GLOBAL SOLUTIONS FOR 2D COUPLED BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS

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1 Electronic Journal of Differential Equations, Vol ), No. 99, pp ISSN: URL: or ftp eje.math.txstate.eu GLOBAL SOLUTIONS FOR D COUPLED BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS HONGJUN GAO, LIN LIN, YAJUN CHU Abstract. In this article, we stuy the perioic initial-value problem of the D couple Burgers-complex-Ginzburg-Lanau Burgers-CGL) equations. Applying the Brezis-Gallout inequality which is available in D case an establishing some prior estimates, we obtain the existence an uniqueness of a global solution uner certain conitions. 1. Introuction In this article, we are concerne with the perioic initial-value problem for the following D couple Burgers-complex-Ginzburg-Lanau Burgers-CGL) equations: P t = ξp iu) P 1 + iv) P P P Q rp Q + fx), x, t) R +, 1.1) Q t = m Q 1 Q ω P + gx), x, t) R +, 1.) P x i + L, t) = P x i, t), Qx i + L, t) = Qx i, t), x, t) R +, 1.3) P x, 0) = P 0 x), Qx, 0) = Q 0 x), x, 1.4) where = [ L, L] [ L, L], L is the perio, fx) an gx) are given real functions. The complex function P x, t) is the rescale amplitue of the flame oscillations, the real function Qx, t) is the eformation of the first front. The Lanau coefficients u, v are real an the coefficients ξ, m are positive, while the coupling coefficients ω > 0 an r = r 1 + ir. The couple Burgers-CGL equations was erive from the nonlinear evolution of the couple long-scale oscillatory an monotonic instabilities of a uniformly propagating combustion wave governe by a sequential chemical reaction, having two flame fronts corresponing to two reaction zones with a finite separation istance between them see [, 9, 1, 16, 17, 18]). It escribes the interaction of the excite oscillatory moe an the ampe monotonic moe, that is ξ > 0, m > 0. Let us recall some previous works which are relate to our results. If there are no terms involve Q in 1.1), then 1.1) reuces to the well-known complex Ginzburg-Lanau equation CGL) that escribes the weakly nonlinear evolution of 010 Mathematics Subject Classification. 35M13, 35D35. Key wors an phrases. Burgers-CGL equation; prior estimate; global solution; Gagliaro-Nirenberg inequality; Höler inequality. c 015 Texas State University. Submitte June, 015. Publishe 7,

2 H. GAO, L. LIN, Y. CHU EJDE-015/99 a long-scale instability see [15]). Many results on the well-poseness an global attractors for the CGL equation have been establishe, one can fin the etails in [6, 7, 8, 13, 14, 19]. An if taking the coupling coefficient ω = 0, 1.1) woul be the well-known Burgers equation see [3]). There are also many works concerning the Burgers equation, one can see [4, 0] an so on. So far, the mathematical analysis an physical stuy about the couple Burgers-CGL have been one by a few researchers. For the perioic initial-value problem of the 1-D Burgers-Ginzburg- Lanau equation, well-poseness an global attractors are obtaine in [10]. By some prior estimates an so-calle continuity metho, the authors in [11] firstly showe the global existence of solutions an attractors to 1-D couple Burgers- CGL equations for flames governe by sequential reaction, an then obtaine the existence of the global attractor A H ) H 3 ) for the problem. In aition, they establishe the estimates of the upper bouns of Hausorff an fractal imensions for the attractors. However, to our knowlege, there is few work on the D couple Burgers-CGL equations. We will prove global well-poseness of problem 1.1)-1.4) as conjecture in [11]. The metho use here is well establishe but nee to be applie skillfully to get the esire result. The purpose of this article is to stuy the global solutions of 1.1)-1.4). The existence an uniqueness of the global solutions to 1.1)-1.4) is obtaine by virtue of the Brezis-Gallout inequality, which is available in the D case. The main result in our paper is as follows. Theorem 1.1. Suppose that P 0 x) H ), Q 0 x) H 3 ), fx) H 1 ) an gx) H ), for any given T > 0, then 1.1)-1.4) has the unique solution P x, t) an Qx, t) satisfying P x, t) L 0, T ; H ) ), P t x, t) L 0, T ; L ) ), Qx, t) L 0, T ; H 3 ) ), Q t x, t) L 0, T ; H 1 ) ). The rest of this article is organize as follows. In section, we briefly give some preliminaries. Some prior estimates for the solutions to 1.1)-1.) are establishe in section 3. Then, by the stanar metho, we can exten the local solutions to global solutions an subsequently the proof of Theorem 1.1 is presente.. Preliminaries In this section, We firstly recall some concepts an conventional notation. Let W k,p ) enote the usual kth orer Sobolev space with its norm 1/p, u W k,p ) := α k Dα u x) p 1 p <, α k ess sup D α u, p =. When p =, we enote u Hk ) = u W k, ). The inner prouct in L ) is efine by u, v) = Re ux) vx)x. For simplicity, we write u = u L. Without any ambiguity, we enote a generic positive constant by C which may vary from line to line. The following inequalities play an important role in the proof of the main results in R.

3 EJDE-015/99 BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS 3 Lemma.1 Gagliaro-Nirenberg inequality [5]). Let be a boune omain with in C m, an let u be any function in W m,r ) L q ), 1 q, r. For any integer j, 0 j m, an for any number a satisfying j/m a 1, set 1 p = j n + a 1 r m n If m j n/r is not a non-negative integer, then D j u L p ) + 1 a) 1 q. C u a W m,r u 1 a L q..1) If m j n/r is a non-negative integer, then.1) hols for a = j/m. The constant C epens only on, r, q, j an a. Lemma. Brezis-Gallout inequality [1]). Let be a boune omain in R, an let u be any function in H ). If u H 1 1, then there hols u L C 1 + log1 + u H ) )..) 3. Existence an uniqueness of solutions In this section, we firstly establish some prior estimates for the solutions to 1.1)-1.4) which guarantee the existence of the global solutions. Lemma 3.1. Assume that gx) L ), then for the problem 1.1)-1.4), it hols that t Q ) E log1 + P H + Q H 3) ) P H + Q H 3 ) + E1, where E 0, E 1 are constants. Proof. Multiplying 1.) by Q an integrating with respect to x over gives t Q = m Q Q Q x ω P Q x + gq x. 3.1) Applying the Brezis-Gallout inequality Lemma.) an Höler inequality, we euce that Q Q x Q L Q C 1 + ) log1 + Q H ) Q 3.) an ω C 1 + log1 + Q H )) Q H 1 P Q x ω Q L P C 1 + ) log1 + Q H ) P C 1 + log1 + Q H )) P. 3.3) Since gx) L ), obviously gq x Q + g Q + C )

4 4 H. GAO, L. LIN, Y. CHU EJDE-015/99 Substituting 3.)-3.4) back into 3.1), we infer that t Q m Q + C 1 + log1 + Q H )) Q H 1 + C 1 + log1 + Q H )) P + Q + C 1 C1 + log1 + P H + Q H 3)) P H + Q H 3 ) + C1 = E log1 + P H + Q H 3)) P H + Q H 3 ) + E1. Thus, the proof is complete. 3.5) Lemma 3.. Assume that fx) L ) an gx) L ), then for 1.1)-1.4), we have P + Q ) E 1 + log1 + P H + Q H 3)) P H t + ) Q H + 3 E3, where E, E 3 are constants. Proof. By ifferentiating 1.) with respect to x an setting Equations 1.1)-1.) become W = Q, 3.6) P t = ξp iu) P 1 + iv) P P P W rp iv W + f, 3.7) W t = m iv W ) 1 W ) ω P ) + g. 3.8) Multiplying 3.7) by P, integrating with respect to x over an taking the real part gives 1 t P = ξ P P P 4 L Re P W P x 4 3.9) Re r P iv W x + Re fp x, where Re P W P x = 1 W P ) x = 1 P iv W x. 3.10) While multiplying 3.8) by W an integrating over yiels 1 t W = m iv W 1 W ) W x + g W x ω P ) W x, where 3.11) ω P ) W x = ω P iv W x. 3.1)

5 EJDE-015/99 BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS 5 Combining 3.9) with 3.1), we arrive at t P + W ) = ξ P P m iv W P 4 L x 4 + ω + 1 r 1 ) P iv W x W ) W x + Re fp x + g W x =: 5 I i, i=1 3.13) where I 1 = ξ P P m iv W P 4 L, 4 I = ω + 1 r 1 ) P iv W x, I 3 = W ) W x, I 4 = Re fp x, I 5 = g W x. 3.14) We now estimate each term on the right-han sie of 3.13). By the Brezis-Gallout inequality an Höler inequality, I an I 3 can be estimate as follows I ω + 1 r 1 P L 1/ iv W m iv W + C P 4 L m iv W + C1 + log1 + P H )) 4 m iv W + C1 + log 1 + P H )) 3.15) m iv W + C P H + C C1 + log1 + P H + W H )) P H + W H ) + m iv W + C an I 3 W L W iv W m iv W + C W L W m iv W + C1 + log1 + W H )) W 3.16) C1 + log1 + P H + W H )) P H + W H ) + m iv W. For I 4 an I 5, one can euce that I 4 P + f P H + C 1 + log1 + P H + W H )) P H + W H ) + C 3.17) an I 5 g iv W m iv W + C 3, 3.18)

6 6 H. GAO, L. LIN, Y. CHU EJDE-015/99 provie fx) L ) an gx) L ). Substituting the above estimates 3.15) 3.18) back into 3.13) leas to P + W ) t ξ P P P H + W H ) + C + C + C 3 P 4 x + C 1 + log1 + P H + W H ) ) = E 1 + log1 + P H + W H )) P H + W H ) + E ) Noting 3.6), we finally obtain P + Q ) t E 1 + log1 + P H + Q H 3)) P H + ) Q H + 3 E3, which is the esire result. 3.0) To obtain the higher orer estimates with the complex amplitue of the flame oscillations P x, t) an the eformation of the first front Qx, t), we can use the transforme equations 3.7)-3.8) an have the following lemmas. Lemma 3.3. Assume that fx) L an gx) H 1 ), then for the problem 1.1)-1.4), we have P + Q ) t E log1 + P H + Q H 3)) P H + ) Q H + 3 E5, where E 4, E 5 are constants. Proof. Multiplying 3.7) by P ), integrating with respect to x over an taking the real part, we obtain 1 t P = ξ P P + Re 1 + iv) P P P x + Re P W P x + Re rp iv W P x 3.1) Re f P x. On the other han, multiplying 3.8) by iv W )) an integrating with respect to x over yiels 1 t iv W = m iv W ) + 1 W ) iv W ) x 3.) + ω P ) iv W ) x g iv W ) x.

7 EJDE-015/99 BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS 7 Aing 3.) an 3.1), one has P + iv W ) t = ξ P P m iv W ) + Re 1 + iv) P P P x + Re P W P x + Re rp iv W P x + W ) iv W ) x + ω P ) iv W ) x Re f P x g iv W ) x = 8 J i, i=1 3.3) where J 1 = ξ P P m iv W ), J = Re 1 + iv) P P P x, J 3 = Re P W P x, J 4 = Re rp iv W P x, J 5 = W ) iv W ) x, J 6 = ω P ) iv W ) x, J 7 = Re f P x, J 8 = g iv W ) x. 3.4) We now estimate J i i =,..., 8) in 3.3). From Lemma., it follows that J 1 + iv P L P P C P L P H C 1 + log1 + P H )) P H C 1 + log1 + P H + W H )) P H + W H ), 3.5) an J 3 W L P P W L P H C1 + log1 + W H )) P H C1 + log1 + P H + W H )) P H + W H ) 3.6) J 4 r P L iv W P C P L P + iv W C 1 + ) log1 + P H ) P + iv W C 1 + log1 + P H )) P H + C W H C 1 + log1 + P H + W H )) P H + W H ), 3.7)

8 8 H. GAO, L. LIN, Y. CHU EJDE-015/99 where we use the Brezis-Gallout an Höler inequalities. By irect computations, we have J 5 W L W iv W ) 1 m W L W + m iv W ) C 1 + log1 + W H )) W H + m iv W ) C 1 + log1 + P H + W H )) P H + W ) H + m iv W ) an J 6 4ω P L P iv W ) 8ω m P L P + m iv W ) C 1 + log1 + P H )) P H + m iv W ) C 1 + log1 + P H + W H )) P H + W ) H + m iv W ). In aition, J 7 an J 8 can be estimate as follows an 3.8) 3.9) J 7 P + f P H + C log1 + P H + W H )) P H + W H ) + C4 3.30) J 8 g iv W ) m iv W ) + m g m iv W ) + C 5. Substituting the above estimate 3.5)-3.31) into 3.3) leas to t P + iv W ) ξ P P + C + 1) 1 + log1 + P H + W H ) ) P H + W H ) + C 4 + C 5 E log1 + P H + W H )) P H + W H ) + E 5. From the transformation 3.6), we finally obtain t P + Q ) E log1 + P H + Q H 3)) P H + Q H 3) + E 5, which completes the proof of this lemma. 3.31) 3.3) 3.33) Lemma 3.4. Assume that fx) H 1 ) an gx) H ), then for the problem 1.1)-1.4), we have P + Q ) t E log1 + P H + Q H 3)) P H + ) Q H + 3 E7, where E 6 an E 7 are constants.

9 EJDE-015/99 BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS 9 Proof. Multiplying 3.7) by P, integrating with respect to x over an taking the real part gives 1 t P = ξ P P Re 1 + iv) P P P x Re P W P x Re rp iv W P x 3.34) + Re f P x. Taking the inner prouct of 3.8) with iv W ) over, one has 1 iv W ) t = m iv W ) 1 W ) iv W ) x ω P ) iv W ) x + g iv W ) x. Aing 3.35) an 3.34), we have where t P + iv W ) ) = ξ P P m iv W ) Re 1 + iv) P P P x Re P W P Re rp iv W P x W ) iv W ) x ω P ) iv W ) x + Re f P x + g iv W ) x =: 8 K i, K 1 = ξ P P m iv W ) K = Re 1 + iv) P P P x, K 3 = Re P W P, K 4 = Re rp iv W P x, K 5 = W ) iv W ) x K 6 = ω P ) iv W ) x, K 7 = Re f P x, K 8 = g iv W ) x. i=1 3.35) 3.36) 3.37) Next we analyze the each integran in 3.36). It follows from Lemmas.1. that K iv P L P P 1 P + C P 4 L P 1 P + C 1 + log 1 + P H ) ) P P

10 10 H. GAO, L. LIN, Y. CHU EJDE-015/99 1 P + C1 + P H ) P P H 1 P + C P H + C 1 P L P H 1 P + C 1 + log1 + P H )) P H C 1 + log1 + P H + W H )) P H + W H ) + 1 P. 3.38) With respect to the term K 3, by irect computation, one has K 3 = Re D P W P + P W P ) x W L D P P + P L 4 W L 4 P 1 P + 4 W L D P + P L 4 W L ) 1 P + 4 W L D P + 1 P 4 L W 4 L 4. Applying the Gagliaro-Nirenberg inequality, it is sufficient to see that P 4 L P 4 P, 3.40) which enable us to estimate K 3 as K 3 1 P + 4 W L D P + 1 P H P L + 1 W H W L 1 P + C 1 + log1 + W H )) P H 3.41) + C 1 + log1 + P H )) P H + C 1 + log1 + W H )) W H C 1 + log1 + P H + W H )) P H + W H ) + 1 P. Similarly to 3.41), we have K 4 = Re riv W P P + P iv W ) P ) x r iv W L 4 P L 4 P + r P L iv W ) P 1 P + C W L 4 P L + C P 4 L iv W ) 3.4) 1 P + C W 4 L 4 + C P 4 L 4 + C P L W H 1 P + C W H W L + C P H P L + C P L W H C 1 + log1 + P H + W H )) P H + W H ) + 1 P.

11 EJDE-015/99 BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS 11 To estimate K 5, we enote W = W 1, W ) for simplicity. Note that W ) iv W ) x = Wx 1 1 Wx W 1 Wx 1 1x 1 + Wx 1 Wx 1 + W Wx 1x 1 + Wx 1 Wx 1 + W 1 Wx 1 x + Wx Wx + W Wx x ) iv W ) x Wx 1 1 Wx Wx 1 Wx 1 + Wx 1 Wx 1 + Wx Wx ) iv W ) x + W 1 Wx 1 1x 1 + W Wx 1x 1 + W 1 Wx 1 x + W W x x ) iv W ) x 8 W L 4 iv W ) + 8 W L D W iv W ). 3.43) By 3.43), we fin that K 5 = W ) iv W ) x 8 W L 4 iv W ) + 8 W L D W iv W ) m 3 iv W ) + 96 m W H W L + 96 m W L W H C 1 + log1 + W H )) W H + m iv W ) 3 C 1 + log1 + P H + W H )) P H + W ) H + m 3 iv W ). 3.44) For the last three terms of 3.36), we have K 6 = ω P P + P P + P P ) iv W ) x 4ω P L P iv W ) + 4ω P L4 iv W ) m 3 iv W ) + C P L P H + C P H P L C 1 + log1 + P H + W H )) P H + W H ) + m 3 iv W ), 3.45) K 7 f P 1 P + C ) an K 8 g iv W ) m 3 iv W ) + C )

12 1 H. GAO, L. LIN, Y. CHU EJDE-015/99 because fx) H 1 ) an gx) H ). Substituting 3.38)-3.47) into 3.36), we obtain the following estimates P + iv W ) ) t C 1 + log1 + P H + W H )) P H + W ) H + ξ P 3.48) + C 6 + C 7 E log1 + P H + W H )) P H + W H ) + E7, where P P H an E 6 = ξ + C, E 7 = C 6 + C 7. Uner the conitions of Lemmas , we have the following result. Lemma 3.5. Assume that fx) H 1 ) an gx) H ), then for the problem 1.1)-1.4), there exist positive constants E, E, E, E such that for any T > 0, P H E exp 1 +E P eet 0 H + Q 0 H 3 ) ), Q H 3 E exp 1 +E P eet 0 H + Q 0 H 3 ) ). Proof. From Lemmas , it is easy to show that Q + P + Q + P + Q + P + Q ) t E 0 + E + E 4 + E 6 ) 1 + log1 + P H + Q H 3)) P H + ) Q H ) + E 3 + E 5 + E 7 = E log1 + P H + Q H 3)) P H + Q H 3 ) + E9. Denote Inequality 3.49) becomes Q + Q + Q + Q = [Q] H 3, P + P + P = [P ] H. t [P ] H + [Q] H 3) E log1 + P H + Q H 3)) P H + Q H 3) + E ) We always use the notation H s = 1/, α s α L ) thus [ ]H an [ ] H 3 are equivalent norms to H an H 3 respectively. Furthermore, there exists a positive constant C 8 such that H s C 8 [ ] H s, s =, 3. Applying Cauchy inequality, from 3.50) it follows that [P ] t H + [Q] ) H 3 E log1 + C 8 [P ] H + C 8 [Q] H 3)) C8[P ] H + C 8[Q] ) H + 3 E9 C 8E log1 + C 8 + [P ] H + C 8 + [Q] H 3)) [P ] H + [Q] H 3 ) + E9 C 8E log1 + C 8 ) ) 1 + log1 + [P ] H + [Q] H 3)) [P ] H + [Q] H 3 ) + E 9 = E log1 + [P ] H + [Q] H 3)) [P ] H + [Q] H 3 ) + E )

13 EJDE-015/99 BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS 13 If there exists a positive constant N such that [P ] H + [Q] H 3 N for all T > 0, then Lemma 3.5 hols. Otherwise, there exists a positive constant E such that t [P ] H + [Q] H 3) E[P ] H + [Q] H 3) log[p ] H + [Q] H3), 3.5) which implies that Therefore, Namely, log log[p ] H + [Q] H 3) ET + log log[p ] H 0) + [Q] H 30)) This proof is complete. ET + E P 0 H + Q 0 H 3). [P ] H exp 1 eet +E P 0 H + Q 0 H 3 ) ), [Q] H 3 exp 1 eet +E P 0 H + Q 0 H 3 ) ). P H E exp 1 eet +E P 0 H + Q 0 H 3 ) ), Q H 3 E exp 1 eet +E P 0 H + Q 0 H 3 ) ). Base on the previous lemmas, we are reay to prove the main result. 3.53) Proof of Theorem 1.1. From Lemmas , when P 0 x) H ), Q 0 x) H 3 ), fx) H 1 ), gx) H ) an for any T > 0, we obtain that P H E exp 1 eet +E P 0 H + Q 0 H 3 ) ), Q H 3 E exp 1 eet +E P 0 H + Q 0 H 3 ) ), where the positive constant C epens only on P 0 H, Q 0 H 3 an T. Furthermore, by the stanar metho, we can exten the local solutions P x, t) an Qx, t) of the perioic initial value problem 1.1)-1.4) to global solutions. Thus, the proof is complete. Acknowlegments. The work wass supporte in part by China NSF Grant Nos , ), PAPD of Jiangsu Higher Eucation Institutions, the Jiangsu Collaborative Innovation Center for Climate Change an the NSF of the Jiangsu Higher Eucation Committee of China No. 14KJB References [1] Brezis, H.; Gallouet, T.; Nonlinear Schröinger evolution equations, Nonlinear Anal., 41980), [] Buckmaster, J. D.; Lufor, G. S. S.; Theory of Laminar Flames, Cambrige University Press, Cambrige, UK, 198. [3] Burgers, J. M.; A mathematical moel illustrating the theory of turbulence, Av. Appl. Mech., 11948), [4] Chiella, S. R.; Yaav, M. K.; Large time asymptotics for solutions to a nonhomogeneous Burgers equation, Appl. Math. Mech. -Engl. E., ), [5] Frieman, A.; Partial Differential Equations, Holt, Reinhart an Winston, New York 1969). [6] Gao, H. J.; Bu C.; Dirichlet inhomogeneous bounary value problem for the n + 1 complex Ginzburg-Lanau equation, J. Differential Equations, ),

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