1 Introduction

As we know,the uniformly propagating planar premixed flame fronts can become unstable leading to kinds of flame structures,such as pulsating flames,spinning and spiral flames,and flames with traveling or standing waves on the flame fronts ^[1] . Premixed gaseous flames generally involve many reactants and many reactions,and the simplest model is the case of a single limiting reactant and called one-stage chemical reaction. In premixed gaseous flames with a one-stage chemical reaction,there are two kinds of diffusional thermal instability,the first one is called the monotonic instability,whose spatio-temporal evolution is governed by the KuramotoSivashinsky (KS) equation ^[2] ,the other one is relative to the oscillatory instability which occurs with a non-zero wave number and frequency at the instability threshold,and its nonlinear evolution is described by complex Ginzburg-Landau(CGL) equations ^[3] .

However,the actual chemical kinetics governing the structure of flames can be quite complex, and the effect of complex chemical kinetics on the dynamics of propagating flame fronts is governed by the two-stage sequential reaction,which leads to the occurrence of two flame fronts with various regimes of propagation ^{[4, 5]} . One of the possible regimes of propagation is that of two planar uniformly propagating fronts with a constant separation distance between them. A linear stability analysis of this regime for the sequential reaction problem showed that,as in the case of a one-stage reaction,the flame fronts can become unstable with respect to either monotonic or oscillatory instabilities ^[6] . The nonlinear interaction between the monotonic and oscillatory modes of instability of the two uniformly propagating flame fronts can be governed by the following coupled KS-CGL equations ^[7] :

The system of coupled equations (1) and (2) describes the interaction of the evolving monotonic and oscillatory modes of instability of the uniform flame fronts. Specifically,when m > 0 and ξ = 1,both monotonic and oscillatory modes are excited. In the case of m > 0 and ξ = −1,the oscillatory mode is excited and the monotonic mode is damped. However,when the monotonic mode is excited and the oscillatory mode is damped,m< 0,and g = 0,and the interaction between the excited oscillatory mode and the damped monotonic mode will be described by a complex Ginzburg-Landau equation for P coupled to a Burgers equation for Q, the system equations of coupled KS-CGL equations (1) and (2) are simplified to the following Burgers-CGL equations ^[7] :

where P (x,t) is the rescaled complex amplitude of the flame oscillations,and Q(x,t) is the deformation of the first front. u and v are the Landau coefficients,and r₁ and w are the coupling coefficients. The parameters r₁ is complex,r₁ = r_1r + i_r1i ,and all other parameters are real.

If there are no coupling with the monotonic mode (terms with Q in (3)),(3) would be the well-known complex Ginzburg-Landau equation that usually describes the weakly nonlinear evolution of a long-scale instability ^[8] . And if the coupling coefficient w is zero in (4),then (4) would be the well-known Burgers equation ^[9] . There are so many works concerning the CGL equation ^{[10, 11, 12, 13, 14]} and Burgers equation ^{[15, 16, 17]} physically and mathematically.

However,there is few work to consider the existence,the uniqueness,and the long time behavior of the solutions for the coupled Burgers-CGL equations (3)−(4) mathematically. In this paper,we consider the following periodic initial value problem for the Burgers-CGL equations in one-dimensional version:

where Ω = [−L,L],and 2L is the period. ξ > 0,w > 0,f (x),and g(x) are given real functions.

In what following,we will study the existence and uniqueness of the global smooth solutions to the periodic initial value problem (5)−(8),and then we will prove the existence of the global attractor and establish the estimates of the upper bounds of Hausdorff and fractal dimensions for the attractor. The rest of paper is organized as follows. In Section 2,we briefly give some notations and preliminaries. In Section 3,we establish a prior estimates for the solutions to the periodic initial value problem (5)−(8) and obtain the existence for the solutions. In Section 4,the global attractor A ⊂ H ² (Ω) × H ³ (Ω) for the periodic initial value problem are obtained. In Section 5,we establish the estimates of the upper bounds of Hausdorff and fractal dimensions for the attractor. In the last Section 6,we make some conclusions. 2 Notations and Preliminaries

Throughout this paper,we denote the spaces of complex valued functions and real valued functions by the same symbols,and denote by ||·|| the norm of L ² (Ω) with the usual inner product (·,·),||·|| L ^p denotes the norm of L ^p (Ω) for 1 ≤ p ≤ ∞ (||·|| = ||·|| ₂). W ^k,p (Ω) is the usual kth order Sobolev space with its norm ||u|| W ^{k,p (Ω)} . When p = 2,we write ||·|| H ^k =||·|| W ^{k,2 (Ω) .} Generally,||·||X denotes the norm of any Banach space X. L ^∞ (0,T ; T) denotes the space of functions u(x,t) which belong to X as functions of x for every t (0 ≤ t ≤ T ). Without any ambiguity,we denote a generic positive constant by C which may vary from line to line,and C_i ,E _i ,k _i ,and κ _i (i = 1,2,· · · ) are the positive constants depending on the known values.

In the following sections,we will use the following inequalities.

Lemma 2.1 (Gagliardo-Nirenberg inequality) ^[18] Let Ω be a bounded domain with ∂Ω in C ^m ,and let u be any function in W ^m,r (Ω) ∩L ^q (Ω),1 ≤ q,r ≤ ∞. For any integer j ,0 ≤ j < m, and for any number a in the interval j/m ≤ a ≤ 1,set

If m − j − n/r is not a non-negative integer,then If m − j − n/r is a non-negative integer,then (9) holds for a = j/m. The constant C depends only on Ω,r,q,j ,and a.

Lemma 2.2 (Generalization of the Sobolev-Lieb-Thirring inequality) ^[19] Assume that Ω ⊂ R ⁿ is of class C ^2m or it enjoys the prolong property,and we denote by |Ω| the volume of Ω. Let φj ,1 ≤ j ≤ N,be a finite family of H ^m (Ω) which is orthonormal in L ² (Ω) and set,for almost every x ∈ Ω,

Then,for every p satisfying

here exists a constant κ₁ such that

The constant κ₁ depends on m,n,and p,it also depends on the shape of Ω but not on its size. It is independent of the family φj and of N .

Especially,when and m = 1,

and when m = 2, where κ₂ andκ₃ are the positive constants only depending on κ₁. 3 Existence of solutions

In this section,we obtain the existence and uniqueness of the solutions for the periodic initial value problem (5)−(8). Firstly,we establish some a prior estimates for the solutions.

Lemma 3.1 Assume that P₀(x) ∈ L ² (Ω),Q₀(x) ∈ H ¹ (Ω),f (x) ∈ L ² (Ω),g (x) ∈ L ² (Ω), and suppose that 2r_1r − 1 > 0. Then for the solutions of the problem (5)−(8),we have

where the constants E₀ and E₁ depend on ||P (0)|| and ||Q_x (0)||.

Proof Firstly,we differentiate (6) with respect to x once and set

then (5) and (6) are changed into Then,multiplying (14) by ,integrating with respect to x over Ω and taking the real part,we can get where From (16) and (17),we have

On the other hand,multiplying (15) by W and integrating over Ω,we have

where Then from (19),(20),and (21),there holds

Multiplying (18) by 2ω and (22) by (2r_1r − 1) respectively,and adding theses two equations together,we can get

Noting the condition 2r_r1 − 1 > 0,we have and Combining (23)−(25) together yields

From Hölder’s inequality,2w||P|| ² ≤ 2w∫_Ω |P| ⁴ dx + C(|Ω|). Since∫_Ω W dx = ∫ _-L ^L Q_x dx = Q| _-L ^L = 0,applying Poincar´e’s inequality,we can get ||W|| ² ≤CW||W_x|| ² ,where CW is a positive constant. Therefore,we have

which implies where

By Gronwall’s inequality,we infer that

Therefore, Combining the transformation (13),the proof of Lemma 3.1 is completed.

Lemma 3.2 Under the conditions of Lemma 3.1,and assume that P₀(x) ∈ H ¹ (Ω),Q₀(x) ∈ H ² (Ω),f (x) ∈ L ² (Ω),g(x) ∈ H ¹ (Ω). Then for the solutions of the problem (5)−(8),we have

where the constant E₂ depends on ||P_x (0)|| and ||Q_xx (0)||.

Proof Similarly to the first step in the proving the Lemma 3.1,we can make use of the transformed equations. Firstly,multiplying (14) by (_xx ),integrating with respect to x over Ω and taking the real part,we can get

where Secondly,we take the inner product of (15) with (−W_xx ) over Ω,then we can obtain Combining (32)−(34),we can get According to the Gagliardo-Nirenberg inequality and Lemma 3.1,we have or Similarly,we have where ε ₁ and ε ₂ are positive constants. Thus,choosing in (37),we can obtain Similarly,from the Gagliardo-Nirenberg inequality,Lemma 3.1,and (37),there holds and On the other hand,from (36),we can also have and For the next term,we can compute that in the same way, For the last two terms,we use Hölder’s inequality to obtain Substituting (39)−(45) into (35),we have From (37) and (38) again,we find Thus,combining the transformation (13),we have By Gronwall’s inequality,we get where the constant E₂ depends on ||P_x (0)|| and ||Q_xx (0)||.

Furthermore,combining (48) and (49),we can also obtain that

where C is a positive constant. The proof of Lemma 3.2 is completed.

Corollary 3.1 Under the conditions of Lemma 3.1 and Lemma 3.2,the solutions of the problem (5)−(8) satisfy

where C are positive constants.

Proof According to Agmon’s inequality,we can conclude the corollary easily.

Lemma 3.3 Under the conditions of Lemma 3.1 and Lemma 3.2,then for the solutions of the problem (5)−(8),we have

where C are positive constants. Proof Considering (6),we multiply this equation by Q and integrate over Ω,then we have By the previous lemmas and the corollary (51),we can get and Substituting (53) and (54) into (52),we find that By Gronwall’s inequality and Agmon’s inequality,we can complete the proof of Lemma 3.3.

Lemma 3.4 Under the conditions of Lemma 3.2,assume that P₀(x) ∈ H ² (Ω),Q₀(x) ∈ H ³ (Ω),f (x) ∈ H ¹ (Ω),and g(x) ∈ H ² (Ω). Then,we have the following estimates:

where the constant E₃ depends on ||P_xx (0)|| and ||Q_xxx (0)||.

Proof In the same way,taking inner product of (14) with x_xxxx and taking the real part, and multiplying (15) by W_xxxx and integrating with respect to x,then adding together yields

According to the Gagliardo-Nirenberg inequality,Lemma 3.1,and Lemma 3.2,we have Similarly,we have where ε ₃ and ε ₄ are positive constants. Thus,by choosing in (58),we can obtain According the above lemmas and Corollary 3.1,there is From Lemma 3.2 and Corollary 3.1,we know that ||W_x || = ||Q_xx || ≤ C and ||W || _L^∞ = ||Q_x|| _L^∞≤ C,then according to the Gagliardo-Nirenberg inequality,we have From (59),we have Similarly to getting (63),we can also obtain that For the next term,we have At last,we can compute the last two terms by Hölder’s inequality as Then combining (57)−(66),there holds From (58) and (59) again,we find Thus,by virtue of the transformation (13),there holds According to Gronwall’s inequality,we can obtain where the constant E₃ depends on ||P_xx (0)|| and ||Q_xxx (0)||.

Furthermore,combining (69) and (70),we can also obtain that

where C is a positive constant. Thus,the proof of Lemma 3.4 is completed.

Corollary 3.2Under the conditions of Lemma 3.1 and Lemma 3.4,the solutions of the problem (5)−(8) satisfy

where C are positive constants.

From the results of Lemma 3.1−Lemma 3.4,we find that

where C depends only on ||P₀|| H ² ,||Q₀|| H ³ ,and T . Thus,by the standard methods,we can extend the local solutions P (x,t),Q(x,t) of the problem (5)−(8) to global solutions,that is,we have the following theorem.

Theorem 3.1Suppose that P₀(x) ∈ H ² (Ω),Q₀(x) ∈ H ³ (Ω),f (x) ∈ H ¹ (Ω),g(x) ∈ H ² (Ω) and suppose that 2r_1r − 1 > 0. Then the problem (5)−(8) has unique solutions P (x,t),Q(x,t) satisfying

4 Existence of global attractor

In this section,we construct the global attractor for the problem (5)−(8). We first note that by Theorem 3.1,there exists a dynamical system S(t)(t≥0) which maps H ² (Ω) × H ³ (Ω) to H ² (Ω) × H ³ (Ω)such that S(t)(u 0,Q₀) = (P (t),Q(t)),the solutions of problem (5)−(8).

In order to obtain the existence of the global attractor associated with semigroup S(t),we need uniform a priori estimates in time. From (29),when ||P₀ || ≤ R and ||Q₀ || H ¹ ≤ R,we have

where depends on the data (f,g,R) when ||P₀ || ≤ R ,and ||Q₀ || H ¹ ≤ R

Applying (73),(74) and repeating the procedure of Lemma 3.2,we can derive that

where C₁ is a constant depending on the data (f,g) and the known parameters.

By Gronwall’s inequality and (75),we find that when t ≥t₁,there is

where depends on the data (f,g,R) when ||P₀ || H ¹ ≤ R,||Q₀ || H ²≤R.

Applying (73),(76),and repeating the procedure of the proof of Lemma 3.4,we obtain that

where C₂depends on the data (f,g) and the known parameters.

Similarly to (76),by Gronwall’s inequality and Lemma 3.4,we can derive that

where depends on the data (f,g,R) when ||P₀ || _{H ²}≤ R,||Q₀ || _{H ³} ≤ R. We observe that (73),(74),(76),and (78) imply that where K is a positive constant.

In what follows,we are going to show that the semigroup S(t) : H ² (Ω) × H ³ (Ω) → H ² (Ω) × H ³ (Ω) is compact for large t.

Lemma 4.1Assume that the conditions of Theorem 3.1 hold,and suppose that f ∈ H ² (Ω),g ∈ H ³ (Ω). Then,for the solutions of the problem (5)−(8),we have

where the constant E₄ depends on ||f _xx ||,||g _xxx ||,and R when ||P₀ || _{H ²}≤R,||Q₀ || _{H ³}≤R.

Proof Firstly by virtue of the transformed equations (14)−(15),taking the inner product of (14) with _xxxxxx over Ω,taking the real part,we can obtain

Secondly,multiplying (15) by −W_xxxxxx and integrating with respect to x over Ω,then we can get Adding (81) and (82) together yields According to the previous lemmas and the corollaries,we have Similar to getting (85),we can obtain On the other hand,for the next two terms,we have and Finally,by Hölder’s inequality,we have Combining (86)−(89),we have where C₇ = max(2ξ + C₃ + C₆,C₄ + C₅) is a constant.

Remembering the transformation (13),this implies that

On the other hand,integrating (77) between tand t + 1,and applying (78),we find that Thus,by the uniform Gronwall lemma,it follows from (91) and (92) that which concludes the proof of Lemma 4.1.

In order to prove the existence of global attractor of problem (5)−(8),we need the following result:

Theorem 4.1 ^[10] We assume that H is a metric space and that the nonlinear operator S(t) of H into itself for t≥0 satisfies

And also S(t)is continuous and uniformly compact for large t. That means for every bounded set B₀,there exists t₀ ,which may depend on B₀ that S(t)B₀ is relatively compact in H. We also assume that there exists an open setU and a bounded set B of U such that B is absorbing in U .

Then the ω-limit set of B: is a compact attractor,which attracts the bounded set of U . It is the maximal bounded attractor in U .

Theorem 4.2Assume that the conditions of Theorem 3.1 hold,and suppose that f ∈ H ² (Ω),g ∈ H ³ (Ω). Then there exists a global attractor A ⊂ H ² (Ω) × H ³ (Ω) for the periodic initial problem (5)−(8),i.e.,there is a set A such that

(i) S_tA = A,t ∈ R ⁺ ;

(ii) dist(S(t)B,A) = 0,for any bounded set B ⊂ H ² (Ω) × H ³ (Ω),where

and S(t)(P₀,Q₀) is a semigroup operator generated by the problem (5)−(8).

Proof On account of the result of Theorem 4.1,we will prove this theorem by checking the conditions in Theorem 4.1.

We observer that (79) shows that the ball

is an absorbing set of S(t)in H ² (Ω) × H ³ (Ω). In addition,Lemma 4.1 implies the dynamical system S(t) is uniformly compact for large t. Thus,according to the Theorem 4.1,we can conclude that the ω-limit set of B: A = ω(B) = is a compact attractor on H ² (Ω) × H ³ (Ω),where the closure is taken in H ² (Ω) × H ³ (Ω).

This completes the proof of Theorem 4.2. 5 Dimension of global attractor

In this section,we prove that the global attractor A as a compact subset of H ² (Ω) × H ³ (Ω) has a finite Hausdorff dimension and a fractal dimension. To this end,we consider the first variation equation of (5)−(8) as follows:

with the initial value and the periodic boundary conditions where (U₀,V₀) ∈ H ² (Ω) × H ³ (Ω),and (P (t),Q(t)) = S(t)(P₀,Q₀) is the solution of (5)−(8) with (P₀,Q₀) ∈ A.

Given (P₀,Q₀) ∈ A and the solution S(t)(P₀,Q₀) ∈H ² (Ω) × H ³ (Ω),by standard methods we can easily prove that for any (U₀,V₀) ∈ H ² (Ω) × H ³ (Ω),the linear initial boundary value problem (94)−(97) possesses a unique solution

and thus (U (t),V (t)) ∈ C(0,T ;H ² (Ω) × H ³ (Ω)).

Let Y₀ = (P₀,Q₀),Z₀ = (U₀,V₀ ),denote by Y (t) = S(t)Y₀,Z (t) = (U (t),V (t)) = DS(t)Z₀ the solution of (94)−(97). We now show S(t) is uniformly differentiable on the bounded set of H ² (Ω) × H ³ (Ω). More precisely,we have

Lemma 5.1 Let R and T be two positive numbers. Then,there exists a constant C depending on R and T such that for every Y₀,Z₀,t satisfying ||Y₀ || _{H ² ×H ³} ≤ R,||Y₀ + Z₀ || _{H ² ×H ³} ≤ R,0 ≤ t ≤ T ,we have

The proof of this result,which is lengthy but classical,is omitted here. For the techniques used,we refer readers to ^[19] and references therein. Note that Lemma 5.1 shows that S(t) is uniformly differentiable on A.

Now,we study the transformation of m-dimensional volumes inH ² (Ω) × H ³ (Ω)by the operator DS(t)Y₀,Y₀ ∈ A. Denote by Z₁(t),Z₂(t),· · · ,Z_N (t) the solution of linear equations (94)−(97) corresponding respective to the initial data

here ξ _j ∈ = L ² (Ω) × H ¹ (Ω),j = 1,2,· · · ,N . By the simple computation,we can deduce that where L(Y (t)) = L(S(t)Y₀ ) is a linear map L(t,Y₀)ξ = Z (t). ∧ denotes the exterior product, Tr the trace of the operator,and Q_N(t) the orthogonal projection of space to the spanning subspace generated by Z₁(t),Z₂(t),· · · ,Z_N(t). Therefore,we can then estimate

At a given time τ ,let Φ_j = (φ_j ,ψ_j ) ^T be orthonormal basis ofsuch that

Thus,we have where the scalar product

Omitting temporarily the variable τ ,we see that According the results in the previous section,we can estimate that and In the similar way,we can also obtain that and For the last two terms,there holds Combining (102)−(108) together yields that where k ₁₁ and k ₁₂ are two positive constants depending on the known values.

From (101) and (109),we know that

From (10) and (11),the results of the generalization of the Sobolev-Lieb-Thirring inequality, we find that

Substituting (111) into (110),we can get where the last two terms with Young’s inequality have been handled,and κ ₄ is a positive constant.

Assuming that Y₀ belong to the global attractor A,we can majorize the quantity

as follows: We see that if N is defined by then qN ≤ −κ ₄ < 0.

By application of Proposition V.2.1 and Theorem V.3.3 in Ref. ^[19],we have thus proved the following theorem.

Theorem 5.1 Assume that the conditions of Theorem 4.2 hold,and and we consider the dynamical system associated with the periodic initial problem (5)−(8). We denote by A the corresponding global attractor whose existence is stated in Theorem 4.2,and let N be defined by (114). Then,the Hausdorff dimension of A is less then or equal to N ,and its fractal dimension is less then or equal to 2N . 6 Conclusions

The coupled Burgers-CGL equations are derived from the nonlinear evolution of the coupled long-scale oscillatory and monotonic instabilities of a uniformly propagating combustion wave governed by a sequential chemical reaction. Thus,it is an interesting topic for mathematicians and other researchers. In this paper,we study the periodic initial value problem (5)−(8) for these coupled equations mathematically. Firstly,the existence and uniqueness of the global smooth solutions are obtained by using some subtle transforms,delicate a priori estimates and so-called continuity method,which assure the well-posedness of the rescaled complex amplitude of the flame oscillations P (x,t)) and the deformation of the first front Q(x,t) in the time [0,T]. Secondly,we consider the long time behavior for these solutions,which study the motions of P (x,t)) and Q(x,t) as t → ∞. We obtain the existence of the global attractor A ⊂ H ² (Ω) × H ³ (Ω) for problem (5)−(8) and establish the estimates of the upper bounds of Hausdorff and fractal dimensions for the attractor. More precisely,the Hausdorff dimension of A is less then or equal to N ,and its fractal dimension is less then or equal to 2N ,where N is defined in (114). For the physical significance,we know that the rescaled complex amplitude of the flame oscillations P (x,t)) and the deformation of the first front Q(x,t) attract the bounded sets of H ² (Ω) × H ³ (Ω) as t → ∞ under the initial conditions P₀(x)and Q₀(x),and the dimensions of the bounded sets are bounded.