Digital Object Identifier (DOI) 10.1007/s00220-006-1534-7Commun. Math. Phys. (2006) Communications in

MathematicalPhysics

On the Motion of a Viscous CompressibleRadiative-Reacting Gas

Donatella Donatelli 1, Konstantina Trivisa2

1 Dipartimento di Matematica Pura and Applicata, Universita di L’Aquila, 67100 L’Aquila, Italy.E-mail: donatell@univaq.it

2 Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA.E-mail: trivisa@math.umd.edu

Received: 22 July 2005 / Accepted: 2 October 2005Published online: 9 March 2006 – © Springer-Verlag 2006

Abstract: A multidimensional model is introduced for the dynamic combustion of com-pressible, radiative and reactive gases. In the macroscopic description adopted here, theradiation is treated as a continuous field, taking into account both the wave (classical)and photonic (quantum) aspects associated with the gas [20, 36]. The model is formu-lated by the Navier-Stokes equations in Euler coordinates, which is now expressed bythe conservation of mass, the balance of momentum and energy and the two specieschemical kinetics equation. In this context, we are dealing with a one way irreversiblechemical reaction governed by a very general Arrhenius-type kinetics law. The analysisin the present article extends the earlier work of the authors [17], since it now coversthe general situation where, both the heat conductivity and the viscosity depend on thetemperature, the pressure now depends not only on the density and temperature but alsoon the mass fraction of the reactant, while the two species chemical kinetics equation isof higher order.

The existence of globally defined weak solutions of the Navier-Stokes equations forcompressible reacting fluids is established by using weak convergence methods, com-pactness and interpolation arguments in the spirit of Feireisl [26] and P.L. Lions [35].

1. Introduction

A multidimensional model is introduced for the dynamic combustion of a viscous, com-pressible, radiative-reactive gas for higher order kinetics. In the macroscopic descriptiona gas can be viewed as a continuum occupying at a given time t ∈ R a certain domain� ∈ R

N . The state of the gas is completely characterized by the density ρ = ρ(t, x), thevelocity u = u(t, x), the temperature θ = θ(t, x), and the mass fraction of the reactantZ = Z(t, x). Here x ∈ � ⊂ R

N, N = 3, denotes the spatial position in the Eulerianreference system.

The motion of the gas is governed by the Navier-Stokes equations, which representthe conservation of mass, the balance of momentum and energy and the two species

D. Donatelli, K. Trivisa

chemical kinetics equation for higher order kinetics,

∂tρ + div(ρu) = 0, (1.1)

∂t (ρu)+ div(ρu ⊗ u)+ ∇p = divS + ρg, (1.2)

∂t (ρe)+ div(uρe)+ divQ = S : ∇u − pdivu + qKf (ρ, θ)Zm, (1.3)

∂t (ρZ)+ div(ρuZ) = −Kf (ρ, θ)Zm + divF. (1.4)

Here, the viscous stress tensor S, the pressure p = p(ρ, θ, Z), the specific internalenergy e = e(ρ, θ, Z), the heat flux Q = Q(θ,∇θ,∇Z) and the species diffusionflux F are related to the macroscopic variables through various constitutive relations,which provide in a certain sense a qualitative description of the physical properties ofthe fluid. In the above system, K represents the reaction rate, f (ρ, θ) the rate function,while g = g(t, x) is a given function representing the external force density. In order tosimplify the species diffusion velocities, we assume that they are given by Fick’s law,namely

F = ρd ∇Z,which also requires that the reactant flux diffusion coefficientD = ρd is a function onlyof the absolute temperature.

In this article we consider an approximation of a single irreversible exothermic reac-tion. These type of reactions, though simple, are qualitatively interesting, since severalphenomena can be modeled by one reaction scheme. More precisely, for the chemicalmodel we consider two phases present: the reactant (unburnt gas) and the product (burntgas) and the reactant is converted to product species via a one way irreversible chemicalreaction.

The reaction function f determines the nature (speed) of the combustion and isassumed to satisfy a very general Arrhenius-type law, namely

f (ρ, θ) ={

0, 0 ≤ θ ≤ θI ,

c0ρm−1θre−c1/(θ−θI ), θ > θI ,

(1.5)

where c0, c1 > 0, r ≤ 4, m ≥ 1 is the kinetic order and θI ≥ 0 is the ignition tem-perature. As it is expected, combustion will occur when the temperature rises above theignition temperature resulting in phase transition which here yields the conversion ofsome or all of the mass of the reactant (unburnt gas) to product species (burnt gas).

1.1. Radiation effects. In the macroscopic description adopted here, the radiation istreated as a continuous field, and both the wave (classical) and photonic (quantum)aspects are taken into account. In the quantum case, the total pressure p in the gas isaugmented, due to the presence of photon gas, by a radiation component pR related tothe absolute temperature θ through the Stefan-Boltzmann law,

pR = a

3θ4, with a > 0 a constant.

The underlying assumption here (cf. [20, 28, 36]) is that the high temperature radiation,is at thermal equilibrium with the fluid. As a result, the specific internal energy of thefluid must be augmented, as we are going to see in the sequel, by the term

eR = eR(ρ, θ) = a

ρθ4.

Compressible Radiative-Reacting Gas

We remark that radiation effects are of particular interest in astrophysical models wherestars can be viewed as gaseous objects in R

3, whose dynamics are often determined byhigh temperature radiation effects [13].

1.2. Constitutive relations. Taking into account the above discussion, the pressure p ofthe gas obeys a general equation of state;

p = p(ρ, θ, Z) = pe(ρ)+ Zθ pθ (ρ)+ a

3θ4, (1.6)

pe, pθ ∈ C[0,∞) ∩ C1(0,∞),

stand for the elastic and thermal pressure respectively.The last term on the right-hand side of (1.6) accounts for the effect of the radiation

with a > 0 being the Stefan-Boltzmann constant.In this article we concentrate on Newtonian fluids for which the viscous stress tensor

S depends linearly on the symmetric part Dx of the velocity gradient,

Dx(u) = 1

2(∇xu + ∇xut ),

and is given by the Newton’s viscosity formula

S = µ(θ)

(∇u + ∇uT − 2

3divu I

)+ ζ(θ) divu I, (1.7)

where the shear viscosity µ and the bulk viscosity ζ are supposed to be nonnegative andcontinuously differentiable functions of the absolute temperature.

The heat flux Q is given by the following law

Q = −κ(θ)∇θ − qD(θ)∇Z, (1.8)

where q represents the difference in heats between the reactant and the product, κ > 0,the heat conductivity coefficient, which is a function of the absolute temperature, arequirement essential in the present context as we are dealing with very high tempera-tures. In other words, the heat flux is given as the sum of the fluxes

QF = −κ(θ)∇θ, Qd = −qF = −qD(θ)∇Z, (1.9)

the first given by the Fourier’s Law and the second given as a multiple of the speciesdiffusion flux.

The presence of the flux Qd in the energy equation is physically relevant in the presentcontext. In a certain sense, we view the reactant and the product as separated fluids, eachone of which having its own density, but both having the same velocity and temperature.Each species is characterized by its own density and heats, namely ρ1 = ρZ, c1, q1for the reactant, ρ2 = ρ(1 − Z), c2, q2 for the product. If one considers the specificheat cv and the heat q as constants the flux Qd (being the sum of fluxes correspondingto the reactant and product species multiplied either by cv or q) often vanishes in theenergy equation. In our case the heats of the two species differ, therefore the presence ofthis term is physically relevant. For further remarks on admissible constitutive laws forcombustion models we refer the reader to Williams [39]. The last term on the right side

D. Donatelli, K. Trivisa

of the energy equation {qKf (ρ, θ)Zm} is the sum of the terms {(−1)j qjf (ρ, θ)Zm}(j = 1, 2 corresponding to the reactant and the product) which represent the rate ofenergy lost to the reactant or gained by the product as a result of the chemical reaction.For further discussion we refer the reader to the article by Chen, Hoff and Trivisa [10].

Considering p = p(ρ, θ, Z) and e = e(ρ, θ, Z) as explicit functions of the density,the absolute temperature and the mass fraction of the reactant and using the generalthermodynamic relation,

θDs = De + pD(

1

ρ

)− qDZ, (1.10)

where D denotes the total differential, we derive the entropy equation, which now reads,

∂t (ρs)+ div(ρus)+ div

(QF

θ

)= S : ∇u

θ− QF · ∇θ

θ2 + 2qKf (ρ, θ)Zm

θ, (1.11)

for suitable entropy s. In the above relation QF is given by the Fourier’s law, while therighthandside of the entropy equation (1.11)

r = S : ∇uθ

− QF · ∇θθ2 + 2qKf (ρ, θ)Zm

θ(1.12)

is typically known as the entropy production.In the present context, we regard the internal energy as a function of the density ρ,

the temperature θ and the reactant mass fractionZ that satisfies the constitutive relation,

e(ρ, θ, Z) = Pe(ρ)+ a

ρθ4 + C(θ, Z), (1.13)

where Pe(ρ) is given by Maxwell’s relationship,

Pe(ρ) =∫ ρ

1

pe(z)

z2 dz (1.14)

and C is a function of the temperature θ and the mass fraction of the reactant Z.In particular, the quantity

cv(θ, Z) = ∂

∂θC(θ, Z),

is the so called specific heat at constant volume. For the sake of simplicity, we shallassume cv to be only a function of Z. Therefore,

e(ρ, θ, Z) = Pe(ρ)+ a

ρθ4 + cv(Z)θ. (1.15)

Multiplying the conservation of mass equation in (1.1) by (ρPe(ρ))′ we obtain

∂t (ρPe(ρ))+ div(ρPe(ρ)u)+ pe(ρ)divu = 0 (1.16)

and so the energy equation (1.3) yields,

∂t (aθ4 + cv(Z)ρθ)+ div

[(cv(Z)ρθ + aθ4)u

]+ divQ

= S : ∇u − θZpθ (ρ)divu − a

3θ4divu + qKf (ρ, θ)Zm. (1.17)

Compressible Radiative-Reacting Gas

We assume that the mixture occupies a bounded domain� ⊂ RN,N = 2, 3 of class

C2+ν, ν > 0, on the boundary of which the following boundary conditions hold

u|∂� = 0, Q|∂� = 0, ∇Z · n|∂� = 0, (1.18)

namely, the velocity satisfies a no-slip boundary condition, while the system is assumedto be thermally insulated.

We consider the following initial conditions:ρ(0, ·) = ρ0,

(ρ u)(0, ·) = m0,

(ρ θ)(0, ·) = ρ0θ0,

(ρZ)(0, ·) = ρ0Z0,

(1.19)

together with the compatibility condition:

m0 = 0 on the set {x ∈ �| ρ0(x) = 0}. (1.20)

The objective of this work is to establish the global existence of weak solutions to thisinitial boundary value problem with large initial data.

This work extends the earlier work of the authors (cf. Donatelli and Trivisa [17])on combustion models since it now covers a more general setting, that captures thephase transition during the combustion process more accurately. More specifically, thepressure law p = p(ρ, θ, Z) now depends on the mass fraction of the reactant and itis a nonlinear function of θ , the heat flux Q depends also on the concentration Z, thetwo species chemical kinetics equation is of higher order, the rate function f = f (ρ, θ)

is allowed to be unbounded both with respect to ρ and θ , while the heat conductivityk = k(θ), the shear and bulk viscosity parameters µ = µ(θ) and ζ = ζ(θ) depend onthe absolute temperature.

A relevant one dimensional combustion model was introduced by Chen, Hoff andTrivisa [10] for the investigation of viscous, compressible, polytropic gases. In that set-ting, the pressure of the mixture was assumed to satisfy the Dalton’s Law, that is thepressure of the mixture was the sum of the pressure of each one of the species andtherefore the specific heat was assumed to be a linear function of the mass fraction ofthe reactant having the property cv(Z) = c1Z + c2(1 − Z), with c1, c2 denoting thespecific heats of the reactant and the product respectively. In the present article, and inan effort to offer a precise description of the change of phase in the multidimensionalsetting, we require that the pressure p = p(ρ, θ, Z) is a function of the mass fraction ofthe reactant satisfying a rather general pressure law. This implies that the specific heat atconstant volume cv = cv(Z) should depend on Z where cv is a Lipschitz function (seeSect. 2).

The outline of this article is as follows. In Sect. 2 we present the general setting ofthe problem, we state the main hypothesis on the system and the constitutive relationsand present the main results. Our approach relies on the concept of a variational solu-tion, which allows us to find the appropriate weak formulation of the problem that willguarantee the necessary compactness of our approximate solution sequence (see also[17, 26, 20]).

In Sect. 3 we introduce a new modified three level approximating scheme, whichinvolves a system of regularized equations (see also [17, 20, 26]) and we resolve theresulting system via a Faedo-Galerkin approximating procedure. Having obtained the

D. Donatelli, K. Trivisa

necessary apriori estimates we obtain the local existence of solutions and we proceedestablishing uniform estimates yielding the appropriate compactness results.

We remark that the constitutive laws presented here are in agreement with the fun-damental principles of continuum physics and combustion theory. The dependence ofthe pressure and the heat flux on the mass fraction of the reactant Z captures quite accu-rately the physical setting offering a better description of the phase transition during thecombustion process. This necessary (in terms of modeling) addition of Z in the pressurelaw, complicates the mathematical analysis since it effects both the constitutive relationsand the equations of our system in a significant way. As a result new energy estimates,apriori estimates, compactness and interpolation arguments are needed in our analy-sis as appear in Sects. 3 and 4, starting with the construction of a new approximatingscheme and the treatment of new energy inequalities (see Sect. 3.1). Moreover, in certainimportant issues such as in proving strong convergence of the density ρ, one needs toobtain boundedness of the oscillation defect measure, which is a quantity expressed interms of certain cut-off functions (Sect. 5). The choice of these cut-off functions dependson whether or not the viscosity parameters depend on θ, as well as on the constitutiverelation for the pressure p and is different from the treatment in [17]. Also, special con-sideration has to be given to higher order terms inZ connected to the modified chemicalkinetics equation and the thermal energy equation, as well as to the fact that the pressureis, in the present context, a nonlinear function of θ due to the radiation effects. To dealwith these new features new interpolation estimates are of use.

In Sect. 4 we let the artificial viscosity ε go to zero, while in Sect. 5 we recoverthe original system by letting δ go to zero. Both processes are very delicate due to theoscillation effects on ρ and concentration effects on the temperature θ and pressure p.To deal with these difficulties we employ a variety of techniques developed by Feireisl[26] and P. L. Lions [35] by accommodating them appropriately to the new context.

In Sect. 6 we present a model arising in astrophysics, which describes the evolutionof gaseous stars and present the notion of variational solution in that setting. The resultof global existence of at least one variational solution is obtained as a consequence ofthe earlier analysis (see also [20, 28]).

Remarks on the equation of state for the pressure and its physical relevance to com-bustion models are presented in Sect. 7.

Existence results for combustion models as far as the one dimensional case is con-cerned are presented in a series articles (see Bebernes and Bressan [4], Bebernes andEberly [5], Bressan [6], Chen [7], Chen, Hoff and Trivisa [9–11], Ducomet [18, 19], Du-comet and Zlotnik [22], Zlotnik [40] and the references therein). Global existence resultsfor weak solutions to a multidimensional combustion model formulated by the Navier-Stokes equations for viscous, compressible, reacting gases are presented by Donatelliand Trivisa [17]. For related articles in the literature we refer the reader to Ducomet andFeireisl [20], Feireisl [27] and Feireisl and Novotny [28]. For a survey on the mathemat-ical theory of combustion models we refer the reader to the manuscripts by Buckmaster[3] and Williams [39].

2. Main Result

If the motion is smooth, the momentum equation (1.2) multiplied by u yields

∂t

(1

2ρ|u|2

)+ div

(1

2ρ|u|2u

)+ div(pu) = div(Su)+ pdivu

−S : ∇u + ρ g · u. (2.1)

Compressible Radiative-Reacting Gas

with{ 1

2ρ|u|2} being the kinetic energy.For a weak variational formulation of the momentum equation (1.2) one should use

a kinetic inequality instead of (2.1), namely

∂t

(1

2ρ|u|2

)+ div

(1

2ρ|u|2u

)+ div(pu) ≤ div(Su)+ pdivu

−S : ∇u + ρ g · u. (2.2)

As a consequence, in order to find the appropriate weak formulation for our problem weneed also to replace the thermal energy equation (1.17) by two inequalities:

∂t (aθ4 + cv(Z)ρθ)+ div

[(cv(Z)ρθ + aθ4)u

]+ divQ ≥ S : ∇u − θZpθ (ρ)divu

−a3θ4divu + qKf (ρ, θ)Zm,

and

E[ρ,u, θ, Z](τ ) ≤ E[ρ,u, θ, Z](0)+∫ τ

0

∫�

ρ g·u dx dt, for τ ≥ 0, (2.3)

with the total energy E defined by

E(ρ,u, θ, Z) =∫�

1

2ρ|u|2 + ρPe(ρ)+ aθ4 + cv(Z)ρθ + qρZdx,

where the elastic potential Pe is given by (1.14). These two inequalities can be viewedas a weak formulation of Eq. (1.3).

In a similar way, the weak variational formulation of the entropy production is givenby

∫ T

0

∫�

ρs∂tφ + ρsu · ∇φ + QF

θ· ∇φ dx dt

≤∫ T

0

∫�

(QF · ∇θθ2 − S : ∇u

θ− qKf (ρ, θ)Zm

θ

)φ dx dt,

(2.4)

for any nonnegative function φ ∈ D((0, T )× RN).

We emphasize that in the framework of weak solutions placing an “inequality” inthe position of the (at least in the formal level) classical equality is not surprising. Theunderlying idea is that part of the kinetic energy may disappear in the form of a positivemeasure and become part of the domain. We refer the reader to [26] for further remarks.

Motivated by the earlier discussion, we introduce now the notion of a variationalsolution to the initial boundary value problem (1.1)-(1.4) together with (1.14) and (1.10).

2.1. Variational solutions.

Definition 2.1. We say that (ρ,u, θ, Z) is a variational solution of the initial boundaryvalue problem (1.1)-(1.4) on the interval (0, T ) if it satisfies the following properties:

D. Donatelli, K. Trivisa

(a) The density ρ is a nonnegative function,

ρ ∈ C([0, T ];L1(�)) ∩ L∞(0, T ;Lγ (�)), ρ(0, ·) = ρ0

satisfying the integral identity:∫ T

0

∫�

ρ ∂tψ + ρ u · ∇ψ dx dt = 0,

for any ψ ∈ C∞([0, T ] × �), ψ(0) = ψ(T ) = 0. In addition, we require that ρis a “renormalized solution” of the continuity equation (1.1) in the sense that theintegral relation∫ T

0

∫�

b(ρ)∂tψ + b(ρ)u · ∇ψ + (b(ρ)− b′(ρ)ρ) divuψ dx dt = 0, (2.5)

holds for any b ∈ C1[0,∞) such that b′(ρ) = 0 for all ρ large enough, and any testfunction

ψ ∈ C∞([0, T ] × �), ψ(0) = ψ(T ) = 0.

(b) The velocity u belongs to the class

u ∈ Lb(0, T ;W 1,b0 (�)), b > 1, ρu(0, ·) = m0,

and the momentum equation (1.2) holds in D′((0, T )×�) in the sense that∫ T

0

∫�

ρ u ∂tψ + ρ(u ⊗ u) : ∇ ψ + p divψ dx dt =∫ T

0

∫�

S : ∇ψ dx dt

−∫ T

0

∫�

ρ gψ dx dt,

for all ψ ∈ [D((0, T )×�)]N.(c) The temperature θ is a nonnegative function,

θ, log(θ) ∈ L2(0, T ;W 1,2).

The entropy ρs as well as the terms in (2.4) are integrable on (0, T ) × � and theinequality (2.4) holds for any nonnegative function φ ∈ D((0, T )× R

3). Moreover,

ess limt→0+

∫�

ρs(t)φ dx ≥∫�

ρ0s0φ dx, for any nonnegative φ ∈ D(�),

where

ρ0s0 = 4a

3θ3

0 − ρ0Z0Pθ(ρ0)+ cv(Z0)ρ0 log(θ0)+ ρ0cv(Z0)− q

θ0ρ0Z0.

(d) The equation of the chemical kinetics holds in D′ in the sense that∫ T

0

∫�

ρ Z ∂tψ + ρ uZ · ∇ψdx dt =∫ T

0

∫�

(Kf (ρ, θ) ρZm)ψ dx dt

+∫ T

0

∫�

D(θ)∇Z · ∇ψ dx dt,

for all ψ ∈ [D((0, T )×�)]N, with Z belonging to L2(0, T ;W 1,2(�)).

Compressible Radiative-Reacting Gas

(e) The energy inequality (2.3) holds for almost all τ ∈ (0, T ) with

E(ρ,u, θ, Z)(0) =∫�

1

2

|m0|2ρ0

+ ρ0Pe(ρ0)+ aθ40 + cv(Z0)ρ0θ0 + qρ0Z0dx.

(f) The functions ρ, ρ u, ρ θ and ρ Z satisfy the initial conditions (1.10) in the weaksense,

ess limt→0+

∫�

ρ(t) ηdx =∫�

ρ0 η dx,

ess limt→0+

∫�

(ρ u)(t) · ηdx =∫�

m0 · η dx,

ess limt→0+

∫�

(ρ θ)(t) ηdx =∫�

ρ0 θ0 η dx,

ess limt→0+

∫�

(ρ Z)(t) ηdx =∫�

ρ0 Z0 η dx,

for all η ∈ D(�).

2.2. Hypothesis.

• Structural conditions on the pressure. The pressure p is assumed to obey the generalpressure law (1.6) where the elastic pressure pe and the thermal pressure pθ arecontinuously differentiable functions of the density. Furthermore,{

pe(0) = 0, p′e(ρ) ≥ a1ρ

γ−1 − c1, pe(ρ) ≤ a2ργ + c2,

pθ (0) = 0, p′θ (ρ) ≥ 0, pθ (ρ) ≤ a3ρ

� + c3,(2.6)

with

γ ≥ 2, γ >4�

3(2.7)

with a1 > 0, a2, b, c1, c2, c3 constants.• Structural conditions on the viscosity parameters. It is well-known that in high tem-

peratures both the viscosity and heatconductivity depend sensitively on the temperature. Here we assume that this depen-dence obeys the rule{

0 < µ(1 + θα) ≤ µ(θ) ≤ µ(1 + θα),

0 < ζθα ≤ ζ(θ) ≤ ζ (1 + θα)(2.8)

for α ≥ 12 .

• Structural conditions on the heat conductivity. Analogously, we set{κ = κC(θ)+ σθ3,

0 < κC ≤ κC(θ) ≤ κC(1 + θ3),(2.9)

where the term {σθ3} with σ > 0 accounts for the radiative effects.• The specific heat at constant volume. We also require that

The specific heat cv is a Lipschitz function ofZ. (2.10)

D. Donatelli, K. Trivisa

• The species diffusion coefficient. The species diffusion coefficientD = ρd is assumedto be a continuously differentiable function depending only on the absolute temper-ature such that

0 < D < D(θ) ≤ D(1 + θ3) (2.11)

for all θ > 0.

Remark 1. As it will be obvious in the forthcoming analysis the presence of the externalforce density g in the momentum equation does not offer any additional difficulty, andit usually appears in the estimates in terms of an extra integral as in (2.3). Therefore, forsimplicity of the presentation we consider from now on that g = 0.

Remark 2. Our analysis applies also in the case where the heat conductivity satisfies themore general condition

κ = κC(θ)+ κ(ρ, θ),

with {0 < κ ≤ κC(θ) ≤ κ(1 + θβ),

κθα ≤ κR(ρ, θ) ≤ κ(1 + θα).

2.3. Main theorem. We are now ready to state our main result.

Theorem 1. Let � ⊂ R3 be a bounded domain with a boundary ∂� ∈ C2+ν, ν > 0.

Suppose that the pressure p is determined by the equation of state (1.6), with a > 0,and pe, pθ satisfying (2.6). In addition, let the viscous stress tensor S be given by (1.7),where µ and ζ are continuous differentiable globally Lipschitz functions of θ satisfying(2.8) for 1

2 ≤ α ≤ 1. Similarly, let the heat flux Q be given by (1.8) with κ satisfying(2.9). Finally, assume that the initial data ρ0,m0, θ0 satisfy

ρ0 ≥ 0, ρ0 ∈ Lγ (�),m0 ∈ [L1(�)]3,

|m0|2ρ0

∈ L1(�),

θ0 ∈ L∞(�), 0 < θ ≤ θ0(x) ≤ θ for a.e. x ∈ �,Z0 ∈ L∞(�), 0 ≤ Z0 ≤ 1 a.e. in �,

|ρ0Z0|2ρ0

∈ L1(�).

(2.12)

Then, for any given T > 0 the initial boundary value problem (1.1)-(1.4) together with(1.18)-(1.19) has a variational solution on (0, T )�.

3. Approximating Scheme

We pursue now following a similar approach as in [17] and we start by introducing athree level approximating scheme which involves a system of regularized equations. Atthis level, it is more convenient to deal with the thermal equation (1.17) instead of theinternal energy equation (1.3). Moreover, taking into account the hypotheses (2.6) onthe pressure law we decompose the elastic pressure component pe(ρ) as

pe(ρ) = pm(ρ)+ pb(ρ), (3.1)

where pm, pb belong to C([0,∞)), pm is a non-decreasing function and pb is boundedon [0,∞). The reason for this decomposition will become apparent in the sequel wherethe properties of the functions pm and pb will appear useful in obtaining a suitableentropy inequality and appropriate estimates for the solution sequence.

Compressible Radiative-Reacting Gas

The approximating scheme now reads

∂tρ + div(ρu) = ε�ρ, (3.2)

∂t (ρu)+ div(ρu ⊗ u)+ ∇(p(ρ, θ, Z)+ δρβ)+ ε∇u · ∇ρ = divS, (3.3)

∂t (aθ4 + cv(Z)ρθ)+div

((aθ4 + cv(Z)ρθ

)u)−div

((κC(θ)+ σθ3)∇θ

), (3.4)

= S : ∇u+ ε|∇ρ|2(p′m(ρ)

ρ+ δβρβ−2

)−(Zθpθ (θ)+ a

3θ4)divu

+ qdiv(D(θ)∇Z)+Kqf (ρ, θ)Zm,

∂t (ρZ)+ div(ρuZ)+ ε∇Z · ∇ρ = −Kf (ρ, θ)Zm + div(D(θ)∇Z). (3.5)

Here we require that the boundary conditions (1.18) hold true, and in addition the fol-lowing boundary condition on ρ is also satisfied:

∇ρ · n|∂� = 0. (3.6)

The initial conditions here are expressed by

ρ(0, ·) = ρ0,δ, (3.7)

ρ u(0, ·) = m0,δ, (3.8)

θ(0, ·) = θ0,δ, (3.9)

Z(0, ·) = Z0,δ. (3.10)

The initial approximation of the densityρ0,δ ∈ C2+ν(�) satisfies the boundary condition(3.6) and at the same time

0 < δ ≤ ρ0,δ ≤ δ− 1

2β on �, (3.11)

and

ρ0,δ → ρ0 in Lγ (�), |{ρ0,δ < ρ0}| → 0 for δ → 0. (3.12)

Moreover, the initial momenta are given by

m0,δ(x) ={

m0 if ρ0,δ(x) ≥ ρ0(x),

0 for ρ0,δ(x) < ρ0(x).(3.13)

The functions θ0,δ ∈ C2+ν(�) satisfy{∇θ0,δ · n|∂� = 0, 0 < θ < θ0,δ ≤ θ on �,

θ0,δ → θ0 in L1(�) δ → 0.(3.14)

Finally, the initial approximations of the mass fraction of the reactant Z0,δ ∈ C2+ν(�)satisfy {

∇Z0,δ · n|∂� = 0, 0 ≤ Z0,δ ≤ 1 on �,

Z0,δ → Z0 in L1(�) δ → 0.(3.15)

D. Donatelli, K. Trivisa

Note that the addition of the extra ε-terms ε∇u∇ρ, ε∇u∇Z in the momentumequation and in the modified chemical reaction equation is necessary in order to guar-antee that certain energy inequalities remain valid. The addition of the extra δ-terms isessential in order to ensure that the pressure estimates are compatible with the vanishingviscosity regularization. We also point out that the parabolic regularization of the conti-nuity equation allows us to overcome the problem of the vacuum, namely even thoughwe do not have uniform bounds on the density ρ itself, the approximating sequence ρnis in fact bounded. Following the same approach as in [17] we will solve the system(3.2)-(3.10) for fixed ε, δ by using a Faedo Galerkin approximating procedure.

3.1. Faedo-Galerkin approximations. The initial boundary value problem (3.2)-(3.15)will be solved via a modified Faedo-Galerkin method. As in [17, 26, 31] we start byintroducing a finite-dimensional space

Xn = span{ηj }nj=1, n ∈ {1, 2, . . . }

with ηj ∈ D(�)N being a set of linearly independent functions, which are dense inC1

0(�,RN). Our aim here is to replace the regularized equation (3.3) by a set of inte-

gral equations, with ρ, θ and Z being exact solutions of (3.2), (3.4) and (3.5). Theapproximate velocities un ∈ C([0, T ];Xn) satisfy a set of integral equations of the form∫

�

ρun(τ ) · η dx −∫�

m0,δ · η

=∫ τ

0

∫�

(ρun ⊗ un − Sn) : ∇η +(pm(ρ)+ Zθpθ (ρ)+ a

3θ4 + δρβ

)divη dxdt

+∫ τ

0

∫�

pb(ρ)divη − ε∇un∇ρη dxdt, (3.16)

for any test function η ∈ Xn, all τ ∈ [0, T ]. As in [17] the density ρ = ρ[u] in (3.2)is determined by u = un as the unique solution of (3.2) with specified boundary andinitial conditions (3.6) and (3.7) respectively. At the same time, θ = θ [ρ,un, Z], withu = un, ρ = ρ[un], Z = [ρ,un] being fixed, is the unique solution of (3.4) underthe boundary and initial conditions (1.18), (3.9). Since the density ρn solves a para-bolic equation for the existence proof we employ standard techniques and we obtain thefollowing bounds for ρn:

(inf�ρ0,δ) exp

(−∫ τ

0‖divun(t)‖L∞dt

)≤ ρn(τ, x) (3.17)

≤ (sup�

ρ0,δ) exp

(−∫ τ

0‖divun(t)‖L∞dt

)

for any τ ≥ 0 and any x ∈ �.For the existence of the temperature θ we note that Eq. (3.4) can be written as a non-

degenerate parabolic equation with respect to U = θ4 with sublinear coefficients. Asfar as the equation of the mass fraction of the reactant (3.5) is concerned let us observethat Eq. (3.5) is a parabolic quasilinear equation with coefficients that lack sufficientregularity in time, therefore we need some special regularization in time (cf. [17]). Atthis point it is possible to apply standard arguments [23], [34] to deduce the existence

Compressible Radiative-Reacting Gas

of a solution to Eq. (3.5). Note that special attention has to be given to the issue ofuniqueness because of the presence of the nonlinear part Zm. Namely, let Z1 and Z2 betwo solutions with the same data. Subtracting the corresponding equations we get

∂t (ρ(Z1 − Z2))+ div(ρu(Z1 − Z2))+ ε∇ρ∇(Z1 − Z2)

= −Kf (ρ, θ)ρ(Zm1 − Zm2 )+ div(D(θ)∇(Z1 − Z2)). (3.18)

Integrating by parts and multiplying (3.18) by sgn(Z1 − Z2) we have∫�

ρ|Z1 − Z2|(τ )dx =∫ τ

0

∫�

|Z1 − Z2|ε�ρdxdt

−K∫ τ

0

∫�

|Zm1 − Zm2 |f (ρ, θ)dxdt

for any τ ∈ [0, T ]. Therefore uniqueness follows by taking into consideration that|Zm1 − Zm2 | ≤ M|Z1 − Z2| and by applying Gronwall’s lemma. Furthermore since allquantities are smooth one can use the maximum principle in order to obtain

0 ≤ Zn(t, x) ≤ 1. (3.19)

By multiplying Eq. (3.5) by Zn and by integrating in space it follows that

d

dt

∫�

1

2ρnZ

2ndx +

∫�

D(θn)|∇Zn|2dx = −K∫�

ρm−1n θrne

−4/θn−θI Zm+1n dx

≤ −K(∫

�

θ4ndx

)r/4(∫

�

ρ4m+r−8

4−rn Z

4m+2r−44−r

n ρnZ2ndx

) 4−r4

,

(3.20)

provided that r ≤ 4. In addition, by using (3.17) and Gronwall’s lemma we get that

Zn is bounded in L∞((0, T )×�) ∩W 1,2((0, T )×�). (3.21)

Having obtained the existence of the sequence of approximate solutions ρn, un, θn, Zn,the next step now is to take the limit as n → ∞. To begin with, we observe that takingin (3.16) η = un(t) we deduce the following kinetic energy equality:

d

dt

∫�

1

2ρn|un|2 + ρnPm(ρn)+ δ

β − 1ρβn dx + ε

∫�

|∇ρ|2(p′m(ρn)

ρn+δβρβ−2

n

)dx

= −∫�

Sn : ∇u dx +∫�

(Znθnpθ (ρn)+ a

3θ4n + pb(ρn)

)divun dx, (3.22)

with

Pm(ρ) =∫ ρ

1

pm(z)

z2 dz.

Integrating in space Eq. (3.5) and using the boundary conditions we get

d

dt

∫�

qρnZndx + ε

∫�

q∇ρn∇Zndx = −∫�

qKf (ρn, Zn)Zmn dx. (3.23)

D. Donatelli, K. Trivisa

Integrating in space (3.4) and adding the resulting equation to the above relations giverise to an energy equality of the form

d

dt

∫�

1

2ρn|un|2 + ρnPm(ρn)+ δ

β − 1ρβn + qρnZn + aθ4

n + cv(Zn)ρnθndx

=∫�

pb(ρn)divundx − ε

∫�

q∇ρn∇Zndx. (3.24)

Considering that, at this stage of the approximation, the temperature θn is strictly positivewe can rewrite Eq. (3.4) as an entropy inequality

∂t

(4a

3θ3n + cv(Z)ρn log(θn)

)+ div

((4a

3θ3n + cv(Z)ρn log(θn)

)un

)

−div(κC(θn)+ σθ3

n

θn∇θn + q

D(θn)∇Znθ

)

≥ −Znpθ (ρn)divun + Sn : ∇unθn

+ κC(θn)+ σθ3n

θ2n

|∇θn|2

+qD(θn)∇Zn∇θnθ2n

−Kqf (ρn, θn)

θnZmn + ε(log(θn)− 1)cv(Zn)�ρn

+c′v(Zn)(−εdiv(∇ρn∇Zn)−Kf (ρn, θn)Z

mn + div(D(θn)∇Zn)

). (3.25)

Moreover Eq. (3.2) multiplied by ρn and integrated over � yields:

d

dt

∫�

1

2ρ2ndx + ε

∫�

|∇ρn|2dx = −1

2

∫�

ρ2ndivundx. (3.26)

Now (3.24), (3.25), (3.26) give rise to

d

dt

∫�

1

2ρn|un|2 + ρnPm(ρn)+ δ

β − 1ρβn + aθ4

n + cv(Z)ρnθn + qρnZndx

+ d

dt

∫�

1

2ρ2n − 4a

3θ3n − cv(Zn)ρn log(θn)dx + ε

∫�

q∇ρn∇Zndx

+∫�

Sn : ∇unθn

+ κC(θn)+ σθ3n

θ2n

|∇θn|2 + ε|∇ρn|2dx

+∫�

qD(θn)∇Zn∇θnθ2n

+∫�

εc′′v(Zn)∇ρn|∇Zn|2dx

≤∫�

(Znpθ (ρn)+ pb(ρn)− 1

2ρ2n

)divundx + ε

∫�

cv(Zn)∇θn∇ρndx

+∫�

ε(log(θn)− 1)c′v(Zn)∇Zn∇ρn + c′′v(Zn)D(θn)|∇Zn|2dx

+∫�

K

(q

θn+ c′v(Zn)

)f (ρn, θn)Z

mn dx. (3.27)

Let us observe now that hypotheses (2.8) yield

Sn : ∇unθn

≥ θα−1|∇un + ∇utn|2. (3.28)

Compressible Radiative-Reacting Gas

By Holder’s inequality we get

|∇un + ∇utn|b ≤ c(θα−1n |∇un + ∇utn|2 + θ4

n

), where b = 8

5 − α. (3.29)

Furthermore in accordance with hypotheses (2.9),

∫�

|∇ log(θn)|2 + |∇θ32n |2dx ≤

∫�

κC(θn)+ σθ3n

θ2n

|∇θn|2dx. (3.30)

Now taking into consideration (3.17), (3.19), (3.28), (3.29), we get the following esti-mates:

supt∈[0,T ]

(‖ρn(t)‖Lβ(�) + ‖ρn(t)|un(t)|2‖L1(�)

)≤ c(δ), (3.31)

supt∈[0,T ]

(‖cv(Zn)ρn(t)θn(t)‖L1(�) + ‖θn(t)‖L4(�)

) ≤ c(δ), (3.32)

supt∈[0,T ]

‖cv(Zn)ρn(t) log(θn)(t)‖L1(�) ≤ c(δ), (3.33)

∫ T

0

∫�

Sn : ∇unθn

+ |∇ log(θn)|2 + |∇θ32n |2 + ε|∇ρn|2dxdt ≤ c(δ), (3.34)

and

‖un‖Lb(0,T ;W 1,b0 (�))

≤ c(δ) with b = 8

5 − α. (3.35)

The first level of approximate solutions are constructed as a limit of ρn, un, θn andZn for n → ∞. By following a similar line of arguments as in [29] we get

ρn −→ ρ in C([0, T ], Lβweak(�)).

By using the estimates obtained in the previous steps we can assume

un −→ u weakly in Lb(0, T ;W 1,b0 (�)),

ρnun −→ ρu ∗-weakly in L∞(0, T ;L 2ββ+1 (�)),

where ρ, u satisfy Eq. (3.16) together with the boundary conditions (3.31) in the senseof distribution. Actually better estimates are available for the density, namely

∂tρn, �ρn are bounded in Lp((0, T )×�), p > 1,

which allow us to conclude that ρ, u satisfy (3.2) a.e. on (0, T )×�)whereas the bound-ary condition (3.6) and initial condition hold in the sense of traces. In order to continuewe have to show the pointwise convergence of the temperature. To this end we applythe following lemma.

D. Donatelli, K. Trivisa

Lemma 2. Let � ⊂ RN , N ≥ 2 be a bounded Lipschitz domain and � ≥ 1 a given

constant. Let ρ ≥ 0 be a measurable function satisfying

0 < M ≤∫�

ρdx,

∫�

ρβ ≤ K for β >2N

N + 2.

Then there exists a constant c = c(M,K) such that

‖v‖L2(�) ≤ c(M,K)

(‖∇v‖L2(�) +

(∫�

ρ|v| 1�

)�)

for any v ∈ W 1,2(�).

Proof. For the proof we refer the reader to Lemma 5.1 in [20].

Using Lemma 2 and the estimates (3.31)- (3.34) it is possible to extract a subsequenceof θn such that

θn −→ θ weakly in L2(0, T ;W 1,2(�)), (3.36)

θn −→ θ weakly-* in L∞(0, T ;L4(�)), (3.37)

log(θn) −→ log(θ) weakly in L2(0, T ;W 1,2(�)). (3.38)

Moreover we have

Zn −→ Z weakly in L2(0, T ;W 1,2(�)).

In order to get the strong convergence we need one more auxiliary result.

Lemma 3. Let� ⊂ RN ,N ≥ 2 be a bounded Lipschitz domain. Let {vn} be a sequence

of functions bounded in

L2(0, T ;Lq(�)) ∩ L∞(0, T ;L1(�)) q >2N

N + 2.

Furthermore assume that

∂tvn ≥ gn in D′((0, T )×�),

where the distributions gn are bounded in the space L1(0, T ;W−m,p(�)), for m ≥ 1,p > 1. Then

vn −→ v in L2(0, T ;W−1,2(�))

passing into a subsequence as the case may be.

Proof. The proof is given in Lemma 6.3 of Chapter 6 in [26].

Compressible Radiative-Reacting Gas

Using now the fact that cv(Zn)ρn log(θn) satisfies the entropy inequality (3.25) andthat cv(Zn) verifies (2.10) we get

ρn log(θn) bounded in L∞(0, T ;L1(�)) ∩ L2(0, T ;L 6ββ+6 (�))

and

ρnun log(θn) bounded in L2(0, T ;L 6β4β+3 (�)).

By a direct application of Lemma 3 and taking into account (2.10) we get

4a

3θ3+cv(Zn)ρn log(θn)−→ 4a

3θ3+cv(Z)ρlog(θ) weakly inL2(0, T ;W−1,2(�)).

(3.39)

By using now (3.36) and (3.38) we can conclude∫ T

0

∫�

(4a

3θ3n + cv(Zn)ρn log(θn)

)θndxdt

−→∫ T

0

∫�

(4a

3θ3 + cv(Z)ρlog(θ)

)θdxdt. (3.40)

Since the function y → 4ay3/3 + cv(Z)ρ log(y) is nondecreasing we have

θn −→ θ strongly in L1((0, T )×�). (3.41)

Now by interpolation arguments we have that

θn −→ θ strongly in Lp((0, T )×�) for p > 4 (3.42)

and

Sn −→ S weakly in Lq((0, T )×�) for q > 1, (3.43)

where

S = µ

(∇u + ∇uT − 2

3divu I

)+ ζ divu I.

Similarly we get

ρn −→ ρ in Lp((0, T )×�) for p > β. (3.44)

By using the same argument as in [17] we have

ρnun −→ ρu in C([0, T ];L2ββ+1weak(�)),

which allows us to pass into the limit and to get that the limit function ρ, u, θ satisfy(3.3) in D′((0, T )×�). Moreover, we have

ρnZn −→ ρZ ∗-weakly in L∞(0, T ;L 2ββ+1 (�)),

ρnZn −→ ρZ in C([0, T ];L2ββ+1

weak(�)),

ρnunZn −→ ρuZ weakly in L2(0, T ;L 2NβN+2β(N−1) (�)).

D. Donatelli, K. Trivisa

So we can pass into the limit in the Eqs. (3.4) and (3.5). Finally multiplying inequality(3.20) by a function ψ ∈ C∞[0, T ], ψ(0) = 1, ψ(T ) = 0, ∂tψ ≤ 0 and integrating byparts we infer

∫ T

0

∫�

(−∂tψ)1

2ρ|Z|2dxdt +

∫ T

0

∫�

ψD(θ)|∇Z|2dxdt

= −K∫ T

0

∫�

f (ρ, θ)ψ |Z|m+1dxdt +∫�

1

2ρ0|Z0|2dx. (3.45)

In the same way we can let n → ∞ in the energy inequality (3.24) in order to get

−∫ T

0

∫�

∂tψ

(1

2ρ|u|2 + ρPm(ρ)+ δ

β − 1ρβ + aθ4 + cv(Z)ρθ + qρZ

)dxdt

=∫�

1

2

m0,δ

ρ0,δ+ ρ0,δPm(ρ0,δ)+ δ

β − 1ρβ0,δ + aθ4

0,δ + cv(Z)ρ0,δθ0,δ + qρ0,δZ0,δdx

+∫ T

0

∫�

ψ (pb(ρ)divu − εq∇ρ∇Z) dxdt (3.46)

for any ψ ∈ C∞[0, T ], ψ(0) = 1, ψ(T ) = 0, ∂tψ ≤ 0.The following two lemmas will be useful in the sequel.

Lemma 4. Let � ⊂ RN be a bounded Lipschitz domain. Suppose that ρ is a given

nonnegative function satisfying

0 < M ≤∫�

ρ dx,

∫�

ρβ dx < K, β >2N

N + 2.

Then the following two statements are equivalent:i) The function θ is strictly positive a.e. on �,

ρ| log(θ)| ∈ L1(�).

ii) The function log(θ) belongs to the Sobolev space W 1,2(�). Moreover, if this is thecase, then

∇ log(θ) = ∇θθ, a.e. on �.

Proof. For the proof we refer the reader to [20].

Lemma 5. Let θn → θ inL2((0, T )×�), and log(θn) → log(θ)weakly inL2((0, T )×�). Then θ is strictly positive a.e. on (0, T )×�, and log(θ) = log(θ).

Proof. For the proof we refer the reader to [20].

Compressible Radiative-Reacting Gas

Using Lemmas 4 and 5 and the estimates (3.31), (3.32) and (3.34) we can pass intothe limit in the entropy inequality (3.25) to get

∫ T

0

∫�

∂tϕ

(4a

3θ3 + cv(Z)ρ log(θ)

)+((

4a

3θ3 + cv(Z)ρ log(θ)

)u)

∇ϕdxdt

−∫ T

0

∫�

(κC(θ)+ σθ3

θ∇θ + D(θ)∇Z

θ

)∇ϕdxdt

≤∫ T

0

∫�

ϕ

(Zpθ(ρ)divu − S : ∇u

θ+ κC(θ)+ σθ3

θ2 |∇θ |2 + D(θ)∇Z∇θθ2

)dxdt

+∫ T

0

∫�

ε∇ (ϕ(log(θ)− 1)cv(Z))∇ρ + ∇(ϕc′v(Z)) (−ε∇ρ∇Z +D(θ)∇Z) dxdt

+∫ T

0

∫�

ϕ(K(c′v(Z)+ q

θ

)f (ρ, θ)Zm

)dxdt

−∫�

ϕ(0)

(4a

3θ3

0,δ + cv(Z0,δ)ρ0,δ log(θ0,δ)

)dx, (3.47)

for any test function ϕ, ϕ ∈ C∞([0, T ] ×�), ϕ ≥ 0, ϕ(T ) = 0.

4. Vanishing Viscosity Limit

Our next goal in this section is to take the limit as ε → 0 in the family of approximatesolutions {ρε,uε, θε, Zε} constructed in the previous section. We point out that sincethe estimates obtained in Sect. 3 are independent of the parameter n, they are still validfor the quantities {ρε,uε, θε, Zε}. Nevertheless, this part will not be without difficulties,namely by sending ε to zero, we will lose spatial regularity of ρε due to the presence ofthe viscosity term ε�ρε. The main difficulty is to establish the strong compactness ofthe density ρε in the space L1((0, T )×�).

4.1. Pressure estimates. The estimates obtained in the previous section yield that thepressure p is bounded only in the non-reflexive space L∞(0, T , L1(�)). We can obtainbetter estimates via the multipliers technique introduced in [30, 35]. In that spirit we usethe following quantities

ϕ(t, x) = ψ(t)B[ρνε ] ψ ∈ D(0, T )

as test functions in the weak formulation of the momentum equation (3.3). Here B[v] isa suitable branch of solutions to the problem (see [30])

div (B[v]) = v − 1

|�|∫�

vdx in �, B[v]|∂� = 0.

After a lengthy but straightforward computation we get the following integral identity

∫ T

0

∫�

(pe(ρε)+ Zεθεpθ (ρε)+ a

3θ4ε + δρβε

)ρνε dxdt =

7∑j=1

Ij , (4.1)

D. Donatelli, K. Trivisa

where ν is a positive constant and

I1 =∫ T

0ψ

(∫�

pe(ρε)+ Zεθεpθ (ρε)+ a

3θ4ε + δρβε dx

)dt,

I2 =∫ T

0ψ

∫�

Sε : ∇B

[ρε − 1

|�|]dxdt,

I3 = −∫ T

0ψ

∫�

[ρεuε ⊗ uε] : ∇B

[ρε − 1

|�|]dxdt,

I4 = ε

∫ T

0ψ

∫�

(∇uε∇ρε) · B

[ρε − 1

|�|]dxdt,

I5 =∫ T

0∂tψ

∫�

ρεuεc[ρε − 1

|�|]dxdt,

I6 = −ε∫ T

0ψ

∫�

ρεuε · B[�ρε]dxdt,

I7 =∫ T

0ψ

∫�

ρεuε · B[div(ρεuε)]dxdt.

Now, as the estimates (3.28)-(3.31) remain valid for {ρε,uε, θε, Ze} we can check thatthe integrals I1-I2 are bounded. Let us point out that estimating the integral I1 we usethe fact that Zε is bounded, namely 0 ≤ Zε ≤ 1. So, by following a similar line ofarguments as in [20, 26], accommodating them appropriately in the new context, it ispossible to show that

δρβ+νε is bounded in L1((0, T )×�), ν > 1. (4.2)

4.2. Strong compactness of the temperature. Taking into consideration the estimates ofthe previous section we may now assume that

θε −→ θ weakly in L2(0, T ;W 1,2(�)), (4.3)

θε −→ θ weakly-* in L∞(0, T ;L4(�)), (4.4)

log(θε) −→ log(θ) weakly in L2(0, T ;W 1,2(�)), (4.5)

Zε −→ Z weakly in L2(0, T ;W 1,2(�)), (4.6)

ρε −→ ρ in C([0, T ], Lβweak(�)), (4.7)

uε −→ u weakly in Lb(0, T ;W 1,b0 (�)), (4.8)

ρεuε −→ ρu in C([0, T ], L2ββ+1 (�)). (4.9)

Combining (4.3), (4.4), (4.5) and (4.9) we obtain

ρε log(θε)uε −→ ρlog(θ)u weakly in Lp((0, T )×�) for p > 1. (4.10)

Following a similar procedure to the one of the previous section we end up with

θε −→ θ strongly in L2((0, T )×�). (4.11)

Compressible Radiative-Reacting Gas

4.3. Convergence for ρ. Our aim now is to prove the strong convergence for ρε. Inparticular we have to control the oscillation of the sequence ρε by proving boundnessof the defect measure

df t[ρε − ρ] =∫�

ρ log ρ(t)− ρ log ρ(t)dx. (4.12)

Now, by using the renormalized version of the regularized continuity equation (3.2),namely

∂tb(ρε)+ div(b(ρε)uε)+ (b′(ρε)ρε − b(ρε))divuε= εdiv(1�∇b(ρε))− ε1�b

′′(ρε)|∇ρε|2

in D′((0, T )× R3), with b ∈ C2[0,∞), b(0) = 0, and b′, b′′ bounded functions and b

convex, and by suitably approximating z → zlogz by smooth functions in the spirit of[20] we get in the limit∫

�

ρ log(ρ)− ρ log(ρ)(τ )dx ≤∫ τ

0

∫�

ρdivu − ρdivudxdt (4.13)

for a.e. τ ∈ [0, T ].In the sequel we employ the multipliers technique as in Feireisl [26] and Lions [35],

that is, we use the quantities

ϕ(t, x) = ψ(t)η(x)(∇�−1)[ρε], ψ ∈ D(0, T ), η ∈ D(�)

as a test function in the approximate momentum equation (3.3) and we end up after arather lengthy computation (see also [20]) with the following relation:

limε→0

∫ T

0

∫�

ψη

[pe(ρε)+θεZεpθ (ρε)+δρβε −

((ζ(θ)− 2

3

)+2µ(θ)

)divu

]ρεdt

=∫ T

0

∫�

ψη

[pe(ρ)+θZpθ (ρ)+δρβ −

((ζ(θ)− 2

3

)+ 2µ(θ)

)divu

]ρ dt

+(J 1 − limε→0

J 1ε )+ 2( lim

ε→0J 2ε − J 2), (4.14)

with

J 1 =∫ T

0

∫�

ψηu · (ρR[ρu] − R[ρ](ρu))dx dt,

J 1ε =

∫ T

0

∫�

ψηuε · (ρR[ρεuε] − R[ρε](ρεuε))dx dt,

J 2 =∫ T

0

∫�

ψ(R[ηµ(θ)∇u] − ηµ(θ)R[∇u])ρdx dt,

J 2ε =

∫ T

0

∫�

ψ(R[ηµ(θε)∇uε] − ηµ(θε)R[∇uε])ρεdx dt,

where

R[A] =∑i,j

Ri,j [Ai,j ], R = Ri,j [v] = F−1ξ→x

[ξiξj

|ξ |2 Fx→ξ [v]

].

D. Donatelli, K. Trivisa

Using now the continuity property of the bilinear form

[v,w] → vR[w] − R[v]w

one obtains as in [29, 26, 35] that

limε→0

J 1ε = J 1.

The convergence

limε→0

J 2ε = J 2

is obtained following the analysis presented in Feireisl [26, 21] in the spirit of Coifmanand Meyer [12].

Now relation (4.14) together with the strong convergence of {θε} yields

ρdivu − ρdivu≤ 1

ζ(θ)− 23 +µ(θ)

[(pe(ρ)ρ − pe(ρ)ρ

)+θ

(pθ(ρ)Zρ − pθ(ρ)Zρ

)]

+ 1

ζ(θ)− 23 + µ(θ)

[δ(ρβρ − ρβ+1

)]≤ I1 + I2 + I3.

At this point, let us remark again that Zε verifies a parabolic equation. Now, using themaximum principle and the initial condition (3.15) we have 0 ≤ Zε ≤ 1. This togetherwith the fact that pθ is a nondecreasing function of ρ yields I2 ≤ 0. Since also I3 ≤ 0we can follow the same path of [20] and we obtain using (4.13),∫

�

ρ log(ρ)− ρ log(ρ)(τ )dx ≤ �

µ

∫ τ

0

∫�

ρ log(ρ)− ρ log(ρ)dx.

Consequently ρ log(ρ) = ρ log(ρ) that means

ρε −→ ρ in L1((0, T )×�). (4.15)

4.4. Passing into the limit (ε → 0). Having established all necessary estimates we arenow ready to let ε → 0. First of all we have

εdiv(1�∇ρε) → 0 in L2(0, T ;W−1,2(RN)) for ε → 0,

and we get the limit functions ρ, u satisfy the continuity equation (1.1) in D′((0, T )×RN), provided they were extended to be zero outside �. From the previous energy

estimates we have,

ε∇uε∇ρε → 0, ε∇ρε∇Zε → 0 in L1(0, T ;L1(�)),

and making use of (4.6) - (4.8) we obtain

ρεuε → ρu, ρεZε → ρZ in C([0, T ];L2ββ+1weak(�)).

Compressible Radiative-Reacting Gas

The limit function ρ, u, θ and Z satisfy in D′((0, T )×�) the momentum equation

∂t (ρu)+ div(ρu ⊗ u)+ ∇(pe(ρ)+ θZpθ (ρ)+ a

3θ4 + δρβ

)= divS. (4.16)

Finally the relations (4.6), (4.15) yield

ρm−1ε Zmε −→ ρm−1Zm in D′((0, T )×�),

and so the equation of the mass fraction of the reactant

∂t (ρZ)+ div(ρuZ) = −Kf (ρ, θ)Zm + div(D(θ)∇Z), (4.17)

is verified in D′((0, T )×�) by the limit function ρ, u, θ and Z. Now passing into thelimit in the energy equality (3.46) we recover the total energy balance.

−∫ T

0

∫�

∂tψ

(1

2ρ|u|2 + ρPe(ρ)+ δ

β − 1ρβ + aθ4 + cv(Z)ρθ + qρZ

)dxdt

=∫�

(1

2

m0,δ

ρ0,δ+ ρ0,δPe(ρ0,δ)+ δ

β − 1ρβ0,δ

)dx

+∫�

(aθ4

0,δ + cv(Z0,δ)ρ0,δθ0,δ + qρ0,δZ0,δ

)dx (4.18)

for any ψ ∈ C∞[0, T ], ψ(0) = 1, ψ(T ) = 0, ∂tψ ≤ 0. Similarly sending ε → 0 in(3.47)

∫ T

0

∫�

∂tϕ

(4a

3θ3 + cv(Z)ρ log(θ)

)+((

4a

3θ3 + cv(Z)ρ log(θ)

)u)

∇ϕdxdt

−∫ T

0

∫�

(κC(θ)+ σθ3

θ∇θ + D(θ)∇Z

θ

)∇ϕdxdt

≤∫ T

0

∫�

ϕ

(Zpθ(ρ)divu − S : ∇u

θ+ κC(θ)+ σθ3

θ2 |∇θ |2 + D(θ)∇Z∇θθ2

)dxdt

+∫ T

0

∫�

∇(ϕc′v(Z)) (D(θ)∇Z) dxdt+∫ T

0

∫�

ϕ(K(c′v(Z)+ q

θ

)f (ρ, θ)Zm

)dxdt

−∫�

ϕ(0)

(4a

3θ3

0,δ + cv(Z0,δ)ρ0,δ log(θ0,δ)

)dx, (4.19)

for any test function ϕ, ϕ ∈ C∞([0, T ] ×�), ϕ ≥ 0, ϕ(T ) = 0.

5. Recovering the Original System (δ → 0)

In this last part we pass into the limit for δ → 0 in the sequence, ρδ , uδ , θδ , Zδ of theapproximate solutions constructed in the previous section and we recover the variationalsolutions. Again in this part the central issue is to recover strong compactness for ρδ andθδ . For simplicity we divide the proof in different steps.

D. Donatelli, K. Trivisa

Step 1. Energy estimates. By the energy equality (4.18) we have

ρδ bounded in L∞(0, T ;Lγ (�)), (5.1)√ρδuδ bounded in L∞(0, T ;L2(�)), (5.2)

√ρδZδ bounded in L∞(0, T ;L2(�)), (5.3)

cv(Zδ)ρδθδ bounded in L∞(0, T ;L1(�)), (5.4)

θδ bounded in L∞(0, T ;L4(�)). (5.5)

Moreover as in Sect. 3 we get

θ3/2δ bounded in L2(0, T ;W 1,2(�)), (5.6)

log(θδ) bounded in L2(0, T ;W 1,2(�)), (5.7)

Sδ bounded in La(0, T ;Ls(�)) with a = 8

5 − α, s = 8

7 − α. (5.8)

By applying now the same procedure as in Sect. 4.1 we get the following refined estimatefor ρδ

ργ+νδ + δρ

β+νδ is bounded in L1((0, T )×�), ν > 1. (5.9)

Step 2. Convergence. Now by virtue of (5.1)-(5.7) we can suppose

ρδ −→ ρ in C([0, T ], Lγweak(�)), (5.10)

uδ −→ u weakly in Lb(0, T ;W 1,b0 (�)), (5.11)

where ρ, u satisfy Eq. (1.1) in D′((0, T )× R3). We have also

ρδuδ −→ ρu in C([0, T ], Lγγ+1 (�)), (5.12)

log(θδ) −→ log(θ) weakly in L2(0, T ;W 1,2(�)), (5.13)

ρδ log(θδ) −→ ρlog(θ) weakly in L2(0, T ;L 6γ6+γ (�)), (5.14)

ρδ log(θδ)uδ −→ ρlog(θ)u weakly in L2(0, T ;L 6γ3+4γ (�)). (5.15)

Step 3. Pointwise convergence for the temperature. By applying Lemma 3 to the entropyinequality (4.19) we obtain

4

3aθ4δ −ρδZδPθ (ρδ)+ρδcv(Z delta) log(θδ)−→ 4

3aθ4+ρPθ (ρ)Z + ρcv(Z)log(θ),

(5.16)

in L2(0, T ;W−1,2(�)). In particular we have∫ T

0

∫�

(4

3aθ4δ − ρδZδPθ (ρδ)+ ρδcv(Zδ) log(θδ)

)θδdxdt

−→∫ T

0

∫�

(4

3aθ4 + ρPθ (ρ)Z + ρcv(Z)log(θ)

)θdxdt,

which implies

θδ −→ θ in L2((0, T )×�).

Compressible Radiative-Reacting Gas

Step 4. Pointwise convergence for the density. In order to pass into the limit we need thestrong convergence of the density. The main part consists in showing that the oscillationdefect measure oscβ+1[ρδ → ρ] defined by

oscβ+1[ρδ→ρ]((0, T )×�)=supk≥1

(lim supδ→0

∫ T

0

∫�

|Tk(ρδ)−Tk(ρ)|β+1dxdt

),

(5.17)

where Tk(ρ) are cut-off functions

Tk(y) = T(yk

)with T ∈ C∞(R) - a concave function,

T (y) ={y, for 0 ≤ y ≤ 12 if y ≥ 3

is bounded. We remark that this choice of cut-off functions differs from the one usedin our earlier work [17] and accommodates appropriately the complexity of the currentmodel, namely the dependence of the viscosity parameters on the absolute temperatureand the dependence of the pressure on the species concentration.

Taking into account that the reactant mass fraction is bounded we estimate the ampli-tude of oscillations using a similar line of argument as in [20] (see also [26]), namelywe write

pe(ρ) = p(c)e (ρ)+ p(m)e (ρ)+ p(b)e (ρ),

withp(b)e uniformly bounded on [0,∞), p(m)e nondecreasing, andp(b)e a convex function

satisfying

p(c)(ρ) ≥ aργ , with a > 0.

Next, we take into consideration the property of the monotone components

pθ(ρ)Tk(ρ) ≥ pθ(ρ)Tk(ρ), p(m)e (ρ)Tk(ρ) ≥ p

(m)e (ρ)Tk(ρ),

and we conclude following the line of argument presented in [20, 26] first that

oscβ+1[ρδ → ρ]((0, T )×�) < ∞,

and then

ρδ −→ ρ strongly in L1((0, T )×�). (5.18)

Step 5. Conclusion. Now in account of (5.10) and (5.12) we get that the continuityequation (1.1) is satisfied in the sense of distribution. Moreover by using (5.9) we get

δρβ −→ 0 in Lβ+νβ ((0, T )×�),

and we recover the momentum equation (1.2). Using a similar analysis as in the previoussection we can verify the reactant mass fraction equation, as well. Finally, in view of(5.18) and the estimates obtained before, we can pass into the limit in the energy equality(4.18) and in the entropy inequality (4.19).

D. Donatelli, K. Trivisa

6. A Related Model in Astrophysics

In this section we present a model which describes the evolution of gaseous stars. In thespirit of our earlier discussion we think of a star as a continuum, that is a gaseous objectwhich occupies a certain domain in R

3. For related articles on the dynamics of gaseousstars we refer the reader to the articles [20, 28].

The evolution of gaseous stars is governed by the Navier-Stokes-Poisson system whichhere reads

∂tρ + div(ρu) = 0, (6.1)

∂t (ρu)+ div(ρu ⊗ u)+ ∇p = divS + ρ∇�, (6.2)

∂t (ρs)+ div(ρus)+ div

(QF

θ

)= S : ∇u

θ− QF · ∇θ

θ2 + Kqf (ρ, θ)Zm

θ, (6.3)

∂t (ρZ)+ div(ρuZ) = −Kf (ρ, θ)Zm + div(D(θ)∇Z), (6.4)

−�� = Gρ, G > 0. (6.5)

In the above system the pressure p, the viscous stress tensor S, the heat flux QF arerelated to the macroscopic variables through the constitutive relations (1.3), (1.7) and(1.9) as described in Sect. 1. The above system can be obtained from (1.1)-(1.4) whenthe gravitational force g in (1.2) is given by

g = −∇�, with −�� = Gρ.

As an immediate consequence of the estimates and the analysis presented in ourearlier discussion (see also [20, 28]) we get the following theorem.

Theorem 6. Let � ⊂ R3 be a bounded domain with a boundary ∂� ∈ C2+ν, ν > 0.

Suppose that the pressure p is determined by the equation of state (1.6), with a > 0,and pe, pθ satisfying (2.6). In addition, let the viscous stress tensor S be given by (1.7),where µ and ζ are continuous differentiable globally Lipschitz functions of θ satisfying(2.8) for 1

2 ≤ α ≤ 1. Similarly, let the heat flux Q be given by (1.8) with κ satisfying(2.9). Finally, assume that the initial data ρ0,m0, θ0 satisfy

ρ0 ≥ 0, ρ0 ∈ Lγ (�),m0 ∈ [L1(�)]3,

|m0|2ρ0

∈ L1(�),

θ0 ∈ L∞(�), 0 < θ ≤ θ0(x) ≤ θ for a.e. x ∈ �,Z0 ∈ L∞(�), 0 ≤ Z0 ≤ 1 a.e. in �,

|ρ0Z0|2ρ0

∈ L1(�).

(6.6)

Then, for any given T > 0 the initial boundary value problem (6.1)-(6.5) together with(1.18)-(1.19) possesses a variational solution on (0, T )�.More precisely, the solutionsatisfies parts (a), (c), (d), (f) in Definition 2.1 and in addition

(b′) The velocity u belongs to the class

u ∈ La(0, T ;W 1,b0 (�)), b > 1, ρu(0, ·) = m0,

and the momentum equation (1.2) holds in D′((0, T )×�) in the sense that∫ T

0

∫�

ρ u ∂tψ + ρ(u ⊗ u) : ∇ ψ + p divψ dx dt =∫ T

0

∫�

S : ∇ψ dx dt

−∫ T

0

∫�

ρ ∇�ψ dx dt,

for all ψ ∈ [D((0, T )×�)]N .

Compressible Radiative-Reacting Gas

(e′) The total energy E defined by

E(ρ,u, θ, Z) =∫�

1

2ρ|u|2 + G

2�−1[ρ]ρ + ρPe(ρ)+ aθ4 + cv(Z)ρθ + qρZdx,

is a constant of motion, specifically

d

dtE[ρ,u, θ, Z](τ ) = 0. (6.7)

Remark 3. The replacement of the energy inequality (2.3) by the conservation of energy(6.7) appears natural taking into consideration that there is no flux of energy through thekinematic boundary.

7. The Equation of State

In the case of polytropic gases the pressure p is related to the macroscopic variablesρ, θ by Boyle’s law

p = Rρθ.

The pressure in a real gas is typically expressed in the terms of a series of the form

p(ρ, θ) = Rθ

∞∑k=1

Bk(θ)ρk,

with Bk denoting the so-called viral coefficients. One of the best known approximationsof that form is the Beattie-Bridgman state equation given by

p(ρ, θ) = Rθρ + β1ρ2 + β2ρ

3 + β3ρ3,

for appropriate constants βi , [2, 26].For a more precise description of the change of phase during combustion it is essen-

tial that the physical property of the material (the conversion from unburnt gas to burntgas) is reflected in the pressure law. The simplest law of that form is (in the literatureof combustion models and in the case of multicomponent reacting ideal gas mixtures)typically given by

p = ρRθ

N∑i=1

(Zi

Wi

),

whereZi represent the mass fraction of species i, andWi the molecular weight of speciesi (cf. Williams [39]).

The pressure law considered here

p(ρ, θ, Z) = pe(ρ)+ Z θ pθ (ρ)+ a

3θ4

is designed to capture both the radiation and reaction effects for one component com-bustion. Moreover, the dependence of the pressure on the mass fraction of the reactantZ assists in providing a more accurate description of the change of phase during theignition process.

D. Donatelli, K. Trivisa

A typical example is a Beattie-Bridgman-type law of the form

p(ρ, θ) = Rρ θZ +n∑k=1

βkρk + a

3θ4.

Acknowledgements. Donatelli was supported in part by the National Science Foundation under Trivisa’sgrant PECASE DMS 0239063 and the EU financial network no. HPRN-CT-2002-00282. Trivisa wassupported in part by the National Science Foundation under the Presidential Early Career Award forScientists and Engineers PECASE DMS 0239063 and an Alfred P. Sloan Foundation Research Fellow-ship. Donatelli gratefully acknowledges the hospitality of the Department of Mathematics, University ofMaryland where this research was performed.

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Communicated by P. Constantin