On one-dimensional compressible Navier-Stokes equations for a reacting mixture in unbounded domains

In this paper we consider the one-dimensional Navier-Stokes system for a heat-conducting, compressible reacting mixture which describes the dynamic combustion of fluids of mixed kinds on unbounded domains. This model has been discussed on bounded domains by Chen (SIAM Jour. Math. Anal., 23 (1992), 609--634) and Chen-Hoff-Trivisa (Arch. Rat. Mech. Anal. 166 (2003), 321--358) among others, in which the reaction rate function is a discontinuous function obeying the Arrhenius Law. We prove the global existence of weak solutions to this model on one-dimensional unbounded domains with large initial data in $H^1$. Moreover, the large-time behaviour of the weak solution is identified and proved. In particular, the uniform-in-time bounds for the temperature and specific volume have been established via energy estimates. For this purpose we utilise techniques developed by Kazhikhov and coauthors ({\it cf.} Siber. Math. Jour. 23 (1982), 44--49; Jour. Appl. Math. Mech., 41 (1977), 273--282), as well as a crucial estimate in the recent work by Li-Liang (Arch. Rat. Mech. Anal. 220 (2016), 1195--1208). Several new estimates are also established, in order to treat the unbounded domain and the reacting terms.


Introduction and Main Results
The equations of motion for the compressible fluids describing chemical reactions and radiative processes have been a heated research topic in fluid dynamics: Cf. [1,2,3,4,6,7,14] and the references cited therein. In the current work we are concerned with the global existence and large-time behaviour of global solutions to the compressible Navier-Stokes equations for a reacting mixture on one-dimensional unbounded domains. Our system describes the physical process of dynamic combustion, for which the reacting rate function is discontinuous and obeys the Arrhenius Law of molecular thermodynamics. Following Chen ([2]) in which the explicit transform from Euler to Lagrangian coordinates has been computed, in this paper our analysis for the compressible Navier-Stokes equations will be carried out in the Lagrangian coordinates, i.e., (1.4) species diffusion, difference in the internal energy of the reactant and the product, the product of Boltzmann's gas constant and the molecular weight, and the reaction rate, respectively.
One distinctive feature of the above system consisting of Eqs. (1.1)-(1.4) is the presence of φ(θ), known as the reaction rate function. Here φ : R → [0, ∞) is a function of the temperature θ determined by the Arrhenius Law: (1.5) where α, A > 0 are thermodynamic constants, and θ ignite > 0 is the threshold temperature which triggers the reaction. In particular, this function is discontinuous at θ ignite . To deal with the reaction rate function φ, we first mollify it and derive uniform bounds for the resulting C 1 functions, and then pass to the limits to recover the discontinuous φ(θ). Here we need the uniform boundedness of φ, which is justified a posteriori via the uniform bounds for the other dynamical variables, i.e., (u, v, Z).
In this work we consider the Cauchy problem on the whole real line Ω = R. More precisely, the initial data is prescribed as follows: (u, v, θ, Z)| t=0 = (u 0 , v 0 , θ 0 , Z 0 ), (1.6) and the following far-field condition is imposed: lim |x|→∞ (u, v, θ, Z)(x, t) = (1, 0, 1, 0) for all t ≥ 0. (1.7) Physically, it means that at the endpoints of the reacting system the density is constant (i.e., no formation of vacuum or density-concentration), and so is the temperature. Also, the endpoints are kept fixed for all the time, with no chemical reaction triggered there.
Moreover, the initial data are assumed to satisfy the following conditions: where m 0 , M 0 are universal constants. The regularity condition in the last line is referred to as the large data condition. Now, let us introduce the notion of weak solutions to the compressible Navier-Stokes system of the reacting mixture, which is our main object of study in this work: u t ∈ L 2 (0, T ; L 2 (R)), v, θ − 1, Z ∈ L ∞ (0, T ; H 1 (R)) ∩ L 2 (0, T ; H 2 (R)); v t , θ t , Z t ∈ L 2 (0, T ; L 2 (R)).
The main results of the paper are summarised as follows: 2 First, assuming the local (in time) existence of weak solutions, we prove the global existence of weak solutions to Eqs. (1.1)-(1.8). Along the way, the uniform bounds (in space-time) for the temperature and the specific volume are established: Meanwhile, the asymptotic states as t → ∞, i.e. the large-time behaviour, of the reacting mixture, can be fully determined: be a global weak solution to Equations (1.1)-(1.8). Then it converges in H 1 to the equilibrium state in the far-field, i.e., The remaining parts of the paper are organised as follows: In §2 we collect several auxiliary conserved quantities and monotonicity formulae for the reacting mixture, which will be used throughout the paper. We also prove 0 ≤ Z ≤ 1. In §3 we establish the upper and lower bounds for the specific volume u. Next, in §4, following the arguments in [1] we derive uniform estimates involving v, θ and their first derivatives. Finally, in §5 we derive the upper and lower bounds for θ uniformly in space-time, together with the uniform bounds for higher derivatives of (u, v, θ, Z), and thus conclude the proof of Theorems 1.2 and 1.3.
Before further development, we point out that the key estimate in this work, i.e., Theorem 4.1, essentially relies on the arguments in the recent paper [1] by J. Li and Z. Liang, which in turn is motivated by the work of Huang-Li-Wang ([8]) on a blowup criterion for compressible Euler equations. The new feature of our work lies in the physical process of dynamic combustions, i.e., the analysis of functions φ and Z, as well as the treatment for unbounded domains.

Conserved Quantity and Entropy Formula
In this section we record the conserved quantity and monotonicity formula of the compressible reacting mixture for future development. First of all, we have: Proof. Let us multiply βZ β−1 with β > 1 to the evolution equation of Z, i.e., Eq. (1.4), to get Thus, in view of 0 ≤ Z ≤ 1 (which shall be established in Lemma 2.2), we integrate from 0 to t and send β → 1 + to obtain Eq. (2.1), using the Dominated Convergence Theorem.
Let us remark that, in Proposition 2.1 above we do not have the conservation of total mass or energy, as they may become unbounded. For instance, let us consider the reacting system of only one type of perfect gas, which it obeys the same γ-law (where γ > 1 is a constant). In this case, the internal energy e = pu/(γ − 1) is proportional to the temperature θ = pu/a, thus the total energy of the reacting gas is However, in view of our far-field condition (Eq. (1.7)), θ ≡ 1 is expected to be a steady state solution, which shall be verified later by the large-time behaviour (cf. Theorem 1.3). Such θ leads to infinite total energy. Similarly, u 0 ≡ 1 gives infinite total mass.
Next, we verify that Z is indeed a ratio, i.e. a number between 0 and 1: Proof. The proof for Z ≥ 0 follows from the maximum principle. We set where β > 0 is to be determined. Then, in view of Eq. (1.4), Y satisfies the following evolution equation: Here the infimum of Y is attained on R, thanks to the far-field condition lim |x|→∞ Y (·, x) = 0.
which contradicts Eq. (2.4). Thus we get Y ≥ 0, which is equivalent to Z ≥ 0. As a remark, here we need the requirement Z t ∈ L 2 (0, T ; L 2 (R)) in Definition 1.1 to ensure that Y t is well-defined.
To prove the upper bound for Z, let us invoke again Eq. (2.2) (reproduced below): Since the right-hand side and the second term on the left-hand side are non-positive, the L β -norm of Z is decreasing in time for all β ∈ [1, ∞). Thus, using the initial condition 0 ≤ Z 0 ≤ 1 and sending β → ∞, one immediately deduces Z ≤ 1. Hence the assertion follows.
Now we establish an important monotonicity formula, which is interpreted as the entropy/energy formula for the reacting mixture, referred to as the "entropy inequality" or "entropy formula" in the sequel. In physics, the expressions consist of the relative entropy, which obeys the Clausius-Duhem inequality of thermodynamics. We refer the readers to the appendix in [3] for a discussion on the relevant physical backgrounds.
Proof. First we derive an alternative version of the evolution equation for temperature: by substituting the mass and momentum equations (1.1)(1.2) into Eq. (1.3), one obtains: Then, for the right-hand side, we observe that R ∂ ∂x î µvvx−avθ u + (1 − 1 θ ) κθx u + av ó dx = 0 holds due to the far-field condition (1.7). In light of Proposition 2.1 we then have which completes the proof once Eq. (2.7) is integrated over [0, T ] × R.

Uniform Bounds for the Specific Volume u
In this section we establish the uniform (in space-time) upper and lower bounds for u. The proof is an adaptation of the classical argument by Kazhikhov and coauthors, cf. [11,12] and the references cited therein. It relies on an explicit representation formula for u in terms of the other dynamical variables, which are in turn controlled by the entropy formula, i.e., Eq. (2.5).
Before stating and proving further results, let us first explain the notations and conventions adopted in the rest of the paper: • We use C i , i ∈ {0, 1, 2, 3, . . .}, to denote the positive constants depending only on the initial data and the fluid. More precisely, It is crucial that C i 's are independent of the uniform norm of φ ′ . • We denote by ǫ the generic small constants appear in the estimates. They only depend on the constants of the fluid, unless otherwise specified. 5 • In § §3-6 we assume momentarily that the reacting rate function φ(θ) is C 1 , and the following bounds are valid: It is crucial for M to be independent of δ, which enables us to apply (♣) to the mollified versions of φ and derive uniform estimates. As a consequence, in §6 we can pass to the limits to recover the estimates for discontinuous φ.
The main theorem in this section is as follows: Then, there exists a universal constant C 0 such that A key ingredient of the proof is the following lemma, which is a modification of the nowstandard "localisation trick" in [11,12]: for all k ∈ Z and t > 0; here I k = [k, k + 1]. Moreover, given any such t and k, we can find b k (t) ∈ I k so that Proof for Lemma 3.2. Let us denote by which is a convex function on [0, ∞). Then, on each space interval I k = [k, k +1], k ∈ Z, applying the entropy formula (2.5) and Jensen's inequality we deduce that Moreover, as ψ (i.e. is monotonically decreasing from infinity to zero on (0, 1] and monotonically increasing from zero to infinity on [1, ∞)), we can find two positive constants γ 1 , γ 2 such that, for all k ∈ Z, t > 0, This prove the first part of the lemma.
For the second part, we fix a small constant ǫ ∈ (0, 1/2). Then, we take any t > 0 and consider the "exceptional" set: By investigating the graph of ψ we note the following: On S k (t), either ψ(θ) or aψ(u) is greater than some large numberK =K(γ 1 , γ 2 ) ≥ 1. Thus, employing Eq. (2.5) and the 6 Chebyshev's inequality, we deduce that where for a Borel set B ⊂ R its one-dimensional Lebesgue measure is denoted as |B|.
Now, we observe thatK increases if either γ 2 increases or γ 1 decreases. Hence, by suitably choosing γ 1 , γ 2 which depend only on a, q, E 0 , we obtain the bound: With Lemma 3.2, we are at the stage of proving our main theorem in this section. The proof is a straightforward adaptation of the estimates in [9][10] by S. Jiang. In fact, similar estimates have been obtained in [3][11] [13] and several other works, but not uniformly in time.
The crucial observation in [9][10] is that, Throughout the following proof, let us write N for universal constants independent of t, x, k. In particular, the independence of k will be justified at the end of the argument.
Proof for Theorem 3.1. The proof is divided into three steps: Testing against the momentum equation (1.2), one obtains: Here, σ is the effective viscous flux, defined as Starting with Eq. (3.9), an integration over [0, t] gives us: Then, we take the exponential of both sides to derive that Now, introduce the following short-hand notations in the above expression: We multiply the above equation by aµ −1 θ(t, x) and integrate over t to obtain: This leads to an explicit representation formula for the specific volume, namely 2. In this step we derive uniform bounds for u based on the above representation formula. First, by sup 0≤t≤T R v 2 (t, x) dx ≤ 2qE 0 (which is an immediate consequence of the entropy formula, i.e., Eq. (2.5)), one concludes that (3.14) Next, for any 0 < s < t ≤ T , a lower bound can be derived for t s θ(τ, x)dτ on I k uniformly in k. For this purpose, we first employ Jensen's inequality to estimate Lemma 3.2), and the concavity of log. Then, we have: for which one utilises Jensen's inequality, the lower bound on t s θdτ , Eq. (2.5), as well as Lemma 3.2. Hence, for arbitrary 0 ≤ τ ≤ t the following holds: 3. Using the bounds in Eqs. (3.14) and (3.17), the representation formula (3.13) and the localisation trick (Lemma 3.2), we now conclude that On the other hand, we have a reverse inequality which bounds θ in terms of u: again due to Lemma 3.2.
Finally, in view of Eq. (2.5), an application of Grönwall lemma to Eqs. (3.19) and (3.18) gives us the uniform-in-time upper and lower bounds for u. In particular, the constants N are independent of k. Thus the proof is now complete.

The Crucial Estimate for v and θ
In this section we establish a key estimate involving v, θ, v x , θ x and suitable powers of them. This inequality is an adaptation of the key estimate in [1] (cf. Lemma 2.2 therein). However, due to the presence of the chemical reaction processes, extra work needs to be done in order to control the variable Z.
Our main result in the current section is summarised as follows: Then there exists C 1 > 0, depending only on the initial data, such that To simplify the presentation, let us collect several simple algebraic identities, which are to be repetitively invoked in the subsequent development: (4.5)
(1)-(3) follow from straightforward algebraic computations; we omit the details here. Let us only comment that in (1), the following choice of constant satisfies the requirement, as ψ(s) has a double zero at 1; also, in (3) any B > 4 3 works. Finally, (4) is the standard Sobolev inequality corresponding to the embedding H 1 (R) ֒→ C 0 (R).
Proof for Theorem (4.1). We divide our arguments in four steps.

1.
We start by deriving an energy estimate for the temperature equation, in the form of Eq. (2.6). The aim is to bound the L 2 norm of θ in the "high-temperature region", in terms of other dynamical variables.
For this purpose let us multiply (θ − 2) + to Eq. (2.6). This gives us Noticing that Ä θ(t, x) − 2 ä + → 0 as |x| → ∞, we integrate over [0, T ] × R to derive: On the other hand, multiplying 2v(θ − 2) + to the momentum equation (1.2) yields that (4.8) Hence, integrating over [0, T ] × R, we obtain as follows: Adding the above two integral expressions together, evaluating [(θ − 2) + ] t on the level set Σ 2 (t), and employing the evolution equation (2.6) for θ, we now arrive at: 2. Now, our task is to estimate I 1 , I 2 , . . . , I 6 term by term. To this end, we use Young's inequality (or Cauchy-Schwarz in the simplest form) repeatedly to separate each I j into one "small" and one "large" part: the "small" part can be absorbed by the left-hand sides, and the "large" part can be controlled via the uniform bounds established in §2, together with the uniform boundedness of u (cf. Theorem 3.1).
• For I 1 , using Eqs. (4.3)(4.5), we estimate as follows: (4.10) • For I 2 , notice that Again, we use Eq. (4.5) and Eq. (2.5) to derive that • For I 3 , let us directly bound x dxdt is a term with special structure. By a standard trick, we integrate against a test function ϕ(θ): Hence, choosing a sequence of test functions ϕ η ∈ C ∞ [0, ∞) such that ϕ η (θ) ≡ 0 for θ ≤ 2, ϕ η (θ) ≡ 1 for θ ≥ 2 + η, and ϕ ′ η (θ) ≥ 0, we immediately get: (4.13) • I 5 is simple: by Eqs. (2.5) and (4.5), (4.14) • Finally let us deal with I 6 , which is the term involving Z. In view of the boundedness of φ in the C 0 -topology, sup 0≤t≤T R Z(t, x) dx ≤ C (cf. Proposition 2.1), and that (θ − 2) + ≤ (θ − 3/2) 2 + (cf. Lemma 4.2), we achieve at the following: On the other hand, by Eq. (2.5) and the identity in Eq. (4.6), we have: Thus Now, we combine the previous estimates in Eqs. (4.10)-(4.15) to control the right-hand side of Eq. (4.9). Indeed, selecting ǫ 1 = 1 2 µ and ǫ 2 = ǫ 3 = ǫ 4 = 1 4 κ proves the existence of a universal constant C 2 > 0, depending only on the initial data, µ, κ, a, q, K, φ L ∞ and C 0 in Theorem 3.1, so that for all 0 ≤ t ≤ T the following holds: 3. In the third step we estimate the term on the right-hand side of Eq. (4.18). 12 To wit, the identity (4.3) and the entropy formula (2.5) imply that where Cauchy-Schwarz and the uniform boundedness of u are used in the last line. Moreover, observe that the first term on right-hand side can be bounded by Eq. (2.5), and by choosing ǫ 5 = κ 4 , the second term can be absorbed into the left-hand side of Eq. (4.18). Thus, there is a universal constant C 3 > 0 such that for all 0 ≤ t ≤ T we have: 4. Finally it remains to bound the right-hand side of Eq. (4.21). For this purpose, we multiply (v 3 ) to the momentum equation (1.2) and investigate the evolution of the L 4 norm of v, as in ). In this manner we obtain: Hence, integrating over [0, T ] × R, we find that To estimate the last term on the right-hand side, one makes use of the following observation in [1]: (u − 1) is square-integrable due to the boundedness of u and the integrability of ψ(u) = u − 1 − log u (cf. Theorem 3.1 and Lemma 4.2). Hence, we consider and estimate K 1 , K 2 , K 3 as follows: 13 • For K 1 , we bound thanks to items (1)(4) in Lemma 4.2 and the entropy formula, namely Eq. (2.5). On the other hand, by Cauchy-Schwarz one has • Similarly, to deal with K 2 , Lemma 4.2 gives us thus one readily derives via analogous arguments. • Finally, K 3 is bounded as follows: where in the final line one utilises Eq. (4.20).
Finally, we select ǫ 6 , ǫ 7 , ǫ 8 , ǫ 9 so small that the corresponding terms get absorbed into the left-hand side of Eq. (4.21). The proof is completed by putting K 1 , K 2 , K 3 together.

Completion of the Proof of Theorems 1.2 and 1.3
With the above preparations, we finally arrive at the stage of proving the main results of the paper, i.e., Theorems 1. This final section is organised as follows: First, let us derive some uniform bounds for the higher derivatives of (u, v, θ, Z). As a by-product, the temperature θ is uniformly bounded from the above. Then, employing these bounds and investigating the limiting process T → ∞, we are able to deduce the large-time behaviour, i.e., Theorem 1.3. Thus the uniform lower bound for θ can be deduced, which agrees with the physical law that the absolute zero temperature cannot be reached. As both the upper and the lower bounds for θ are at hand, our local (in time) estimates can be extended globally. Finally, the global existence of weak solutions are derived as a corollary of the estimates aforementioned.
Lemma 5.1. There exists a universal constant C 5 such that the following estimate holds for the weak solution on [0, T ) × R: Moreover, θ is uniformly bounded from above: Proof. Before carrying out the estimates, we notice that the terms in Eq. (5.1) involving u 2 xt , v 2 t , θ 2 t , Z 2 t are bounded by the other terms in the same equation: This is an immediate consequence of Eqs. (1.1) -(1.4). Therefore, we only need to bound the spatial derivatives, which is shown in the following five steps: Then, multiplying Ä log(u) ä x to both sides, we obtain: In view of Theorem 4.1 and the entropy formula (2.5), we integrate over [0, T ] × R to get So, by choosing suitably small ǫ 10 , the above estimates give us 2. Now we estimate the derivatives of v by multiplying v xx to the momentum equation (1.2). In this way one gets Thus we obtain: The last three terms on the right-hand side are bounded by the entropy formula (2.5), Theorem 4.1 and Eq. (5.4) in Step 1 of the same proof. Thus, choosing ǫ 11 suitably small, we arrive at the following: 3. Next, let us estimate the derivatives of Z, which is specific to our problem of the reacting mixture. We multiply Z xx to Eq. (1.4) to get Now recall that 0 ≤ Z ≤ 1 always holds (Lemma 2.2); so, thanks to the Sobolev inequality in Eq. (4.6), the following estimates are valid: On the other hand, multiplying Z to Eq. (1.4) leads to: As shown in Lemma 2.2, the L 2 norm of Z decreases in time; thus which leads to the conclusion as follows: 4. In this step we establish the bounds for derivatives of θ. As before, multiplying θ xx to the temperature equation (2.6) yields: to derive the following inequality: In the sequel let us bound each of the four terms on the right-hand side of the preceding expression. For the first term, we consider where we have used Young's inequality as well as Eq. (5.4) and the entropy formula (2.5).
The second term is easily bounded as follows: For the third term we compute as follows: Finally, for the fourth term, we employ again Eq. (4.6) to derive that which again is based on the Cauchy-Schwartz inequality and the entropy formula (2.5).
Therefore, using the previous estimates, we choose suitable ǫ 14 , ǫ 15 , ǫ 16 and ǫ 17 to get: 5. Finally we conclude the uniform upper boundedness of θ in space-time. Notice that, by the Sobolev inequality (4.6), for any 0 ≤ t ≤ T there holds Then, using Theorem 4.1, it can be deduced that (θ − 2) + (t, ·) L 2 (R) ≤ √ C 1 for any t, while θ x (t, ·) L 2 (R) is estimated by Eq. (5.12). Hence we get In particular, by comparing the growth rate at infinity, we get: θ(·, ·) ≤ C 11 . (5.14) Completing the squares and writing the first two terms on right-hand side in the full divergence form, we obtain that Then, we restrict to the finite spatial interval [−L, L], and multiply (2pζ 2p−1 ) to the previous equation with p > 3 2 . Integrating by parts and using the periodic boundary condition on [−L, L], we arrive at d dt Now, applying Hölder's inequality together with the uniform lower bound u ≥ C −1 0 in Theorem 3.1, one deduces: Here, it is crucial to choose L depending on p: Indeed, we take which is a uniform estimate in L and p. Thus, for any fixed T * > 0, we can send p, L to infinity and apply the Grönwall lemma to conclude that ζ(t, x) ≤ Ce CT * . Finally, to promote the local (in time) bound to a global bound, we make use of the result in Step 1 above: there we have shown that θ → 1 uniformly as t → ∞. As a result, choose a T * ∈ (0, ∞) such that 0.99 ≤ θ(t, x) ≤ 1.01 whenever t ≥ T * and x ∈ R. Thus, together with the local lower bound of θ (Eq. (5.26)), we readily conclude the global lower bound for θ. Now we are able to conclude the proof of Theorems 1.2 and 1.3, subject to the condition (♣).
3. Finally let us remove the condition (♣) and establish the theorems for generic discontinuous functions obeying the Arrenhius' law.
Therefore, the proof of Theorems 1.2 and 1.3 is now complete.
At the end of the paper we make four concluding remarks: (1) First of all, the physical meaning of the results in this paper is natural: For a onedimensional reacting mixture on unbounded domains, if the far-field condition is imposed as in Eq. (1.7), then the chemical reaction will occur and proceed toward completion as time approaches infinity, regardless of the detailed structure of the reaction-rate function.
In this process, the density and temperature of the reacting mixture will be uniformly bounded away from zero and infinity. (2) Next, in this work we only consider the global existence of weak solutions, but we have not addressed the issues of classical (e.g. C ∞ or C 2,α ) solutions at all. Indeed, due to the discontinuity of φ(θ) (viewed as a part of the coefficients of the evolution equation (1. 3)), we should not expect the existence of classical solutions. These boundary conditions are also considered in [1] for one-dimensional heat-conducting compressible fluids without reaction terms.
Here, we claim that the same statements for Theorems 1.2 and 1.3 remain valid, subject to boundary conditions (I) or (II). This can be proved by precisely the same arguments, as long as the integration by parts arguments still hold. Indeed, Eqs. (4.7) (4.8) (4.22) (4.20) (5.3) (5.5) and (5.7) remain valid; also, under the condition (I), Eq. (5.11) stays the same, while subject to the condition (II) we can make simple modifications to recover Eq. (5.12). (4) In the end, we emphasize that the arguments in [12] for the lower boundedness of temperature are not valid on unbounded domains (u −1 / ∈ L p (R) for p ≥ 1), and the arguments in [1] for the large-time behaviour cannot be applied without modifications in presence of the Z term. In our work, new estimates have been developed to cope with unbounded domains and the chemical reaction terms.