On the problem of singular limit of the Navier-Stokes-Fourier system coupled with radiation or with electromagnetic field

Transcription

1 On the problem of singular limit of the Navier-Stokes-Fourier system coupled with radiation or with electromagnetic field Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague Univ. of Ohio, 2015

2 Diffusion limits in a model of radiative flow (B. Ducomet, S.N.) Low Mach number limit in a model of radiative flow (B. Ducomet, S.N.) Low Mach number limit for a model of accretion disk (D.Donatelli, B.Ducomet,S.N.) Inviscid incompressible limits on expanding domains (E.Feireisl, S.N., Y.Sun)

3 General questions Compressible vs. incompressible Is air compressible? Is it important? Is the physical space bounded or unbounded? Viscous vs. inviscid What is turbulence? Do extremely viscous fluids exhibit turbulent behavior? Effect of rotation Does it matter that the Earth rotate? Is the rotation fast or slow? Is it important?

4 The scaling effect Characteristic values and scaling X X X char Scaling of derivatives t 1 T char t x 1 L char x

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 Thermostatic variable mass density ϱ = ϱ(t, x) Motion macroscopic velocity u = u(t, x)

7 (compressible) Navier-Stokes system Mass conservation Claude Louis Marie Henri Navier [ ] t ϱ + div x (ϱu) = 0 Momentum balance t (ϱu) + div x (ϱu u) + x p(ϱ) = div x S + ϱf George Gabriel Stokes [ ] Isaac Newton [ ] Newton s rheological law ( S = µ x u + t xu 2 ) 3 div xui + ηdiv x ui

8 Compressible Navier-Stokes system with radiation Equation of continuity t ϱ + div x (ϱu) = 0 Equation of motion t (ϱu) + div x (ϱu u) + x p(ϱ) = µ x u + λ x div x u + S F

9 Radiative transfer equation Radiative intensity 1 c ti + ω x I = S, c is speed of light ( ) 1 S = σ a (B I ) + σ s I d ω I 4π S 2 SF = (σ a + σ s ) ωi d ω dν. 0 S 2 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 0.

10 Hypothesis Isotropy. The coefficients σ a, σ s are independent of ω. Grey hypothesis The coefficients σ a, σ s are independent of ν. B = B(ν, ϱ) measures the departure from equilibrium is a barotropic equivalent of the Planck function b the frequency average of B(ν, ϱ) b(ϱ) := 0 B(ν, ϱ) dν.

11 boundary conditions u Ω = 0 { } I (t, x, ω, ν) = 0 for (x, ω) (x, ω) (x, ω) Ω S 2, ω n 0,

12 Scaled equations Scaling X X X char Mass conservation [Sr] t ϱ + div x (ϱu) = 0 Momentum balance [ ] 1 [Sr] t (ϱu) + div x (ϱu u) + Ma 2 x p(ϱ) [ ] 1 = ( x u + λ x div x u) + (external forces) Re

13 Transport of radiative intensity Sr C ti + ω x I = S = ( ) 1 = Lσ a (B I ) + LL s σ s I dω I. 4π S 2

14 Characteristic numbers - Strouhal number Čeněk Strouhal [ ] Strouhal number length [Sr] = 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

15 Mach number Mach number Ernst Mach [ ] [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

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

17 Radiation dimensionless numbers C = c U ref L = L ref σ a,ref, L s = σ s,ref σ a,ref P = L ref ν ref S ref c ρ ref U 2 ref,

18 Target system Incompressible limit Low Mach number compressible incompressible Fast rotation Low Rossby number 3D motion 2D motion Inviscid limit High Reynolds number viscous flow inviscid flow Diffusion limit compressible Navier-Stokes system with radiation compressible Navier-Stokes system with diffusion

19 Diffusion limit Mass conservation t ϱ + div x (ϱu) = 0 Momentum balance t (ϱu) + div x (ϱu u) + x p(ϱ) = div x S( x u)+ ( εσ a + 1 ) ε σ s ωi dω dν. 0 S 2

20 Transport of radiative intensity ε t I + ω x I = εσ a (B I ) + 1 ε σ s Asymptotic limit(formal) c 1 ε, σ a εσ a (ϱ), σ s 1 ε σ s(ϱ), ( 1 ) 4π I dω I S 2

21 Limit system Continuity equation t ϱ + div x (ϱu) = 0 Momentum equation t (ϱu) + div x (ϱu u) + x ( p(ϱ) N ) = µ u + (λ + µ) x div x u Diffusion equation 1 t N div x ( σ s (ϱ) xn) = σ a (ϱ)(b(ϱ) N), b(ϱ) = 0 B(ϱ, ν) dν

22 Preparing the initial data Ill prepared initial data ϱ(0, ) = ϱ + εϱ (1) 0,ε, u(0, ) = u 0,ε { } ϱ (1) 0,ε bounded in ε>0 L2 L {u 0,ε } ε>0 bounded in L 2 Well prepared initial data ϱ (1) 0,ε 0 in L2 as ε 0 u 0,ε u 0 in L 2 as ε 0, div x u 0 = 0

23 Fundamental issues Solvability of the primitive system The primitive system should admit (global) in time solutions for any choice of the scaling parameters and any admissible initial data Solvability of the target system The target system should admit solutions, at least locally in time; the solutions are regular Stability The family of solutions to the primitive system should be stable with respect to the scaling parameters

24 Assumptions Assumption on the pressure p is a C 1 function on [0, ) such that p(0) = 0, p (ρ) > 0 for all ρ > 0, such that p (z) ρ γ 1 = p > 0, γ > 3 2. Assumptions on radiative quantities 0 σ s (ϱ), σ a (ϱ) c 1, σ a (ϱ)b m (ν, ϱ) h(ν), h L 1 (0, ) for m = 1, 2, for any ϱ 0.

25 Weak formulation Renormalized continuity equation T (( ) ) ϱ + β(ϱ) t ψ dx dt T (( + ϱ + β(ϱ) 0 Ω 0 Ω ) u x ψ + ( ) ) β(ϱ) β (ϱ)ϱ div x uψ ( ) = ϱ 0 + β(ϱ 0 ) ψ(0, ) dx Ω satisfied for any ψ Cc ([0, ) Ω), and any β C [0, ), β Cc [0, ). dx dt

26 The momentum equation ϱu(τ, )φ(τ, ) dx ϱ 0 u 0 φ(0, ) dx = τ 0 Ω Ω Ω ϱu t φ + ϱu u : x φ + pdiv x φ S : x φ S F φ dx dt, for any φ C 1 ([0, T ] Ω; R 3 ) with φ Ω = 0, any τ [0, T ].

27 Definition of weak solution of primitive system Weak solution the density ϱ is a non negative measurable function, ρ C weak (0, T ; L γ (Ω)) u L 2 (0, T ; W 1,2 (Ω)), ϱu C weak (0, T ; L 2γ γ+1 (Ω; R 3 )), p L 1 ((0, T ) Ω), I L ((0, T ) Ω S 2 (0, )), I L (0, T ; L 1 (Ω S 2 (0, ))),

28 Existence of primitive system Existence of weak solution Ω R 3 be a bounded domain of class C 2,ν, ν > 0. Assumptions on p, the transport coefficients σ a, σ s and the equilibrium function B are satisfied Let (ϱ,, u, I ) be a weak solution to radiative Navier-Stokes system for (t, x) [0, T ] Ω, and (ω, ν) S 2 R + in the sense of previous definition

29 Class of regularity of primitive system the density ϱ is a non negative measurable function, ρ C weak (0, T ; L γ (Ω)) u L 2 (0, T ; W 1,2 (Ω)), ϱu C weak (0, T ; L 2γ γ+1 (Ω; R 3 )), p L 1 ((0, T ) Ω), I L ((0, T ) Ω S 2 (0, )), I L (0, T ; L 1 (Ω S 2 (0, ))),

30 A finite energy weak solution [ ] 1 Ω 2 ϱ u 2 (τ) + Π(ϱ)(τ) + E R (τ) dx τ [ + µ x u 2 + (λ + µ) div x u 2] dx dt, Ω for a.e. τ (0, T ). 0 Ω [ 1 q 2 + Π(ϱ 0 ) + E0 R 2 ϱ 0 E R (t, x) = 1 c S 2 Π(ϱ) = ϱ 0 ϱ 1 ] T dx dt + S F u 0 Ω I (t, x, ω, ν) dω dν p(z) z 2 dz

31 satisfying the integral identities for the continuity equation and the momentum equations and the transport of radiative intensity

32 Global weak solutions Barotropic case P. L. Lions (98) p(ϱ) = ϱ γ, γ 9/5 Generalization to a larger class of exponents γ > 3/2 E. Feireisl, A. Novotný and H. Petzeltová

33 Stability result Main Theorem(B.Ducomet, Š.N.) Ω R 3 be a bounded domain of class C 2,ν Assumptions on p,radiative quantities (ϱ ε, u ε, I ε ) be a weak solution of rescaled system of equations ϱ 0,ε ϱ 0 in L γ (Ω), Ω (ϱu) 0,ε ϱ 0,ε dx c, I 0,ε (, ν) h(ν), h L 1 L (0, ).

34 Convergence Then up to subsequences ϱ ε ϱ in C([0, T ]; L 1 (Ω)) and in C weak ([0, T ]; L γ (Ω)), u ε u weakly in L 2 (0, T ; W 1,2 (Ω; R 3 )), I ε I weakly in *L (0, T ; Ω S 2 (0, ))

35 Limit system where ϱ, u, I is a weak solution satisfying t ϱ + div x (ϱu) = 0 t (ϱu) + div x (ϱu u) + x ( p(ϱ) N ) = µ u + (λ + µ) x div x u ( ) 1 t N div x σ s (ϱ) xn = σ a (ϱ)(b(ϱ) N), b(ϱ) = 0 B(ϱ, ν) dν.

36 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.

37 Full system The continuity equation t ϱ + div x (ϱu) = 0 in (0, T ) Ω, The momentum equation t (ϱu) + div x (ϱu u) + x p(ϱ, ϑ) = div x S + S F in (0, T ) Ω, The energy equation ( )) ( ( ) ) 1 1 t (ϱ 2 u 2 + e(ϱ, ϑ) + div x ϱ 2 u 2 + e(ϱ, ϑ) u ) +div x (pu + q Su = S E in (0, T ) Ω,

38 The radiative intensity Sources 1 c ti + ω x I = S in (0, T ) Ω (0, ) S 2. S := S a,e + S s, where ) ( 1 S a,e = σ a (B(ν, ϑ) I, S s = σ s 4π S E = S(, ν, ω) dν dω, S 2 SF (t, x) = 1 c 0 0 S 2 ωs dω dν, S 2 I (, ω) dω I ),

39 Full system Maxwell s equation Stress tensor S = µ e ϱ = 1 ϱ 2 ( p(ϱ, ϑ) ϑ p ). ϑ ( x u + t xu 2 ) 3 div xu + η div x u I, the shear viscosity coefficient µ = µ(ϑ) > 0 the bulk viscosity coefficient η = η(ϑ) 0 are effective functions of the temperature Fourier s law q = κ x ϑ, the heat conductivity coefficient κ = κ(ϑ) > 0

40 Radiation quantities the absorption coefficient σ a = σ s (ν, ϑ) 0, the scattering coefficient σ s = σ s (ν, ϑ) 0 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,

41 Hypothesis on pressure P : [0, ) [0, ) p(ϱ, ϑ) = ϑ 5/2 P ( ϱ ϑ 3/2 ) + a 3 ϑ4, a > 0, P C 1 [0, ), P(0) = 0, P (Z) > 0 for all Z 0, 0 < 5 3 P(Z) P (Z)Z < c for all Z 0, Z P(Z) lim Z Z = p 5/3 > 0. a 3 ϑ4 - equilibrium radiation pressure.

42 Assumptions on viscosities 0 < c 1 (1 + ϑ) µ(ϑ), µ (ϑ) < c 2, 0 η(ϑ) c(1 + ϑ), Assumption on heat conductivity coefficient for any ϑ 0. 0 < c 1 (1 + ϑ 3 ) κ(ϑ) c 2 (1 + ϑ 3 ) Assumptions on radiation quantities 0 σ a (ν, ϑ), σ s (ν, ϑ), ϑ σ a (ν, ϑ), ϑ σ s (ν, ϑ) c 1, 0 σ a (ν, ϑ)b(ν, ϑ), ϑ {σ a (ν, ϑ)b(ν, ϑ)} c 2, σ a (ν, ϑ), σ s (ν, ϑ), σ a (ν, ϑ)b(ν, ϑ) h(ν), h L 1 (0, ).

43 Weak formulation of renormalized continuity equation T (b(ϱ) t ϕ + b(ϱ)u x ϕ) dx dt + 0 Ω T 0 Ω (( ) ) b(ϱ) b (ϱ)ϱ div x uϕ = b(ϱ 0 )ϕ(0, ) dx Ω dxdt satisfied for any ϕ Cc ([0, T ) Ω), and any b C [0, ), b Cc [0, )

44 Weak formulation of the momentum equation T (ϱu t ϕ + ϱu u : x ϕ + pdiv x ϕ) dx dt = 0 T 0 Ω Ω S : x ϕ + S F ϕ dx dt (ϱu) 0 ϕ(0, ) dx Ω for any ϕ C c ([0, T ) Ω; R 3 ). u L 2 (0, T ; W 1,2 0 (Ω; R 3 )),

45 Entropy inequality T ( ϱs t ϕ + ϱu x ϕ + q ) ϑ xϕ dx dt 0 Ω T 1 0 Ω ϑ T Ω (ϱs) 0 ϕ(0, ) dx ( S : x u q xϑ ϑ for any ϕ C c ([0, T ) Ω), ϕ 0. 0 Ω ) ϕ dx dt 1 ) (u S F S E ϕ dx dt ϑ

46 The total energy balance τ + 0 Ω ( 1 2 ϱ u 2 + ϱe(ϱ, ϑ) + E R Ω S 2, ω n 0 0 ) (τ, ) dx ω ni (t, x, ω, ν) dν dω ds x dt ( ) 1 = (ϱu) (ϱe) 0 + E R,0 dx, Ω 2ϱ 0 E R (t, x) = 1 I (t, x, ω, ν) dω dν. c S 2 0 E R,0 = I 0 (, ω, ν) dω dν S 2 0

47 Definition of weak solution We say that ϱ, u, ϑ, I is a weak solution of problem if ϱ 0, ϑ > 0 for a.a. (t, x) Ω, I 0 a.a. in (0, T ) Ω S 2 (0, ), ϱ L (0, T ; L 5/3 (Ω)), ϑ L (0, T ; L 4 (Ω)), u L 2 (0, T ; W 1,2 0 (Ω; R 3 )), ϑ L 2 (0, T ; W 1,2 (Ω)), I L ((0, T ) Ω S 2 (0, )), I (t, ) L (0, T ; L 1 (Ω S 2 (0, ))), and if ϱ, u, ϑ, I satisfy their weak formulation, together with the transport equation.

48 Full system - the Navier - Stokes - Fourier system Global existence of weak solution E. Feireisl et al. Singular limits of full system for the Navier type of boundary conditions-e. Feireisl, A. Novotný Concept of weak- strong uniqueness- E. Feireisl, A. Novotný, Y. Sun, B. J. Jin

49 Theorem(stability) (B. Ducomet, E.Feireisl,Š. N.) Let Ω R3 be a bounded Lipschitz domain. Assume that the thermodynamic functions p, e, s,the transport coefficients µ, λ, κ, σ a, and σ s satisfy the hypothesis. Let {ϱ ε, u ε, ϑ ε, I ε } ε>0 be a family of weak solutions to our problem in the sense of Definition of weak solution such that Ω ϱ ε (0, ) ϱ ε,0 ϱ 0 in L 5/3 (Ω), ( ) 1 Ω 2 ϱ ε u ε 2 + ϱ ε e(ϱ ε, ϑ ε ) + E R,ε (0, ) dx ( ) 1 (ϱu) 0,ε 2 + (ϱe) 0,ε + E R,0,ε dx E 0, 2ϱ 0,ε Ω [ϱ ε s(ϱ ε, ϑ ε ) + s R (I ε )](0, ) dx (ϱs + s R ) 0,ε dx S 0, Ω and

50 0 I ε (0, ) I 0,ε ( ) I 0, I 0,ε (, ν) h(ν) for a certain h L 1 (0, ). Then and ϱ ε ϱ in C weak ([0, T ]; L 5/3 (Ω)), u ε u weakly in L 2 (0, T ; W 1,2 0 (Ω; R 3 )), ϑ ε ϑ weakly in L 2 (0, T ; W 1,2 (Ω)), I ε I weakly-(*) in L ((0, T ) Ω S 2 (0, )), at least for suitable subsequences, where {ϱ, u, ϑ, I } is a weak solution of our problem.

51 Simplified model S F = 0 The entropy of a photon gas s R = 2k c 3 ν 2 [n log n (n + 1) log(n + 1)] dωdν, 0 S 2 n = n(i ) = c2 I 2hν 3 The radiative entropy flux q R = 2k c 2 Entropy is the occupation number 0 t s R + div x q R = k h S 2 ν 2 [n log n (n + 1) log(n + 1)] ω dωdν, 0 1 S ν log n n + 1 S dωdν =: ςr. 2

52 Total Entropy ( t ϱs + s R ) ( + div x ϱsu + q R ) ( q ) + div x = ς + ς R. ϑ S E ϑ k h k h S ν 2 S 2 1 ν ς R =: [ ] n(i ) log n(i ) + 1 log n(b) σ a (B I ) dωdν n(b) + 1 [ ] n(i ) log n(i ) + 1 log n(ĩ ) n(ĩ ) + 1 σ s (Ĩ I ) dωdν, ς =: 1 ϑ ( S : x u q xϑ ϑ ) S E ϑ,

53 L ref, T ref, U ref, ρ ref, ϑ ref, p ref, e ref, µ ref, κ ref, the reference hydrodynamical quantities (length, time, velocity, density, temperature, pressure, energy, viscosity, conductivity) I ref, ν ref, σ a,ref, σ s,ref, the reference radiative quantities (radiative intensity, frequency, absorption and scattering coefficients).

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

55 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 dω I, 4π S 2 t ϱ + div x (ϱu) = 0, t (ϱu) + div x (ϱu u) + x p(ϱ, ϑ) div x T = 0.

56 Entropy inequality ( q ) t (ϱs + εs R )+div x (ϱus + q R )+ div x 1 ϑ ϑ + 1 ε +ε d dt Ω S ν 2 S 2 1 ν [ log [ log ( 1 2 ϱ u 2 + ϱe + E R n(i ) n(i ) + 1 log n(b) n(b) + 1 ] n(i ) n(i ) + 1 log n(ĩ ) n(ĩ ) + 1 ) dx + 1 ε 0 ( S : x u q ) xϑ ϑ ] σ a (I B) dωdν σ s (I Ĩ ) dωdν, Γ + ω n I dγ + dν = 0.

57 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 dω I, 4π S 2 t ϱ + div x (ϱu) = 0, t (ϱu) + div x (ϱu u) + x p(ϱ, ϑ) div x T = 0.

58 ( q ) t (ϱs + εs R ) + div x (ϱus + q R ) + div x 1 ϑ ϑ +ε + 1 ε 1 0 S ν 2 0 S 2 1 ν [ log [ log n(i ) n(i ) + 1 log n(b) n(b) + 1 ] n(i ) n(i ) + 1 log n(ĩ ) n(ĩ ) + 1 ( S : x u q ) xϑ ϑ ] σ a (I B) dωdν σ s (I Ĩ ) dωdν.

59 The limit system equilibrium system Continuity equation t ϱ + div x (ϱu) = 0, The momentum equation t (ϱu) + div x (ϱu u) = div x T(ϱ, ϑ), The energy equation t (ϱe(ϱ, ϑ)) + div x (ϱe(ϱ, ϑ)u) + div x (K(ϱ, ϑ) x ϑ) = S(ϱ, ϑ) : x u p(ϱ, ϑ)div x u, Radiative transfer I = B(ν, ϑ).

60 Boundary condition u Ω = 0, ϑ n Ω = 0, Initial condition (ϱ(x, t), u(x, t), ϑ(x, t)) t=0 = ( ϱ 0 (x), u 0 (x), ϑ 0 (x) ), The compatibility conditions u 0 (x) Ω = 0, ϑ 0 n Ω = 0.

61 E(ϱ, ϑ) = e(ϱ, ϑ) + B(ϑ) 1 ϱ, and K(ϑ) = κ(ϑ) 3σ a(ϑ) ϑb(ϑ). existence of a global solution for the small data - Matsumura and Nishida

62 (ϱ, 0, ϑ) be a given constant state with ϱ > 0 and ϑ > 0. e 0 := ϱ 0 ϱ L (Ω) + u 0 H1 (Ω) + ϑ 0 ϑ H1 (Ω) + T 0 L2 (Ω) + V 0 L4 (Ω), where V 0 is the initial vorticity V ij = j u i i u j ), E 0 := e 0 + x ϱ 0 L 2 (Ω) + x ϱ 0 L α (Ω) + x T 0 L 2 (Ω), for an arbitrary fixed α such that 3 < α < 6.

63 The non-equilibrium diffusion regime, Navier-Stokes-Rosseland system Continuity equation t ϱ + div x (ϱu) = 0, The momentum equation t (ϱu) + div x (ϱu u) = div x T(ϱ, ϑ), Energy equation t (ϱe(ϱ, ϑ)) + div x (ϱe(ϱ, ϑ)u) + div x (κ(ϑ) x ϑ) = S(ϱ, ϑ) : x u p(ϱ, ϑ)div x u σ a (ϑ)[b(ϑ) N],

64 Diffusion equation t N 1 3 div x Boundary equations ( ) 1 σ s (ϑ) xn = σ a (ϑ) (B(ϑ) N). u Ω = 0, ϑ n Ω = 0, N := 0 I 0 dν N Ω = 0. Initial conditions (ϱ(x, t), u(x, t), ϑ(x, t), N(x, t)) t=0 = ( ϱ 0 (x), u 0 (x), ϑ 0 (x), N 0 (x) ),

65 The compatibility conditions. u 2 Ω = 0, ϑ 0 n Ω = 0, N 0 Ω = 0. N 0 (x) = I 0 (x, ν, ω) dω dν 0 S 2

66 Strong solution of the limit system for small data Let (ϱ, 0, ϑ, N) be a given constant state with ϱ > 0, ϑ > 0 and N = B(ϑ). and e 0 := ϱ 0 ϱ L (Ω) + u 0 H 1 (Ω) + ϑ 0 ϑ H 1 (Ω) + N 0 N H 1 (Ω) + T 0 L 2 (Ω) + V 0 L 4 (Ω), E 0 := e 0 + x ϱ 0 L2 (Ω) + x ϱ 0 Lα (Ω) + x T 0 L2 (Ω), for an arbitrary fixed α such that 3 < α < 6.

67 Given three numbers ϱ R +, ϑ R + and E R + we define Oess H the set of hydrodynamical { essential values essh } O H ess := (ϱ, ϑ) R 2 ϱ : 2 < ϱ < 2ϱ, ϑ 2 < ϑ < 2ϑ,O R ess the set of radiative{ essential values essr } O R ess := E R E R : 2 < E R < 2E, witho ess := Oess H Oess, R and their residual counterparts res O H res := (R + ) 2 \Oess, H Ores R := R + \Oess, R O res := (R + ) 3 \O ess.

68 Theorem(Equilibrium case): Ω R 3 be a bounded domain of class C 2,ν. The thermodynamic functions p, e, s satisfy hypotheses, P C 1 [0, ) C 2 (0, ), the transport coefficients µ, λ, κ, σ a, σ s and the equilibrium function B satisfy hypothesis, B C 1. 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) [0, T ] Ω S 2 R +, supplemented with the boundary conditions and the initial conditions (ϱ 0,ε, u 0,ε, ϑ 0,ε, I 0,ε )

69 such that ϱ ε (0, ) = ϱ 0 + εϱ (1) 0,ε, u ε(0, ) = u 0,ε, ϑ ε (0, ) = ϑ 0 + εϑ (1) 0,ε, where (ϱ 0, u, ϑ 0 ) H 3 (Ω) are smooth functions such that (ϱ 0, ϑ 0 ) belong to the set Oess, H where ϱ > 0, ϑ > 0, are two constants and Ω ϱ(1) 0,ε dx = 0, Ω ϑ(1) 0,ε dx = 0.

70 Suppose also that u 0,ε u 0 strongly in L (Ω; R 3 ), ϱ (1) 0,ε ϱ(1) 0 strongly in L 2 (Ω), ϑ (1) 0,ε ϑ(1) 0 strongly in L 2 (Ω), I (1) 0,ε I (1) 0 strongly in L ((0, T ) Ω (0, )). Then up to subsequences ϱ ε ϱ strongly in L (0, T ; L 5 3 (Ω)), u ε u strongly in L 2 (0, T ; W 1,2 (Ω; R 3 )), ϑ ε ϑ strongly in L (0, T ; L 4 (Ω)), where (ϱ, u, ϑ) is the smooth solution of the equilibrium decoupled system on [0, T ] Ω and I (t, x, ν, ω) = B(ν, ϑ(t, x)), with initial data (ϱ 0, u 0, ϑ 0 ).

71 Theorem (Non-equilibrium): Ω R 3 be a bounded domain of class C 2,ν. Assume that the thermodynamic functions p, e, s, P C 1 [0, ) C 2 (0, ), the transport coefficients µ, λ, κ, σ a, σ s satisfy hypothesis together with B C 1. 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) [0, T ] Ω S 2 R +, supplemented with the boundary conditions and the initial conditions (ϱ 0,ε, u 0,ε, ϑ 0,ε, I 0,ε ) such that

72 ϱ ε (0, ) = ϱ 0 + εϱ (1) 0,ε, u ε(0, ) = u 0,ε, ϑ ε (0, ) = ϑ 0 + εϑ (1) 0,ε, I ε (0, ) = I 0 + εi (1) 0,ε, where the functions (ϱ 0, u, ϑ 0 ) and x I 0 (x, ω, ν) belong to H 3 (Ω) and are such that (ϱ 0, ϑ 0, E R (I 0 )) belong to the set O ess, where ϱ > 0, ϱ > 0, E R > 0 are three constants and Ω ϱ(1) 0,ε dx = 0, Ω ϑ(1) 0,ε dx = 0, Ω I (1) 0,ε dx = 0.

73 Suppose also that u 0,ε u 0 strongly in L (Ω; R 3 ), ϱ (1) 0,ε ϱ(1) 0 strongly in L 2 (Ω), ϑ (1) 0,ε ϑ(1) 0 strongly in L 2 (Ω), I (1) 0,ε I (1) 0 strongly in L ((0, T ) Ω (0, )). Then up to subsequences ϱ ε ϱ strongly in L (0, T ; L 5 3 (Ω)), u ε u strongly in L 2 (0, T ; W 1,2 (Ω; R 3 )), ϑ ε ϑ strongly in L (0, T ; L 4 (Ω)),

74 and N ε N strongly in L ((0, T ) Ω), where N ε = I 0 S 2 ε dω dν and (ϱ, u, ϑ, N) is the smooth solution of the Navier-Stokes-Rosseland system on [0, T ] Ω with initial data (ϱ 0, u 0, ϑ 0, N 0 ).

75 ballistic free energy H Θ (ϱ, ϑ) = ϱe(ϱ, ϑ) Θϱs(ϱ, ϑ), radiative ballistic free energy H R Θ(I ) = E R (I ) Θs R (I ). relative entropy E(ϱ, ϑ r, Θ) := H Θ (ϱ, ϑ) ϱ H Θ (r, Θ)(ϱ Θ) H Θ (r, Θ).

76 τ + 0 Ω τ + 0 Ω Ω Ω ( ) 1 2 ϱ ε u ε U 2 + E (ϱ ε, ϑ ε r, Θ) + εh R (I ε ) (τ, ) dx τ + ω n x I ε (t, x, ω, ν) dγ dν dt 0 Γ + τ ( Θ + S ε : x u ε q ) ε x ϑ ε dx dt 0 Ω ϑ ε ϑ ε [ Θ n(i ε ) log ν n(i ε ) + 1 log n(b ε ) n(b ε ) S 2 S 2 Θ ν [ log n(i ε ) n(i ε ) + 1 log n(ĩε) n(ĩε) + 1 ] ] σ a (j) ε (B ε I ε ) dω dν dx dt σ s (j) ε (Ĩ ε I ε ) dω dν dx dt, 1 ( ϱ0,ε u 0,ε U(0, ) 2 + E (ϱ 0,ε, ϑ 0,ε r(0, ), Θ(0, )) + εh R (I 0,ε ) ) dx+r 2

77 ( 1 Ω 2 ϱ ε u ε u 2 + E (ϱ ε, ϑ ε ϱ, ϑ) ) { ( ( 1 Ω 2 ϱ0,ε u 0,ε u(0, ) 2 + E (ϱ 0,ε, ϑ 0,ε ϱ(0, ), θ(0, )) + ) εh R (I 0,ε ) dx + e 0 }e K3t

78 Suppose that e 0 Cε 2 and the initial data of the primitive system and any of the target systems are close in the following sense ϱ 0,ε ϱ 0 L2 (Ω) Cε, ϑ 0,ε ϑ 0 L2 (Ω) Cε, ϱ 0,ε (u 0,ε u) L2 (Ω;R 3 ) Cε.

79 Semi-relativistic model Definition of function B 0 α(ϑ) 1 Remark: If u c B(ν, ω, u, ϑ) = 2h ν ( 3 c 2 hν kϑ e equilibrium Planck s function B(ν, ϑ) = 2h 1 α ω u c ) 1 << 1 one recovers the standard ν 3 c. 2 e kϑ hν 1, Berthon, Buet, Coulombel, Depres, Dubois, Goudon, Morel, Turpault M1 Levermore model α = σ a + σ s σ a + 2σ s,

80 σ a (ϑ, u) = χ( u ) σ a (ϑ) 0 and σ s (ϑ) 0 { 1 if s c, χ(s) = 0 if s c + β, for an arbitrary β > 0. 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.

81 Entropy inequality t (ϱs) + div x (ϱsu) + div x ( q ϑ) ( 1 ϑ =: ς, S : x u q x ϑ ϑ ) S E ϑ + S F u ϑ where the ( first term of the) right hand side ς m := 1 ϑ S : x u q x ϑ ϑ is the (positive) matter entropy production.

82 The formula for the entropy of a photon gas s R = 2k c 3 ν 2 [n log n (n + 1) log(n + 1)] dωdν, 0 S 2 n = n(i ) = c2 I 2hα 3 ν 3 The radiative entropy flux q R = 2k c 2 0 is the occupation number. The radiative transfer equation t s R + div x q R = k h S 2 ν 2 [n log n (n + 1) log(n + 1)] ω dωdν, 0 S 2 1 ν log n n + 1 S dωdν =: ςr.

83 The equilibrium-diffusion regime- Limit system Ma = Sr = Pe = Re = P = 1, C = ε 1, L s = ε 2 and L = ε 1, Continuity equation t ϱ + div x (ϱu) = 0, Momentum equation t (ϱu) + div x (ϱu u) + x p = div x S, Energy equation ( 1 t 2 ϱ u ϱe ) (( + div 1 x 2 ϱ u 2 + ϱe + p ) u + q Su ) = 0,

84 Entropy equation ( ) q t (ϱs) + div x (ϱsu) + div x = 1 ϑ ϑ Radiation I = B(ν, ϑ), ( S : x u q ) xϑ, ϑ where p(ϱ, ϑ) = p(ϱ, ϑ) + a 3 ϑ4, e(ϱ, ϑ) = e(ϱ, ϑ) + a ϱ ϑ4, k(ϑ 0 ) = κ(ϑ) + 4a 3σ a ϑ 3, q = k(ϑ) x ϑ and ϱs = ϱs aϑ3.

85 Boundary conditions u Ω = 0, ϑ n Ω = 0, Initial conditions (ϱ(x, t), u(x, t), ϑ(x, t)) t=0 = ( ϱ 0 (x), u 0 (x), ϑ 0 (x) ), for any x Ω, Compatibility conditions u 0 (x) Ω = 0, ϑ 0 n Ω = 0.

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

87 The non-equilibrium diffusion regime - limit system Ma = Sr = Pe = Re = P = 1, C = ε 1, L = ε 2 and L s = ε 1. Continuity equation t ϱ + div x (ϱu) = 0, Momentum equation t (ϱu) + div x (ϱu u) + x p = div x S, Total Energy ( ) (( ) ) 1 1 t 2 ϱ u ϱe + div x 2 ϱ u 2 + ϱe + p u + q Su = 0,

88 Entropy equation t (ϱs) + div x (ϱsu) + div x ( q ϑ) = ( 1 ϑ S : x u q x ϑ ϑ ) + 1 x N u 3 ϑ σa(ϑ) ( ϑ aϑ 4 N ),

89 Diffusion equation t N 1 3 div x ( ) 1 σ s (ϑ) xn = σ a (ϑ) ( aϑ 4 N ), where p = p N, e = e + N ϱ and q = κ xϑ + 1 3σ s x N Boundary conditions u Ω = 0, ϑ n Ω = 0, N Ω = 0, Initial conditions (ϱ(x, t), u(x, t), ϑ(x, t), N(x, t)) t=0 = ( ϱ 0 (x), u 0 (x), ϑ 0 (x), N 0 (x) ), for any x Ω, with N 0 (x) = I 0 (x, ν, ω) dω dν 0 S 2

90 Compatibility conditions u 0 Ω = 0, ϑ 0 n Ω = 0, N 0 Ω = 0. N = aθ 4 r.

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

92 Crucial Lemma Let (ϱ (e), u (e), θ (e) ) be the solution of problem full N-S-F with radiation satisfying the conditions of Theorem 1 (equilibrium case) and let (ϱ (ne), u (ne), θ (ne), θ r (ne) ) be the solution of problem N-S-F with radiation satisfying the conditions of Theorem 2 (non equilibrium case) and choose (r, U, Θ) = (ϱ (e), u (e), θ (e) ) in the equilibrium case or (r, U, Θ, Θ r ) = (ϱ (ne), u (ne), θ (ne), θ r (ne) ) in the non equilibrium case.

93 Relative entropy inequality ( ) 1 Ω 2 ϱ ε u ε U 2 + E (ϱ ε, ϑ ε r, Θ) + H R (I ε ) (t, ) dx 1 [ ( 1 Ce 0 + ε Ω 2 ϱ 0,ε u 0,ε U(0, ) 2 + E (ϱ 0,ε, ϑ 0,ε r(0, ), Θ(0, )) + H R (I 0,ε ) )dxe C where C and C are positive constant depending on (r, U, Θ, Θ r ) and e 0 is the same as in Theorems 1 and 2. ε t,

94 NSFR with Electromagnetic field t ϱ + div x (ϱu) = 0 in (0, T ) Ω, (1) t (ϱu) + div x (ϱu u) + x p + ϱ χ u = div x S + ϱ Ψ ϱ x χ x 2 + j B in (0, T ) Ω, (2) t (ϱe)+div x (ϱeu)+div x q = S : x u pdiv x u+ j E S E in (0, T ) Ω, (3) 1 c ti + ω x I = S in (0, T ) Ω (0, ) S 2. (4) t B + curlx ( B u) + curl x (λ curl x B) = 0 in (0, T ) Ω. (5) Ψ = 4πG(ηϱ + g) in (0, T ) Ω ɛ. (6)

95 electric current - j electric field E 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, λ = λ(ϑ) > 0 is the magnetic diffusivity of the fluid. We also assume that the system is globally rotating at uniform velocity χ around the vertical direction e 3 Ψ(t, x) = G K(x y)(ηϱ(t, y) + g(y)) dy, Ω where K(x) = 1 x, and the parameter η may take the values 0 or 1: for η = 1 selfgravitation is present and for η = 0 gravitation only acts as an external field

96 Boundary conditions: u Ω = 0, q n Ω = 0, B n Ω = 0, E n Ω = 0, (7) I (t, x, ν, ω) = 0 for x Ω, ω n 0, (8) where n denotes the outer normal vector to Ω.

97 Primitive system ( ) 1 ε t I + ω x I = σ a (B I ) + σ s I dω I, (9) 4π S 2 t ϱ + div x (ϱu) = 0, (10) t (ϱu) + div x (ϱu u) + 1 ε 2 xp(ϱ, ϑ) + ϱ χ u = div x S + 1 ε ϱ Ψ ϱ x χ x ε 2 j B t ( ϱe + εe R ) + div x ( ϱeu + F R ) + div x q = ε 2 S : x u pdiv x u + j E (12)

98 ( t ϱs + εs R ) ( + div x ϱsu + q R ) ( q ) + div x ς ε, (13) ϑ with + + ς ε = 1 ϑ 1 0 S ν 2 0 S 2 1 ν ( ε 2 S : x u q xϑ + λ ) ϑ ζ curl xb 2 [ log [ log n(i ) n(i ) + 1 log n(b) n(b) + 1 ] n(i ) n(i ) + 1 log n(ĩ ) n(ĩ ) + 1 ] σ a (I B) dωdν σ s (I Ĩ ) dωdν,

99 t B + curlx ( B u) + curl x (λ curl x B) = 0, (14) ( d 1 dt Ω 2 ε2 ϱ u 2 + ϱe + εe R + 1 2ζ B εϱψ + 1 ) ϱ χ x 2 2 dx + 0 Γ + ω ni dγ + dν = 0 (15) where Γ + = {(x, ω) Ω S 2 : ω n x > 0}

100 We consider the pressure in the form ( ϱ ) p(ϱ, ϑ) = ϑ 5/2 P + a ϑ 3/2 3 ϑ4, a > 0, (16) where P : [0, ) [0, ) is a given function with the following properties: P C 1 [0, ), P(0) = 0, P (Z) > 0 for all Z 0, (17) 0 < 5 3 P(Z) P (Z)Z < c for all Z 0, (18) Z P(Z) lim Z Z = p 5/3 > 0. (19)

101 the specific internal energy e is e(ϱ, ϑ) = 3 2 ϑ ( ϑ 3/2 and the associated specific entropy reads ϱ ( ϱ ) s(ϱ, ϑ) = M ϑ 3/2 ) ( ϱ ) P + a ϑ4 ϑ 3/2 ϱ, (20) + 4a ϑ 3 3 ϱ, (21) with M (Z) = P(Z) P (Z)Z Z 2 < 0.

102 0 < c 1 (1 + ϑ) µ(ϑ), µ (ϑ) < c 2, 0 η(ϑ) c(1 + ϑ), (22) 0 < c 1 (1 + ϑ 3 ) κ(ϑ), λ(ϑ) c 2 (1 + ϑ 3 ) (23) for any ϑ 0. Moreover we assume that σ a, σ s, B are continuous functions of ν, ϑ such that 0 σ a (ν, ϑ), σ s (ν, ϑ), ϑ σ a (ν, ϑ), ϑ σ s (ν, ϑ) c 1, (24) 0 σ a (ν, ϑ)b(ν, ϑ), ϑ {σ a (ν, ϑ)b(ν, ϑ)} c 2, (25) σ a (ν, ϑ), σ s (ν, ϑ), σ a (ν, ϑ)b(ν, ϑ) h(ν), h L 1 (0, ). (26) for all ν 0, ϑ 0, where c 1,2,3 are positive constants.

103 Target system div x U = 0, (27) ϱ( t U+div x (U U))+ x Π = div x (2µ D( U))+ 1 ζ curl x B B + F (28) ω x I 1 = t B + curlx ( B U) + curl x (λ curl x B) = 0, (29) div x B = 0, (30) ϱ c P ( t Θ + div x (ΘU)) div x (κ Θ) = G, (31) ) ω x I 0 = σ a (B I 0 ) + σ s (Ĩ0 I 0, (32) ( ) σ a ϑ B + ϑ σ a (B I 0 )+ ϑ σ s (Ĩ0 I 0 ) )Θ σ a I 1 +σ s (Ĩ1 I 1, (33)

104 We finally consider the boundary conditions U Ω = 0, Θ n Ω = 0, B n Ω = 0, curl x B n Ω = 0 (34) for (27)-(31) and I 0 (x, ν, ω) = 0 for x Ω, ω n 0 (35) I 1 (x, ν, ω) = 0 for x Ω, ω n 0 (36) for (32) and (33), and the initial conditions U t=0 = U 0, Θ t=0 = Θ 0, B t=0 = B 0, I 0 t=0 = I 0,0, I 1 t=0 = I 1,0. (37)

105 Theorem: Hypotheses and stability result Let Ω R 3 be a bounded domain of class C 2,ν. Assume that the thermodynamic functions p, e, s satisfy hypotheses (16-21) with P C 1 [0, ) C 2 (0, ), and that the transport coefficients µ, η, κ, λ, σ a, σ s and the equilibrium function B comply with (22-26). Let (ϱ ε, u ε, ϑ ε, B ε, I ε ) be a weak solution of the scaled system (1-6) for (t, x, ω, ν) [0, T ] Ω S 2 R +, supplemented with the boundary conditions (7-8) and initial conditions (ϱ 0,ε, u 0,ε, ϑ 0,ε, B 0,ε, I 0,ε ) given by ϱ ε (0, ) = ϱ+εϱ (1) 0,ε, u ε(0, ) = u 0,ε, ϑ ε (0, ) = ϑ+εϑ (1) 0,ε, I ε(0, ) = I +εi (1) 0,ε, Ω B ε (0, ) = εb (1) 0,ε, where ϱ > 0, ϑ > 0, I > 0 are constants and ϱ (1) 0,ε dx = 0, ϑ (1) 0,ε dx = 0, I (1) 0,ε dx = 0, Ω Ω Ω B (1) 0,ε dx = 0 for all ε > 0.

106 Assume that ϱ (1) 0,ε ϱ(1) 0 weakly ( ) in L (Ω), u (1) 0,ε U 0 weakly ( ) in L (Ω; R 3 ), ϑ (1) 0,ε ϑ(1) 0 weakly ( ) in L (Ω), I (1) 0,ε I (1) 0 weakly ( ) in L (Ω S 2 R + ), B (1) 0,ε B(1) 0 weakly ( ) in L (Ω; R 3 ),

107 Then ess and up to subsequences ϑ ε ϑ ε sup t (0,T ) ϱ ε (t) ϱ L 4 3 (Ω) Cε, (38) u ε U weakly ( ) in L 2 (0, T ; W 1,2 (Ω; R 3 )), (39) = ϑ (1) Θ weakly ( ) in L 2 (0, T ; W 1,2 (Ω)), (40) I ε I 0 weakly ( ) in L 2 (0, T ; L 2 (Ω S 2 R + )), (41) and I ε I ε B ε ε = B(1) B weakly ( ) in L 2 (0, T ; W 1,2 (Ω; R 3 )), (42) = I (1) I 1 weakly ( ) in L 2 (0, T ; L 2 (Ω S 2 R + )), (43) where (U, Θ, B, I 0, I 1 ) solves the system (27)-(33).

108 compressible Navier-Stokes system: t ϱ + div x (ϱu) = 0, (44) t (ϱu) + div x (ϱu u) + 1 ε 2 xp(ϱ) = νdiv x S( x u), (45) S( x u) = x u + t xu 2 3 div xui, (46) the no-slip boundary condition u ΩM = 0, (47) Ω M R 3 is a smooth, bounded, simply connected domain.

109 Kelliher, Lopes, and Nussenzveig-Lopes the inviscid limit of the incompressible Navier-Stokes system on a family of domains Ω M = MΩ, M,

110 We consider a family of domains {Ω M } M>0 enjoying the following properties: Ω M R 3 are simply connected, bounded C 2 domains, uniformly for M ; there exists ω > 0 such that {x R 3 x < ωm } Ω M ; (48) there exists β > 0 such that Ω M 2 βm 2, (49) where 2 denotes the standard two-dimensional Hausdorff measure.

111 Our goal is to identify the triple singular limit, where ε 0, ν 0, while M.

112 p C[0, ) C 3 (0, ), p(0) = 0, p (ϱ) > 0 for ϱ > 0, where p (ϱ) lim ϱ ϱ γ 1 = p, (50) γ > 3 2. (51)

113 We consider the ill-prepared initial data in the form ϱ(0, ) = ϱ 0,ε = 1 + εϱ (1) 0,ε, u(0, ) = u 0,ε, (52) ϱ (1) 0,ε ϱ(1) 0 in L 2 (R 3 ), u 0,ε u 0 in L 2 (R 3 ; R 3 ) as ε 0. (53)

114 We expect that ϱ 1, u v, where v is a solution of the incompressible Euler system t v + v x v + x Π = 0, div x v = 0 in R 3. (54)

115 The principal difficulties of a rigorous proof of such a scenario are: The target Euler system is defined on R 3 while the primitive system (44-53) on Ω M, the solution v is not an admissible test function in the relative entropy inequality. The same problem occurs with the solutions of the associated acoustic system.

116 The class of finite energy weak solutions of the compressible Navier-Stokes system (44-47) satisfying, besides the standard weak formulation of the equations (44-46), the energy inequality Ω M [ 1 2 ϱ u ε 2 H(ϱ) Ω M ] τ (τ, ) + ν S( x u) : x u dt (55) 0 Ω M [ 1 2 ϱ 0,ε u 0,ε ] ε 2 H(ϱ 0,ε) for a.a. τ > 0, where we have set ϱ p(z) H(ϱ) = ϱ 1 z 2 dz.

117 Solutions of the target system u 0 C m (R 3 ; R 3 ) for a certain m > 4, supp[u 0 ] compact in R 3. v(0, ) = v 0 = H[u 0 ], (56) where H denotes the standard Helmholtz projection onto the space of solenoidal functions, possesses a smooth solution v C k ([0, T max ); W m k,2 (R 3 ; R 3 )), k = 1,.., m 1 (57) defined on a maximal time interval [0, T max ), T max > 0.

118 Acoustic system Lighthill s acoustic analogy ε t ϱ 1 ε + div x (ϱu) = 0, ε t (ϱu) + p ϱ 1 (1) x = ε ( ε [νdiv x S( x u) div x (ϱu u) x p(ϱ) p (1) ϱ 1 )] p(1), ε

119 The acoustic system ε t s + Ψ = 0, ε t x Ψ + a x s = 0, a = p (1) > 0, (58) the initial data s(0, ) = ϱ (1) 0, xψ(0, ) = x Ψ 0 = u 0 H[u 0 ]. (59)

120 Main results Hypotheses and stability result Theorem: Let the pressure p satisfy the hypotheses (50), (51). Let {Ω M } M>0 be a family of uniformly C 2 -domains in R 3 such that (48), (49) hold for M = M(ε), εm(ε) as ε 0. (60) Let the initial data [ϱ 0,ε, u 0,ε ] for the compressible Navier-Stokes system (44-47) be of the form ϱ(0, ) = ϱ 0,ε = 1+εϱ (1) 0,ε, u(0, ) = u 0,ε, ϱ (1) 0,ε L 2 L (R 3 )+ u 0,ε L2 (R 3 ;R 3 ) D. (61) In addition, suppose we are given functions u 0, ϱ (1) 0 such that u 0 C m (R 3 ; R 3 ), ϱ (1) 0 C m (R 3 ), u 0 C m (R 3 ;R 3 )+ ϱ (1) 0 C m (R 3 ) D, m > 4, (62) supp[u 0 ], supp[ϱ (1) 0 ] compact in R3. (63)

121 Let T max > 0 be the life-span of the smooth solution v of the Euler system (54), endowed with the initial datum v 0 = H[u 0 ], and let 0 < T < T max. Let [s, Ψ] be the solution of the acoustic system (58), with the initial data (59). Then there exists ε 0 > 0 such that ( ) ( ) ϱ 1 ϱ u x Ψ v (τ, ) + (τ, ) s(τ, ) L 2 (Ω M ;R 3 ) ε [ ϱ c(d, T, α) u 0,ε u 0 L2 (Ω M ;R 3 ) + (1) + ( ν + ε α + 1 ) ] 1/2, εm(ε) τ [0, T ], 0 < α < 1, and 0 < ε ε 0, 0,ε ϱ(1) 0 L2 (Ω M ) L 2 +L γ (Ω M ) (64) for any weak solution [ϱ, u] of the compressible Navier-Stokes system (44-47).

122 Corollary Hypotheses and stability result In addition to the hypotheses of Theorem 1, assume that and Then ϱ (1) 0,ε ϱ(1) 0 in L 2 (R 3 ), u 0,ε u 0 in L 2 (R 3 ; R 3 ) as ε 0, ess ess t (δ,t ) sup t (0,T ) ν = ν(ε) 0 as ε 0. ϱ 1 L2 +L γ (K) 0 as ε 0, sup ( ) ϱ u v (t, ) 0 as ε 0 (65) L2 (K;R 3 ) for any 0 < δ < T and any compact K R 3.

123 Reduction of dimension A straight layer Ω ɛ = ω (0, ɛ), where ω is a 2-D domain. ɛ 0 in N-S and N-S-F case Singular limit on expanding domain with rotation Introducing relative entropy inequality to numerical numerical analysis T.Karper,A.Novotny, E.Feireisl Coupling N-S-F with magnetic field low Mach number limit in domain dependent on time singular limit in fluid-structure interaction