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 J Math Anal 23:609–634, 1992) and Chen–Hoff–Trivisa (Arch Ration Mech Anal 166:321–358, 2003), among others, in which the reaction rate function is a discontinuous function obeying the Arrhenius’ law of thermodynamics. We prove the global existence of weak solutions to this model on one-dimensional unbounded domains with large initial data in H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document}. Moreover, the large-time behaviour of the weak solution is identified. 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–Shelukhin (cf. Kazhikhov in Siber Math J 23:44–49, 1982; Solonnikov and Kazhikhov in Annu Rev Fluid Mech 13:79–95, 1981) and refined by Jiang (Commun Math Phys 200:181–193, 1999, Proc R Soc Edinb Sect A 132:627–638, 2002), as well as a crucial estimate in the recent work by Li–Liang (Arch Ration Mech Anal 220:1195–1208, 2016). 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 central research topic in fluid dynamics, cf. [1][2][3]5,6,10,13,15] 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 [1], in which the explicit transform from the Euler to the 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) In the above system we are solving for the four dynamic variables (u, v, θ, Z), which represent the specific volume, velocity, temperature, and mass fraction of the reactant, respectively. The positive constants μ, κ, q, d, a, and K are the coefficients of bulk viscosity, heat conduction, 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 rate of reaction, respectively.
One distinctive feature of the above system (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: 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 regularise 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 are 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: 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: 4 , Z 0 ∈ L 1 (R), (1.8) where m 0 , M 0 are uniform 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: The main results of the paper are summarised as follows. First, we prove the global existence of weak solutions to Eqs. (1.1)-(1.8). Along the way uniform bounds (in space-time) for the temperature and specific volume are established: Moreover, there is a uniform constant The key point of Theorem 1.2 above is that C 0 is independent of T . Furthermore, the asymptotic states as t → ∞, i.e., the large-time behaviour of the reacting mixture, can be fully determined: The remaining parts of the paper are organised as follows: In Sect. 2 we collect several auxiliary conserved quantities and monotonicity formulae for the reacting mixture, which will be used throughout the paper. We also introduce a regularisation of the system (1.1)-(1.8). In Sect. 3 we establish the upper and lower bounds for the specific volume u, adapting the methods by Kazhikhov-Shelukhin [11,12] and Jiang [8,9]. Next, in Sect. 4, following the arguments by Li-Liang [13] we derive uniform estimates involving v, θ and their first derivatives. Finally, in Sect. 5 we obtain upper and lower bounds for θ uniformly in space-time, together with the bounds for the 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 [13] by Li and Liang, which in turn is motivated by the work of Huang-Li-Wang [7] on a blow-up 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 unboundedness of the spatial domains.

Regularisation, conserved quantity, and entropy formula
In this section we first introduce the regularised system of Eqs. (1.1)-(1.8): we replace the discontinuous function φ therein by φ δ ∈ C ∞ ([0, ∞)), which is required to satisfy Then, as in Chen [1], by an application of the contraction mapping principle and Schauder estimates for the linearised parabolic equations, one can prove the local existence of strong solutions in the function Here the infimum of Y is attained on R, thanks to the far-field condition lim |x|→∞ Y (·, which contradicts Eq. (2.4). Thus, we get Y ≥ 0, which is equivalent to Z ≥ 0.
To prove the upper bound for Z, let us multiply pZ p−1 for p ≥ 1 to obtain d dt As Z ≥ 0, the L p norm of Z is decreasing in time for all p ∈ [1, ∞). Thus, using the initial condition 0 ≤ Z 0 ≤ 1 and sending p → ∞, we complete the proof.
An immediate corollary of the proof of Lemma 2.2 is the following Proposition 2.3. Notice that it also holds for the non-regularised system: to prove Proposition 2.3 we only need the non-negativity of φ(θ), rather than any regularity properties.
Proof. Send p → 1 + in Eq. (2.5) and use the dominated convergence theorem.
Let us remark that in Proposition 2.3 above we do not have the conservation of total mass or energy, as they may become unbounded. For instance, consider the reacting system of only one type of perfect gas. In this case the total energy is R θ(t, x) + v(t,x) 2 2 + qZ(t, x) dx. However, in view of the far-field condition (1.7), θ ≡ 1 is expected to be a steady-state solution, which shall be verified later by the large-time behaviour (Theorem 1.3). Similarly, u 0 ≡ 1 gives us infinite total mass. 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 (u − 1 − log(u)) and (θ − 1 − log(θ)) consist of the relative entropy, which obeys the Clausius-Duhem inequality of thermodynamics. We refer the readers to the appendix in [2] for a discussion on the relevant physical backgrounds.
Then, for the right-hand side, note that R ∂ ∂x μvvx−avθ u + (1 − 1 θ ) κθx u + av dx = 0 due to the far-field condition (1.7). In light of Proposition 2.3, we then have Now, by the condition on the initial data, R v 2 0 dx is finite. In view of the assumption that 0 < m 0 ≤ u 0 (·), θ 0 (·) ≤ M 0 < ∞, it remains to establish the following claim: Here C depends only on m 0 and M 0 . Indeed, consider the auxiliary function whose Taylor expansion around 1 is for someŝ between 1 and s.
from which the claim follows directly. This proves the entropy inequality.
As a remark, Proposition 2.4 is valid for both the regularised and non-regularised systems.

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 Shelukhin, 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. (2.7). 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 the C i 's are independent of the uniform norm of φ . • We denote by the generic small constants that appear in the estimates. They only depend on the constants of the fluid, unless otherwise specified. The main result of this section is as follows: A key ingredient of the proof is the following "localisation trick" in [11,12] by Kazhikhov-Shelukhin. For self-containedness we include the proof below: 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 which is a convex function on [0, ∞). Then, on each space interval I k = [k, k + 1], k ∈ Z, applying the entropy formula (2.7) and Jensen's inequality we deduce that . Moreover, as ψ 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 ∈ (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. (2.7) and the Chebyshev's inequality, we deduce thatK 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 can bound uniformly in time. Therefore, for each t ∈ [0, T ), let us pick an arbitrary b k (t) ∈ I k \S k (t) to complete the proof.
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 [8,9] by Jiang. In fact, similar estimates have been obtained in [2,11,14] and several other works, but not uniformly in time. The crucial observation in [8,9] is that, although 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. (3.8), the integration over [0, t] gives us: Then, we take the exponential of both sides to derive that (3.10) Now, introduce the following short-hand notations in the above expression: (3.11) 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 the uniform bounds for u, based on the above representation formula. First, since sup 0≤t≤T R v 2 (t, x) dx ≤ C (which is an immediate consequence of the entropy formula, i.e., Eq. (2.7)), one concludes that (3.13) 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

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ZAMP On one-dimensional compressible Navier-Stokes equations Page 9 of 24 106 which holds in view of Eq. (2.7), I k u(t, x) dx ≤ γ 2 (due to Lemma 3.2), as well as the concavity of log.
Thus, we have: 3. Using the bounds in Eqs. (3.13) and (3.16), the representation formula (3.12), and the localisation trick (Lemma 3.2), we now conclude that (3.17) On the other hand, we have a reverse inequality which bounds θ in terms of u: To simplify the presentation, let us collect several simple algebraic identities that shall be used repetitively in the subsequent development: Proof.
(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. (2.8). 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. (2.8). This gives us Noticing that (θ(t, x) − 2) + → 0 as |x| → ∞, we integrate over [0, T ] × R to derive: On the other hand, multiplying 2v(θ − 2) + to Eq. (1.2) yields that Hence, integrating over [0, T ] × R, we obtain Adding the above two integral expressions together, evaluating [(θ − 2) + ] t on the level set Σ 2 (t), and employing the evolution equation (2.8) for θ, we now arrive at: (4.9)

2.
Our task in the sequel is to estimate I 1 , I 2 , . . . , I 6 term by term. To this end, we use Young's inequality (or Cauchy-Schwarz) repeatedly to separate each I j into "small" and "large" parts: the small part can be absorbed into the left-hand sides, and the large part can be controlled via the uniform bounds established in §2, and also the uniform boundedness of u (Theorem 3.1).
• For I 1 , using Eqs. (4.3) and (4.5), we estimate as follows: (4.10) • For I 2 , notice that Again, we use Eqs. (4.5) and (2.7) to derive that • For I 3 , let us directly bound v 2 κθx u 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 • I 5 is simple: by Eqs. (2.7) 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 3), and that (θ − 2) + ≤ (θ − 3/2) 2 + (cf. Lemma 4.2), we achieve the following: On the other hand, by Eq. (2.7) 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 1 = 1 2 μ and 2 = 3 = 4 = 1 4 κ proves the existence of a 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: 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. (2.7), and by choosing 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 Kazhikhov-Shelukhin [12]. 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 [13]: (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, let us consider and estimate K 1 , K 2 , K 3 as follows: • For K 1 , we bound thanks to items (1) and (4) in Lemma 4.2 and the entropy formula, namely Eq. (2.7). On the other hand, by Cauchy-Schwarz one has hence, • 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 6 , 7 , 8 , 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 (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 is derived as a corollary of the estimates aforementioned.
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. Then, multiplying (log(u)) x to both sides, we obtain In view of Theorem 4.1 and the entropy formula (2.7), we integrate over [0, T ] × R to get So, by choosing suitably small 10 , the above estimates give us 2. Now we estimate the derivatives of v by multiplying v xx to the momentum equation (1.2). This gives us from which we obtain that The last three terms on the right-hand side are bounded by the entropy formula (2.7), Theorem 4.1, and Eq. (5.4) in Step 1 of the same proof. Thus, choosing 11 suitably small, we get 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 106 Page 18 of 24 S. Li ZAMP We recall that 0 ≤ Z ≤ 1 (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: Thus, integrating over space-time, we obtain

4.
In this step we establish the bounds for derivatives of θ. As before, multiplying θ xx to the temperature equation (2.8) yields: Now, we integrate over [0, T ] × R and repetitively use Eq. (2.7), Theorem 4.1, Eq. (4.6), Young's inequality, as well as Eqs. (5.4) (5.6) and (5.10) in the previous steps of the same proof, 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 Here, in the first line we use the Cauchy-Schwarz inequality and the obvious interpolation; in the second Eq. (5.4); in the third the identity |θ x ) dx , in the fourth Cauchy-Schwarz again, and in the last line we invoke Theorem 4.1.
The second term is easily bounded as follows: For the third term, we compute: where we have used the Cauchy-Schwarz inequality, the Young's inequality ab ≤ 2 3 a 3/2 + 1 3 b 3 for a, b ≥ 0, and Theorem 4.1 in each line, respectively.
Finally, for the fourth term, we employ again Eq. (4.6) to derive that which again is based on the Cauchy-Schwarz inequality and the entropy formula (2.7). Therefore, using the previous estimates, we choose suitable 14 , 15 , 16 , and 17 to get:
Based on Lemma 5.1, we are now ready to establish the global existence and the large-time behaviour of the weak solutions, which are the main results of the paper.
Proof of Theorems 1.2 and 1.3. The arguments are divided into four steps.

1.
First, let us prove the large-time behaviour under the temporary assumption (♣) introduced in §2, i.e., the regularisation. We can easily deduce that Indeed, by the Cauchy-Schwarz inequality and integration by parts, there holds which is bounded by C 5 in Lemma 5.1; then, we send T → ∞. The treatment for the other two terms is similar.

2.
In this step we establish the uniform lower bound for θ, based on the large-time behaviour established in Step 1 of the same proof for C 1 reaction rate functions. For this purpose, we first obtain a lower bound for θ up to some given time T * > 0 on the compact domain [−L, L] for some finite number L > 0. Let us denote by ζ := θ −1 . Then, multiplying (−θ −2 ) to the temperature equation (1.3), we arrive at the following evolution equation for ζ: 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 >