ON THE NAVIER-STOKES EQUATIONS FOR EXOTHERMICALLY REACTING COMPRESSIBLE FLUIDS

Transcription

1 ON THE NAVIER-STOKES EQUATIONS FOR EXOTHERMICALLY REACTING COMPRESSIBLE FLUIDS GUI-QIANG CHEN DAVID HOFF KONSTANTINA TRIVISA Abstract. We analyze mathematical models goerning planar flow of chemical reaction from unburnt gases to burnt gases in certain physical regimes in which diffusie effects such as iscosity and heat conduction are significant. These models can be then formulated as the Naier-Stoes equations for exothermically reacting compressible fluids. We first establish the existence and dynamic behaior, including stability, regularity, and large-time behaior, of global discontinuous solutions of large oscillation to the Naier-Stoes equations with constant adiabatic exponent γ and specific heat c. Our approach for the existence and regularity is to combine the difference approximation techniques with the energy methods, total ariation estimates, and wea conergence arguments to deal with large ump discontinuities; and for large-time behaior is an a posteriori argument directly from the wea form of the equations. The approach and ideas we deelop here can be applied to soling a more complicated model where γ and c ary as the phase changes; and we then describe this model in detail and contrast the results on the asymptotic behaior of the solutions of these two different models. We also discuss other physical models describing dynamic combustion.. Introduction We are concerned with mathematical models goerning chemical reacting fluids from unburnt gases to burnt gases in certain physical regimes in which diffusie effects such as iscosity and heat conduction are significant. These models can be then formulated as the Naier-Stoes equations for exothermically reacting compressible fluids. We establish the existence and dynamic behaior, including stability, regularity, and large-time behaior, of discontinuous solutions to the Naier- Stoes equations for one-dimensional reacting compressible fluids with discontinuous initial data of large oscillation. Let, u, θ, and Z represent the specific olume, the elocity, the temperature, and the reactant mass fraction. Let ɛ and λ be fixed positie iscosity parameters; q the difference in the heats of the formation of the reactant and the product, while K denotes the rate of the reactant. Then, in Lagrangian coordinates, the Naier-Stoes equations goerning planar flow of reacting compressible fluids are gien by t u x =, u t +p(, θ) x = ( ɛu x )x (.), E t +(up(, θ)) x = ( ) ɛuu x+λθ x, x Z t + Kφ(θ)Z =, where E = e + u + qz is the total specific energy. The rate function φ(θ) is a Lipschitz function typically determined by the Arrhenius law, which requires that φ(θ) = exp A/θ, with the actiation energy A, for θ sufficiently larger than θ I (the ignition temperature), while φ(θ) = for θ<θ I. The internal energy e is gien ia the following thermodynamic equation of state: (.) e = p/(γ ) = c θ, where θ is the temperature, γ>is the adiabatic exponent, and c = a γ where a = RM > with R the Boltzmann s gas constant and M the molecular weight. 99 Mathematics Subect Classification. 35B4, 35D5, 76N, 35B45. Key words and phrases. Global discontinuous solutions, discontinuous initial data, large oscillation, eolution of large ump discontinuities, asymptotic behaior, combustion, Naier-Stoes equations, difference approximations, energy estimates, total ariation estimates, uniform bounds, posteriori.

2 GUI-QIANG CHEN DAVID HOFF KONSTANTINA TRIVISA The model under consideration describes exothermically reacting compressible flow which transforms the reactant: unburnt gases (Z = ) to the product: burnt gases (Z = ), ia an irreersible chemical reaction goerned by Arrhenius inetics. Between these two different phases, there is a region describing this change of phase ( <Z<). We consider the initial-boundary alue problem for system (.) with x in a bounded interal x and subect to the Dirichlet-Neumann mixed boundary conditions for t>: (.3) u(i, t) =, θ x (i, t) =, i =,. Let the initial data (.4) (, u, θ, Z) t= =(,u,θ,z )(x), x, be gien, satisfying C (x) C, θ (x) C, Z (x), (.5) θ L + u L 4+ L + Z L C, for a constant C >. We first establish the existence, regularity, stability, and large-time behaior of discontinuous solutions of the initial-boundary alue problem (.) (.4) with large discontinuous initial data (,u,θ,z )(x) satisfying (.5), with emphasis on the large-time behaior of the solutions. The main obectie is to obtain certain uniform estimates of solutions that are independent of time, een with large discontinuous initial data, which allow to determine the large-time behaior of discontinuous solutions. We also analyze the conditions for the complete burning asymptotically. Our approach for the existence and regularity is to combine the difference approximation techniques with the energy methods, total ariation estimates, and wea conergence arguments to deal with large ump discontinuities; and for large-time behaior is an a posteriori argument directly from the wea form of the equations. Our existence results can be extended with little difficulty to the Cauchy problem as in [3, 5, 6]. The approach and ideas we deelop here can be applied to soling a more complicated model where γ and c ary when the phase changes. We then describe this model in detail and contrast the results on the asymptotic behaior of the solutions for the first model with those for the second model. We also discuss other physical models describing dynamic combustion. The approach presented here can be also applied to other initial-boundary alue problems as in Chen [3]. Regarding early wors which are closely related to our results, we refer to Chen [3] for the existence and asymptotic behaior of global non-discontinuous solutions for the reacting, compressible Naier-Stoes equations with diffusion of chemical species, and to Chen-Hoff-Triisa [4] on the well-posedness and asymptotic behaior of global discontinuous solutions for non-reacting compressible Naier-Stoes equations (.) with large discontinuous initial data. Also see Matsumura- Yanagi [8] for the isothermal case, Amoso-Zlotnic [, ] for the existence and uniqueness of wea solutions and some of their time-dependent estimates for certain initial-boundary data, and Hoff [,, 3] for the global well-posedness and large-time behaior of solutions with small discontinuous initial data for non-reacting Naier-Stoes equations. For the exothermically reacting, compressible Euler equations, we refer to Chen-Wagner [6] for global discontinuous solutions in BV for the Cauchy problem with initial data in BV. We remar that all results for (.) can be conerted to equialent statements for the Naier- Stoes equations in Euler coordinates (cf. [3]): ρ t +(ρu) x =, (ρu) t +(ρu + p) x =(ɛu x ) x, (.6) (ρe) t +(u(ρe + p)) x =(ɛuu x ) x +(λθ x ) x, (ρz) t +(ρuz) x = Kφ(θ)ρZ. Corresponding statements concerning continuous dependence are more subtle, howeer, owing the fact that the change of ariables from Lagrangian to Eulerian coordinates is solution-dependent, and our solutions are only minimally regular.

3 NAVIER-STOKES EQUATIONS FOR REACTING COMPRESSIBLE FLUIDS 3 The outline of this paper is as follows. In Section, we state the main theorems and gie seeral remars. In Section 3, we describe the main ingredient of our approach, that is, we construct semidiscrete difference approximations and obtain all the energy and regularity estimates for the approximate solutions. One significant point in our analysis in Section 3 is the establishment of time-independent bounds for the approximate solutions and their higher-order difference quotients. These estimates enable us to establish the global existence and dynamic behaior of discontinuous solutions in Section 4. In Section 5, we describe a more complicated model which taes the change phase during the ignition process into consideration by allowing the dependence of γ and c, as well as the pressure p, on the reactant mass fraction Z. The approach and ideas we deelop in Sections 3 and 4 can be applied to soling the more complicated model. We contrast the results on the qualitatie behaior of discontinuous solutions of these two different models. The detailed analysis of the results on the asymptotic analysis of solutions to this new model will be presented in detail in [5]. We also discuss other models describing dynamic combustion.. Main Theorems In this section, we describe the main theorems and gie seeral remars on the Naier-Stoes equations with constant adiabatic exponent γ and specific heat c. Energy and regularity estimates for arious quantities appearing in (.) are central in our analysis. For this purpose, we introduce the following functionals: ( E(t) = sup σ(s) ux (,s) +σ (s) θ x (,s) ) (.) (.) F(t) = s t + ( (u x,θ x )(,s) +σ(s) u s (,s) +σ (s) θ s (,s) )ds, ( sup σ (s) u s (,s) +σ 3 (s) θ s (,s) ) s t + (σ (s) u xs (,s) +σ 3 (s) θ xs (,s) )ds, where σ(s) = min(s, ), and denotes the norm in L (, ). Our results on the well-posedness and qualitatie behaior of discontinuous solutions of the Cauchy problem (.) (.5) are stated as follows. Theorem. (Existence and Regularity). Gien the initial data (,u,θ,z )(x) satisfying (.5), there exists a global discontinuous solution (, u, θ, Z)(x, t) of (.) (.5) such that, u, Z C([, ); L ),θ C((, ); L ) with θ(,t) θ wealy in L when t. Furthermore, there is a constant M>independent of t, depending only on the system parameters and C, such that, for all t [, ) and x (, ), M (x, t) M, Z(x, t), (.3) M θ(x, t) Mσ (t), E(t)+F(t) M. Remar.. Theorem. indicates the existence and regularity of discontinuous solutions of (.) (.5) with large discontinuous initial data. We emphasize that the bound constant M> in the estimates in (.3) is independent of time, een with large discontinuous initial data, which allows us to determine the large-time behaior of discontinuous solutions in Section 4. We show that the elocity, the internal energy, the density, and the pressure always decay asymptotically, while the discontinuities of the reactant mass fraction may in general persist all the time, een asymptotically. We recognize the sufficient conditions of initial data for the complete burning and the decay of all discontinuities when t. The conditions are also necessary for certain initial data.

4 4 GUI-QIANG CHEN DAVID HOFF KONSTANTINA TRIVISA Let (,Z )(x) be piecewise smooth, haing ump discontinuities at isolated points y < < y N. Then, by applying the Ranine-Hugoniot condition to (.) (together with the hypothesis that u(x, t) and θ(x, t) are continuous in positie time), we find at the heuristic leel that discontinuities in, p, Z, u x, and θ x occur only at x = y and satisfy the following ump conditions: (.4) [ p(, θ) ɛu x ] =, [ θx ] =, [Z] t +Kφ(θ)[Z] =, where [f] denotes the ump of function f across x = y :[f(t)] := f(y +,t) f(y,t). The following result on the discontinuities and their asymptotic decay depends crucially on the fact that the pointwise bounds for and θ in Theorem. are independent of time. Theorem. (Discontinuities and Asymptotic Decay). Let (,Z )(x) be piecewise in H, haing isolated ump discontinuities at points y <y < <y N, in addition to the hypotheses of Theorem.. Then, the quantities (,t),p(,t),z(,t),u x (,t), and θ x (,t) hae one-sided limits at each point of discontinuity x = y for t>, and the ump conditions in (.4) hold pointwise. Furthermore, (.5) while (.6) [log ](t) = [log ]() exp [Z](t) =[Z]() exp α (s)θ (s)ds, with α (t) = a[ ] (t) ɛ[log ](t),θ (t)=θ(y,t), Kφ(θ (s))ds. Moreoer, there is a constant M depending on C, but independent of t and N, such that, when t, ( ) (.7) [, p, u x,θ x ](t) Mmin exp M t /,σ(t) 3/ exp M t. Remar.. Theorem. indicates that the large umps of initial discontinuities for (u x,θ x,,p) decay exponentially. Obsere that, if θ(x, t) has the large-time asymptotic state θ <θ I, then [Z](t) is constant when t is large. Howeer, in general, [Z](t) may not decay when t, due to the initial ump of Z (x) and the ignition effect of chemical reaction, in comparison with with Theorem. in [4]. Theorem.3 (Large-Time Behaior). Let (, u, θ, Z)(x, t) be the solution constructed in Theorem.. Then there exists a function Z (x) such that, when t, Z(x, t) Z (x), pointwise, (.8) (,t) L r (,), r<, (u(,t),θ(,t) θ ) H (,), where E = ( (c θ + u + qz )(x)dx, θ = c E q ) Z (x)dx, and = (x)dx. The next natural question is what the conditions are on the initial data for the complete burning Z (x) =. Theorem.4. If (.9) E >c θ I +q Z (x)dx, then Z (x) =, and the time asymptotic states (,u,θ,z ) of the solutions are = (x)dx, u =, θ = E /c, Z. In the case that the solution is piecewise continuous as in Theorem., the umps [], [u x ], [e], [θ x ], [Z], when t

5 NAVIER-STOKES EQUATIONS FOR REACTING COMPRESSIBLE FLUIDS 5 at each discontinuity point x = y. On the other hand, if Z (x) is not zero a.e. and Z (x) a.e., then (.) E c θ I. Remar.3. Theorem.4 indicates that physical ariables (, p, u, θ) always decay asymptotically in L, and condition (.9) is a sufficient condition for the complete burning and the decay of large umps of initial discontinuities for Z asymptotically. Uniqueness is a rather delicate issue for solutions which are as general as those of Theorem., owing to the absence of uniform regularity in the initial layer near t =. By imposing slightly stronger conditions on the initial data, howeer, we can improe the smoothing rates implicit in the definitions of E(t) and F(t) sufficiently to proe that solutions are in fact unique and depend continuously on their initial data. In the following theorem, we mae the additional regularity precise and state continuous dependence results in arious topologies. Theorem.5 (Regularity and Continuous Dependence). (i) Assume that the initial data (,u,θ,z )(x) hae some additional regularity, namely, C (x) C, θ (x) C, Z (x), TV( )+TV(u )+TV(θ )+TV(Z ) C. Then there exists a constant M>, independent of t, such that the solution of Theorem. also satisfies (.) u x (,t) Mσ /4 (t), θ x (,t) Mσ /4 (t). (ii) Solutions satisfying the bounds in (i) are unique and depend continuously on their initial data in the sense that, if (,u,θ,z ) and (,u,θ,z ) are any two such solutions and if S(t) is defined by S(t) = ( )(,t) + (u u )(,t) α + (θ θ )(,t) β + (Z Z )(,t), where α and β are small and positie ( r denotes the norm in the negatie Sobole space H r (, )), then, gien T>, there is a constant C(T ) such that, for t T, S(t) C(T )S(). (iii) The result of (ii) also holds for the functional S(t) = (,u u,θ θ,z Z )(t) + Var( )(t). For the non-reacting system (.), the regularity result (i) and the continuous dependence result (iii) are proed in Hoff [] for the Cauchy problem, and (ii) is proed in Hoff [] for the initial-boundary alue problem, in which the ariable Z is absent. There is no difficulty in adapting the analysis to the present context. We note, howeer, that the results of [] require that the initial data be small in a suitable sense. This smallness assumption is applied in the proofs of (i) and (iii) only to accommodate a fairly general dependence of temperature on internal energy (see [], (3.7) (3.8), and the subsequent discussion). In our case, this dependence is linear and no smallness assumption is needed. The continuous dependence result (iii) is somewhat unsatisfactory since it requires that perturbations in the initial specific olume be measured in an unsuitably strong topology. This deficiency is remedied in (ii), but at the expense of weaening somewhat the topologies in which perturbations in u and θ are measured. Of course, the estimates in L for the perturbations in u and θ can be recoered by interpolating the H r estimates in (ii) with the H estimates in (i) or (.). Howeer, the resulting L bounds will depend on t for t near.

6 6 GUI-QIANG CHEN DAVID HOFF KONSTANTINA TRIVISA 3. Difference Approximations and A-Priori Estimates In this section we describe the main ingredient of our approach, that is, we construct semidiscrete difference approximations for (.) (.5) and derie arious a-priori estimates for these approximations required for the subsequent analysis. Let h be an increment in x such that Qh = for some Q = Q(h) Z +, x = h for,,,q, and x = h for,3,,q. Approximations (,u,θ,z )(t) to ((x,t),u(x,t),θ(x,t),z(x,t)) are then constructed as follows: (3.) = δu, ( ) δu (3.) u + δp = ɛδ, ė + p δu = ɛ (δu ) ( ) δθ (3.3) + λδ + qkφ(θ )Z, (3.4) Ż + Kφ(θ )Z =. Here p = p(,θ ), e = c θ, and is taen to be the aerage = + + with, 3,,Q and,,,q, and δ is the operator defined by δw l = w l+ w l, h l =, or. For the time being we assume only that initial data (,u,θ,z )() for (3.) (3.4) hae been specified and satisfy (3.5) u = u Q =, δθ = δθ Q =, and (3.6) C () C, θ () C (), u4 ()h + θ ()h C. We also assume that there are distinguished points < x < x < < x N <,N = N(h),N 4 h, such that (3.7) ( ( [ i ()] + [Z i ()] )+ δ () + δz () ) h C. = i i Clearly, the initial alue problem (3.) (3.7) has a unique solution (,u,θ,z )(t), defined at least for small time. The a-priori bounds to be deried in this section will show that these solutions exist globally in time, and will proide sufficient compactness both to extract limiting solutions as h and to determine their asymptotic behaior of the solutions. Here and in what follows, M>will denote a generic constant independent of h and t. Let (,u,θ,z )(t) be the solutions of (3.) (3.7). As a first step, we establish the basic energy estimates, which will be crucial in our analysis. Lemma 3.. For the approximations (,u,θ,z )(t) for t (, ), (i) Z (t) ; (ii) (t)h =; (iii) Z (t)h + (i) (c θ (t)+qz (t))h + () Z (t)h + (i) E(t)+ (3.8) Kφ(θ (s))z (s)hds = Z ()h; u (t)h = (c θ () + qz ())h + Kφ(θ (s))z (s)hds = Z ()h; (V(s)+W(s))ds = E() <, with E(t) = (c (θ log θ )+a( log )) h + V (t) = ( ) ɛ(δu) θ h + λ ( ) (δθ ) θ + θ h, W (t) = qk ( ) θ θ φ(θ )Z. u ()h; u h,

7 NAVIER-STOKES EQUATIONS FOR REACTING COMPRESSIBLE FLUIDS 7 Proof: The result (i) is obtained by using the difference equation (3.4) and the properties of the rate function φ = φ(θ). The results of (ii) () are obtained by summing and integrating appropriately the difference equations (3.) (3.4), using the boundary conditions, and following the line of argument presented in [4]. To show (i), we differentiate the energy E = E(t) and use (3.) (3.3) and (i) (), and then integrate the resulting identity to obtain ( ) E(t) E() = q Z (t)h V (s)ds qk φ(θ )Z (s)hds θ = The result now follows. Z ()h V (s)ds + q θ ( ) θ Kφ(θ )Z hds. The result (i) is fundamental in the analysis since it is the one establishing the presence of time-independent bounds of solutions and their higher order deriaties. Next, we obtain pointwise bounds for the specific olume = (t), a lower bound for the internal energy θ = θ (t), and a further energy bound, with the aid of Lemma 3.. Lemma 3.. There exists M>such that (i) M (t) M< ; (ii) θ (t) M(t+) ; (iii) For û determined by u 3 u 3 + =3û (u + u ), ((θ ) + u Z )h + + (δθ ) hds ( Kφ Z +û θ +û (δu ) ) hds M. Proof: The proof for (i) is achieed by deriing two different representations for the specific olume. The pointwise bounds are obtained by combining the analysis gien in Chen-Hoff- Triisa [4] with an idea presented in Kazhio-Sheluhin [7]. Recall that [7] refers to regular solutions and clearly the deelopment of new analytical techniques is required in order to deal with discontinuous solutions. These new techniques were deeloped in [4] for the study of the Naier-Stoes equations for non-reacting compressible fluids and can be applied here by taing the special features of the new system into consideration. Note that the desired representations of the specific olume ( ) can be deried by starting from the momentum equation and by obtaining appropriate bounds from Lemma 3. and the boundary conditions, while taing the special characteristics of the model into consideration (see [4]). The result (ii) is obtained by using the difference equation (3.3) and the energy bounds resulting from Lemma 3.. The result (iii) is obtained by combining the energy estimates from the momentum equation and the results of Lemma 3. with the part (i) of Lemma 3.. This estimate will be useful in establishing estimates on the ariation of Z (see Lemma 3.3). Remar 3.. Lemmas 3. and 3. in combination with the bounds (3.9) θ i (t) θ (t)h h Mh, u i(t) u (t)h h Mh show that the system of ordinary differential equations (3.) (3.4) is solable for all t>for fixed h>.

8 8 GUI-QIANG CHEN DAVID HOFF KONSTANTINA TRIVISA Next we derie some a-priori estimates, which proide essential information on the eolution of large umps of discontinuities. More specifically, the next lemmas show that the magnitude of any ump of discontinuity of = ( ) and Z = Z ( ): [Z ]=Z + Z, [ ]= +, at time t can be controlled by the magnitude of umps of discontinuity of = ( ) and Z = Z ( ) at time t =.Without ambiguity, we denote f =(f) = f + +f for all other ariables except u = u(x,t). Lemma 3.3. (i) For any,,,q, (3.) [Z ](t) = exp Kφ dτ [Z ]() exp s Kφ dτ KZ (s)[φ ](s) ds; (ii) There exists M>such that, for the distinguished discontinuities, <x <x < < x N <, (3.) sup δz (t) h + δz (t) h M. t =i i Proof: (i) Equation (3.4) implies [Z ] t = K[(φZ) ]= Kφ [Z ] KZ [φ ]. Then we hae d s s exp Kφ dτ [Z ] = exp Kφ dτ KZ [φ ]. ds Integrating the last relation on the interal [,t] yields (3.). Now, (ii). First, using (3.4), we hae δz (t) δz () + KZ () τ exp exp := I + I. τ τ exp φ Kφ( θ )ds ( θ ) δθ dτ φ Kφ( θ )ds δθ ( θ ) φ Kφ( θ )ds ( θ ) δθ dτ. θ + θ ( θ + Mh δθ )dτ Summing oer all, we obtain (3.) δz (t) h δz () h + Z ()(I + I )h. Now, using the behaior of function φ(θ), Z ()I h δθ Z () θ + θ dτh + M Z ()h, while δθ Z ()I h M θ + θ hdτ. Using (3.) and Lemma 3., we obtain sup t = i δz (t) h M.

9 NAVIER-STOKES EQUATIONS FOR REACTING COMPRESSIBLE FLUIDS 9 Similarly, we hae = i δz (t) h M = i δz () h + M = i Z () Using Lemma 3. again, we obtain sup t τ exp = i δz h M. φ Kφ( θ )ds ( θ ) δθ dτh. Lemma 3.4. There exists M>such that, for each t>, the following estimates hold. (i) On the distinguished discontinuities, <x <x < <x N <, (3.3) where µ (t) = exp [log ](t) =µ α (s)θ (s)ds (t)[log ]() + µ (t) µ (s)r,h (s)ds, = i, i N,, α (t)= a[ ] (t) ɛ[log ](t), R,h(t) = a ɛ [e ](t) ( ) (t)+ h ɛ u (t). (ii) Away from the distinguished discontinuities, (3.4) sup (δ ) h + ( + θ )(δ ) hds + (δu ) hds M( + Nh / t). t i i Proof: (i) Fix a ump point = i and set w =(log). Then [ɛw ] t = ɛ The momentum equation (3.) yields = ɛδu =[p ]+h u. [( ) ] ( ) (3.5) [p ]=aθ + a[θ ]. Thus, (3.6) [ẇ ]=α (t)[w ]+R,h (t). Let µ (t) = exp t ɛ α (s)e (s)ds. Then d dt (µ [w ]) = µ [ẇ ]+µ ( α (t)e (t)) [w ]=µ R,h. Integrating with respect to t yields (3.3). (ii). Let x <x < <x N be distinguished nodes, at which discontinuities in (,t) are modeled, and [f i ] the ump f i+ f i in a sequence f. Here and in what follows, we denote f f. i We now obtain an estimate for the quantity (δ ) h. Notice that (3.) implies that ɛδẇ = u +δp. Multiplying the aboe relation by δw and using (.), we obtain ɛ (δw ) h t = u δw h (3.7) t u δẇ hds + δp δw hds. Notice that δp = aθ δ + + aδθ ( ),

10 GUI-QIANG CHEN DAVID HOFF KONSTANTINA TRIVISA which yields Therefore, (δw ) h + aθ δw δ + hds M + M u (t)h + (δw ) (t)h + δu ẇ h u i [ẇ i ]ds + M i (δ ) h + M θ (δ ) hds M + M (δu ) h Using [ẇ i ]=aθ i [w i ]+R i,h, one has M i u i [ẇ i ]hds + M δw δθ hds. δ δθ hds. i u i [ẇ i ]hds = i u i (aθ i (s)[w i ](s)+r i,h(s)) hds. The result then follows by using the same line of argument as in [4]. Remar 3.. By the mean-alue theorem, we hae which implies [(log ) ] = [log + ] [log ]= ṽ [ ], (3.8) [ ] = ṽ [w ] M [w ]. Next, using Hölder s inequality, the estimates δθ h or δθ M h, and similar line of arguments presented in [4], we obtain Therefore, µ (t) µ (s)r,h (s)ds Mh /. (3.9) while (3.) [ (t)] Mµ (t) [ ()] + Mh /, ( ) [u x,e x ](t) Mmin exp M t /,σ(t) 3/ exp M t, as t, where σ(t) = min(t, ),t>. Recall that the initial data are discontinuous, the auxiliary function σ = σ(t) = min(t, ), t >, will sere as a weight for the following regularity estimates. The estimates deried in this section will be also crucial in the study of the large-time behaior of the solutions to (.) (.5) in Section 4.

11 NAVIER-STOKES EQUATIONS FOR REACTING COMPRESSIBLE FLUIDS Lemma 3.5. (i) sup σ(t) (δu ) (t)h + σ(s) u (s)h ds M( + Nh / t), t ( (ii) sup σ(t) ) (δθ ) (t)h + σ (s) θ (s)h ds M( + Nh / t), t (iii) sup σ (t) u (t)h + (δu ) 4 (t)h + σ (s) (δ u ) (s)h ds M( + Nh / t), t (i) sup σ 3 (t) θ (t)h + σ 3 (s) ( (δ θ ) ) (s)h ds M( + Nh / t), t () sup ( ) Ż (t)h + Kφ(θ )Ż (s)hds M( + Nh / t). t Proof: The proof of statements (i) (i) is standard and follows similar line of arguments as in [4,, ] with the aid of Lemmas Special attention has been taen to accommodate the large discontinuous initial data and the special character of the system. The result () is a direct corollary of equation (3.4), Lemma 3., and the fact that the rate function φ(θ) is typically bounded. Lemma 3.6. There exists M > such that, for all t (,T] and for distinguished discontinuities <x <x < <x N <, (3.) [Z i (t)] M( [Z i ()] +(Th) / ). Proof: Lemma 3.3 yields that [Z i (t)] M [Z i ()] + M Using Lemma 3.5, we hae as before that which implies (3.). [θ ](t) M h t, [θ i ](s) ds. 4. Existence and Dynamic Behaior of Solutions 4.. Existence and Regularity of Solutions: Proof of Theorem.. We diide the proof into four steps. Step. First we start with assuming that (,Z )(x) are functions of bounded ariation. Existence and regularity statements in this case can be deried by the technique of Hoff [] [] and Chen-Hoff-Triisa [4], with the aid of Lemmas Briefly, we begin with initial data as in Theorem., that is, with (,Z )(x) piecewise H. Difference approximations (,u,θ,z )(t) are constructed as in Section, and these mesh functions are used to construct approximate solutions ( h,u h,θ h,z h )(x, t) by a suitable interpolation procedure. The estimates of Lemmas , which are uniform in h, then apply to show that these approximate solutions are appropriately compact, that their limits are indeed wea solutions, and that these wea solutions inherit all the properties asserted in Theorem.. We then obtain a uniform total ariation estimate for (, Z)(x, t). We apply this estimate in order to complete the solution operator to more general data, for which (,Z )(x) are of bounded ariation. This entire construction requires a fairly lengthy, but straightforward analysis. The details for the present case are nearly identical to those of [] [], in which the important differences are that all of the estimates gien here are independent of time and are alid for large initial data. These details are therefore omitted.

12 GUI-QIANG CHEN DAVID HOFF KONSTANTINA TRIVISA Step. We now proe the main results of this paper for the initial data (,u,θ,z )(x) satisfying (.5). Let (,u δ δ,θ,z δ )(x) δ be smooth approximation to (,u,θ,z )(x), δ>,satisfying (.5) independent of δ, and let ( δ,u δ,θ δ,z δ )(x, t) be the corresponding solutions, guaranteed to exist by our earlier discussion and haing no umps. Our analysis shows that (4.) M δ M, M θ δ Mσ, Z δ, E δ +G δ M, for M > independent of δ. Then we conclude from (4.) that u δ and θ δ are uniformly bounded and uniformly Hölder on any compact set in t >so that, passing to subsequence (still denoted) δ, (4.) u δ u, θ δ θ uniformly on any compact set in t >. Step 3. Now we show that there is a further subsequence (still denoted) δ, δ (,t) (,t) strongly in L r, r<,t. To proe this, define F := ɛu x p, choose δ,δ >, and abbreiate w := log and = δ, = δ,etc. Then which yields (w w ) t = u,x u,x = ɛ (F F + p p ), (w w ) t a θ + θ ( ) ( ) (4.3) (w w ) ɛ w w = ɛ (F F )+ a + (4.4) (θ θ ):= ɛ F F. Fix x and define α(x, t) = a θ +θ ( ) ( ). ɛ w w Then there exists c > such that c α(x, t) c. From (4.4), w (x, t) w (x, t) = exp which implies that +ɛ exp α(x, s)ds (w (x, ) w (x, )) s α(x, τ)dτ ( F (x, s) F (x, s))ds, (4.5) w (,t) w (,t) L w (,) w (, ) L + ɛ F (,s) F (,s) L ds. The first term on the right tends to, as δ,δ, since δ strongly. Concerning the second term, note that ( ɛux F x = u t and F t = )t p = ɛu xt ɛu x p u x p e e t, so that our bounds (4.) for E δ and G δ gie that F δ is uniformly bounded in H ([, ] [,]) for each =,3,. Hence, by a diagonal process, there exists a further subsequence (still denoted)

13 NAVIER-STOKES EQUATIONS FOR REACTING COMPRESSIBLE FLUIDS 3 δ such that F δ F in L ([, ] [,]) for all and therefore in L as well. Finally, note that σ C σ σ F δ (x, t) dxdt C ( u δ x +)dxdt ( / σ (u δ x) dx) dt + Cσ C t / dt + Cσ Cσ /. Thus, F δ F in L ([, ] [,]) for all. Furthermore, we can choose by (4.) a simple subsequence (still denoted) δ such that The bound then implies that θ δ (x, t) θ(x, t) dxdt, for all. σ θ δ (x, s)dxds Cσ θ δ (x, s) θ(x, s) dxds, for all t. Applying this in (4.5), we then get that log δ (,t) is a Cauchy sequence in L ([, ]) for all t. Step 4. We next show that there is a further subsequence (still denoted) δ such that Z δ (,t) Z(,t) strongly in L r, r<,t. Let Z = Z δ,z =Z δ,etc. Then we hae (Z Z ) t = Kφ(θ )Z + Kφ(θ )Z = Kφ(θ )(Z Z )+K(φ(θ ) φ(θ ))Z, so that Z (x, t) Z (x, t) = exp K φ(θ (x, s))ds (Z (x, ) Z (x, )) + exp Since φ(θ) and φ(θ) is Lipschitz in θ, (4.6) K Z (,t) Z (,t) L Z (,) Z (, ) L + C s φ(θ (x, τ)dτ (φ(θ ) φ(θ ))Z (x, s)ds. θ (,s) θ (,t) L ds. The first term on the right tends to as δ,δ because Z δ Z strongly. Again, we can choose by (4.) a simple subsequence (still denoted) δ such that θ δ θ dxds, for all t. Hence, (4.6) implies that Z δ (,t) Z(,t) for all t. 4.. Large-Time Behaior of the Solutions: Proof of Theorems..5. The large-time behaior results of Theorem.3 can be now deried as an a posteriori consequence of the wea form of the equations in (.) and the uniform estimates (.3) in Theorem. with M independent of t. Proof of Theorem.3: Let α(t) = u x(x, t)dx. Then α(t)dt < and Var(α)= α(t) dt u x u xt dxdt <. Thus, u x(x, t)dx ast. Since u(,t)=,u(,t) inl,hence in L, and then (4.7) u(,t) H, t.

14 4 GUI-QIANG CHEN DAVID HOFF KONSTANTINA TRIVISA For θ(x, t), the argument is the same, except that, without a boundary condition, we now only that θ(,t) θ (t) H, where θ (t) = θ(x, t)dx. Since Z t, there exists Z (x) such that (4.8) Z(x, t) Z (x), pointwise. The conseration of energy implies that (c θ + u + qz)dx = E, which yields (4.9) θ (t) = ) (E q Z (x)dx = θ, c which is a constant, and then (4.) θ(,t) θ H. Now choose p a constant and define by aθ = p, or = aθ. p We claim that, for any r [, ), (4.) (,t) L r, t. Define F = ɛu x p + p, so that F x = u t. Let β(t) = F xdx = u t dx so that β(t) dt <, and (4.) Var(β)= M β(t) dt u t u tt dxdt = u xt ( u xt + u x + t + θ t )dxdt <, by the energy estimates. Thus, β(t) ast, which implies F (,t) F (t), in H, where F (t) = F(x, t)dx. On the other hand, F F =( ɛu x + aθ ( ( ɛu x dx)+a (x, t) dx (x, t) θ θ ). u t F xt dxdt dx θ θ ) The first two terms on the right-hand side of the identity aboe go to zero in L, so that (x, t) α (t) in L, and (,t) α (t), in L, where α (t) =( (x,t) dx). Integrate to get α (t), which implies α (t), t. Therefore, we hae (,t) = (x)dx, in L, and hence in all L r, r<.

15 NAVIER-STOKES EQUATIONS FOR REACTING COMPRESSIBLE FLUIDS 5 Proof of Theorem.4: Now we show that the results in (.8) and (4.3) E >c θ I +q Z (x)dx implies Z (x), the state of the complete burning. Clearly, condition (4.3) implies that θ >θ I, and therefore the last equation in (.) yields Z =.On the other hand, if Z and Z =, then θ θ I, and so E c θ I. 5. A More General Model We now discuss the following more general model for combustion: t u x =, u t +p(, e, Z) x = ( ɛu x )x (5.), E t +(up(, e, Z)) x = ( ) ɛuu x+λθ x, x Z t + Kφ(θ)Z =. a(z) γ(z) Here γ(z) > is as before the adiabatic exponent and c (Z) =. The thermodynamic equation of state implies that a(z)θ =(γ(z) )e. This model taes into consideration the change of phase during the ignition process allowing γ and c, as well as the pressure p, to ary with respect to the reacting mass fraction Z, which is important for certain physical situations. More specifically, for the chemical interaction, we consider different phases: the reactant (unburnt gases) (Z= ), the product (burnt gases) (Z= ), and the phase in between where the reactant is transformed to the product by a one-step irreersible chemical reaction goerned by Arrhenius inetics; in this region, <Z<.. The reactant (unburnt gas) is described by the parameters γ, c = c, Z =;. The product (burnt gas) is described by the parameters γ, c = c,z =. For simplicity, we assume, for each phase, a perfect-gas γ-law (e = p /(γ ), =,) and the Dalton law for the pressure of the mixture, that is, p = p + p, which lead to the conclusion [] that the parameters γ,γ,c,c are related to each other through the following conditions: (5.) γ(z) = γ c Z+γ c ( Z), c (Z)=c Z+c ( Z). c Z + c ( Z) This model system, under Dalton s law, is equipped with a physical entropy. More precisely, we hae Theorem 5.. Under the Dalton law, system (.) is endowed with a physical entropy η := c (Z) log θ + a(z) log + h(z), with appropriate function h = h(z), which satisfies the Clausius-Duhem inequality: ( ) λθx η t κ qφ(θ)z, θ x θ expressing the second law of thermodynamics. An interesting obseration is that η = η(, u, E, Z) is not in general a conex function. Howeer, if and θ hae uniform upper and lower bound, then there is h(z) such that η(, u, E, Z) is uniformly conex, which is considered as such only under ery special consideration. This is the main reason that the estimates in our analysis are not in general time-independent for this model. The approach and techniques we deelop in 3 and 4 can be applied to soling the existence and dynamic behaior of discontinuous solutions of (5.) and (5.) with discontinuous initial data of large oscillation. In this section, we describe some results in our recent efforts and contrast the results on the asymptotic analysis of solutions to the model (.) with the corresponding results for (5.) and (5.). The qualitatie behaior of the solutions corresponding to these two systems are significantly different. We refer the reader to [5] for the detailed analysis for (5.) and (5.).

16 6 GUI-QIANG CHEN DAVID HOFF KONSTANTINA TRIVISA Theorem 5. (Existence and Regularity). Gien the initial data (,u,θ,z )(x) satisfying (.5), there exists a global discontinuous solution (, u, θ, Z)(x, t) of (5.) (5.) and (.3) (.5) such that, u, Z C([, ); L ), θ C((, ); L ) with θ(,t) θ wealy in L as t. Furthermore, for each T >, there is a constant M = M(T ) > depending only on the system parameters, C, and T such that, for all t (,T],x (, ), M (x, t) M, Z(x, t), (5.3) M θ(x, t) Mσ (t), E(t)+F(t) M. Theorem 5. establishes the existence and regularity of discontinuous solutions of (5.) (5.) and (.3) (.5) with large discontinuous initial data. Theorem 5.3. Assume that (,Z )(x) are piecewise H, haing isolated ump discontinuities at points y < <y N, in addition to the hypotheses of Theorem 5.. Then, the quantities (,t),p(,t),z(,t),u x (,t), and θ x (,t) hae one-sided limits at each point of discontinuity x = y for t>, and the ump conditions (.4) hold pointwise. It is useful for the analysis to sole the ump condition (.4). For example, at the point of discontinuity x =(y,t), we hae d (5.4) dt [log ](t) =α (t)θ (t)[log ](t)+β (t)[z](t), and hence (5.5) for µ (t) = exp [log ](t) =µ (t)[log ]() + µ (t) µ (s)β (s)[z](s)ds, α (s)θ (s)ds, α (t)= a(z) (t) [ ] (t) ɛ[log ](t), β (t)=aθ (t) ( ) (t). There is no hope of describing the large-time dynamics of these umps unless M in Theorem 5. is independent of T. Howeer, een in this case, if θ has the large-time asymptotic state θ <θ I, then [Z] does not conerge to as t, hence by (5.5), neither does [log ]. Theorem 5.4 (Large-Time Behaior). Let (, u, θ, Z)(x, t) be a solution satisfying the bounds in Theorem 5. with M independent of t. Then there exist a constant θ > and functions (,Z )(x), (5.6) θ = E q Z (x)dx c, (Z )dx (x)= (γ(z (x)) ) (x)dx (γ(z, (x)) )dx such that, as t, Z(x, t) Z (x), pointwise, e(x, t) c (Z (x))θ, pointwise, (5.7) (u(,t),θ(,t) θ ) H (,), (,t) ( ) L r (,), r<, where E = (e + u + qz )(x)dx. The next natural questions are whether the complete burning (Z (x) = ) occur and whether the singularities disappear in the time-asymptotic limit. It is easy to see from (5.6) that, if E is sufficiently large, then θ >θ I and the Z-equation yields that Z = (complete burning), the bounds in Theorem 5.3 imply that, in the case of piecewise-smooth solutions, all ump discontinuities decay to zero. Conersely, if Z (x) = and

17 NAVIER-STOKES EQUATIONS FOR REACTING COMPRESSIBLE FLUIDS 7 Z, then it is easy to see that θ θ I, and hence that E c θ I. The precise statements are as follows. Theorem 5.5. (i) If (5.8) E > maxc θ I, (c +(c c ) Z )θ I +q Z, then the time-asymptotic state (,u,e,z ) is gien by (x) = (x)dx, u =, (5.9) e =E, and θ = E c, Z (x), and, in the case that the solution is piecewise smooth as in Theorem 5.3, [], [u x ], [e], [θ x ], [Z], when t at each point x = y. (ii) On the other hand, if Z (x) and Z a.e., then (5.) E c θ I. Remar 5.. When θ < θ I, as may certainly happen when E is small, the singularities described in Theorem 5.4 do not decay, and (x) and Z (x) may be discontinuous. Then the mapping (,u,e,z ) (,u,e,z ) need not be smooth for or Z (hence for e ). That is, low-energy solutions may display a failure of asymptotic compactness. Contrast this with Naier-Stoes flows for non-reacting fluids in which hyperbolic smoothing insures the compactness of the aboe map, thereby allowing for a global attractor theory. Also contrast this with the results concerning the reacting model for the case γ = γ. As it is nown, the constitutie equations of a real gas in (.) and (.) are fairly well approximated within moderate ranges of θ and by the model of a polytropic ideal gas, in which e = c θ, σ = p(, θ)+ ɛu x, Q = λθ x (5.) with suitable constants c,ɛ,λ. Howeer, under ery high temperatures and densities, the equations in (5.) may become inadequate, since in particular specific heat, conductiity, and iscosity may ary with temperature and density. The model (5.) and (5.) is certainly more realistic in certain physical situations, since it taes into consideration the dependence of c = c (Z),γ =γ(z), and p = p(, θ, Z) onz. An een more realistic model that (.) would be a linearly iscous gas (or Newtonian fluid) ɛ(, θ) σ(, θ, u x,z)= p(, θ, Z)+ u x, satisfying the Fourier s law of heat flux λ(, θ) (5.) Q(, θ, θ x )= θ x. In certain physical regimes, the diffusion of chemical species may also play a role. It would be interesting to deelop the approach and ideas present here to sole these models and their extension to the multidimensional case. Appendix A. Proof of Theorem 5. We start with a system of the form t u x =, u t +σ x =, (A.) ( e+ u )t +qz (uσ) x = Q x, Z t + KφZ =.

18 8 GUI-QIANG CHEN DAVID HOFF KONSTANTINA TRIVISA Here, u, θ, e, E are described as before, while σ and Q denote the stress and the heat flux, respectiely. In establishing the existence of a physical entropy for system (A.), we will also specify what appropriate choices are for the stress σ and the heat flux Q. Here the internal energy, stress, and heat flux are determined through the constitutie relations: e =ê(, θ, θ x,z), (A.) σ = ˆp(, θ, θ x,z,z x )+ ɛux, Q = ˆQ(, θ, u x,θ x ), while φ = φ(θ). The constitutie ariables are subect to restrictions arising from the second law of thermodynamics. We see a physical entropy η so that the Clausius-Duhem inequality ( ) Q (A.3) η t K qφ(θ)z θ x θ is satisfied, expressing the second law of thermodynamics. Using the Clausius-Duhem inequality (A.3), the momentum equation yields (A.4) Set Then (A.4) yields e t u x σ θη t + Qθ x θ. Ψ=e θη. (A.5) Ψ t + ηθ t + pu x ɛu x Qθ x θ. Choose Ψ= ˆΨ(, θ, Z, u x,θ x ). Then (A.5) implies that (A.6) (Ψ θ + η)θ t + (Ψ +p)u x +Ψ Z Z t +Ψ θx θ xt +Ψ ux u xt +ηθ t + pu x ɛu x Qθ x (A.7). θ At this point, we hae to require certain conditions to guarantee the sign in (A.7) for any solution. We choose η = Ψ θ, p = Ψ, (A.8) Ψ ux =, Ψ θx =, Ψ Z >, Q = λθx where Q is a multiple of θ x and hence satisfies the Fourier s law of heat flux. The conditions in (A.8) yield Ψ= ˆΨ(, θ, Z), which implies that one has to loo for η, e, and θ such that η θ = e θ θ, (θη) = p + e, (A.9) (θη) z e z, η >. Now, we choose (A.) e θ = c (Z), which, by Dalton s law, yields the relation e θ = c Z + c ( Z). Therefore,, η θ = θ (c Z + c ( Z)) and (A.) η = c (Z) log θ + f(, Z).

19 NAVIER-STOKES EQUATIONS FOR REACTING COMPRESSIBLE FLUIDS 9 Now, by Dalton s law, (θη) = p = e (γ(z) ) = e (γ )c Z+(γ )c ( Z) c Z+c ( Z), (A.) (θη) Z c (Z)θ =(c c )θ. Therefore, (θf) = (θ(c Z + c ( Z) log θ)) + θ ((γ )c Z +(γ )c ( Z)), that is, f(, Z) =((γ )c Z +(γ )c ( Z)) log + h(z), with ω(θ, Z) h(z) = θ(c Z + c ( Z)) log θ, θ independent of θ by choosing ω(θ, Z). The relation (θη) Z c (Z)θ is equialent to the condition h (Z) (c c ) log θ +((γ )c (γ )c ) log +(c c ), = (c c )( + log θ +((γ )c (γ )c ) log. Therefore, the entropy we are seeing is of the form: η = c (Z) log θ + a(z) log + h(z), for an appropriate function h = h(z). Acnowledgments Gui-Qiang Chen s research was supported in part by the National Science Foundation under Grants DMS , INT , and INT Daid Hoff s research was supported in part by the National Science Foundation under Grant DMS Triisa was supported in part by the National Science Foundation under Grants DMS-7496 and INT and the Alfred P. Sloan Foundation Research Fellowship. References [] A. A. Amoso and A. A. Zlotnic, A semidiscrete method for soling equations of the one-dimensional motion of a non-homogeneous iscous heat-conducting gas with nonsmooth data, Iz. Vyssh. Uchebn. Zaed. Mat. 997, 3 9 (Russian); transl. in Russian Math. (Iz. VUZ), 4 (997), 7. [] A. A. Amoso and A. A. Zlotnic, On Stability of Generalized Solutions to the Equations of One-Dimensional Motion of a Viscous Heat Conducting Gas. Siberian Math. J. 38, No. 4, (997) [3] G.-Q. Chen, Global solutions to the compressible Naier-Stoes equations for a reacting mixture, SIAM J. Math. Anal. 3 (99), [4] G.-Q. Chen, D. Hoff, and K. Triisa, Global solutions of the compressible Naier-Stoes equations with large discontinuous initial data, Commun. Partial Diff. Eqs. 5 (), [5] G.-Q. Chen, D. Hoff, and K. Triisa, Global solutions to the Naier-Stoes equations for exothermically reacting compressible fluids with large discontinuous initial data. Preprint, Northwestern Uniersity,. [6] G.-Q. Chen and D. Wagner, Global entropy solutions to exothermically reacting, compressible Euler equations, J. Diff. Eqs. (to appear). [7] R. Courant and K. O. Friedrichs, Supersonic Flow and Shoc Waes, Interscience: New Yor, 948. [8] C. M. Dafermos, Global smooth solutions to the initial boundary alue problem for the equations of one dimensional nonlinear thermoiscoelasticity, SIAM J. Math. Anal. 3 (98), [9] J. Glimm, The continuous structure of discontinuities, Lecture Notes in Physics 344 (989), [] E. Godlewsi and P. Raiart, Numerical Approximation of Hyperbolic Systems of Conseration Laws, Appl. Math. Sc. 8, Springer-Verlag: New Yor, 996. [] D. Hoff, Global well posedness of the Cauchy problem for nonisentropic gas dynamics with discontinuous initial data, J. Diff. Eqs. 95 (99), [] D. Hoff, Discontinuous solutions of the Naier Stoes equations for compressible flow, Arch. Rational Mech. Anal. 4 (99), [3] D. Hoff. Discontinuous solutions of the Naier-Stoes equations for multidimensional heat-conducting fluids, Arch. Rational Mech. Anal. 39 (997),

20 GUI-QIANG CHEN DAVID HOFF KONSTANTINA TRIVISA [4] D. Hoff and D. Serre. The failure of continuous dependence on the initial data for the Naier Stoes equations of compressible flow. SIAM J. Appl. Math. 5 (99), [5] Y. Kanel, On a model system of equations of one dimensional gas motion, J. Diff. Eq. 4 (968), [6] A. V. Kazhiho. On the theory of initial boundary alue problems for the equations of one dimensional nonstationary motion of a iscous heat-conductie gas. Din. Sploshnoi Sredy 5 (98), 37 6 (Russian). [7] A. V. Kazhiho and V. V. Sheluhin, Unique global solution with respect to time of initial-boundary-alue problems for one dimensional equations of a iscous gas, J. Appl. Math. Mech. 4 (977), [8] A. Matsumura and S. Yanagi, Uniform boundedness of the solutions for a one-dimensional isentropic model system of compressible iscous gas, Commun. Math. Phys. 75 (996), Gui-Qiang Chen Department of Mathematics, Northwestern Uniersity Eanston, IL 68-73, USA gqchen@math.northwestern.edu Daid Hoff Department of Mathematics, Indiana Uniersity Bloomington, IN , USA hoff@indiana.edu Konstantina Triisa Department of Mathematics, Uniersity of Maryland College Par, MD 74-45, USA triisa@math.umd.edu