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Volume Viscosity and Internal Energy Relaxation: Error Estimates
Vincent Giovangigli, Wen Yong
To cite this version:
Vincent Giovangigli, Wen Yong. Volume Viscosity and Internal Energy Relaxation: Error Estimates.
Nonlinear Analysis: Real World Applications, Elsevier, 2018. �hal-02364644�
Volume Viscosity and Internal Energy Relaxation:
Error Estimates
Vincent GIOVANGIGLI a,∗ , Wen An YONG b,c
a
CMAP, CNRS, Ecole Polytechnique, 91128 Palaiseau cedex, France
b
ZCAM, Tsinghua University, Beijing, 100084, CHINA
c
Beijing Computational Science Research Center, Beijing 100084, CHINA
Abstract
We investigate the fast relaxation of internal energy in nonequilibrium gas mod- els derived from the kinetic theory of gases. We establish uniform a priori estimates and existence theorems for symmetric hyperbolic-parabolic systems of partial differential equations with small second order terms and stiff sources.
We prove local in time error estimates between the out of equilibrium solution and the one-temperature equilibrium fluid solution for well prepared data and justify the apparition of volume viscosity terms.
Keywords: Thermodynamics, Nonideal, Supercritical, Mathematical Structure, Kinetics, Stability, Flames
1. Introduction
The kinetic theory of polyatomic gases shows that the volume viscosity co- efficient is related to the time required for the internal and translational tem- peratures to come to equilibrium [6, 17, 39, 40, 3, 4, 5]. We establish in this paper local in time error estimates between the solution of an out of equilib- rium two-temperature model and the solution of a one-temperature equilibrium model—including volume viscosity terms—when the relaxation time goes to zero.
The system of partial differential equations modeling fluids out of thermody- namic equilibrium as derived from the kinetic theory of gases is first summarized [3, 4]. This system and its symmetrizability properties have been investigated in our previous work [26]. The symmetrizing normal variable w of the out of equilibrium model is taken in the form
w = ρ, v, 1
T tr − 1 T in
, − 1 T
t
, (1.1)
∗
Corresponding author
Email addresses: vincent.giovangigli@polytechnique.fr (Vincent GIOVANGIGLI),
wayong@mail.tsinghua.edu.cn (Wen An YONG)
where ρ denotes the gas density, v the fluid velocity, T tr the translational tem- perature, T in the internal temperature, and T the local equilibrium temperature.
The resulting system of partial differential equations is in the general form A 0 (w)∂ t w + X
i∈D
A i (w)∂ i w − ǫ d
X
i,j∈D
∂ i B ij (w)∂ j w + 1
ǫ L(w)w = ǫ d b(w, ∂
xw
ii), (1.2) where ∂ t denotes the time derivative operator, ∂ i the space derivative oper- ator in the ith direction, D = { 1, . . . , d } the spatial directions, d the space dimension, ǫ, ǫ d ∈ (0, 1] two positive parameters and w = (w
i, w
ii) t is decom- posed into its hyperbolic components w
iand parabolic components w
ii. The matrix A 0 is symmetric positive definite and bloc-diagonal, A i are symmetric, B t ij = B ji , B ij have nonzero components only into the right lower B
ii,iiij blocs,
B
ii,ii= P
i,j∈D B
ii,iiij (w)ξ i ξ j is positive definite for ξ ∈ Σ d−1 , L is positive semi- definite with a fixed nullspace E , and b(w, ∂
xw
ii) is quadratic in the gradients.
Denoting by π the orthogonal projector onto E
⊥, the normal variable w is such that we have the commutation relation πA 0 = A 0 π. The source term is also naturally in quasilinear form as is typical in a relaxation framework and often encountered in mathematical physics [52]. The small parameter ǫ is associated with energy relaxation and the small parameter ǫ d with second order dissipative terms.
We establish uniform a priori estimates for linearized symmetric hyperbo- lic-parabolic systems with small dissipation and stiff sources obtained from the nonlinear equations (1.2). Symmetrized forms are important for analyzing hy- perbolic as well as hyperbolic-parabolic systems of partial differential equations modeling fluids [29, 18, 49, 38, 32, 37, 33, 36, 34, 7, 46, 21, 19, 54, 8, 9, 47, 31, 15, 51, 42, 35, 20, 22, 2, 16, 55, 47, 25, 9, 24, 43, 56, 44]. A priori esti- mates are obtained uniformly with respect to the parameters ǫ d ∈ (0, 1] and ǫ ∈ (0, 1]. The differences with the estimates established by Kawashima [32]
are the inclusion of extra terms associated with the fast variables πw/ǫ and πw/ √
ǫ as well as the coupling with the estimates for time derivatives. De- noting by w ⋆ a constant equilibrium state and ¯ τ a positive time, we estimate w − w ⋆ in the space C 0 [0, τ], H ¯ l
as well as ∂ t w and πw/ǫ in L 2 (0, τ), H ¯ l−1 for l ≥ [d/2] + 2 where H l = H l ( R d ) denotes the usual Sobolev space when the initial solution is close to the equilibrium manifold. A priori estimates require the commutation between the mass matrix and the orthogonal projector onto the fast manifold πA 0 = A 0 π. These estimates lead to local existence theorems for well prepared initial conditions on a time interval independent of both pa- rameters ǫ d ∈ (0, 1] and ǫ ∈ (0, 1]. Key points for local existence are notably to take into account stiff sources in the linearized equations in order to build approximated solutions, the new estimates for time derivatives, and the con- vergence rate of successive approximations that may depends on ǫ. Stronger estimates for ∂ t w in C 0 [0, τ ¯ ], H l−2
as well as for π∂ t w/ǫ in L 2 (0, τ), H ¯ l−3
with l ≥ [d/2] + 4 are also established when the initial time derivative is close
to the equilibrium manifold. These theorems yield the first existence results for
the out of equilibrium two-temperature model derived in [3] and symmetrized in [26]. On the other hand, the situation of ill prepared initial data lay beyond the scope of this work and we refer the reader to [28]. In the same vein, only local existence results are investigated and we refer to [27] for global existence results.
We finally investigate the singular limit ǫ, ǫ d → 0 in the system modeling flu- ids out of thermodynamic equilibrium. Various relaxation models have also been investigated in the literature in different physical and mathematical contexts [36, 7, 8, 10, 35, 41, 50, 16, 53, 43, 44]. In order to investigate the asymptotic behavior of solutions as ǫ, ǫ d → 0 we combine a priori estimates out of thermo- dynamic equilibrium with stability results associated with the equilibrium limit model. The fast variable notably corresponds to the rescaled temperature dif- ference with (T tr − T in )/ǫ = − T tr T in πw/ǫ and we use that perturbed hyperbolic- parabolic systems with small second order terms and perturbing right hand sides admit local solutions that depend continuously on perturbations. Denoting by w e = (ρ e , v e , − 1/T e ) t the solution of the equilibrium one-temperature model including the volume viscosity terms and by ϕw = (ρ, v, − 1/T ) t the projection on the slow manifold of the normal variable w out of equilibrium, we establish that ϕw − w e = O ǫ(ǫ + ǫ d )
. This justifies the addition of the volume viscosity term − κ e ( ∇ · v e )I in the viscous tensor Π e at equilibrium
Π e = − κ e ( ∇ · v e )I − η e ∇ v e + ( ∇ v e ) t − 2 3 ( ∇ · v e )I ,
where κ e and η e denote the equilibrium volume and shear viscosities, discard- ing O ǫ(ǫ + ǫ d )
Burnett type residuals. In the situation where ǫ = ǫ d , it has also been established that the equilibrium system corresponds to a second order Chapman-Enskog expansion for small relaxation times [26] and the error esti- mates of ϕw − w e yields a rigorous jusitification of the second order accuracy. To the author’s knowledge, it is the first time that the apparition of volume viscosity terms is justified rigorously with an error estimate. Note incidentally that ex- perimental measurements [45, 48] as well as theoretical calculations [6, 17, 39, 3]
have shown that the volume viscosity coefficient is of the same order as the shear viscosity coefficient for polyatomic gases and the impact of volume viscosity in fluid mechanics has also been established [12, 13, 11, 30, 1, 3].
The nonequilibrium two-temperature model and its symmetrization are sum- marized in Section 2. A priori estimates and local existence results are estab- lished in Section 3. Stability for equilibrium models and convergence of the nonequilibrium model towards the one-temperature model is established in Sec- tion 4.
2. Governing equations
The system of equations modeling fluids out of thermodynamic equilibrium
as derived from the kinetic of gases is summarized and recast into a convenient
normal form [3, 4, 26]. The local equilibrium temperature, the volume viscosity
coefficient, and the equations at equilibrium are also discussed.
2.1. Conservation equations
In a nonequilibrium gas with internal degrees of freedom, the conservation of mass, momentum, internal energy and total energy may be written in the form [3]
∂ t ρ + ∇ · (ρv) = 0, (2.1)
∂ t (ρv) + ∇ · (ρv ⊗ v + pI) + ∇ · Π = 0, (2.2)
∂ t (ρe in ) + ∇ · (ρve in ) + ∇ · Q in = ω in , (2.3)
∂ t ρ(e tr + e in + 1 2 | v | 2 )
+ ∇ · ρv(e tr + e in + 1 2 | v | 2 ) + vp
+ ∇ · (Q tr + Q in + Π · v) = 0, (2.4) where ∇ denotes the space derivative operator, ρ the mass density, v the fluid velocity, ⊗ the tensor product symbol, p the pressure, Π the viscous tensor, I the unit tensor in the physical space R d , e in the internal energy of internal origin per unit mass, Q in the heat flux of internal origin, ω in the energy exchange rate, e tr the internal energy of translational origin per unit mass, and Q tr the heat flux of translational origin. The components of v and ∇ are written v = (v 1 , . . . , v d ) t and ∇ = (∂ 1 , . . . , ∂ d ) t where v i denotes the velocity in the ith spatial direction,
∂ i the derivation in the ith spatial direction and bold symbols are used for vector or tensor quantities in the physical space R d . The equations (2.2)–(2.4) have to be completed by relations expressing the thermodynamic properties e in , e tr , and p, the rate of energy exchange ω in , and the transport fluxes Π, Q in and Q tr .
2.2. Thermodynamics
The pressure p, the total internal energy per unit mass e, the internal energy of translational origin per unit mass e tr , and the internal energy of internal origin per unit mass e in are in the form
p = ρrT tr , e = e tr + e in , e tr = c v,tr T tr , e in = e in,st + Z T
inT
stc in (T
′) dT
′, (2.5) where r denotes the gas constant per unit mass, c v,tr = 3 2 r the translational heat at constant volume per unit mass, T tr the translational temperature, c in the internal heat per unit mass, T in the internal temperature, T st the standard tem- perature, and e in,st the internal formation energy at the standard temperature.
We will also use in the following the translational heat at constant pressure per unit mass c p,tr = 5 2 r and the formation energy at zero temperature e 0 in = e in (0).
The rate of energy exchange between the translational and internal degrees of freedom ω in may also be written [3]
ω in = ρc in
τ in
(T tr − T in ), (2.6)
where τ in denotes the energy exchange time.
2.3. Transport fluxes
In the framework of the kinetic theory of polyatomic gases out of thermo- dynamic equilibrium, the translational and internal heat fluxes are in the form [3]
Q tr = − λ tr,tr ∇ T tr − λ tr,in ∇ T in , (2.7) Q in = − λ in,tr ∇ T tr − λ in,in ∇ T in , (2.8) where λ tr,tr , λ tr,in , λ in,tr , and λ in,in denote thermal conductivities. On the other hand, the viscous tensor is given by
Π = − η ∇ v + ( ∇ v) t − d 2
′( ∇ · v)I
, (2.9)
where η denotes the shear viscosity and d
′the dimension of the velocity space in the underlying kinetic framework. It will be assumed in the following that the dimension of the kinetic velocity space d
′is such that 2 ≤ d
′and d ≤ d
′. The assumption 1 ≤ d ≤ d
′means that the spatial dimension d of the model has eventually been reduced. The assumption 2 ≤ d
′is natural since d
′= 3 in our physical world and since Π is identically zero when d
′= 1.
The thermal conductivities λ tr,tr , λ tr,in , λ in,tr , and λ in,in and the shear vis- cosity η are obtained from the kinetic theory of non equilibrium gases [3]. From the expression (2.9) it is also noted that the viscous tensor Π does not present a volume viscosity term and our aim is to investigate the apparition of such a contribution in the one-temperature equilibrium limit model as the relaxation time τ in goes to zero.
2.4. Mathematical assumptions
The mathematical assumptions for the thermodynamic properties, the en- ergy exchange rate, and the transport coefficients are the following where κ ≥ 3 denotes the regularity class [23, 19, 3, 26].
(T
1) The formation energy e in,st and formation entropies s tr,st and s in,st are real constants. The mass per unit mole m, the gas constant R, and the gas constant per unit mass r = R/m are positive. The internal species heat per unit mass c in (T in ) is a C
κ−1function over [0, ∞ ) and there exist constants c and c such that 0 < c 6 c in (T in ) 6 c for all T in > 0.
(T
2) The energy exchange rate τ in (p, T tr , T in ) is in the form τ in = ǫ¯ τ in = ǫ p st τ ¯ in st
p , (2.10)
where ǫ ∈ (0, 1] denotes a positive parameter, τ ¯ in (p, T tr , T in ) = p st τ ¯ in st /p the rescaled energy exchange time and ¯ τ in st (T tr , T in ) the rescaled energy exchange time at the standard pressure p st which only depends on T tr
and T in . The rescaled time τ ¯ in st is a positive C
κfunction of the two
temperatures T tr , T in ∈ (0, ∞ ).
(Tr
1) The coefficients η, λ tr,tr , λ tr,in , λ in,tr , and λ in,in are in the form η = ǫ d η, ¯ λ tr,tr =ǫ d λ ¯ tr,tr , λ tr,in = ǫ d λ ¯ tr,in ,
λ in,tr = ǫ d ¯ λ in,tr , λ in,in = ǫ d ¯ λ in,in , (2.11) where ǫ d ∈ (0, 1] denotes a positive parameter, and η, ¯ λ ¯ tr,tr , λ ¯ tr,in , λ ¯ in,tr , and ¯ λ in,in the rescaled transport coefficients. The rescaled coefficients η, ¯ λ ¯ tr,tr , ¯ λ tr,in , λ ¯ in,tr , and λ ¯ in,in are C
κfunctions of the two temperatures T tr , T in ∈ (0, ∞ ).
(Tr
2) For any T tr , T in ∈ (0, ∞ ), the matrix
"
T in 2 λ ¯ in,in T tr 2 λ ¯ in,tr T in 2 ¯ λ tr,in T tr 2 ¯ λ tr,tr
#
, (2.12)
is symmetric positive definite. In the viscous tensor (2.9), the coefficient η is positive and the dimension d
′of the kinetic velocity space is such that max(2, d) ≤ d
′.
The rescaled energy exchange time ¯ τ in as well as the rescaled transport co- efficients ¯ η, ¯ λ tr,tr , ¯ λ tr,in , ¯ λ in,tr , and ¯ λ in,in have been introduced in order to in- vestigate the fast relaxation limit.
2.5. Volume viscosity
The local thermal equilibrium temperature is defined as the unique scalar T such that
e tr (T ) + e in (T ) = e tr (T tr ) + e in (T in ), (2.13) keeping in mind that e tr (T ) + e in (T ) is an increasing function of T . The tem- perature T is a C
κfunction of (T tr , T in ) and is the temperature that would be obtained at local thermal equilibrium T tr = T in assuming that the internal energy e tr + e in is kept fixed. Letting e c in = R 1
0 c in T in + s(T − T in )
ds, we may write e in (T ) − e in (T in ) = (T − T in ) e c in so that (T tr − T )c v,tr = (T − T in ) e c in and (T tr − T ) e c v = (T tr − T in ) e c in where e c v = c v,tr + e c in (T, T in ). Letting c v (T in ) = c v,tr + c in (T in ) and
κ = κ(T tr , T in ) = r e c in pτ in
c v e c v
= ǫ r e c in p st τ ¯ in st c v e c v
, (2.14)
the following relation is obtained after some algebra ρr(T tr − T ) = − κ ∇ · v − κ
p
Π:∇ v + ∇ · Q tr − c v,tr
c in
∇ · Q in
+ ρ∂ t (T tr − T in ) + ρv · ∇ (T tr − T in )
. (2.15)
Note that we have κ = ǫ¯ κ with ¯ κ = r e c in p¯ τ in /(c v e c v ) from assumption (2.10).
Equation (2.15) is a relaxation equation that yields formally ρr(T tr − T ) =
− κ ∇ · v + O ǫ(ǫ + ǫ d )
so that both temperatures T tr and T in should converge towards the local equilibrium temperature T . In the momentum equation, the pressure tensor ρrT tr I + Π is thus asymptotically in the form
ρrT tr I + Π = ρrT I − κ ( ∇ · v)I − η ∇ v + ( ∇ v) t − d 2
′( ∇ · v)I
+ O ǫ(ǫ + ǫ d ) . This is in agreement with classical one-temperature models where the pressure ρrT is evaluated at the thermal equilibrium temperature T and the viscous tensor Π includes a volume viscosity term − κ( ∇ · v)I. Such a physically in- tuitive derivation may be found in many physics papers and books either in a molecular framework or in a macroscopic fluid framework usually around equi- librium states [6, 17, 39, 40, 3, 4, 5]. Numerical simulations using Boltzmann equation have consistently established that the limit one-temperature model is an accurate description of the two temperature fluid when the relaxation time is small [3]. In our previous work [26], it has further been established that the Chapman-Enskog method exactly yields the one-temperature fluid equations with the volume viscosity terms at second order in the fast relaxation limit. The goal of this paper is to justify with an error estimate both the above physically in- tuitive approximation as well as the accuracy of the two term Chapman-Enskog expansion [26].
2.6. Quasilinear forms
Letting n = d+3, the conservative variable u ∈ R n associated with equations (2.1)–(2.4) is found to be
u = ρ, ρv, ρe in , ρ(e tr + e in + 1 2 | v | 2 ) t
,
and the natural variable z ∈ R n is defined by z = ρ, v, T in , T tr t
. For conve- nience, the velocity components of vectors in R n = R × R d × R 2 are generally written as vectors of R d . We introduce the corresponding open sets O
uand O
zof R n given by
O
u= n
u = u ρ , u
v, u in , u tr + u in + 1 2 | u
v| 2 u ρ
t
; u ρ > 0, u in > u ρ e 0 in , u tr > 0 o ,
and O
z= (0, ∞ ) × R d × (0, ∞ ) 2 . The following proposition has been established in our previous work [26].
Proposition 2.1. Assuming that (T
1) holds, the map z 7−→ u is a C
κdiffeo- morphism from the open set O
zonto the open set O
uand the open set O
uis convex.
The equations modeling fluids out of thermodynamic equilibrium may then be written in the compact form
∂ t u + X
i∈D
∂ i F i + ǫ d
X
i∈D
∂ i F diss i − 1
ǫ Ω = 0, (2.16)
where F i denotes the convective flux in the ith direction, ǫ d the Knudsen number, F diss i the rescaled dissipative flux in the ith direction, ǫ the relaxation parameter, and Ω the rescaled source term.
From the governing equations (2.1)–(2.4) the convective flux F i in the ith direction is given by
F i = ρv i , ρvv i + pe i , ρe in v i , (ρe tr + ρe in + 1 2 ρ | v | 2 + p)v i
t
, (2.17)
where e i denotes the basis vectors of R d . Similarly, the dissipative flux ǫ d F diss i is given by
ǫ d F diss i = 0, Π i , Q in,i , Q tr,i + Q in,i + Π i · v t
, (2.18)
where Q tr = (Q tr,1 , . . . , Q tr,d ) t , Q in = (Q in,1 , . . . , Q in,d ) t , Π ij , 1 ≤ i, j ≤ d, are the components of the viscous tensor Π, and Π i = (Π 1i , . . . , Π di ) t . The source term is finally given by
1
ǫ Ω = 0, 0, ω in , 0 t
, (2.19)
where the rate of enegy exchange ω in is defined by (2.6). From the expressions of the viscous tensor and of the heat fluxes we deduce that the dissipative fluxes F diss i are linear expression in terms of spatial derivatives of z and may thus be written in the form F diss i = − P
j∈D B b ij (z)∂ j z. Using Proposition 2.1, we may then write that F diss i = − P
j∈D B ij (u)∂ j u where the dissipation matrix B ij
is defined as B ij = b B ij ∂
uz. Further introducing the Jacobian matrices of the convective fluxes A i = ∂
uF i the governing equations are finally rewritten into the compact form
∂ t u + X
i∈D
A i (u)∂ i u − ǫ d
X
i,j∈D
∂ i B ij (u)∂ j u
− 1
ǫ Ω(u) = 0. (2.20) In our previous work [26], all possible normal variables leading to a symmet- ric hyperbolic-parabolic structure have been shown to be F
i(ρ), F
ii(v, T in , T tr ) t
where F
iand F
iiare diffeomorphisms in R and R d+2 , respectively. The natural variable z is in particular a normal variable but for convenience the following normal variable will be used
w = ρ, v, 1
T tr − 1 T in
, − 1 T
t
. (2.21)
The density w
i= ρ is the hyperbolic variable, w
ii= (v, T 1
tr− T 1
in, − T 1 ) t the parabolic variable, and the corresponding normal form has been evaluated [26].
The third component of w goes to zero with the relaxation time and the other components ρ, v, − 1/T t
are expected to converge towards the corresponding normal variable at thermodynamic equilibrium w e = ρ e , v e , − 1/T e
t
.
Theorem 2.2. Assume that the assumptions (T
1)(T
2) and (Tr
1)(Tr
2) hold.
Then the map u → w is a C
κ−1diffeomorphism from the open set O
uonto the
open set O
w= (0, ∞ ) × R d × R × ( −∞ , 0). The system written in the w variable is in the normal form with a source term in quasilinear form
A 0 (w)∂ t w + X
i∈D
A i (w)∂ i w − ǫ d
X
i,j∈D
∂ i B ij (w)∂ j w + 1
ǫ L(w)w = ǫ d b(w, ∂
xw
ii), (2.22) and the matrices A 0 , A i , i ∈ D , B ij , i, j ∈ D , L, as well as the quadratic residual b and the source term Ω are detailed in previous work [26]. The matrix A 0 is symmetric positive definite, A i , i ∈ D , are symmetric, we have B t ij = B ji , i, j ∈ D , L is positive semi-definite with a fixed nullspace E , and b(w, ∂
xw
ii) is quadratic in the gradients. Using the bloc structure induced by the partitioning between hyperbolic and parabolic variable, A 0 is bloc diagonal, B ij has nonzero coefficients only in the right lower bloc B
ii,iiij and for any ξ in the sphere Σ d−1 the matrix B
ii,ii(w, ξ) = P
i,j∈D B ij (w)ξ i ξ j is positive definite. The matrices B ij have the structure B ij = 1 r B λ δ ij + rT η ¯
trB η ij where B λ is associated with thermal conductivities and B η ij with shear viscous effects. The equilibrium linear manifold with respect to the normal variable is the fixed subspace E = R × R d ×{ 0 }× R and the normal variable w is quasilinear on the fast manifold E
⊥= R e d+2 . Finally, the normal variable is compatible with the fast manifold so that πA 0 = A 0 π.
2.7. Equations at equilibrium
In order to investigate the fast relaxation limit ǫ → 0 we will need to es- tablish a stability theorem for the equations governing fluids at thermodynamic equilibrium that are summarized in this section. The equations modeling one- temperature fluids are in the form [6, 17, 19]
∂ t ρ e + ∇ · (ρ e v e ) = 0, (2.23)
∂ t (ρ e v e ) + ∇ · (ρ e v e ⊗ v e + p e I) + ∇ · Π e = 0, (2.24)
∂ t (ρe e + 1 2 ρ | v e | 2 ) + ∇ · v e (ρe e + 1 2 ρ | v e | 2 + p e )
+ ∇ · (Q e + Π e · v e ) = 0, (2.25) where the subscript e denotes thermodynamic equilibrium, ρ e the mass density, v e the fluid velocity, p e the pressure, Π e the viscous tensor involving the volume viscosity, e e the internal energy per unit mass, and Q e the heat flux.
The pressure p e and the internal energy per unit mass e e are in the form p e = ρ e rT e and e e = e e,st + R T
eT
stc v (T
′) dT
′where c v (T e ) = c v,tr + c in (T e ) denotes the heat at constant volume per unit mass, T e the equilibrium temperature, e e,st
formation energy at the standard temperature and we have e e (T e ) = e tr (T e ) + e in (T e ). The equilibrium viscous tensor is in the form
Π e = − κ e (T e ) ( ∇ · v e )I − η e (T e ) ∇ v e + ( ∇ v e ) t − d 2
′( ∇ · v e )I
, (2.26)
where η e (T e ) = η(T e , T e ) and κ e (T e ) = κ(T e , T e ) so that κ e = r c in pτ in /c 2 v , and
the heat flux is given by Q e = − λ e (T e ) ∇ T e , with λ e (T e ) = λ tr,tr (T e , T e ) +
λ tr,in (T e , T e ) + λ in,tr (T e , T e ) + λ in,in (T e , T e ).
Letting n e = d + 2, the conservative variable u e ∈ R n
eassociated with equations (2.23)–(2.25) is u e = ρ e , ρ e v e , ρe e + 1 2 ρ e v e · v e t
and the corresponding natural variable reads z e = ρ e , v e , T e t
. The corresponding open sets are given by O
ue=
u e = (u ρ , u
v, u tl ) t ∈ R n
e; u ρ > 0, u tl > f e (u ρ , u
vwhere f e (u ρ , u
v) = u ρ e 0 e + 1 2
uvu·uvρ
and O
ze= (0, ∞ ) × R d × (0, ∞ ). The map z e → u e is easily shown to be a C
κdiffeomorphism from O
zeonto O
ue. Introducing the convective and dissipative fluxes of the equilibrium fluid model (2.23)–(2.25)
F e i = ρ e v ei , ρ e v e v ei + p e e i , (ρe e + p e + 1 2 ρ e | v e | 2 )v ei t
, (2.27)
ǫ d F e diss i = 0, Π ei , Q ei + Π ei · v e
t
, (2.28)
using straighforward notation, the system at equilibrium may be rewritten in quasilinear form
∂ t u e + X
i∈D
A e i (u e )∂ i u e − ǫ d
X
i,j∈D
∂ i B e ij (u e )∂ j u e
= 0, (2.29)
where A e i , i ∈ D , denote the Jacobian matrices A e i = ∂
ueF e i and B e ij , i, j ∈ D , the dissipation matrices at equilibrium with F e diss i = − P
j∈D B e ij ∂ j u e [32, 34, 19].
The equations of the one-temperature equilibrium model may also be written in normal form [26] with the normal variable
w e =
ρ e , v e , − 1 T e
t
, (2.30)
where the density w ei = ρ e is the hyperbolic variable and w eii = (v e , − T 1
e) t the parabolic variable.
Theorem 2.3. Assume that (T
1)(T
2) and (Tr
1)(Tr
2) hold. Then the map v e → w e is a C
κ−1diffeomorphism from the open set O
veonto the open set O
we= (0, ∞ ) × R d × ( −∞ , 0). The system written in the w e variable is of the normal form
A e 0 (w e )∂ t w e + X
i∈D
A e i (w e )∂ i w e − ǫ d
X
i,j∈D
∂ i B e ij (w e )∂ j w e
= ǫ d b e (w e , ∂
xw
iie), (2.31) where A e 0 , A e i , i ∈ D , B e ij , are detailed in [26] and have regularity at least κ − 1, and b e is a quadratic residual. The matrices at equilibirum are related to the analog matrices out of equilibrium with the relations A e 0 = ψ t A 0 (ψw e )ψ, A e i = ψ t A i (ψw e )ψ with
ψ =
1 0 1,d 0 0 d,1 I 0 d,1
0 0 1,d 0 0 0 1,d 1
. (2.32)
Moreover, the dissipation matrices B e ij have the structure B e ij = 1 r B λ,e δ ij +
κ ¯
erT
eB κ,e ij + rT η ¯
eeB η,e ij with B λ,e = ψ t B λ (ψw e )ψ and B η,e ij = ψ t B η ij (ψw e )ψ whereas the matrices B κ,e ij , i, j ∈ D , are given in [26].
Denoting by ϕ the linear operator ϕ = ψ t where ψ is the rectangular matrix (2.32), one of the goal of this paper is to establish that the equilibrium projection ϕw of the normal variable w out of thermodynamic equilibrium is close to the normal variable w e at thermodynamic equilibrium so that ϕw − w e = O ǫ(ǫ + ǫ d )
.
3. Hyperbolic-parabolic systems with stiff source terms
We investigate in this section local existence theorems for hyperbolic-para- bolic systems of partial differential equations in normal form with small second order terms and stiff sources. We generally follow the elegant formalism and methods of proof of Kawashima [32], the differences being the inclusion of extra terms associated with the fast variables πw/ǫ and πw/ √
ǫ, the coupling with the estimates for time derivatives, and the derivation of estimates uniformly with respect to the parameters ǫ d ∈ (0, 1] and ǫ ∈ (0, 1].
3.1. Preliminaries
We consider an abstract hyperbolic-parabolic system with small second order terms and stiff sources in normal form. The system is written
A 0 (w)∂ t w + X
i∈D
A i (w)∂ i w − ǫ d
X
i,j∈D
∂ i B ij (w)∂ j w + 1
ǫ L(w)w = ǫ d b(w, ∂
xw
ii), (3.1) where w = (w
i, w
ii) t ∈ O
w, O
wis an open set of R n , w
iare the hyperbolic components, w
iithe parabolic components, and ǫ d , ǫ ∈ (0, 1] are two positive parameters. The dimensions of the hyperbolic and parabolic components are denoted by n
iand n
iirespectively so that n = n
i+ n
ii. The matrices A 0 , A i , B ij , and L are assumed to have at least regularity κ − 2. We will generally assume that κ is as large as required by the various theorems in the following, in particular that κ − 3 ≥ l + 1 ≥ l 0 + 2 where l 0 = [d/2] + 1. The matrix A 0 is symmetric positive definite, the matrices A i are symmetric, B ij satisfy B t ij = B ji , and L is positive semi-definite with a fixed nullspace E . The matrices A 0 and B ij , i, j ∈ D , have the bloc structure
A 0 =
"
A
i,i0 0 n
i,n
ii0 n
ii,n
iA
ii,ii0
#
, B ij =
"
0 n
i,n
i0 n
i,n
ii0 n
ii,n
iB
ii,iiij
# ,
and B
ii,ii(w, ξ) = P
i,j∈D B
ii,iiij (w)ξ i ξ j is positive definite for w ∈ O
wand ξ ∈ Σ d−1 . The quadratic source term is also in the form
b(w, ∂
xw
ii) = X
i,j∈D
m ij (w)∂ i w∂ j w = 0, X
i,j∈D
m
ii,ii,iiij (w)∂ i w
ii∂ j w
ii, t
(3.2)
where m ij are third order tensors depending on w with at least regularity κ − 3.
Denoting by π the orthogonal projector onto the orthogonal of the equilibrium manifold E
⊥, we assume that A 0 satisfies the compatibility condition
π A 0 (w) = A 0 (w) π, w ∈ O
w. (3.3) We are only interested in well prepared initial data in this paper, that is, we assume that the initial condition w 0 is close to the equilibrium manifold E in such a way that πw 0 is small, and we refer to [28] for the situation of ill prepared initial data. In the same vein, only local existence is investigated and we refer to [27] for global existence results.
We denote by u ⋆ and w ⋆ corresponding constant equilibrium states in the u, and w variables respectively, so that w ⋆ ∈ O
w∩ E and π w ⋆ = 0. We denote by
| • | l the norm in the Sobolev space H l = H l ( R d ) and otherwise k • k A in the functional space A. If α = (α 1 , . . . , α d ) ∈ N d is a multiindex, we denote as usual by ∂ α the differential operator ∂ 1 α
1· · · ∂ d α
dand by | α | its order | α | = α 1 + · · · +α d . The square of k th derivatives of a scalar function φ, like T , ρ, or v i , 1 ≤ i ≤ d, is defined by
| ∂ k φ | 2 = X
|α|=k