GLOBAL SOLUTIONS TO A MODEL FOR EXOTHERMICALLY

REACTING, COMPRESSIBLE FLOWS WITH LARGE DISCONTINUOUS INITIAL DATA

Gui-Qiang Chen  David Hoff  Konstantina Trivisa

Abstract. We prove the global existence of solutions of the Navier-Stokes equations describing the dynamic combustion of a compressible, exothermically reacting fluid, and we study the largetime behavior of solutions, giving necessary and sufficient conditions for complete combustion in certain cases. The adiabatic constants and specific heats of the burned (product) and unburned (reactant) fluids may differ, and the initial data may be large and discontinuous.

1. Introduction.

We prove the global existence of solutions of the Navier-Stokes equations describing the dynamic combustion of a compressible, exothermically reacting fluid, and we study the large-time behavior of solutions, giving necessary and sufficient conditions for complete combustion in certain cases. The adiabatic constants and specific heats of the burned (product) and unburned (reactant) fluids may differ, and the initial data may be large and discontinuous.

In the present context, the Navier-Stokes equations express the conservation of mass, the balance of momentum and energy, and two-species chemical kinetics as follows:

(1.1)  \begin{cases}v_{t} − u_{x} = 0, \\ u_{t} + p(v, θ, Z)_{x} = (\frac{εu_{x}}{v} )_{x}, \\ (e + \frac{1}{2}u^{2} + qZ)_{t} + (up(v, θ, Z))_{x} = (\frac{λθ_{x} + εuu_{x}}{v} )_{x}, \\ Z_{t} + Kϕ(θ)Z = 0. \end{cases}

Here v = 1/ρ, u, θ, and Z represent specific volume, velocity, temperature, and reactant mass fraction, respectively, and are the basic unknown functions of x and t; e and p are the specific internal energy and pressure, and ϕ is the reaction-rate function (the precise assumptions we make on the state functions p(v, θ, Z), e(θ, Z) and ϕ(θ) will be given below); ε, λ, K, and q are positive constants describing viscosity, thermal conduction, reaction rate, and the difference in heats between reactant and product. The above equations are written in Lagrangian coordinates, so that x = constant corresponds to the trajectory of a fixed fluid particle.

First, in Theorem 1.1 we establish the global-in-time existence of solutions with quite general initial data, obtaining a number of results concerning the regularity of solutions, as well as bounds which in certain cases are independent of time. In Theorem 1.2 we give a result concerning the propagation of singularities in solutions for which v and Z are initially piecewise H^{1}. These results show that, while there is a sort of “parabolic regularization” of u and θ, singularities in v, p, Z, e, u_{x}, and θ_{x} persist for all time, convecting along particle trajectories, and evolving according to certain dynamics derivable from the corresponding Rankine-Hugoniot conditions. In Theorem 1.3 we discuss the large-time behavior of solutions under the assumption that the regularity estimates of Theorem 1.1 are independent of time, showing that the solution converges as t → ∞ to an equilibrium determined by the initial mass and energy. Finally, in Theorem 1.4 we give both necessary conditions and sufficient conditions that complete combustion occurs, that is, that Z ≡ 0 in the equilibrium state, in terms of the total initial energy.

1991 Mathematics Subject Classification. 35B40, 35D05, 76N10, 35B45.

Key words and phrases. Global discontinuous solutions, large discontinuous initial data, evolution of large jump discontinuities, asymptotic behavior, combustion, Navier-Stokes equations, difference approximations, energy estimates, total variation estimates, uniform bounds.

We begin by describing some of the physical background and deriving the specific forms of the state functions p(v, θ, Z) and e(θ, Z). In this discussion, x will be the usual (Eulerian) spatial coordinate in physical space, and ρ, u, θ, and Z will be unknown functions of x and t. The density ρ should satisfy the standard continuity equation

(1.2)  ρ_{t} + (ρu)_{x} = 0,

and the reactant mass fraction of a fixed fluid particle will decrease according to standard chemical kinetics

(1.3)  Z_{t} + uZ_{x} = −Kϕ(θ)Z.

Here K is a rate constant, and the reaction function ϕ is assumed to be an increasing, Lipschitz function on [0, ∞) satisfying

(1.4)  C^{−1}ϕ(θ) ≤ θ^{2}ϕ′(θ) ≤ Cϕ(θ)

for a constant C, and

(1.5)  \begin{cases}ϕ(θ) = 0, 0 ≤ θ ≤ θ_{i}, \\ ϕ(θ) > 0, θ > θ_{i}, \end{cases}

where θ_{i} > 0 is the ignition temperature. These conditions are satisfied, for example, by the Arrhenius function ϕ(θ) = e^{−A/θ}, θ ≫ θ_{i}. Thus, when Z = 1, for a given fluid particle, that particle consists entirely of reactant. When its temperature rises above the ignition temperature, combustion will occur, resulting in the conversion of some or all of its mass to the product species, and decreasing Z accordingly. When Z = 0, complete combustion has occurred, and the particle consists entirely of the product species.

We may regard the reactant and product as separate fluids with respective densities ρ_{1} = Zρ and ρ_{2} = (1 − Z)ρ, having the same velocities and temperatures at each point, and characterized by polytropic equations of state

(1.6)  \begin{cases}P_{j} = c_{j}(γ_{j} − 1)ρθ, \\ e_{j} = c_{j}θ,  j = 1, 2. \end{cases}

Here P_{j} and e_{j} are pressures and specific internal energies, γ_{j} > 1 are adiabatic constants, and c_{j} are (constant) specific heats. The two fluids satisfy the momentum and energy equations

(1.7)  (ρ_{j}u)_{t} + (ρ_{j}u^{2} + P_{j})_{x} = ε_{j}u_{xx},
(1.8)  (ρ_{j}E_{j})_{t} + (ρ_{j}E_{j}u + P_{j}u)_{x} = λ_{j}θ_{xx} + ε_{j}(uu_{x})_{x} + (−1)^{j}q_{j}ρϕ(θ)Z,

where E_{j} = e_{j} + u^{2}/2, and ε_{j}, λ_{j}, and q_{j} are positive constants. The last term on the right side of (1.8) represents the rate of energy lost to the reactant or gained by the product as a result of the chemical reaction.

Now define macroscopic pressure and energy by

(1.9)  \begin{cases}P = P_{1} + P_{2} (Dalton’s law), \\ ρE = ρ_{1}E_{1} + ρ_{2}E_{2}, \\ e = E − u^{2}/2, \end{cases}

and system parameters

ε = ε_{1} + ε_{2}, λ = λ_{1} + λ_{2}, q = q_{2} − q_{1}.

Summing in (1.7) and (1.8), we then obtain that

(1.10)  (ρu)_{t} + (ρu^{2} + P)_{x} = εu_{xx},

and

(1.11)  (ρE)_{t} + (ρEu + P u)_{x} = λθ_{xx} + qρϕ(θ)Z.

We define a variable specific heat c(Z) by

(1.12)  c(Z) = c_{1}Z + c_{2}(1 − Z),

so that, by (1.6) and (1.9),

(1.13)  e(θ, Z) = Ze_{1} + (1 − Z)e_{2} = c(Z)θ.

We also define γ(Z) by

(1.14)  γ(Z) = \frac{Zc_{1}γ_{1} + (1 − Z)c_{2}γ_{2}}{Zc_{1} + (1 − Z)c_{2}} ,

so that, again by (1.6) and (1.9),

(1.15)  P (ρ, θ, Z) = a(Z)ρθ,

where a(Z) = c(Z)(γ(Z) − 1). Thus the functions of state P and e for the composite fluid are exactly those of a polytropic fluid, as in (1.6), but with adiabatic “constant” γ and specific heat c depending on Z. The equations (1.2), (1.3), (1.10), and (1.11), together with the state equations (1.13) and (1.15), then comprise a closed system for the unknown functions ρ, u, θ, and Z. Rewriting in terms of the Lagrangian coordinate, still denoted by x, and taking

(1.16)  p(v, θ, Z) = P (v^{−1}, θ, Z) = a(Z)θ/v,

we then obtain the system (1.1).

In the present paper we consider a fluid of finite total mass confined to a bounded region, so that, without loss of generality, 0 < x < 1. We impose boundary conditions

(1.17)  u(x, t) = 0, θ_{x}(x, t) = 0, x = 0, 1.

The solutions we obtain will be weak solutions in the usual sense: writing the system (1.1) in the form

w_{t}^{j} + f^{j}(w)_{x} = (b^{jk}(w)w_{x}^{k})_{x} + g^{j}(w),

we say that w is a weak solution provided that w is suitably integrable, and

(1.18)  ∫_{0}^{1} w^{j}(x, ·)ψ(x, ·)dx|_{t_{1}}^{t_{2}} = ∫_{t_{1}}^{t_{2}} ∫_{0}^{1}(w^{j}ψ_{t} + (f^{j} − b^{jk}w_{x}^{k})ψ_{x} + g^{j}ψ)dxdt

for all t_{2} ≥ t_{1} ≥ 0 and all ψ ∈ C^{1}([0, 1] × [0, ∞)), with the exception that, when w^{j} = u, (1.18) is required to hold only for ψ satisfying the boundary conditions ψ(0, t) = ψ(1, t) = 0.

We consider initial data

(1.19)  (v, u, θ, Z)|_{t=0} = (v_{0}, u_{0}, Z_{0}, θ_{0})

satisfying

(1.20)  \begin{cases}C_{0}^{−1} ≤ v_{0}(x) ≤ C_{0}, θ_{0}(x) ≥ C_{0}^{−1}, 0 ≤ Z_{0}(x) ≤ 1, \\ ||θ_{0}||_{L^{2}} + ||u_{0}||_{L^{4}} ≤ C_{0} \end{cases}

for a constant C_{0} > 0, which may be arbitrarily large. Finally, we define the following functionals for a given weak solution (v, u, θ, Z) of (1.1), (1.17), and (1.19):

D(t) = sup_{0≤s≤t} (||u(·, s)^{2}||^{2} + ||θ(·, s)||^{2})

+ ∫_{0}^{1} ∫_{0}^{t} (θ^{−1}u_{x}^{2} + u^{2}u_{x}^{2} + θ_{x}^{2} + ϕ(θ)Z^{2})dxdt,
(1.21)  E(t) = sup_{0≤s≤t} (σ(s)||u_{x}(·, s)||^{2} + σ^{2}(s)||θ_{x}(·, s)||^{2})

+ ∫_{0}^{1} (||(u_{x}, θ_{x})(·, s)||^{2} + σ(s)||u_{t}(·, s)||^{2} + σ^{2}(s)||θ_{t}(·, s)||^{2}) ds,
F(t) = sup_{0≤s≤t} (σ^{2}(s)||u_{t}(·, s)||^{2} + σ^{3}(s)||θ_{t}(·, s)||^{2})

+ ∫_{0}^{t} (σ^{2}(s)||u_{xt}(·, s)||^{2} + σ^{3}(s)||θ_{xt}(·, s)||^{2}) ds,

where σ(t) = min{1, t}, and || · || denotes the usual norm in L^{2}([0, 1]).

The following theorem then gives our main results concerning the existence and regularity of solutions of the system (1.1).

Theorem 1.1. Assume that the functions p(v, θ, Z), e(θ, Z), and ϕ(θ) are as described above in (1.4), (1.5), and (1.12)–(1.16), and let initial data (v_{0}, u_{0}, θ_{0}, Z_{0}) be given, as in (1.20). Then the initial-boundary value problem (1.1), (1.17), and (1.19) has a global weak solution (v, u, θ, Z) for which

(1.22)  \begin{cases}v ∈ C([0, ∞); L^{p}([0, 1])), & 1 ≤ p < ∞, \\ u ∈ C([0, ∞); L^{p}([0, 1])), & 1 ≤ p < 4, \\ θ ∈ C((0, ∞); L^{2}([0, 1])), & θ(·, t) ⇀ θ_{0} weakly in L^{2} as t → 0, \\ Z ∈ C([0, ∞); L^{p}([0, 1])), & 1 ≤ p ≤ ∞. \end{cases}

Also, given T > 0, there is a constant M depending on T, C_{0}, and on the system parameters such that, for t ∈ (0, T],

(1.23)  \begin{cases}M^{−1} ≤ v(x, t) ≤ M, a.e. x ∈ [0, 1], \\ M^{−1} ≤ θ(x, t) ≤ M max\{1, t^{−1}\}, x ∈ [0, 1], \\ 0 ≤ Z(x, t) ≤ 1, a.e. x ∈ [0, 1], \end{cases}

and

(1.24)  D(T) + E(T) + F(T) ≤ M.

In particular, u and θ are Hölder continuous on compact sets in [0, 1] × (0, ∞), and

u(·, t), θ(·, t), (\frac{εu_{x}}{v} − p)(·, t) ∈ H^{1}([0, 1])

for t > 0. Finally, in the case that c_{1} = c_{2} and γ_{1} = γ_{2} (see (1.12) and (1.14)), the bounds in (1.23) and (1.24) hold with a constant M which is independent of T.

We prove Theorem 1.1 in Sections 2 and 3, by first deriving certain a priori bounds for suitable approximate solutions, then applying these bounds to obtain the solutions of the theorem as strong limits of the approximate solutions. Both aspects of the analysis involve interesting features. The most important of the a priori estimates are the pointwise bounds for the temperature and the specific volume, and are achieved by a modification of the analysis in Chen, Hoff, and Trivisa [5] for nonreacting flows and Chen [4] for reacting flows, which was motivated by an idea of Kazhikov and Shelukhin [20]. These pointwise bounds depend crucially on the existence of an entropy, which is a convex function of the dependent variables satisfying the Clausius-Duhem inequality, the latter guaranteeing that the integral of the entropy over space is monotone in time. We show in the appendix that such an entropy can indeed be constructed, but only for v, θ in a compact set in (0, ∞) × (0, ∞); we also show that there is in fact no single entropy with the required properties for all (v, θ) ∈ (0, ∞) × (0, ∞). In other words, the existence of a suitable entropy, which is required for the derivation of pointwise bounds for v and θ, depends on these very pointwise bounds. This circularity leads to apparently unavoidable Gronwall inequalities, resulting in the time-dependence of the constant M in Theorem 1.1. These difficulties do not occur if c_{1} = c_{2} and γ_{1} = γ_{2}, however, and in this case time-independent pointwise bounds can be obtained for v and θ.

Once these pointwise bounds have been established, time independent or not, the higher-order regularity estimates in (1.21) and (1.24) can be derived by straightforward but rather long and technical energy estimates. All these a priori bounds are given in Section 2.

The other noteworthy aspect of the proof of Theorem 1.1 is the proof of strong compactness of approximate solutions in the absence of uniform regularity for the variables Z and v. Now, since Z can be represented in terms of θ by integrating its evolution equation along particle trajectories, compactness for approximate Z’s can be obtained from the corresponding compactness of approximate θ’s. For the specific volume v, the argument is much more subtle, and strong compactness is shown to result from certain structural properties of approximate solutions derived from considerations relating to the cancellation, persistence, and evolution of singularities in solutions. These compactness arguments are given in Section 3.

The basic facts concerning the propagation of singularities in solutions of (1.1) are best understood at the level of piecewise smooth solutions. These facts are given in Theorem 1.2 below and are similar to corresponding results for nonreacting flows (see [14] example). Still, there are interesting new features which arise, due to the variability of the mass fraction. To describe these results, we first suppose that (v, u, θ, Z) is a piecewise smooth solution, having one-sided limits along a curve (y(t), t), and we derive at the heuristic level the various conditions that guarantee that (v, u, θ, Z) is a weak solution across (y(t), t). First, in light of the regularity estimates in Theorem 1.1 for u and θ,

(1.25)  [u(y, t)] = [θ(y, t)] = 0,

where [u(y, t)] = u(y+, t) − u(y−, t), etc. Next, applying the standard Rankine-Hugoniot conditions to the mass equation in (1.1), we find that ẏ(t)[v(y, t)] = [u(y, t)] = 0. Combining this with the observation that singularities in Z must remain at fixed points of x−space, we see that y(t) = y is constant. We then conclude that singularities remain stationary in Lagrangian space, and therefore convect along particle trajectories in physical space. Applying the Rankine-Hugoniot condition to the other equations in (1.1), we then find that, at (y, t),

(1.26)  [\frac{εu_{x}}{v} − p] = 0, [\frac{θ_{x}}{v} ] = 0, [Z]_{t} + Kϕ(θ)[Z] = 0.

We solve these equations formally as follows. First,

(1.27)  \frac{d}{dt}[log v(y, t)] = [\frac{v_{t}}{v} ] = [\frac{u_{x}}{v} ] = ε^{−1}[p] = α[log v] + β[Z],

where

(1.28)  α(y, t) = ε^{−1}θ(y, t) \frac{c_{+}(γ_{+} − 1) + c_{−}(γ_{−} − 1)}{2} \frac{[v^{−1}(y, t)]}{[log v(y, t)]},

and

(1.29)  β(y, t) = ε^{−1} \frac{v_{+} + v_{−}}{2v_{+}v_{−}} (c_{1}(γ_{1} − 1) + c_{2}(γ_{2} − 1))θ(y, t)

(here c_{+} = c(Z(y+, t)), etc.). Solving the ode’s in (1.26) and (1.27), we then obtain

[log v(y, t)] = e^{− ∫_{0}^{t} α(y,s)ds}[log v_{0}(y)]

+ ∫_{0}^{t} e^{− ∫_{s}^{t} α(y,τ)dτ} β(y, s)[Z(y, s)]ds, (1.30)

and

(1.31)  [Z(y, t)] = e^{− ∫_{0}^{t} Kϕ(θ(y,s))ds}[Z_{0}(y)].

Explicit expressions for [u_{x}] and [θ_{x}] can then be obtained from (1.26). Observe that α is strictly negative, so that the strengths of jump discontinuities in v and Z, hence in u_{x} and θ_{x}, decrease in time. Unlike the case of nonreacting flows, however, these strengths may not decrease to zero as t → ∞, because θ(y, t) may be asymptotically less than θ_{i}. We shall discuss this point further, following the statement of Theorem 1.4. Also see [3,7] for the asymptotic behavior of solutions for inviscid flows.

These observations are made precise in the following theorem.

Theorem 1.2. Assume that the hypotheses of Theorem 1.1 are in force, and suppose in addition that v_{0} and Z_{0} are piecewise in H^{1}, having simple jump discontinuities at points y_{1} < y_{2} < ⋯ < y_{n} in (0, 1) (thus v_{0}, Z_{0} ∈ H^{1}((y_{j−1}, y_{j})) for each j). Then, for the solution (v, u, θ, Z) of Theorem 1.1, the quantities v(·, t), p(·, t), Z(·, t), u_{x}(·, t), and θ_{x}(·, t) are piecewise in H^{1} for t > 0 and therefore have one-sided limits at each y_{j} for t > 0, and the jump conditions (1.25) and (1.26) and the representations (1.30) and (1.31) hold in a strict pointwise sense at each y = y_{j} for t > 0.

Next we turn to the study of the large-time behavior of solutions. This will require the assumption that the estimates of Theorem 1.1 hold independently of time, and as indicated in Theorem 1.1, this time-independence is known so far only for the special case that c_{1} = c_{2} and γ_{1} = γ_{2}. We emphasize, however, that the large-time results given below in Theorems 1.3 and 1.4 do not depend directly upon the latter assumptions in any way, but instead are consequences of the weak forms of the equations (1.1) and time-independent estimates (1.23) and (1.24). In the following theorem, we apply these estimates to show that the solution tends to a time-asymptotic steady-state solution, and we characterize this steady-state in terms of the asymptotic massfraction and the initial mass and energy, given by

(1.32)  v̄_{0} = ∫_{0}^{1} v_{0}(x)dx,

(1.33)  E_{0} = ∫_{0}^{1} (c(Z_{0}(x))θ_{0}(x) + \frac{1}{2}u_{0}(x)^{2} + qZ_{0}(x))dx.

Theorem 1.3. Let (v, u, θ, z) be a weak solution of (1.1) as in Theorem 1.1, satisfying (1.23) and (1.24) with M independent of T, and let v̄_{0} and E_{0} be as above in (1.32) and (1.33). Then there is a function Z_{∞} such that, as t → ∞,

Z(x, t) → Z_{∞}(x), a.a. x;

and, if v_{∞}(x) and θ_{∞} are defined by

(1.34)  θ_{∞} = \frac{E_{0} − q ∫_{0}^{1} Z_{∞}(x)dx}{∫_{0}^{1} c(Z_{∞}(x))dx},

and

(1.35)  v_{∞}(x) = \frac{γ(Z_{∞}(x)) − 1}{∫_{0}^{1}(γ(Z_{∞}(x)) − 1)dx} v̄_{0},

then, as t → ∞,

(1.36)  ||(u(·, t), θ(·, t) − θ_{∞})||_{H^{1}([0,1])} → 0,
(1.37)  ||v(·, t) − v_{∞}||_{L^{p}([0,1])} → 0, 1 ≤ p < ∞.

Of particular interest are the conditions that guarantee that the steady-state (v_{∞}, 0, θ_{∞}, Z_{∞}) is constant in space and is characterized by complete combustion, Z_{∞} ≡ 0. In view of the fourth equation in (1.1), this should be true when the asymptotic temperature θ_{∞} is greater than the ignition temperature θ_{i}, and the latter should hold when the total energy E_{0} in the system is sufficiently large. This is indeed the case; the precise statements are given in the following theorem.

Theorem 1.4. Let the hypotheses and notations of Theorem 1.3 be in force and assume that the initial mass-fraction Z_{0} is not identically zero.

(a) If Z_{∞} ≡ 0, then E_{0} ≥ c_{2}θ_{i}.

(b) Conversely, if

(1.38)  E_{0} > max{c_{2}θ_{i}, (c_{2} + (c_{1} − c_{2}) ∫_{0}^{1} Z_{0}dx)θ_{i} + q ∫_{0}^{1} Z_{0}dx},

then Z_{∞} ≡ 0 and v_{∞} ≡ v̄_{0}.

The gap between the necessary condition in (a) and the sufficient condition in (b) can be closed for the special class of solutions for which v = v_{0} is constant, u = 0, and θ, Z, and e depend only on t. In this case, the fluid has neither kinetic energy nor elastic potential energy, and all effects are thermal:

\frac{de}{dt} = Kqϕ(θ)Z,
\frac{dZ}{dt} = −Kϕ(θ)Z,
θ = \frac{e}{(c_{1} − c_{2})Z + c_{2}} .

The solution (e, Z) thus remains on the line segment

e + qZ = E_{0} = constant,

so that

e(t) = E_{0} − qZ(t) and θ(e(t), Z(t)) = \frac{E_{0} − qZ(t)}{(c_{1} − c_{2})Z(t) + c_{2}} .

The trajectory {(e(t), Z(t)) : 0 ≤ t < ∞} is therefore disjoint from the set of points {(e, Z) : θ(e, Z) ≤ θ_{i}} if and only if

E_{0} − qZ > (c_{2} + (c_{1} − c_{2})Z)θ_{i}

for Z ∈ (Z_{∞}, Z_{0}]. It then follows easily that Z_{∞} = 0 if and only if

E_{0} ≥ c_{2}θ_{i} and E_{0} > (c_{2} + (c_{1} − c_{2})Z_{0})θ_{i} + qZ_{0}.

Of course, the results of the present paper apply to nonreacting flows as well: if Z_{0} ≡ 0, then Z(·, t) ≡ 0 for all t, c(Z) ≡ c_{2}, and γ(Z) ≡ γ_{2}. The bounds in (1.23) and (1.24) are then independent of time, and the time-asymptotic state given in Theorem 1.3 is simply (v̄_{0}, 0, E_{0}/c_{2}, 0). In particular, singularities disappear in the time-asymptotic limit (see (1.30)). This same “asymptotic compactness” is known even when an external force is added, at least for the case of barotropic flows, and is the basis for the existence of a global attractor (Hoff and Ziane [17]). For the more general case of reacting flows considered here, the situation is quite a bit more complicated: when θ_{∞} < θ_{i} the steady-state will not be constant in space, and, more importantly, singularities in solutions will not disappear in the time-asymptotic limit (see (1.30) and (1.31)).

The existence and asymptotic behavior of global smooth solutions for reacting, compressible Navier-Stokes flows, with constant adiabatic exponent and additional species diffusion, was first established by Chen [4]. The case of nonreacting flows in one space dimension has been treated by many authors. See, for example, Chen, Hoff, and Trivisa [5] on the well-posedness and asymptotic behavior of global solutions with large discontinuous initial data, Matsumura and Yanagi [23] for the isothermal case, Amosov and Zlotnick [1] for the existence of weak solutions for certain initialboundary data, and Hoff [13]–[15] for the global well-posedness and large-time behavior for small discontinuous initial data, including the case of several space variables.

We also remark that the results of the present paper carry over to the initial boundary value problem with Dirichlet boundary data for the temperature θ for (1.1), as well as to the corresponding system written in Eulerian coordinates, (1.2), (1.10), and (1.11). Some related discussions for dynamic combustion of reacting fluids can be found in [2,7,8,9,11,12,21,22,24].

2. A Priori Estimates for Approximate Solutions

In this section we begin the proof of the existence result, Theorem 1.2, by constructing approximate solutions based upon a suitable semidiscrete difference scheme. We shall show in a sequence of lemmas that these approximate solutions satisfy all the estimates of Theorems 1.2, independently of the spatial discretization. These estimates will then be applied in Section 3 to obtain the solution of Theorems 1.2 as the strong limit of these approximate solutions, and the more general solutions of Theorem 1.1 will be obtained as the limits of smooth solutions corresponding to mollified initial data.

Let h be an increment in x such that Qh = 1 for some Q = Q(h) ∈ ℤ_{+}, x_{k} = kh for k ∈ \{0, 1, ⋯, Q − 1\}, and x_{j} = jh for j ∈ \{\frac{1}{2}, \frac{3}{2}, ⋯, Q − \frac{1}{2}\}. Approximations (v_{j}, u_{k}, θ_{j}, Z_{j})(t) to (v(x_{j}, t), u(x_{k}, t), θ(x_{j}, t), Z(x_{j}, t)) are then constructed as follows:

v̇_{j} = δu_{j}, (2.1)

u̇_{k} + δp_{k} = εδ (\frac{δu}{v})_{k} , (2.2)

c_{v}(Z_{j})θ̇_{j} + p_{j}δu_{j} = ε\frac{(δu_{j})^{2}}{v_{j}} + λδ (\frac{δθ}{v} )_{j} + Kϕ(θ_{j})Z_{j}(q + c′_{v}(Z_{j})θ_{j}), (2.3)

Ż_{j} + Kϕ(θ_{j})Z_{j} = 0. (2.4)

Here p_{j} = p(v_{j}, θ_{j}, Z_{j}), v_{k} is taken to be the average v_{k} = \frac{v_{k+ \frac{1}{2}} +v_{k− \frac{1}{2}}}{2} with j ∈ \{\frac{1}{2}, \frac{3}{2}, ⋯, Q − \frac{1}{2}\} and k ∈ {0, 1, ⋯, Q − 1}, and δ is the operator defined by

δw_{l} = \frac{w_{l+ \frac{1}{2}} − w_{l− \frac{1}{2}}}{h} , l = k, or j.

For the time being we assume only that initial data (v_{j}, u_{k}, θ_{j}, Z_{j})(0) for (2.1)–(2.4) have been specified and satisfy

u_{0} = u_{Q} = 0, δθ_{0} = δθ_{Q} = 0, (2.5)

and

\begin{cases}C_{0}^{−1} ≤ v_{j}(0) ≤ C_{0}, θ_{j}(0) ≥ C_{0}^{−1}, 0 ≤ Z_{j}(0) ≤ 1, \\ ∑_{k} u_{k}^{4}(0)h + ∑_{j} θ_{j}^{2}(0)h ≤ C_{0}. \end{cases} (2.6)

We also assume that there are distinguished points 0 < x_{k_{1}} < x_{k_{2}} < ⋯ < x_{k_{N}} < 1, N = N(h), N^{4}h ≤ 1, such that

∑_{k=k_{i}} (|[v_{k_{i}}(0)]| + |[Z_{k_{i}}(0)]|) + ∑_{k≠k_{i}} (|δv_{k}(0)|^{2} + |δZ_{k}(0)|^{2}) h ≤ C_{0}. (2.7)

Standard theory of ordinary differential equations applies to show that the initial value problem (2.1)–(2.7) has a unique solution (v_{j}, u_{k}, θ_{j}, Z_{j})(t), defined at least for small time. The a-priori bounds to be derived in this section will show that these solutions exist globally in time, and will provide sufficient compactness both to extract limiting solutions as h → 0 and to determine their asymptotic behavior.

Here and in what follows, C > 0 will denote a generic constant independent of h and t, M = M(m_{0}) is a constant independent of h but depending on T, for a given interval [0, T], only through a parameter m_{0} = m_{0}(T), which in turn is also independent of h and N.

Using the difference equations (2.1)–(2.7) and the boundary conditions, we derive some basic but important energy estimates described in Lemmas 2.1–2.2.

Lemma 2.1.

∑_{j} v_{j}(t)h = 1, (2.8)

0 ≤ Z_{j}(t) ≤ 1, (2.9)

∑_{j} Z_{j}(t)h + ∫_{0}^{t} ∑_{j} Kϕ(θ_{j}(s))Z_{j}(s)hds = ∑_{j} Z_{j}(0)h, (2.10)

∑_{j} (c_{v}(Z_{j})θ_{j}(t) + qZ_{j}(t)) h + \frac{1}{2} ∑_{k} u_{k}^{2}(t)h (2.11)

= ∑_{j} (c_{v}(Z_{j}(0))θ_{j}(0) + qZ_{j}(0)) h + \frac{1}{2} ∑_{k} u_{k}^{2}(0)h,

∑_{j} \frac{1}{2}Z_{j}^{2}(t)h + ∫_{0}^{t} ∑_{j} Kϕ(Z_{j}(s))Z_{j}^{2}(s)hds = ∑_{j} \frac{1}{2}Z_{j}^{2}(0)h. (2.12)

Proof: From equation (2.1), we obtain \frac{d}{dt} ∑_{j} v_{j}h = ∑_{j} δu_{j}h = u_{Q} − u_{0} = 0, which yields (2.8). Regarding Z_{j}, the set [0, 1] is invariant for equation (2.4), for each j, since ϕ(θ_{j}) ≥ 0; and then (2.9) follows. Summing (2.4) and integrating the resulting equation on the interval [0, t], we conclude (2.10). Summing (2.3) and integrating the resulting equation on [0, t], we obtain

∑_{j} (c_{v}(Z_{j}(t))θ_{j}(t) + qZ_{j}(t)) h + ∫_{0}^{t} ∑_{j}(p_{j}δu_{j})hds

= ∑_{j} (c_{v}(Z_{j}(0))θ_{j}(0) + qZ_{j}(0)) h + ∫_{0}^{t} ∑_{j} \frac{ε(δu_{j})^{2}}{v_{j}} hds. (2.13)

Now, summing (2.2) and integrating the resulting equation on [0, t], we have

\frac{1}{2} ∑_{k} u_{k}^{2}(t)h − ∫_{0}^{t} ∑_{j}(p_{j}δu_{j})hds = \frac{1}{2} ∑_{k} u_{k}^{2}(0)h − ε ∫_{0}^{t} ∑_{j} \frac{(δu_{j})^{2}}{v_{j}} hds. (2.14)

Adding (2.13) and (2.14) together yields (2.11). Finally, using (2.4), we have

\frac{d}{dt} (∑_{j} \frac{1}{2}Z_{j}^{2}(t)h) = ∑_{j} Z_{j}(t)Ż_{j}(t)h = −K ∑_{j} ϕ(θ_{j}(t))Z_{j}^{2}(t)h.

Integrating this last equation on the interval [0, t], we obtain (2.12). ■

Lemma 2.2. There exists m_{0} = m_{0}(T), independent of h and N, such that, for each T > 0 and any t ∈ [0, T],

E(t) + ∫_{0}^{t}(V (s) + W (s))ds ≤ m_{0} < ∞, (2.15)

with

\begin{cases}E(t) = ∑_{j} (c_{v}(Z_{j})(θ_{j} − 1 − log θ_{j}) + a(Z_{j})(v_{j} − 1 − log v_{j})) h + \frac{1}{2} ∑_{k} u_{k}^{2}h, \\ V (t) = ∑_{j} (\frac{ε(δu_{j})^{2}}{v_{j}e_{j}} ) h + λ ∑_{k} (\frac{(δθ_{k})^{2}}{v_{k}θ_{k+ \frac{1}{2}} θ_{k− \frac{1}{2}}} ) h, \\ W (t) = −K ∑_{j} (\frac{θ_{j}−1}{θ_{j}}) ϕ(θ_{j})Z_{j}(q + c′_{v}(Z_{j})θ_{j}). \end{cases}

In the case that γ and c_{v} are independent of Z, that is, γ_{1} = γ_{2} and c_{1} = c_{2}, the parameter m_{0} is independent of T.

Proof: The choice of the energy E = E(t) under consideration is inspired by the analysis in the appendix which leads us to a precise form of a physical entropy for (1.1). Differentiating E = E(t) with respect to t and taking into consideration the relation (1.2) and the momentum equation (2.3), we obtain

\frac{d}{dt}E(t) = ∑_{j} (c_{v}′(Z_{j})(θ_{j} − 1 − log θ_{j}) + a′(Z_{j})(v_{j} − 1 − log v_{j})) Ż_{j}h

− ∑_{j} \frac{1}{θ_{j}} (\frac{ε(δu_{j})^{2}}{v_{j}} + λδ (\frac{δθ}{v} )_{j} + Kϕ(θ_{j})Z_{j}(q + c′_{v}(Z_{j})θ_{j})) h

+ ∑_{j} (c_{v}(Z_{j})θ̇_{j} + a(Z_{j})v̇_{j}) h + ∑_{k} u_{k}u̇_{k}h.

Integrating the last relation on the interval [0, t], we get

E(t) − E(0) = ∑_{l=1}^{4} A_{l}(t),

10 NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE, REACTING FLOW

where

A_{1}(t) = ∑_{j} ∫_{0}^{t} a(Z_{j})v̇_{j}ds h,

A_{2}(t) = − ∑_{j} ∫_{0}^{t} \frac{1}{θ_{j}} (\frac{ε(δu_{j})^{2}}{v_{j}} + λδ (\frac{δθ}{v} )_{j}) ds h,

A_{3}(t) = ∑_{j} ∫_{0}^{t} (c_{v}(Z_{j})θ̇_{j} − \frac{1}{θ_{j}} Kϕ(θ_{j})Z_{j}(q + c′_{v}(Z_{j})θ_{j})) ds h + ∫_{0}^{t} ∑_{k} u_{k}u̇_{k}hds,

A_{4}(t) = ∑_{j} ∫_{0}^{t} (c_{v}′(Z_{j})(θ_{j} − 1 − log θ_{j}) + a′(Z_{j})(v_{j} − 1 − log v_{j})) Ż_{j}ds h.

Now, A_{2}(·) is equivalent to

A_{2}(t) = − ∫_{0}^{t} (∑_{j} \frac{ε(δu_{j})^{2}}{v_{j}θ_{j}} h + λ ∑_{k} \frac{(δθ_{k})^{2}}{v_{k}θ_{k+ \frac{1}{2}} θ_{k− \frac{1}{2}}} h) ds.

Using Lemma 2.1, we have

A_{3}(t)= ∑_{j} (c_{v}(Z_{j}(t))θ_{j}(t) − c_{v}(Z_{j}(0))θ_{j}(0)) h + (\frac{1}{2} ∑_{k} u_{k}^{2}(t)h − \frac{1}{2} ∑_{k} u_{k}^{2}(0)h)

+ ∑_{j} ∫_{0}^{t} (−c_{v}′(Z_{j})Ż_{j}θ_{j} − \frac{1}{θ_{j}} Kϕ(θ_{j})Z_{j} (q + c′_{v}(Z_{j})θ_{j})) dsh

= − ∑_{k} qZ_{j}(t)h + ∑_{k} qZ_{j}(0)h + ∑_{j} ∫_{0}^{t} Kϕ(θ_{j})Z_{j} (c_{v}′(Z_{j})θ_{j} − \frac{1}{θ_{j}} (q + c′_{v}(Z_{j})θ_{j})) dsh

= ∑_{j} ∫_{0}^{t} KZ_{j}ϕ(θ_{j})\frac{θ_{j} − 1}{θ_{j}} (q + c′_{v}(Z_{j})θ_{j}) ds h.

Therefore, by Lemma 2.1, we obtain (2.16) E(t) + ∫_{0}^{t} (V (s) + W (s)) ds

= E(0) + ∑_{j} ∫_{0}^{t} a(Z_{j})v̇_{j}ds h

− ∫_{0}^{t} ∑_{j} (c_{v}′(Z_{j})(θ_{j} − 1 − log θ_{j}) + a′(Z_{j})(v_{j} − 1 − log v_{j})) Kϕ(θ_{j})Z_{j}hds

≤ E(0) + C_{1} ∫_{0}^{t} ∑_{j} (c_{v}(Z_{j})(θ_{j} − 1 − log θ_{j}) + a(Z_{j})(v_{j} − 1 − log v_{j})) hds

+ ∑_{j} a(Z_{j}(t))v_{j}(t)h − ∑_{j} a(Z_{j}(0))v_{j}(0)h − ∫_{0}^{t} ∑_{j} Kϕ(θ_{j})a′(Z_{j})Z_{j}v_{j}h ds.

Hence,

E(t) + ∫_{0}^{t} (V (s) + W (s)) ds

≤ E(0) + C_{1} ∫_{0}^{t} ∑_{j} (c_{v}(Z_{j})(θ_{j} − 1 − log θ_{j}) + a(Z_{j})(v_{j} − 1 − log v_{j})) hds + C_{2}T.

The result (2.15) now follows by using the Gronwall inequality.

Observe that, if one assumes that γ and c_{v} are independent of Z, that is, γ_{1} = γ_{2} and c_{1} = c_{2}, then (2.16) and Lemma 2.1 imply that m_{0} is independent of T. The detailed analysis of the qualitative behavior of the solutions in this case including the stability results are presented in [6]. Some further discussion on the model and its variants is presented in the appendix.

■

Next we obtain pointwise bounds for the specific volume v = v_{j}(t) with the aid of Lemmas 2.1–2.2. This estimate will be crucial in the forthcoming analysis.

Lemma 2.3. There exists M = M(m_{0}) > 0, independent of h, such that, for each T > 0,

M^{−1} ≤ v_{j}(t) ≤ M < ∞, for all t ∈ [0, T]. (2.17)

Proof: The proof is based on the derivation of two different representations for the specific volume v. The pointwise bounds are obtained by combining the analysis given in Chen, Hoff, and Trivisa [5] and Chen [4] with an idea in Kazhikov and Shelukhin [20] and using equation (2.2), Lemmas 2.1–2.2, and boundary condition (2.5) with special consideration of the features of the system. In particular, the pointwise bounds are essential to guarantee the continuous dependence of the solutions on the initial data (cf. [16]). Here we present only the main steps of the proof. Step 1. Lemma 2.2 yields that, for any t > 0, there exist j_{1}(t) and positive constants α, β with α < β such that

α ≤ v_{j_{1}}, θ_{j_{1}} ≤ β.

Following now similar line of arguments to the one presented in [5], we get

v_{j}(t) = \frac{1 + \frac{1}{ε} ∫_{0}^{t} a(Z_{j}(s))P (s)Q_{j}(s)θ_{j}(s)ds}{P (t)Q_{j}(t)}, (2.18)

where

\begin{cases}P (t) = v_{j_{1}(t)}(0)exp\{\frac{1}{ε} ∫_{0}^{t} p_{j_{1}(t)}(s)ds\}, \\ Q_{j}(t) = \frac{1}{v_{j}(0)v_{j_{1}} (t)}exp\{\frac{1}{ε} ∑_{(j_{1},j)}(u_{k}(0) − u_{k}(t))h\}. \end{cases} (2.19)

Furthermore, using the momentum equation, we obtain that there exists j_{2} = j_{2}(t) such that

∑_{k<j_{2}(t)} u_{k}(0)h + ∫_{0}^{t} σ_{j_{2}(t)}(s)ds = ∑_{j} (v_{j}(0) ∑_{k<j} u_{k}(0)h) h

− ∫_{0}^{t} ∑_{j} (\frac{u_{j+ \frac{1}{2}}^{2} + u_{j− \frac{1}{2}}^{2}}{2} + a(Z_{j})θ_{j}) hds + O(h^{1/2}), (2.20)

where σ_{j} = (\frac{εδu}{v})_{j} − p_{j}, and O(h^{1/2}) denotes the terms bounded by Mh^{1/2}.

Step 2. Next, we derive (cf. [5]) another representation for v_{j} = v_{j}(t). For any t ≥ 0,

(2.21)

v_{j}(t) = (1 + O(h^{1/2}))D_{j}(t)exp \{−\frac{1}{ε} ∫_{0}^{t} ∑_{l} (\frac{u_{l+ \frac{1}{2}}^{2} + u_{l− \frac{1}{2}}^{2}}{2} + a(Z_{l})θ_{l}) hds\}

× \{1 + \frac{1 + O(h^{1/2})}{ε} ∫_{0}^{t} \frac{a(Z_{j})θ_{j}}{D_{j}(s)} exp \{\frac{1}{ε} ∫_{0}^{s} ∑_{l} (\frac{u_{l+ \frac{1}{2}}^{2} + u_{l− \frac{1}{2}}^{2}}{2} + a(Z_{l})θ_{l}) hdτ\} ds\} ,

where

D_{j}(t) = v_{j}(0)exp \{\frac{1}{ε} ( ∑_{(j_{2}(t),j)} u_{k}(t)h − ∑_{k<j} u_{k}(0)h + ∑_{l} v_{l}(0) (∑_{k<l} u_{k}(0)h) h)\} ,

and j_{2}(t) is determined in (2.20).

This result is obtained by following similar arguments to the one given in Lemma 2.2 in [5].

Step 3. Next we obtain pointwise bounds for v_{j}(·). We first set

m_{v}(t) = min_{x∈[0,1]} v(x, t), m_{θ}(t) = min_{x∈[0,1]} θ(x, t),

M_{v}(t) = max_{x∈[0,1]} v(x, t), M_{θ}(t) = max_{x∈[0,1]} θ(x, t).

Using Lemma 2.2, we conclude that there exist M = M(m_{0}) > 0, β̂ = β̂(m_{0}), and β̃ = β̃(m_{0}) such that

M^{−1} ≤ D_{j}(t), Q_{j}(t) ≤ M,  β̂ ≤ ∑_{j} θ_{j}(t)h ≤ β̃.

The second representation (2.21) for v_{j}(·) implies

M_{v}(t) ≤ Mexp{−a_{1}t} (1 + ∫_{0}^{t} M_{θ}(s)exp{a_{1}s}ds) , (2.22)

where a_{1} = \frac{1}{ε} β̂ min_{0≤Z≤1} a(Z).

Next, we obtain bounds for M_{θ}(·) and m_{θ}(·), namely,

(2.23)  M_{θ}(t) ≤ M(1 + M_{v}(t)B(t)),
(2.24)  m_{θ}(t) ≥ M^{−1} (1 − M_{v}(t)B(t)) ,

where

B(t) = ∑_{k} \frac{(δθ_{k})^{2}}{θ_{k+ \frac{1}{2}} θ_{k− \frac{1}{2}} v_{k}} h.

Substituting (2.23) into (2.22), we get

M_{v}(t) ≤ M exp{−a_{1}t} {1 + ∫_{0}^{t} exp{a_{1}s} (1 + M_{v}(s)B(s)) ds} .

Setting now A(t) = M_{v}(t) exp{a_{1}t} and taking into consideration that ∫_{0}^{t} B(τ)dτ ≤ M, we get

M_{v}(t) = exp{−a_{1}t}A(t) ≤ M. (2.25)

Next, by using (2.24), (2.25), and the second representation (2.21) for v_{j}(·) yields that there exists t_{0} > 0 such that

m_{v}(t) ≥ \frac{M^{−1}}{2} , t ≥ t_{0} > 0. (2.26)

On the other hand,

P (t) = ∑_{j} P (t)v_{j}(t)h = ∑_{j} (\frac{1 + \frac{1}{ε} ∫_{0}^{t} a(Z_{j}(s))θ_{j}(s)P (s)Q_{j}(s)ds}{Q_{j}(t)}) h

≤ M (1 + ∫_{0}^{t} P (s)(∑_{j} θ_{j}(s)h)ds) ≤ M (1 + ∫_{0}^{t} P (s)ds) ,

which gives

P (t) ≤ M exp{Mt}.

Therefore, from (2.18),

v_{j}(t) ≥ \frac{M^{−1}}{P (t)} ≥ M^{−1} exp\{−Mt\} ≥ M^{−1} exp\{−Mt_{0}\}, when 0 ≤ t ≤ t_{0}.

Combining this last relation with (2.26), we obtain the result. ■

The next lemma establishes a lower bound for the temperature θ. It will be clear in the sequel that, away from the initial line {t = 0}, the temperature θ has indeed uniform bounds.

Lemma 2.4. There exists M = M(m_{0}) > 0, with m_{0} independent of h and N, such that, for all t,

θ_{j}(t) ≥ \frac{1}{M(t + 1)}. (2.27)

Proof: Set ω_{j} = \frac{1}{θ_{j}}. Multiplying (2.3) by \{−θ_{j}^{−2}\}, we obtain

c_{v}(Z_{j})ω̇_{j} = a(Z_{j})\frac{ω_{j}δu_{j}}{v_{j}} − \frac{ε ω_{j}^{2}(δu_{j})^{2}}{v_{j}} + λδ (\frac{δω}{v} )_{j} − Kqϕ(θ_{j})Z_{j}ω_{j} (ω_{j} + \frac{c_{v}′(Z_{j})}{q}) ,

which implies

c_{v}(Z_{j})ω̇_{j} ≤ \frac{M}{v_{j}} + λδ (\frac{δω}{v} )_{j} − Kqϕ(θ_{j})Z_{j}ω_{j} (ω_{j} + \frac{c_{v}′(Z_{j})}{q}) . (2.28)

Multiplying the last inequality by {2pω_{j}^{2p−1}}, p > 1, we obtain

\frac{d}{dt}(c_{v}(Z_{j})ω_{j}^{2p}) ≤ 2pM \frac{ω_{j}^{2p−1}}{v_{j}} + 2pλδ (ω^{2p−1} \frac{δω}{v} )_{j} − (2p + 1)K \frac{ϕ(θ_{j})}{θ_{j}}Z_{j}c′_{v}(Z_{j})wω_{j}^{2p−1}.

Summing over all j and using Lemma 2.3, we obtain

∑_{j} \frac{d}{dt}(c_{v}(Z_{j})ω_{j}^{2p})h ≤ Mp ∑_{j} ω_{j}^{2p−1}h.

Now,

(∑_{j} ω_{j}^{2p−1}h) ≤ (∑_{j} ω_{j}^{2p}h)^{\frac{2p−1}{2p}} (∑_{j} h)^{\frac{1}{2p}} ≤ (∑_{j} ω_{j}^{2p}h)^{\frac{2p−1}{2p}} .

Set

g(t) = (∑_{j} c_{v}(Z_{j})ω_{j}^{2p}h)^{\frac{1}{2p}} .

Then

\frac{d}{dt} (g(t)^{2p}) ≤ Mpg(t)^{2p−1}. (2.29)

Therefore,

(∑_{j} ω_{j}(t)^{2p}h)^{\frac{1}{2p}} ≤ C1^{\frac{1}{2p}} (∑_{j} ω_{j}(0)^{2p}h)^{\frac{1}{2p}} + Mt,

where C_{1} = max_{0≤Z≤1} c_{v}(Z)/ min_{0≤Z≤1} c_{v}(Z). Letting now p → ∞, we get

∥ω(t)∥_{L^{∞}} ≤ ∥ω(0)∥_{L^{∞}} + Mt ≤ M(1 + t).

This completes the proof. ■

The next lemma will be useful in establishing estimates on the variation of Z (Lemma 2.7). Note, Lemmas 2.1–2.4, in combination with the bounds

(2.30)

c_{v}(Z_{j})θ_{i}(t) ≤ (∑_{j} c_{v}(Z_{j})θ_{j}(t)h) h^{−1} ≤ Mh^{−1}, u_{i}^{2}(t) ≤ (∑_{j} u_{j}^{2}(t)h) h^{−1} ≤ Mh^{−1},

show that the system of ordinary differential equations (2.1)–(2.4) is solvable for all t > 0 for fixed h > 0.

Lemma 2.5. For û_{j} determined by u_{j+ \frac{1}{2}}^{3} − u_{j− \frac{1}{2}}^{3} = 3û_{j}^{2}(u_{j+ \frac{1}{2}} − u_{j− \frac{1}{2}} ), there exists M = M(m_{0}) such that

∑_{j}(c_{v}(Z_{j})(θ_{j} − 1)^{2} + (θ_{j} − 1)u_{j+ \frac{1}{2}}^{2} + Lu_{j+ \frac{1}{2}}^{4} + Z_{j}^{2})h + ∫_{0}^{t} ∑_{k}(δθ_{k})^{2}hds

(2.31)  + ∫_{0}^{t} ∑_{j} (Kϕ_{j}Z_{j}^{2} + û_{j}^{2}θ_{j}^{2} + û_{j}^{2}(δu_{j})^{2}) hds ≤ M,

where L > (4 min_{0≤Z≤1} c_{v}(Z))^{−1}.

Proof: The result (2.31) is obtained by combining the energy bounds, which are derived by using the momentum equation (2.2), with the results of Lemma 2.1.

Multiplying the momentum difference equation (2.2) by {u_{k}^{3}}, summing over all k, and integrating on [0, t], we obtain

∑_{k} u_{k}^{4}(t)h + ∫_{0}^{t} ∑_{j} û_{j}^{2}(δu_{j})^{2}hds ≤ M + M ∫_{0}^{t} ∑_{j} û^{2}e_{j}^{2}hds. (2.32)

Next, multiplying (2.2) by {u_{k}}, we get

(\frac{1}{2}u_{k}^{2})_{t} + u_{k}δp_{k} = −ε\frac{(δu_{k− \frac{1}{2}} )^{2}}{v_{k− \frac{1}{2}}} + εδ (τ_{−}u\frac{δu}{v})_{k} , (2.33)

where (τ_{−}u)_{j} = u_{j− \frac{1}{2}} . Notice that (2.3) implies that

c_{v}(Z_{j})(θ_{j} − 1)_{t} + p_{j}δu_{j} = \frac{ε(δu_{j})^{2}}{v_{j}} + λδ(\frac{δθ}{v} )_{j} + Kϕ(θ_{j})Z_{j}(q + c′_{v}(Z_{j})θ_{j}). (2.34)

Multiplying (2.34) by {2(θ_{j} − 1)}, we get

c_{v}(Z_{j})((θ_{j} − 1)^{2})_{t} + 2λ\frac{(δθ_{j})^{2}}{v_{j}} − 2λδ ((θ_{j} − 1)\frac{δθ}{v} )_{j} (2.35)

= −2(θ_{j} − 1)p_{j}δu_{j} + 2ε\frac{(θ_{j} − 1)(δu_{j})^{2}}{v_{j}} + 2Kϕ(θ_{j})(θ_{j} − 1)Z_{j}(q + c_{v}′(Z_{j})θ_{j}).

Therefore,

(c_{v}(Z_{j})(θ_{j} − 1)^{2})_{t} + 2λ\frac{(δθ_{j})^{2}}{v_{j}} − 2λδ ((θ_{j} − 1)\frac{δθ}{v} )_{j}

= −2(θ_{j} − 1)δ(uτ_{+}p)_{j} + 2(θ_{j} − 1)u_{j+ \frac{1}{2}} δp_{j+ \frac{1}{2}} + 2ε\frac{(θ_{j} − 1)(δu_{j})^{2}}{v_{j}}

+2Kϕ(θ_{j})(θ_{j} − 1)Z_{j}(q + c_{v}′(Z_{j})θ_{j}) − Kϕ(θ_{j})(θ_{j} − 1)^{2}c′_{v}(Z_{j})Z_{j}.

Using (2.33), we now arrive at

(c_{v}(Z_{j})(θ_{j} − 1)^{2})_{t} + 2λ\frac{(δθ_{j})^{2}}{v_{j}} − 2λδ ((θ_{j} − 1)\frac{δθ}{v} )_{j}

= −2(θ_{j} − 1)δ(uτ_{+}p)_{j} + 2(θ_{j} − 1) (− (\frac{1}{2}u_{j+ \frac{1}{2}}^{2} )_{t} − ε\frac{(δu_{j})^{2}}{v_{j}} + εδ (τ_{−}u\frac{δu}{v})_{j+ \frac{1}{2}} )

+2ε\frac{(θ_{j} − 1)(δu_{j})^{2}}{v_{j}} + Kϕ(θ_{j})Z_{j}(θ_{j} − 1) (2q + c_{v}′(Z_{j})(θ_{j} + 1)) ,

NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE, REACTING FLOW 15

which, using (θ_{j} − 1)(u_{j+ \frac{1}{2}}^{2} )_{t} = ((θ_{j} − 1)u_{j+ \frac{1}{2}}^{2} )_{t} − u_{j+ \frac{1}{2}}^{2} θ̇_{j}, yields that

(c_{v}(Z_{j})(θ_{j} − 1)^{2} + (θ_{j} − 1)u_{j+ \frac{1}{2}}^{2} )_{t} + 2λ\frac{(δθ_{j})^{2}}{v_{j}} − 2λδ ((θ_{j} − 1)\frac{δθ}{v} )_{j} + 2δ((θ_{j} − 1)uτ_{+}p)_{j}

= 2u_{k}p_{k+ \frac{1}{2}} δθ_{k} − 2εu_{j− \frac{1}{2}} \frac{δu_{j}}{v_{j}} δθ_{j− \frac{1}{2}} + Kϕ(θ_{j})Z_{j}(θ_{j} − 1) (2q + c_{v}′(Z_{j})(θ_{j} + 1))

+\frac{u_{j+ \frac{1}{2}}^{2}}{c_{v}(Z_{j})} (−p_{j}δu_{j} + ε\frac{(δu_{j})^{2}}{v_{j}} + λδ (\frac{δθ}{v} )_{j} + Kϕ(θ_{j})Z_{j}(q + c_{v}′(Z_{j})θ_{j})) . (2.36)

Summing (2.36) over all j and integrating over [0, t], we get

∑_{j} (c_{v}(Z_{j})(θ_{j} − 1)^{2} + (θ_{j} − 1)u_{j+ \frac{1}{2}}^{2} ) (t)h

= ∑_{j} (c_{v}(Z_{j})(θ_{j} − 1)^{2} + (θ_{j} − 1)u_{j+ \frac{1}{2}}^{2} ) (0)h − 2ε ∫_{0}^{t} ∑_{j} (u_{j− \frac{1}{2}} \frac{δu_{j}}{v_{j}} δθ_{j− \frac{1}{2}} ) hds

−2λ ∫_{0}^{t} ∑_{j} \frac{(δθ_{j})^{2}}{v_{j}} hds + 2 ∫_{0}^{t} ∑_{j}(u_{k}p_{k+ \frac{1}{2}} δθ_{k})hds − ∫_{0}^{t} ∑_{j} \frac{1}{c_{v}(Z_{j})} (u_{j+ \frac{1}{2}}^{2} p_{j}δu_{j}) hds

+ ∫_{0}^{t} ∑_{j} (u_{j+ \frac{1}{2}}^{2} \frac{(δu_{j})^{2}}{c_{v}(Z_{j})v_{j}} ) hds + ∫_{0}^{t} ∑_{j} (\frac{u_{j+ \frac{1}{2}}^{2}}{c_{v}(Z_{j})}δ (\frac{δθ}{v} )_{j}) hds

+ ∫_{0}^{t} ∑_{j} Kϕ(θ_{j})Z_{j} ((θ_{j} − 1) (2q + c′_{v}(Z_{j})(θ_{j} + 1)) + \frac{u_{j+ \frac{1}{2}}^{2}}{c_{v}(Z_{j})}(q + c_{v}′(Z_{j})θ_{j})) hds. (2.37)

Now, take L such that L > (4 min_{0≤Z≤1} c_{v}(Z))^{−1} and consider the quantity

w_{j}(t) = (c_{v}(Z_{j})(θ_{j} − 1)^{2} + (θ_{j} − 1)u_{j+ \frac{1}{2}}^{2} + Lu_{j+ \frac{1}{2}}^{4} )^{1/2} .

Then, multiplying (2.32) by L adding to (2.36), following the line of arguments as in [5], [13], [14], and noting that u_{j}u_{j− \frac{1}{2}} = u_{j}^{2} − \frac{h}{2}u_{j}δu_{j}, we arrive at the estimate

∑_{j} w_{j}^{2}(t)h + ∫_{0}^{t} ∑_{k}(δθ_{k})^{2}hds + ∫_{0}^{t} ∑_{j}(u_{j}^{2} + û_{j}^{2})(δu_{j})^{2}hds

≤ M + M ∫_{0}^{t} ∑_{k} (|u_{k}p_{k+ \frac{1}{2}} ||δe_{k}| + |u_{k}ū_{k+ \frac{1}{2}} p_{k+ \frac{1}{2}} δu_{k+ \frac{1}{2}} |) hds

+M ∫_{0}^{t} ∑_{j} (h|u_{j}||δu_{j}|^{3} + |u_{j− \frac{1}{2}} ||δu_{j}||δθ_{j− \frac{1}{2}} | + (θ_{j}^{2} + 1 + u_{j+ \frac{1}{2}}^{2} )(θ_{j} + 1)ϕ(θ_{j})Z_{j}) hds,

where ū_{k} = \frac{u_{k+1}+u_{k}}{2} . The last relation implies (cf. [5], [6], [13], [14]) that

∑_{j} (c_{v}(Z_{j})(θ_{j} − 1)^{2} + (θ_{j} − 1)u_{j+ \frac{1}{2}}^{2} + Lu_{j+ \frac{1}{2}}^{4} ) h

+ ∫_{0}^{t} (∑_{k}(δθ_{k})^{2}h + ∑_{j}(u_{j}^{2} + û_{j}^{2})(δu_{j})^{2}h) ds ≤ M. (2.38)

Finally, from Lemma 2.1,

\frac{1}{2} ∑_{j} Z_{j}^{2}h + ∫_{0}^{t} ∑_{j} Kϕ(θ_{j})Z_{j}^{2}hds = \frac{1}{2} ∑_{j} Z_{j}^{2}(0)h.

Adding the last two relations, we obtain the result. ■

Next we derive some a-priori estimates, which provide essential information on the evolution of large jumps of discontinuities. More specifically, the next lemmas show that the magnitude of any jump of discontinuity of v = v_{j}(·) and Z = Z_{j}(·):

[Z_{k}] = Z_{k+ \frac{1}{2}} − Z_{k− \frac{1}{2}} , [v_{k}] = v_{k+ \frac{1}{2}} − v_{k− \frac{1}{2}} ,

at time t can be controlled by the magnitude of jumps of discontinuity of v = v_{j}(·) and Z = Z_{j}(·)

at time t = 0. Without ambiguity, we denote f_{k} = (f)_{k} = \frac{f_{k+ \frac{1}{2}} +f_{k− \frac{1}{2}}}{2} for all other variables except u_{k} = u(x_{k}, t).

Lemma 2.6. For any k ∈ {0, 1, ⋯, Q},

(2.39)

[Z_{k}(t)] = exp {− ∫_{0}^{t} Kϕ_{k}(τ) dτ} [Z_{k}(0)] + ∫_{0}^{t} exp {− ∫_{s}^{t} Kϕ_{k}(τ) dτ} Q_{k}(s)[θ_{k}] ds,

where

(2.40)  Q_{k}(t) = −KZ_{k}(t) ∫_{0}^{1} ϕ′(αθ_{k+ \frac{1}{2}} (t) + (1 − α)θ_{k− \frac{1}{2}} (t))dα.

Proof: Equation (2.4) implies

[Z_{k}]_{t} = −K[(ϕZ)_{k}] = −Kϕ_{k}[Z_{k}] − KZ_{k}[ϕ_{k}].

Now,

[ϕ_{k}] = ϕ(θ_{k+ \frac{1}{2}} ) − ϕ(θ_{k− \frac{1}{2}} ) = \{∫_{0}^{1} ϕ′(αθ_{k+ \frac{1}{2}} + (1 − α)θ_{k− \frac{1}{2}} )dα\} [θ_{k}].

Therefore,

\frac{d}{ds} \{exp \{∫_{0}^{s} Kϕ_{k}(τ)dτ\} [Z_{k}]\} = exp \{∫_{0}^{s} Kϕ_{k}(τ)dτ\} Q_{k}[θ_{k}].

Integrating the last relation on the interval [0, t], we obtain (2.39). ■

Lemma 2.7. There exists M = M(m_{0}) > 0, m_{0} independent of h and N, such that, for the distinguished discontinuities 0 < x_{k_{1}} < x_{k_{2}} < ⋯ < x_{k_{N}} < 1,

sup_{t} ( ∑_{k=k_{i}} |δZ_{k}|h + ∑_{k≠k_{i}} |δZ_{k}|^{2}h) ≤ M. (2.41)

Proof: Step 1. We first prove that sup_{t} ∑_{k=k_{i}} |δZ_{k}|h ≤ M. Notice that

δZ_{k} = \frac{1}{h} (Z_{k+ \frac{1}{2}} (0) − Z_{k− \frac{1}{2}} (0)) (exp \{− ∫_{0}^{t} Kϕ(θ)ds\})_{k}

+\frac{1}{h}Z_{k}(0) (exp \{− ∫_{0}^{t} Kϕ(θ_{k+ \frac{1}{2}} )ds\} − exp \{− ∫_{0}^{t} Kϕ(θ_{k− \frac{1}{2}} )ds\})

= δZ_{k}(0) (exp {− ∫_{0}^{t} Kϕ(θ)ds})_{k}

+Z_{k}(0) exp {− ∫_{0}^{t} Kϕ(θ̃_{k})ds} {− ∫_{0}^{t} Kϕ′(θ̃_{k})δθ_{k}dτ} . (2.42)

NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE, REACTING FLOW 17

Therefore,

|δZ_{k}| ≤ |δZ_{k}(0)| + Z_{k}(0) ∫_{0}^{t} exp {− ∫_{0}^{t} Kϕ(θ̃_{k})ds} K|ϕ′(θ̃_{k})||δθ_{k}|dτ

≤ |δZ_{k}(0)| + Z_{k}(0) ∫_{0}^{t} exp \{− ∫_{0}^{t} Kϕ(θ̃_{k})ds\} K|ϕ′(θ̃_{k})|\frac{|δθ_{k}|}{\sqrt{θ_{k+ \frac{1}{2}} θ_{k− \frac{1}{2}}}} \sqrt{θ_{k+ \frac{1}{2}} θ_{k− \frac{1}{2}}} dτ

≤ |δZ_{k}(0)| + Z_{k}(0) ∫_{0}^{t} exp \{− ∫_{0}^{t} Kϕ(θ̃_{k})ds\} K|ϕ′(θ̃_{k})|\frac{|δθ_{k}|}{\sqrt{θ_{k+ \frac{1}{2}} θ_{k− \frac{1}{2}}}} (θ̃_{k} + ch|δθ_{k}|)dτ.

Summing over all k, we obtain

∑_{k} |δZ_{k}|h ≤ ∑_{k} |δZ_{k}(0)|h + ∑_{k} Z_{k}(0)(I_{1_{k}} + I_{2_{k}})h.

Furthermore,

∑_{k} Z_{k}(0)I_{1_{k}}h

= ∑_{k} Z_{k}(0) ∫_{0}^{t} exp \{− ∫_{0}^{t} Kϕ(θ̃_{k})ds\} ϕ′(θ̃_{k})θ̃_{k} \frac{|δθ_{k}|}{\sqrt{θ_{k+ \frac{1}{2}} θ_{k− \frac{1}{2}}}} dτh

≤ ∑_{k} Z_{k}(0) ∫_{0}^{t} \frac{|δθ_{k}|^{2}}{θ_{k+ \frac{1}{2}} θ_{k− \frac{1}{2}}} dτh + ∑_{k} Z_{k}(0) ∫_{0}^{t} exp \{−2 ∫_{0}^{t} Kϕ(θ̃_{k})ds\} |ϕ′(θ̃_{k})θ̃_{k}|^{2}dτh,

while

∑_{k} Z_{k}(0)I_{2_{k}}h = ∑_{k} Z_{k}(0) ∫_{0}^{t} exp \{− ∫_{0}^{t} Kϕ(θ̃_{k})ds\} |ϕ′(θ̃_{k})|\frac{|δθ_{k}|^{2}}{\sqrt{θ_{k+ \frac{1}{2}} θ_{k− \frac{1}{2}}}} hdτh
= ∑_{k} Z_{k}(0) ∫_{0}^{t} exp \{− ∫_{0}^{t} Kϕ(θ̃_{k})ds\} |ϕ′(θ̃_{k})|\frac{|δθ_{k}|^{2}}{θ_{k+ \frac{1}{2}} θ_{k− \frac{1}{2}}} \sqrt{θ_{k+ \frac{1}{2}} θ_{k− \frac{1}{2}}} hdτh.

The result now follows by using Lemma 2.2.

Step 2. Similarly, we now prove sup_{t} ∑_{k=k_{i}} |δZ_{k}|^{2}h ≤ M.

(δZ_{k})^{2} ≤ M(δZ_{k}(0))^{2} (exp {−2 ∫_{0}^{t} Kϕ(θ)ds})_{k}
+ MZ_{k}^{2}(0) exp {−2 ∫_{0}^{t} Kϕ(θ̃_{k})ds} {∫_{0}^{t} Kϕ′(θ̃_{k})δθ_{k}ds}^{2}
≤ M(δZ_{k}(0))^{2} + CZ_{k}^{2}(0) ∫_{0}^{t} exp {−2 ∫_{0}^{t} Kϕ(θ̃_{k})ds} |ϕ′(θ̃_{k})|^{2}|δθ_{k}|^{2}ds.

Thus,

∑_{k} |δZ_{k}|^{2}h

≤ M ∑_{k} |δZ_{k}(0)|^{2}h + M ∑_{k} Z_{k}^{2}(0) ∫_{0}^{t} exp {−2 ∫_{0}^{t} Kϕ(θ̃_{k})ds} |ϕ′(θ̃_{k})|^{2}|δθ_{k}|^{2}dτh.

Using Lemma 2.5, we obtain the result. ■

For the study of the large jumps of discontinuities, we also need estimates on [v], [u_{x}], and [θ_{x}]. The following lemma addresses this issue.

18 NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE, REACTING FLOW

Lemma 2.8. There exists M = M(m_{0}) > 0, m_{0} independent of h and N, such that the following estimates hold:

(a) For the distinguished discontinuities, 0 < x_{k_{1}} < x_{k_{2}} < ⋯ < x_{k_{N}} < 1, [log v_{k}(t)] = μ_{k}^{−1}(t)[log v_{k}(0)] + μ_{k}^{−1}(t) ∫_{0}^{t} μ_{k}(s)(R_{k,h}(s) + β_{k}(s)[Z_{k}(s)])ds,

|[v_{k}(t)]| ≤ μ_{k}^{−1}(t)|[v_{k}(0)]| + μ_{k}^{−1}(t) ∫_{0}^{t} μ_{k}(s)β_{k}(s)[Z_{k}(s)]ds + Mh^{1/2},

|[u_{x}(t)]| + |[θ_{x}(t)]| ≤ M min (exp{−M^{−1}t^{1/2}}, σ(t)^{−3/2} exp{−M^{−1}t}) , (2.43)

where σ(t) = min(1, t) and

μ_{k}(t) = exp \{− ∫_{0}^{t} α_{k}(s)θ_{k}(s)ds\} , R_{k,h}(t) = \frac{a(Z)_{k}}{ε} [θ_{k}] (\frac{1}{v} )_{k} + \frac{h}{ε}u̇_{k},

α_{k}(t) = \frac{a(Z)_{k} [\frac{1}{v} ]_{k}}{ε[log v_{k}]} , β_{k}(t) = a(Z̃)_{k} (\frac{θ}{v} )_{k}; (2.44)

(b) Away from the distinguished discontinuities,

sup_{t} ∑_{k≠k_{i}}(δv_{k})^{2}h + ∫_{0}^{t} ∑_{k≠k_{i}}(1 + θ_{k})(δv_{k})^{2}hds + ∫_{0}^{t} ∑_{j}(δu_{j})^{2}hds ≤ M(1 + Nh^{1/2}t). (2.45)

Proof: (a) Fix a jump point k = k_{i}. Then, for w_{k} = log v_{k},

[εw_{k}]_{t} = \frac{εv̇_{k}}{v_{k}} = \frac{εδu_{k}}{v_{k}} = [p_{k}] + hu̇_{k}. (2.46)

The momentum equation (2.2) yields

[p_{k}] = a(Z)_{k}θ_{k} [\frac{1}{v} ]_{k} + \frac{a(Z)_{k}}{ε} [θ_{k}] (\frac{1}{v} )_{k} + a′(Z̃_{k})[Z_{k}] (\frac{θ}{v} )_{k} . (2.47)

Therefore, [ẇ_{k}] = α_{k}(t)θ_{k}(t)[w_{k}] + β_{k}(t)[Z_{k}] + R_{k,h}(t), (2.48) where α_{k} = α_{k}(t), β_{k} = β_{k}(t), and R_{k,h} = R_{k,h}(t) are given in (2.44) with

a(Z)_{k} = \frac{a(Z(y_{k}+, t)) + a(Z(y_{k}−, t))}{2}.

Now let μ_{k}(t) = exp{− ∫_{0}^{t} α_{k}(s)θ_{k}(s)ds}. Then

\frac{d}{dt}(μ_{k}(t)[w_{k}]) = μ_{k}(t)[ẇ_{k}] + μ_{k}(t) (−α_{k}(t)θ_{k}(t)) [w_{k}] = μ_{k}(t)(R_{k,h}(t) + β_{k}(t)[Z_{k}]).

Integrating with respect to t, we obtain

[w_{k}(t)] = μ_{k}^{−1}(t)[w_{k}(0)] + μ_{k}^{−1}(t) ∫_{0}^{t} μ_{k}(s) (R_{k,h}(s) + β_{k}(s)[Z_{k}(s)]) ds.

By the mean-value theorem, we have

[(log v)_{k}] = log v_{k+ \frac{1}{2}} − log v_{k− \frac{1}{2}} = \frac{1}{ṽ_{k}} [v_{k}],

which implies that |[v_{k}]| = |ṽ_{k}[w_{k}]| ≤ M|[w_{k}]|. (2.49) Next, we estimate the quantity

{∫_{0}^{t} μ_{k}(s)R_{k,h}(s)ds}

using the Hölder inequality, the estimates ∫_{0}^{t} |δθ_{k}|^{2}h or |δθ_{k}| ≤ C/\sqrt{h}, and the line of arguments presented in [5] and [6] to obtain

μ_{k}^{−1}(t) ∫_{0}^{t} μ_{k}(s)R_{k,h}(s)ds ≤ Mh^{1/2}.

Therefore,

|[v_{k}(t)]| ≤ μ_{k}^{−1}(t)|[v_{k}(0)]| + μ_{k}^{−1}(t) ∫_{0}^{t} μ_{k}(s)β_{k}(s)[Z_{k}(s)]ds + Mh^{1/2}.

(b) Let x_{k_{1}} < x_{k_{2}} < ⋯ < x_{k_{N}} be distinguished nodes at which discontinuities in v(·, t) are modeled and [w_{k_{i}}] the jump w_{k_{i}+ \frac{1}{2}} − w_{k_{i}− \frac{1}{2}} in a sequence \{w_{j}\}. Here and in what follows, we denote

∑_{k}′w_{k} ≡ ∑_{k≠k_{i}} w_{k}.

Step 1. We obtain an estimate for the quantity ∑(δv_{k})^{2}h. Set w = log v. Then (2.2) implies that

εδẇ_{k} = u̇_{k} + δp_{k}.

Multiplying the above relation by δw_{k} and using (1.2), we obtain

\frac{ε}{2} ∑′(δw_{k})^{2}h |_{0}^{t} = ∑′u_{k}δw_{k}h |_{0}^{t} − ∫_{0}^{t} ∑′u_{k}δẇ_{k}hds + ∫_{0}^{t} ∑′δp_{k}δw_{k}hds. (2.50)

Since

δp_{k} = −a(Z)_{k}θ_{k} \frac{δv_{k}}{v_{k+ \frac{1}{2}} v_{k− \frac{1}{2}}} + \frac{a(Z)_{k}}{ε} δθ_{k} (\frac{1}{v} )_{k} + a′(Z̃_{k})δZ_{k} (\frac{θ}{v} )_{k} ,

we have

∑′(δw_{k})^{2}h + ∫_{0}^{t} ∑′ \frac{a(Z)_{k}θ_{k}δw_{k}δv_{k}}{v_{k+ \frac{1}{2}} v_{k− \frac{1}{2}}} hds

≤ M + \frac{1}{η} ∑ u_{k}^{2}(t)h + η ∑′(δw_{k})^{2}(t)h

+ ∫_{0}^{t} ∑_{j} |δu_{j}ẇ_{j}|hds − ∑_{i} ∫_{0}^{t} u_{k_{i}}[ẇ_{k_{i}}]ds + M ∫_{0}^{t} ∑′|δw_{k}| (a(Z)_{k}|δθ_{k}| + a′(Z̃)_{k}θ_{k}|δZ_{k}|) hds,

where η is sufficiently small. Therefore,

∑′(δv_{k})^{2}h + M ∫_{0}^{t} ∑′θ_{k}(δv_{k})^{2}hds

≤ M + M ∫_{0}^{t} ∑_{j}(δu_{j})^{2}h − M′ ∑_{i} ∫_{0}^{t} u_{k_{i}}[ẇ_{k_{i}}]hds

+M ∫_{0}^{t} ∑′|δv_{k}|(a(Z)_{k}|δθ_{k}| + a′(Z̃_{k})θ_{k}|δZ_{k}|)hds,

where M′ is a constant which is similar to M but may be different.

The third term in the above relation can be controlled by using Lemmas 2.1, 2.2, and 2.4. Taking into consideration

[ẇ_{k_{i}}] = α_{k_{i}}(t)θ_{k_{i}}(t)[w_{k_{i}}] + β_{k_{i}}(t)[Z_{k_{i}}] + R_{k_{i},h}(t),

one has

∑_{i} ∫_{0}^{t} u_{k_{i}}[ẇ_{k_{i}}]hds = ∑_{i} ∫_{0}^{t} u_{k_{i}}(s) (α_{k_{i}}(s)θ_{k_{i}}(s)[w_{k_{i}}] + β_{k_{i}}(s)[Z_{k_{i}}] + R_{k_{i},h}(s)) hds.

The first and third terms can be handled by using the line of argument in [5] and [6]. Here we estimate only the reacting term

∫_{0}^{t} ∑_{i} u_{k_{i}}(s)β_{k_{i}}(s)[Z_{k_{i}}]hds = ∫_{0}^{t} ∑_{i} u_{k_{i}}(s)a′(Z̃)_{k_{i}}(s) (\frac{θ_{k_{i}}}{v} ) (s)[Z_{k_{i}}]hds

≤ M ∫_{0}^{t} ∥u∥_{L^{∞}} (∑_{i} |δZ_{k_{i}}|h) ds ≤ M sup_{t} (∑_{i} |δZ_{k_{i}}|h) ∫_{0}^{t} (∑_{j}(δu_{j})^{2}h)^{1/2} ds. (2.51)

The quantity ∫_{0}^{t} ∑_{j}(δu_{j})^{2}hds can be estimated by following the line of argument in [13] and [14]. The result now follows by using Lemma 2.7. ■

Since the initial data is discontinuous, we need to introduce an auxiliary function σ = σ(t) = min(t, 1), t > 0, which will serve as a weight for the following regularity estimates. The estimates derived in this section will be also crucial in the study of the large-time behavior of the solutions to (1.1)–(1.7) in Section 3.

Lemma 2.9. There exists M = M(m_{0}), m_{0} independent of h and N, such that the following estimates hold:

(a) sup_{t} (σ(t) ∑_{j}(δu_{j})^{2}(t)h) +∫_{0}^{t} σ(s) ∑_{k} u̇_{k}^{2}(s)h ds ≤ M(1 + Nh^{1/2}t),

(b) sup_{t} (σ(t) ∑_{k}(δθ_{k})^{2}(t)h) + ∫_{0}^{t} σ^{2}(s) ∑_{j} θ̇_{j}^{2}(s)h ds ≤ M(1 + Nh^{1/2}t),

(c) sup_{t} (σ^{2}(t) (∑_{k} u̇_{k}^{2}(t)h + ∑_{j}(δu_{j})^{4}(t)h)) + ∫_{0}^{t}σ^{2}(s) ∑_{j}(δu̇_{j})^{2}(s)h ds ≤ M(1 + Nh^{1/2}t),

(d) sup_{t} (σ^{3}(t) ∑_{j} θ̇_{j}^{2}(t)h) + ∫_{0}^{t} σ^{3}(s) ∑_{j} (\frac{(δθ̇_{j})^{2}}{v_{j}} ) (s)h ds ≤ M(1 + Nh^{1/2}t),

(e) sup_{t} (∑_{j} Ż_{j}^{2}(t)h) + ∫_{0}^{t} ∑_{j} (Kϕ(θ_{j})Ż_{j}^{2}) (s)hds ≤ M(1 + Nh^{1/2}t).

Proof: The proofs of statements (a)–(d) are standard and follow similar lines of arguments as in [5, 13, 14] with the aid of Lemmas 2.1–2.8. Special attention has been taken to accommodate the large discontinuous initial data and the fact that γ and c_{v} depend on Z and therefore vary in space and time. Result (e) is a direct corollary of equation (2.4), Lemma 2.1, and the boundedness of ϕ(θ). ■

Lemma 2.10. There exists M = M(m_{0}) > 0, m_{0} independent of h and N, such that, for all t ∈ (0, T] and for distinguished discontinuities 0 < x_{k_{1}} < x_{k_{2}} < ⋯ < x_{k_{N}} < 1,

|[Z_{k_{i}}(t)]| ≤ M(|[Z_{k_{i}}(0)]| + T^{1/2}h^{1/2}).

Proof: Lemma 2.6 yields that, for the distinguished discontinuities,

|[Z_{k_{i}}(t)]| ≤ M|[Z_{k_{i}}(0)]|

−K ∫_{0}^{t} exp {− ∫_{s}^{t} Kϕ_{k_{i}}(τ) dτ} (Z_{k_{i}} ∫_{0}^{t} ϕ′(αθ_{k_{i}−} + (1 − α)θ_{k_{i}+})dα) |[θ_{k_{i}}(s)]| ds,

which implies

|[Z_{k_{i}}(t)]| ≤ M|[Z_{k_{i}}(0)]| + M ∫_{0}^{t} |[θ_{k_{i}}]|ds.

Using Lemma 2.9, we have

|[θ_{k_{i}}]| ≤ M\sqrt{\frac{h}{t}}.

Therefore,

|[Z_{k_{i}}(t)]| ≤ M(|[Z_{k_{i}}(0)]| + T^{1/2}h^{1/2}).
■

3. Existence of Solutions: Proofs of Theorems 1.1 and 1.2

In this section we apply the estimates of Section 2 to prove the existence results, Theorems 1.1 and 1.2. We shall prove Theorem 1.2 first, obtaining in particular solutions with smooth initial data. We then construct the more general solutions of Theorem 1.1 as limits of these smooth solutions. The proof of Theorem 1.2, which is quite standard, is nearly identical to that of related constructions in the literature, and therefore will be just sketched. The proof of Theorem 1.1, on the other hand, involves an interesting argument establishing strong convergence of densities in the absence of regularity, and will therefore be presented in detail.

Proof of Theorem 1.2: The construction of solutions here is nearly identical to that carried out in [13, 14], and in earlier work, for non-reacting, viscous flows. Briefly, initial data (v_{j}(0), u_{k}(0), θ_{j}(0), Z_{j}(0)) for the semidiscrete system (2.1)–(2.4) is generated from the initial data (v_{0}, u_{0}, θ_{0}, Z_{0}) in such a way that the constant C_{0} in (2.6)–(2.7) is bounded by the constant C_{0} in the hypothesis of Theorem 1.2. The results of Section 2 then apply to show that the system (2.1)–(2.4) has a global solution (v_{j}(t), u_{k}(t), θ_{j}(t), Z_{j}(t)) defined for all t, and satisfying all the conclusions of Lemmas 2.1–2.10. These mesh functions are then extended to approximate solutions (v^{h}(t), u^{h}(t), θ^{h}(t), Z^{h}(t)), which then satisfy all the bounds in (1.23)–(1.31), independently of h (modulo terms which are 0(h)). The solution (v, u, θ, Z) of Theorem 1.2 is then obtained as the strong limit as h_{i} → 0 for a suitable sequence {h_{i}}. The details, which are routine, are nearly identical to those of [13], Section 3, and therefore are omitted. ■

Proof of Theorem 1.1: First we construct smooth approximations (v_{0}^{δ}, u_{0}^{δ}, θ_{0}^{δ}, Z_{0}^{δ}) to the initial data (v_{0}, u_{0}, θ_{0}, Z_{0}) of Theorem 1.1 by mollifying appropriate extensions of the data with a smoothing kernel of width δ. We then apply the result of Theorem 1.2 to obtain global solutions (v^{δ}, u^{δ}, θ^{δ}, Z^{δ}) of (1.1) with initial data (v_{0}^{δ}, u_{0}^{δ}, θ_{0}^{δ}, Z_{0}^{δ}) which satisfy the bounds in (1.23)–(1.24) independently of δ. We shall obtain the solution (v, u, θ, Z) of Theorem 1.1 as the limit, in an appropriate sense, of these smooth solutions.

First, the uniform bounds in (1.24) for ∫_{0}^{1}(u_{x}^{δ})^{2}(x, t)dx for t > 0 and for ∫_{τ}^{T} ∫_{0}^{1}(u_{t}^{δ})^{2}dxdt for

τ > 0 show that u^{δ} are uniformly Hölder continuous on compact sets in [0, 1] × (0, ∞). There is therefore a subsequence δ_{k} → 0 such that u^{δ_{k}} converges uniformly, say to u, uniformly on these sets. Then u_{x}^{δ}(·, t) → u_{x}(·, t) in D′((0, 1)), and since {u_{x}^{δ}(·, t)} is weakly compact in L^{2}([0, 1]), again by the uniform bounds in (1.23), u_{x}^{δ}(·, t) ⇀ u_{x}(·, t) weakly in L^{2}([0, 1]). In particular, u(·, t) ∈ H^{1}([0, 1]) for t > 0. Similar arguments apply to the sequence θ^{δ}. We have therefore identified a sequence δ_{k} = δ → 0 such that

(3.1)  \begin{cases}u^{δ} → u, θ^{δ} → θ uniformly on compact sets in [0, 1] × (0, ∞); \\ u_{x}^{δ}(·, t) ⇀ u_{x}(·, t), θ_{x}^{δ}(·, t) ⇀ θ_{x}(·, t) weakly in L^{2}([0, 1]) for t > 0; \\ u_{x}^{δ} ⇀ u_{x}, θ_{x}^{δ} ⇀ θ_{x} weakly in L^{2}([0, 1] × [0, T]) for T > 0. \end{cases}

Next, we show that, for this same sequence {δ_{k}}, Z^{δ_{k}}(·, t) converges, say to Z(·, t), strongly in L^{p}([0, 1]) for all t ≥ 0 and all p ∈ [1, ∞). To prove this, we solve the fourth equation in (1.1) for Z^{δ_{j}} − Z^{δ_{k}} to obtain that

\begin{aligned}Z^{δ_{j}}(x, t) & − Z^{δ_{k}}(x, t) = e^{−K ∫_{0}^{t} ϕ(θ^{δ_{j}} (x,s))ds}(Z^{δ_{j}}(x, 0) − Z^{δ_{k}}(x, 0)) \\ & + ∫_{0}^{t} e^{−K ∫_{s}^{t} ϕ(θ^{δ_{j}} (x,τ))dτ} (ϕ(θ^{δ_{k}}(x, s)) − ϕ(θ^{δ_{j}}(x, s)))Z^{δ_{k}}(x, s)ds, \end{aligned}

so that

\begin{aligned}∥Z^{δ_{j}}(·, t) − Z^{δ_{k}}(·, t)∥_{L^{1}} ≤ & ∥Z^{δ_{j}}(·, 0) − Z^{δ_{k}}(·, 0)∥_{L^{1}} \\ & + C ∫_{0}^{t} ∥θ^{δ_{j}}(·, s) − θ^{δ_{k}}(·, s)∥_{L^{1}}ds. \end{aligned} ; (3.2)

The first term on the right here goes to zero as δ_{j}, δ_{k} → 0 because Z^{δ}(·, 0) → Z_{0} in L^{1}, and the second term is bounded by

\begin{matrix}∫_{τ}^{t} ∫_{0}^{1} | & θ^{δ_{j}} − θ^{δ_{k}}|dxds + C ∫_{0}^{τ} ∫_{0}^{1}(θ^{δ_{j}} + θ^{δ_{k}})dxds \\ ≤ ∫_{τ}^{t} ∫_{0}^{1} |θ^{δ_{j}} − θ^{δ_{k}}|dxds + Cτ \end{matrix} ; (3.3)

by the uniform bound in (1.24) for D. This together with (3.1) then shows that the second term on the right side of (3.2) approaches zero as δ_{j}, δ_{k} → 0. This proves that Z^{δ}(·, t) → Z(·, t) in L^{1}; convergence in L^{p} for p < ∞ then follows from the fact that 0 ≤ Z^{δ} ≤ 1.

Finally, we claim that there is a further subsequence δ_{j} → 0 such that v^{δ_{j}}(·, t) converges, say to v(·, t), strongly in L^{p} for all t ≥ 0 and all p ∈ [1, ∞). To prove this, we first define

(3.4)  F^{δ} = \frac{εu_{x}^{δ}}{v^{δ}} − p(v^{δ}, θ^{δ}, Z^{δ}).

Then F_{x}^{δ} = u_{t}^{δ}, so that, by (1.23),

∥F_{x}^{δ}(·, t)∥_{L^{2}} ≤ C(τ, T), τ ≤ t ≤ T,

and

|F_{t}^{δ}| ≤ C(|u_{xt}^{δ}| + |u_{x}^{δ}|^{2} + |u_{x}^{δ}| + |θ_{x}^{δ}| + |Z^{δ}|),

so that, again by (1.23),

(3.5)  ∫_{τ}^{T} ∫ (F_{τ}^{δ})^{2}dxdt ≤ C(τ, T), 0 < τ < T.

Thus, {F^{δ}} is uniformly bounded in H^{1}([0, 1] × [τ, T]), and there is a further subsequence δ_{j} → 0 such that F^{δ_{j}} → F strongly in L^{2}([0, 1] × [1/k, k]) for all k ∈ ℤ_{+}. Note also that

∫_{0}^{τ} ∫_{0}^{1} |F^{δ}|dxdt ≤ C ∫_{0}^{τ} ∫_{0}^{1}(|u_{x}^{δ}| + θ^{δ})dxdt ≤ Cτ^{1/2}.

It therefore follows that

(3.6)  F^{δ_{j}} → F in L^{1}([0, 1] × [0, T]) for T > 0

(the limit F could be identified in terms of v, θ, Z, but the present argument does not require it). Now fix j and k, let v^{δ_{j}} = v^{j}, v^{δ_{k}} = v^{k}, etc., and let L = log v. Then, from the first equation in (1.1) and from the definition of F,

(3.7)  ε\frac{∂}{∂t}(L^{j} − L^{k}) = (F^{j} − F^{k}) + (p^{j} − p^{k}).

Since p = a(Z)θ/v, we have

p^{j} − p^{k} = a^{k}θ^{k}(\frac{(v^{j})^{−1} − (v^{k})^{−1}}{L^{j} − L^{k}} )(L^{j} − L^{k}) + O(|θ^{j}||Z^{j} − Z^{k}| + |θ^{j} − θ^{k}|).

Letting −α denote the coefficient of L^{j} − L^{k} on the right, so that α ≥ 0, and substituting into (3.7), we then obtain that

ε\frac{∂}{∂t}(L^{j} − L^{k}) = −α(L^{j} − L^{k}) + (F^{j} − F^{k}) + O(|θ^{j}||Z^{j} − Z^{k}| + |θ^{j} − θ^{k}|),

and therefore that

\begin{matrix}∥(L^{j} − L^{k})(·, t)∥_{L^{1}} ≤ & ∥(L^{j} − L^{k})(·, 0)∥_{L^{1}} \\ + & C ∫_{0}^{t} (∥(F^{j} − F^{k})(·, s)∥_{L^{1}} + ∥θ^{j}(·, s)∥_{L^{∞}}∥(Z^{j} − Z^{k})(·, s)∥_{L^{1}} \\ + ∥(θ^{j} − θ^{k})(·, s)∥_{L^{1}})ds. \end{matrix} ; (3.8)

The first term on the right here approaches zero as j, k → ∞ because v^{δ}(·, 0) → v(·, 0) in L^{1} as δ → 0, and the first and third terms in the time integral have already been shown above in (3.2), (3.3), and (3.6) to approach zero. For the remaining term, we apply the bounds in (1.23) again to obtain, for t > 0 fixed,

(3.9)  ∥θ^{δ}∥_{L^{∞}} ≤ ( ∫_{0}^{1}(θ^{δ})^{2}dx)^{1/2} + ( ∫_{0}^{1}(θ^{δ})^{2}dx)^{1/4}( ∫_{0}^{1}(θ_{x}^{δ})^{2}dx)^{1/4} ≤ C(1 + t^{−1/2}).

The second term in the time integral in (3.8) therefore goes to zero by the L^{1} convergence Z^{j}(·, t) → Z(·, t) and by the dominated convergence theorem. It therefore follows from (3.8) that {v^{δ_{j}}(·, t)} is strongly Cauchy in L^{1}([0, 1]), hence in L^{p}([0, 1]) for all p ∈ [1, ∞), and for each t ≥ 0.

It is now routine to check that (v, u, θ, Z) is indeed a weak solution of (1.1) with initial data (v_{0}, u_{0}, θ_{0}, Z_{0}), and that (v, u, θ, Z) inherits all the bounds (1.23) and (1.24) in Theorem 1.1 from the smooth solutions (v^{δ}, u^{δ}, θ^{δ}, Z^{δ}). The results in (1.22) concerning regularity in time are then easily derived from the bounds in (1.23), (1.24), and the weak forms of the equations (1.1). ■

4. Large-Time Behavior

In this section we prove the large-time behavior result, Theorem 1.3, for a weak solution of (1.1) which satisfies the bounds in (1.23) and (1.24) with a constant M which is now assumed to be independent of time. As indicated in the statement of Theorem 1.1, M is indeed independent of time when c_{1} = c_{2} and γ_{1} = γ_{2}. The results of Theorem 1.3 do not otherwise depend on these conditions, however. We also prove Theorem 1.4 giving necessary conditions and sufficient conditions for complete combustion in the time-asymptotic solutions.

Proof of Theorem 1.3: We fix a weak solution (v, u, θ, Z) satisfying the conclusions of Theorem 1.1, and we assume that M in (1.23) and (1.24) is independent of time. The bound

∫_{1}^{∞} ∫ (u_{x}^{2} + u_{xt}^{2})dxdt ≤ M

then shows that the function ∫_{0}^{1}(u_{x})^{2}(x, t)dx is in (L^{1}∩BV)([1, ∞)), and consequently approaches

zero as t → ∞. That is, u(·, t) → 0 in H^{1}([0, 1]) as t → ∞.

To describe the large-time behavior of Z, we again let (v^{δ}, u^{δ}, θ^{δ}, Z^{δ}) be the smooth solution of (1.1) corresponding to mollified initial data, so that, for a suitable sequence δ → 0, Z^{δ}(·, t) → Z(·, t) in L^{p}([0, 1]) for p ∈ [1, ∞), and θ^{δ} → θ uniformly on compact sets in [0, 1] × (0, ∞), just as in Section 3. Then, from (1.1),

Z^{δ}(x, t) = Z^{δ}(x, 0)e^{−K ∫_{0}^{t} ϕ(θ^{δ}(x,s))ds}.

The integral in the exponent here converges to ∫_{0}^{t} ϕ(θ(x, s))ds as δ → 0 because ϕ is Lipschitz and ∥θ^{δ}(·, s)∥_{L^{∞}} ≤ C(1 + s^{−1/2}) (see (3.9)). Thus, if E is the set of Lebesgue points of Z_{0}, then, for a particular representative Z(·, t),

(4.1)  Z(x, t) = Z_{0}(x)e^{−K ∫_{0}^{t} ϕ(θ(x,s))ds}, x ∈ E.

Z(x, ·) is therefore a decreasing function of t for x ∈ E, and so converges as t → ∞, say to Z_{∞}(x). To describe the large-time behavior of θ, we apply the bounds

∫_{1}^{∞} ∫_{0}^{1}(θ_{x}^{2} + θ_{xt}^{2})dxdt ≤ M

to conclude that θ(·, t) − θ̄(t) → 0 in H^{1}([0, 1]) as t → ∞, where

θ̄(t) = ∫_{0}^{1} θ(x, t)dx.

On the other hand, the conservation of energy,

(4.2)  ∫_{0}^{1} (c(Z(x, ·))θ(x, ·) + \frac{1}{2}u(x, ·)^{2} + qZ(x, ·))dx |_{0}^{t} = 0

holds for smooth solutions, and therefore for the weak solutions of Theorem 1.1 as well, by the strong convergence of smooth solutions described in Section 3. Taking the limit as t → ∞ in (4.2), we therefore obtain that

(∫_{0}^{1} c(Z_{∞}(x))dx) θ̄(t) + q ∫_{0}^{1} Z_{∞}(x)dx → E_{0},

where E_{0} is the total initial energy, so that

θ̄(t) → \frac{E_{0} − q ∫_{0}^{1} Z_{∞}(x)dx}{∫_{0}^{1} c(Z_{∞}(x))dx} ,

and therefore

(4.3)  θ(·, t) → θ_{∞} ≡ \frac{E_{0} − q ∫_{0}^{1} Z_{∞}(x)dx}{∫_{0}^{1} c(Z_{∞}(x))dx}

in H^{1}([0, 1]) as t → ∞. It then follows that

e(x, t) = c(Z(x, t))θ(x, t) → c(Z_{∞}(x))θ_{∞}

as t → ∞, for x ∈ E.

Before deriving the large-time behavior of v, we examine that of the quantity F = \frac{εu_{x}}{v} − p introduced in Section 3. First, since F_{x} = u_{t}, we have from (1.23) that

∫_{1}^{∞} ∫_{0}^{1} F_{x}^{2}dxdt ≤ M.

Also,

\begin{aligned}∫_{1}^{∞} |\frac{d}{dt} & ∫_{0}^{1} F_{x}^{2}dx|dt = 2 ∫_{1}^{∞} | ∫_{0}^{1} u_{xt}F_{t}dx|dt \\ & ≤ C ∫_{1}^{∞} ∫_{0}^{1} |u_{xt}|(|u_{xt}| + |u_{x}|^{2} + |Z_{t}| + |u_{x}| + |θ_{t}|)dxdt ≤ M \end{aligned}

by routine estimates based on (1.23) and (1.24). Thus ∫_{0}^{1} F_{x}(x, t)^{2}dx ∈ (L^{1} ∩ BV)([1, ∞))

(actually, the above estimates are carried out for the smooth solutions (v^{δ}, u^{δ}, θ^{δ}, Z^{δ}) of Section 3, but the conclusion persists in the limit as δ → 0). Thus, for the weak solution under consideration,

∫_{0}^{1} F_{x}(x, t)^{2}dx → 0, and

(4.4)  F (·, t) − ∫_{0}^{1} F (x, t)dx → 0 in H^{1}([0, 1])

as t → ∞.

To derive the large-time behavior of v, we define

v̄_{0} = ∫_{0}^{1} v_{0}(x)dx = ∫_{0}^{1} v(x, t)dx

and

v_{∞}(x) = \frac{γ(Z_{∞}(x)) − 1}{∫_{0}^{1}(γ(Z_{∞}(x)) − 1)dx}v̄_{0} ,

and we show that v(·, t) → v_{∞} as t → ∞ in L^{p} for p = 2, hence for all p ∈ [1, ∞). We denote by o(1) any term which approaches zero in L^{2} as t → ∞. Then, from (4.4), the definition of F, and the fact that u_{x}(·, t) = o(1),

o(1) = F (·, t) − ∫_{0}^{1} F (x, t)dx

= o(1) + ∫_{0}^{1} (\frac{γ(Z(x, t)) − 1}{v(x, t)})θ(x, t)dx − (\frac{γ(Z(·, t)) − 1}{v(·, t)})θ(·, t)

= o(1) + ∫_{0}^{1} ((\frac{γ(Z) − γ(Z_{∞})}{v})θ + (\frac{γ(Z_{∞}) − 1}{v})(θ − θ_{∞}) + (\frac{γ(Z_{∞}) − 1}{v})θ_{∞})dx

− ((\frac{γ(Z) − γ(Z_{∞})}{v})θ + (\frac{γ(Z_{∞}) − 1}{v})(θ − θ_{∞}) + (\frac{γ(Z_{∞}) − 1}{v})θ_{∞})

= o(1) + ( ∫_{0}^{1} (\frac{γ(Z_{∞}(x)) − 1}{v(x, t)})dx − \frac{γ(Z_{∞}(·)) − 1}{v(·, t)})θ_{∞} ,

since Z(·, t) → Z_{∞} in L^{1} and θ → θ_{∞} in H^{1}. Thus

(4.5)  v(·, t) − \frac{γ(Z_{∞}(·)) − 1}{∫_{0}^{1} (\frac{γ(Z_{∞}(x)) − 1}{v(x, t)})dx} → 0

in L^{2} as t → ∞. Integrating with respect to x, we find that that

∫_{0}^{1} (\frac{γ(Z_{∞}(x)) − 1}{v(x, t)})dx → \frac{1}{v̄_{0}} ∫_{0}^{1}(Z_{∞}(x) − 1)dx,

and, then substituting back into (4.5), that v(·, t) → v_{∞} in L^{2} as t → ∞. ■

Proof of Theorem 1.4: First assume that Z_{∞} ≡ 0. Then θ_{∞} ≥ θ_{i}, for otherwise, since θ(·, t) → θ_{∞} uniformly in x, there would be a time T such that θ(x, t) ≤ θ_{i} and ϕ(θ(x, t)) = 0 for all x and all t ≥ T. It would then follow from (4.1) that Z_{∞} > 0 on a set of positive measure. Thus θ_{∞} ≥ θ_{i}, and so by the conservation of mass (4.2),

E_{0} = ∫_{0}^{1} (c(Z)θ + \frac{1}{2}u^{2} + qZ)dx → c(0)θ_{∞} = c_{2}θ_{∞} ≥ c_{2}θ_{i},

so that E_{0} ≥ c_{2}θ_{i}.

To prove (b) of Theorem 1.4, we first observe from (4.3) that

(4.6)  θ_{∞} > θ_{i} ⟺ E_{0} > c_{2}θ_{i} + [(c_{1} − c_{2})θ_{i} + q] ∫ Z_{∞}dx.

Now, E_{0} > c_{2}θ_{i} by hypothesis, so that in the case that the term in the brackets above is nonpositive, the contingency on the right hand side of (4.6) holds, and θ_{∞} > θ_{i}. In the other case, that

(4.7)  (c_{1} − c_{2})θ_{i} + q > 0,

we obtain from (1.38) that

\begin{aligned}E_{0} & > c_{2}θ_{i} + ((c_{1} − c_{2})θ_{i} + q) ∫_{0}^{1} Z_{0}dx \\ & ≥ c_{2}θ_{i} + ((c_{1} − c_{2})θ_{i} + q) ∫_{0}^{1} Z_{∞}dx \end{aligned}

by (4.7) and the fact that ∫_{0}^{1} Z(x, t)dx is decreasing in time. The fact (4.6) then applies to show

that θ_{∞} > θ_{i} in this case as well. It then follows from (4.1) that Z_{∞} = 0, a.e. ■

Appendix: Existence of Entropies

In this section we establish the existence of a physical entropy for our system (1.1). Let us start with a system of the form

\begin{cases}v_{t} − u_{x} = 0, \\ u_{t} − σ_{x} = 0, \\ (e + \frac{u^{2}}{2} + qZ)_{t} − (uσ)_{x} = Q_{x}, \\ Z_{t} + KϕZ = 0. \end{cases} (4.8)

Here v, u, θ, e are described as before, while σ and Q denote the stress and the heat flux, respectively. In the process of establishing the presence of a physical entropy for system (4.8), we will also specify what some appropriate choices are for the stress σ and the heat flux Q. Here the internal energy, stress, and heat flux are determined through the constitutive relations

\begin{cases}e = ê(v, θ, θ_{x}, Z), \\ σ = −p̂(v, θ, θ_{x}, Z, Z_{x}) + \frac{εu_{x}}{v} , \\ Q = Q̂(v, θ, u_{x}, θ_{x}), \end{cases} (4.9)

while ϕ = ϕ(θ). The constitutive variables are subject to restrictions arising from the second law of thermodynamics. We seek a physical entropy η for which the Clausius-Duhem inequality

η_{t} − (\frac{Q}{θ})_{x} − K \frac{qϕZ}{θ} ≥ 0 (4.10)

is satisfied. Using (4.10), the momentum equation yields

e_{t} + uu_{t} − uσ_{x} − u_{x}σ = Q_{x} + KqϕZ ≤ θη_{t} + \frac{Qθ_{x}}{θ} ,

or

e_{t} − u_{x}σ ≤ θη_{t} + \frac{Qθ_{x}}{θ} . (4.11)

In terms of the Helmholtz free energy Ψ = e − θη, the last relation (4.11) can be rewritten as

Ψ_{t} + ηθ_{t} + pu_{x} − \frac{εu_{x}^{2}}{v} − \frac{Qθ_{x}}{θ} ≤ 0. (4.12)

Assuming that

Ψ = Ψ̂(v, θ, u_{x}, θ_{x}, Z),

then (4.12) implies that

Ψ_{v}v_{t} + Ψ_{θ}θ_{t} + Ψ_{Z}Z_{t} + Ψ_{θ_{x}}θ_{xt} + Ψ_{u_{x}}u_{xt} + ηθ_{t} + pu_{x} − \frac{εu_{x}^{2}}{v} − \frac{Qθ_{x}}{θ} ≤ 0. (4.13)

At this point we have to require certain conditions to guarantee the sign in (4.13), namely,

\begin{cases}η = −Ψ_{θ}, Ψ_{Z} > 0, \\ p = −Ψ_{v}, Ψ_{u_{x}} = 0, \\ Q = \frac{λθ_{x}}{v} , Ψ_{θ_{x}} = 0. \end{cases} (4.14)

Here Q is a multiple of θ_{x} and hence satisfies the Fourier’s law of heat flux. The conditions in (4.14) yield that the Helmholtz free energy Ψ is independent of u_{x} and θ_{x}, that is,

Ψ = Ψ̂(v, θ, Z).

Relations (4.9) and (4.14) imply that one has to look for (η, e, θ) such that

\begin{cases}η_{θ} = \frac{e_{θ}}{θ} , \\ (θη)_{v} = p + e_{v}, \\ (θη)_{z} ≤ e_{z}, \\ η > 0. \end{cases}

Now, by the Dalton law

e_{θ} = c_{v}(Z) ≡ c_{1}Z + c_{2}(1 − Z).

Therefore,

η_{θ} = \frac{1}{θ} (c_{1}Z + c_{2}(1 − Z)) ,

which implies

η = c_{v}(Z) log θ + f(v, Z). (4.15)

Now, by the Dalton law,

\begin{cases}(θη)_{v} = p = (γ(Z) − 1)\frac{e}{v} = \frac{a(Z)}{c_{v}(Z)} \frac{e}{v} , \\ (θη)_{Z} ≤ c_{v}′(Z)θ = (c_{2} − c_{1})θ. \end{cases}

Therefore,

(θf)_{v} = − (c_{v}(Z)θ log θ)_{v} + a(Z)\frac{θ}{v} ,

and then

f(v, Z) = −c_{v}(Z) log θ + a(Z) log v + \frac{ω(θ, Z)}{θ},

or

f(v, Z) = a(Z) log v + h(Z),

with

h(Z) = \frac{ω(θ, Z)}{θ} − c_{v}(Z)θ log θ.

Relation (θη)_{Z} ≤ c_{v}′(Z)θ is equivalent to the condition

h′(Z) ≤ (c_{2} − c_{1}) log θ + ((γ_{1} − 1)c_{1} − (γ_{2} − 1)c_{2}) log v + (c_{2} − c_{1})
= (c_{2} − c_{1})(1 + log θ) + ((γ_{1} − 1)c_{1} − (γ_{2} − 1)c_{2}) log v.

Therefore, the entropy we are seeking is of the form

η = c_{v}(Z) log θ + a(Z) log v + h(Z),

for appropriate function h = h(Z). ■

Clearly η = η(·) is not in general a convex function, a remark closely related to the fact that the estimates in our analysis are not in general time-independent.

Remark 4.1. As is well-known, the constitutive equations of a real gas are fairly well approximated within moderate ranges of θ and v by the model of an ideal gas, in which

e = c_{v}θ, σ = −p(v, θ) + \frac{εu_{x}}{v} , Q = \frac{λθ_{x}}{v} (4.16)

with suitable constants c_{v}, ε, λ. However, as our asymptotic results also indicate, under very high temperatures and densities, the equations in (4.16) become inadequate, since in particular specific heat, conductivity, and viscosity vary with temperature and density. The model introduced in this work is certainly more realistic since it takes into consideration the dependence of c = c_{v}(Z), γ = γ(Z), p = p(v, θ, Z). A more realistic model than (1.1) would be a linearly viscous gas (or Newtonian fluid)

σ(v, θ, u_{x}, Z) = −p(v, θ, Z) + \frac{ε(v, θ)}{v}u_{x},

satisfying the Fourier’s law of heat flux

Q(v, θ, θ_{x}) = \frac{λ(v, θ)}{v}θ_{x}. (4.17)

It would be interesting to investigate such a model and compare the difference of solution behavior between this model and the previous models.

Acknowledgments

Gui-Qiang Chen’s research was supported in part by the National Science Foundation under Grants DMS-0204225, DMS-0204455, and INT-9987378. David Hoff’s research was supported in part by the National Science Foundation under Grant DMS-9986658. Trivisa was supported in part by the National Science Foundation under Grants DMS-0072496 and INT-9987378 and under the Alfred P. Sloan Foundation Research Fellowship.

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Gui-Qiang Chen Department of Mathematics, Northwestern University Evanston, IL 60208-2730, USA gqchen@math.northwestern.edu

David Hoff Mathematics Department, Indiana University Bloomington, IN 47405-4301, USA hoff@indiana.edu

Konstantina Trivisa Department of Mathematics, University of Maryland College Park, MD 20742-4015, USA trivisa@math.umd.edu