 (,T;W 1,2 (Ω; R 3 )), ϑ ε ϑ strongly in L (,T;L 4 (Ω)), where (, u,ϑ) is the smooth solution of the equilibrium decoupled system (2.42)-(2.45) on [,T] Ω and I(t,x,ν, ω) = B(ν,ϑ(t,x)), with initial data (, u,ϑ ).

55 Theorem Let Ω 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, λ, κ, σ a, σ s and the equilibrium function B comply with ( ), B C 1. Let ( ε, u ε,ϑ ε,i ε ) be a weak solution to the scaled radiative Navier-Stokes system for (t,x, ω,ν) [,T] Ω S 2 R +, supplemented with the boundary conditions ( ) and the initial conditions (,ε, u,ε,ϑ,ε,i,ε ) such that

56 ε(, ) = + ε (1),ε, u ε(, ) = u,ε, ϑ ε (, ) = ϑ + εϑ (1),ε, I ε (, ) = I + εi (1),ε, where the functions (, u,ϑ ) and x I (x, ω,ν) belong to H 3 (Ω) and are such that (,ϑ,e R (I )) belong to the set O ess defined in (2.59) where >, >,E R > are three constants and (1) Ω,ε dx =, Ω ϑ(1),ε dx =, Ω I(1),ε dx =.

57 Suppose also that u,ε u strongly in L (Ω; R 3 ), (1),ε (1) strongly in L 2 (Ω), ϑ (1),ε ϑ(1) strongly in L 2 (Ω), I (1),ε I(1) strongly in L ((,T) Ω (, )). Then up to subsequences ε strongly in L (,T;L 5 3 (Ω)), u ε u strongly in L 2 (,T;W 1,2 (Ω; R 3 )), ϑ ε ϑ strongly in L (,T;L 4 (Ω)),

58 and N ε N strongly in L ((,T) Ω), where N ε = S I 2 ε d ω dν and (, u,ϑ,n) is the smooth solution of the Navier-Stokes-Rosseland system (2.49)-(2.52) on [,T] Ω with initial data (, u,ϑ,n ).

59 We establish a relative entropy inequality satisfied by any weak solution (, u, ϑ, I) of the radiative Navier-Stokes system Let us consider a set {r,θ, U} of smooth functions such that r and Θ are bounded below away from zero and U =. Ω We call ballistic free energy the thermodynamical potential given by H Θ (,ϑ) = e(,ϑ) Θ s(,ϑ), and radiative ballistic free energy the potential H R Θ (I) = ER (I) Θs R (I). The relative entropy is then defined by E(,ϑ r,θ) := H Θ (,ϑ) H Θ (r,θ)( Θ) H Θ (r,θ).

60 ( ) 1 2 ε u ε U 2 + E ( ε,ϑ ε r,θ) + εh R (I ε ) (τ, )dx τ + ω n x I ε (t,x, ω,ν) dγdνdt Γ + τ ( Θ + S ε : x u ε q ) ε x ϑ ε dxdt ϑ ε Ω τ + Ω τ + Ω Θ S 2 ν S 2 Θ ν Ω ϑ ε [ log [ log n(i ε ) n(i ε ) + 1 log n(b ε ) n(b ε ) + 1 n(i ε ) n(i ε ) + 1 log n(ĩε) n(ĩ ε ) + 1 ] ] σ a (j) ε (B ε I ε )d ωdνdx σ s (j) ε (Ĩ ε I ε )d ωdνdxd

61 1 τ Ω 2 E()dx + R(x, t) Ω where E() = 1 2 (,ε u,ε U(, ) 2 + E (,ε,ϑ,ε r(, ),Θ(, )) + εh R (I,ε )) dx

62 τ Ω R(x,t) =: + + τ τ τ Ω ε( u ε U) x U ( U uε ) ε (s ε s(r,θ)) Ω ( ε Ω τ τ dx dt ( U uε ) x Θ dx dt ) ( )) ( tu + U xu U uε Ω Ω (p ε div x U Sε : x U ) ( ) εsε R t Θ + q ε R x Θ dx dt dx dt dx dt

63 τ τ τ (( + Ω Ω ( ) ε (s ε s(r,θ)) t Θ dx dt ε (s ε s(r,θ)) U x Θ dx dt Ω τ q ε x Θ dx dt ϑ ε 1 ε r Ω ) t p(r,θ) ε r ) u ε x p(r,θ) dx dt

64 (r = ρ, U = u,θ = ϑ), where (, u,ϑ) is a classical solution of the target system (in the equilibrium case or in the non equilibrium case)

65 Semi-relativistic model α(ϑ) 1 If u c B(ν, ω, u,ϑ) = 2h ν ( 3 c 2 hν kϑ e 1 α ω u c ) 1, (2.6) << 1 one recovers the standard equilibrium Planck s function B(ν,ϑ) = 2h c 2 ν 3 e hν kϑ 1. Berthon, Buet, Coulombel, Depres, Dubois, Goudon, Morel, Turpault M1 Levermore model α = σ a + σ s σ a + 2σ s, (2.61)

66 σ a (ϑ, u) = χ( u ) σ a (ϑ) and σ s (ϑ) χ(s) = { 1 if s c, if s c + β, for an arbitrary β >. The role of this cut-off is to deal with the singularity of B In the over-relativistic regime ( u c) we decide to decouple matter and radiation.

67 ( ) t ( s) + div x ( s u) + div q x ϑ = ( 1 ϑ S : x u q xϑ ϑ ) S E ϑ + S F u ϑ (2.62) =: ς, where the ( first term of the) right hand side ς m := 1 ϑ S : x u q xϑ ϑ is the (positive) matter entropy production.

68 the formula for the entropy of a photon gas s R = 2k c 3 ν 2 [n log n (n + 1)log(n + 1)] d ωdν, (2.63) S 2 where n = n(i) = c2 I is the occupation number. Defining the 2hα 3 ν 3 radiative entropy flux q R = 2k c 2 ν 2 [n log n (n + 1)log(n + 1)] ω d ωdν, S 2 (2.64) and using the radiative transfer equation, we get the equation t s R + div x q R = k 1 h S 2 ν log n n + 1 S d ωdν =: ςr. (2.65)

69 ς R = k h { [ ] 1 S 2 ν log n(i) n(b) n(i)+1 log n(b)+1 σ a (B I) [ ] } + log n(i) n(ĩ) n(i)+1 log σ n(ĩ)+1 s (Ĩ I) d ωdν, + 1 ϑ Choosing now S 2 ( 1 α ω u ) S d ωdν c ασ s SF u σ a + σ s ϑ. α = σ a + σ s σ a + 2σ s, (2.66)

70 We get ς R = k h k h 1 S 2 ν + 1 ϑ S E S F u ϑ. S 2 1 ν [ log n(i) n(i) + 1 log ] n(b) σ a (B I) d ωdν n(b) + 1 [ ] log n(i) n(ĩ) n(i)+1 log σ n(ĩ)+1 s (Ĩ I) d ωdν (2.67)

71 we obtain finally ( t s + s R) ( ( ) + div x s u + q R) q + div x ϑ = 1 ( S : x u q xϑ ϑ ϑ k [ 1 log h S 2 ν ) k h n(i) n(i) + 1 log S 2 1 ν [ log n(ĩ) n(ĩ) + 1 n(i) n(i) + 1 log ] σ s (Ĩ I) d ωdν, ] n(b) σ n(b) + 1 (2.68)

72 Diffusion limits- non-relativistic limits The equilibrium-diffusion regime Ma = Sr = Pe = Re = P = 1, C = ε 1, L s = ε 2 and L = ε 1, t + div x ( u) =, (2.69) t ( u) + div x ( u u) + x p = div x S, (2.7) ( ) (( ) ) 1 1 t 2 u 2 + e +div x 2 u 2 + e + p u + q S u =, (2.71)

73 ( ) q t ( s) + div x ( s u) + div x = 1 ( S : x u q ) xϑ, ϑ ϑ ϑ (2.72) I = B(ν,ϑ), (2.73) where p(,ϑ) = p(,ϑ) + a 3 ϑ4, e(,ϑ) = e(,ϑ) + a ϑ4, k(ϑ ) = κ(ϑ) + 4a 3σ a ϑ 3, q = k(ϑ) x ϑ and s = s aϑ3.

74 We also get boundary conditions u Ω =, ϑ n Ω =, (2.74) initial conditions ( (x,t), u(x,t), ϑ(x,t)) t= = ( (x), u (x), ϑ (x) ), (2.75) for any x Ω, with the following compatibility conditions u (x) Ω =, ϑ n Ω =. (2.76)

75 existence of unique solution for strong solution ( local (small time) or global for (small data)) rigorous proof of singular limit using relative entropy inequality

76 The non-equilibrium diffusion regime Ma = Sr = Pe = Re = P = 1, C = ε 1, L = ε 2 and L s = ε 1. t + div x ( u) =, (2.77) t ( u) + div x ( u u) + x p = div x S, (2.78) ( ) (( ) ) 1 1 t 2 u 2 + e +div x 2 u 2 + e + p u + q S u =, ( ) t ( s) + div x ( s u) + div q x ϑ = ( 1 ϑ S : x u q xϑ ϑ ) + 1 xn u 3 ϑ σa(ϑ) ( ϑ aϑ 4 N ), (2.79) (2.8)

77 t N 1 3 div x ( ) 1 σ s (ϑ) xn = σ a (ϑ) ( aϑ 4 N ), (2.81) where p = p N, e = e + N and q = κ xϑ + 1 3σ s x N with boundary conditions initial conditions u Ω =, ϑ n Ω =, N Ω =, (2.82) ( (x,t), u(x,t), ϑ(x,t),n(x,t)) t= = ( (x), u (x), ϑ (x),n (x) ), (2.83) for any x Ω, with N (x) = S I (x,ν, ω) d ω dν and and 2 the compatibility conditions u 2 Ω =, ϑ n Ω =, N Ω =. (2.84) N = aθ 4 r. (2.85)

78 existence of unique solution for strong solution ( local (small time) or global for (small data)) rigorous proof of singular limit using relative entropy inequality

79 Non-relativistic limit C = O(ε 1 ) A small amount of radiation is present so P = ε. Finally we put Ma = 1, Sr = 1, Pe = 1, Re = 1, L = L s = 1

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

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