Research Article The Coupled Kuramoto-Sivashinsky-KdV Equations for Surface Wave in Multilayered Liquid Films

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1 ISRN Mathematical Physics Volume 3, Article ID , 7 pages Research Article he Coupled Kuramoto-Sivashinsky-KdV Equations for Surface Wave in Mtilayered Liquid Films Maomao Cai, Dening Li, and Chontita Rattanak 3 Department of Mathematics, Weber State University, Ogden, U 8448, USA Department of Mathematics, West Virginia University, Morgantown, WV 656, USA 3 Department of Mathematics, Facty of Science, Mahidol University, Bangkok 4, hailand Correspondence shod be addressed to Maomao Cai; chloecai@weber.edu Received 4 June 3; Accepted 5 Jy 3 Academic Editors: S. C. Lim and W.-H. Steeb Copyright 3 Maomao Cai et al. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the coupled Kuramoto-Sivashinsky-KdV equations describing the surface waves on mtilayered liquid films. A priori energy estimates for linearized problem are derived and local existence of solutions for initial-value problem is established.. Introduction his paper studies a two-dimensional coupled Kuramoto- Sivashinsky-Korteweg-de Vries equation. he model was introduced in [] to describe the surface waves on mtilayered liquid films, and the two-dimensional model was proposed in []. A generalized equation that combines conservative and dissipative effects is a mixed Kuramoto-Sivashinsky- Korteweg-de Vries (KS-KdV) equation, u t +uu x +u xxx = αu xx γu xxxx, () which was first introduced in [3] andisoftencalledthe Benney equation. his equation finds various applications in plasma physics, hydrodynamics, and other fields [4, 5]. Another version of the Benney equation was proposed in [] forarealwavefieldu(x,t),basedontheks-kdv equation, which is linearly coupled to an additional linear dissipative equation for an extra real wave field V (x, t) that provides for the stabilization of the zero background solution. he model is as follows: u t +uu x +u xxx = αu xx γu xxxx +ε V x, () V t +a V x =ΓV xx +ε u x. he system describes the propagation of surface waves in a two-layer liquid film with one layer being dominated by viscosity. Here the coefficients α, γ, Γ >, thecoupling parameters ε,ε >,anda is a group-velocity mismatch between the two waves fields. he linear coupling via the first derivatives is the same as in known models of coupled internal waves propagating in mti-layered fluids [6]. hen, the linear dissipative equation in () implies that the substrate layer is essentially more viscous []. In [], the stability of solutions in the system of () is investigated by treating the gain and the dissipation constants α, γ, Γ as small parameters and making use of the balance equation for the net momentum. In [7], an energy estimate has been derived for the linearized model of (). In this paper, we consider the following two-dimensional version of () for general viscous fluid without the smallness assumptions on α, γ, Γ: u t +uu x +Δu x = αu xx γδ u+ε V x, V t +a V x =ΓΔV +ε u x. he system (3) is proposed in [] in the study of cylindrical solitary pses. One immediately notices that the two space variables (x, y) in (3) are not symmetric. his is because of the underlying nonsymmetric physics; see []. he stability of steady-state solutions is analyzed by perturbation theory and wave mode in[, 8]. Global solutions for the coupled Kuramoto-Sivashinsky-KdV system are studied in [9]. However, the existence of local solution is not available. In (3)

2 ISRN Mathematical Physics this paper, we will use the energy estimate approach to study such solution and establish its linear stability and the local existence of such solution for initial-value problems. hepaperisarrangedasfollows.insection, weinvestigate the linear stability of the solution to (3)byestablishing the energy estimates for its linearization. In Section 3, we use the obtained energy estimate and continuation method to show the existence of the solution for such linearized problem. Finally in Section 4, the existence of solution for the nonlinear problem is obtained by linear iteration method.. Linear Stability and High-Order Estimate Let ( u, Ṽ) be a small perturbation of a bounded C solution (u, V ) of (3): u u +ε u, V V +εṽ, with ε. Substituting (u, V) in (4)into(3)andomittingthehigherorder terms of ε, we obtain a linearized system for ( u, Ṽ): u t +u u x +Δ u x = α u xx γδ u+ε Ṽ x +f, Ṽ t +a Ṽ x =ΓΔṼ +ε u x +g. he linear stability of solution (u, V ) is determined by theenergyestimatefor( u, Ṽ) under small initial perturbation. For simplicity of notation, we will omit the over (u, V) in the following. Hence, we will discuss the following initial-value problem for the linearized system: u t +u u x +Δu x = αu xx γδ u+ε V x +f, (4) (5) heorem. Any solution (u, V) C ([, ]; S(R )) of (6) satisfies the estimate t ( u (t) + V (t) )+ u (t) + V (t) C( u (t) + V (t) + f (t) + g (t) ), t [, ], ( u (t) + V (t) )+ ( u (s) + V (s) )ds (9) C( u + V f (s) + g (s) )ds). () Here and in the following, C always denotes a constant depending only on and coefficients of (6). In particar, the constant C depends upon u in the coefficients of (6) only in its L norm. Proof. ake L inner product of the equations in (6) with (u, V) over R. Noticing that w t w = (/) t w,wehave t ( u + V ) + γδ u, u ΓΔV, V + αu xx,u = u u x,u a V x, V Δu x,u + ε V x, u + f, u + ε u x, V + g,v. () Integrating by parts in (x, y) and noticing that for any δ> u Δu + u, u 4 δ u +C δ u, () V t +a V x =ΓΔV +ε u x +g, u(x,y,)=u (x, y), V (x,y,)=v (x, y). (6) we have γδ u, u ΓΔV, V δ( Δu + V ), Here, the coefficients α, γ, Γ, ε, and ε are all positive constants with possible smooth small perturbations. u is a given bounded smooth function and a is a given bounded smooth function. Let, denote the L inner product in R and let H k (R ) be the usual Sobolev space defined by the norm f k = R Dj f dx dy (7) j k with H (R )=L (R ) and f = f,where D j f= j f j x j. (8) y Let S(R ) be the Schwartz rapidly decaying function space []. We have the following energy estimate for the linearized problem (6). αu xx, u + au x,u + cv x, V + Δu x,u + V x,u + u x, V δ( Δu + V )+C( u + V ), f, u + g, V f + g + u + V. (3) Substituting (3) into(), we obtain (9). o obtain (), we apply Gronwall s theorem []to the inequality t P (t) CP(t) +M(t) (4) with P (t) = u (t) + V (t) t u (s) + V (s) )ds, (5) M (t) = f (t) + g (t).

3 ISRN Mathematical Physics 3 herefore, we have and hence e Ct t P (t) P() e Cs M (s) ds (6) t P (t) C (t) (P () + M (s) ds), (7) and () follows readily. In particar, we notice that the constant C in () depends upon u only in its L norm over [, ] R. Inequalities (9)and()in heorem are obtained for the Cauchy initial data. Obviously, it is also valid for the initialboundary value problems with periodic boundary conditions studied for the periodic waves of KS-KdV system in []. Inequalities (9) and() in heorem can be improved. aking L inner product of the two equations in (6) with (Δu, ΔV) and integrating by parts in (x, y),wehave t ( u + V )+δ( Δu + ΔV ) u u x, Δu Δu x, Δu αu xx, Δu + ε V x, Δu + f,δu a V x,δv + ε u x,δv + g,δv. he right side of (8)canbecontrolledby δ( Δu (t) + ΔV (t) ) +C( u (t) + V (t) + f (t) + g (t) ) (8) (9) with constant C depending on u only in L norm. Combining (8) and(4) with(9) and noticing (), we have the following: t ( u (t) + V (t) )+ u (t) 3 + V (t) C( u (t) + V (t) + f (t) + g (t) ), t [, ]. () Indeed, we can further improve () bytakingl inner product of the first equation in (6)withΔ u to derive t Δu +γ Δ u u u x,δ u Δu x,δ u αu xx,δ u + ε V x,δ u + f, Δ u. he right side of ()canbecontrolledby () γ Δ u +C( u 3 + V + f ). () Combining (), (), and (), we have t ( u (t) + Δu + V (t) )+ u (t) 4 + V (t) C( u (t) + V (t) + f (t) + g (t) ), t [, ]. (3) Applying Gronwall s inequality to () and(3), we can obtain or ( u (t) + V (t) )+ ( u (s) 3 + V (s) )ds C( u + V f (s) + g (s) )ds) (4) ( u (t) + V (t) )+ ( u (s) 4 + V (s) )ds C( u + V f (s) + g (s) )ds). (5) Wecanalsoincludetheestimatefor(u t, V t ) in (5) by using the equations in (6)to obtain ( u (t) + V (t) ) u (s) 4 + V (s) + V t (s) + u t (s) )ds C( u + V f (s) + g (s) )ds). (6) It is easy to see that the constant C in (6) depends upon u in (6) only in its L norm over [, ] R. Since H (R ) canbecontinuouslyimbeddedintoboundedc(r ) by the Sobolev imbedding theorem [], the constant C in (6) hence depends only on u in its L ([, ], H (R )) norm. We can obtain higher-order estimates by taking spatial derivative of (6) and applying (6) to the expanded system. Hencewehavethefollowingtheorem. heorem. Any solution (u, V) C ([, ]; S (R )) of (6) satisfies the estimate, for any integer k, ( u (t) k+ + V (t) k+ ) u (s) k+4 + V (s) k+ + V t (s) k + u t (s) k )ds C k ( u k+ + V k+ f (s) k + g (s) k )ds). (7)

4 4 ISRN Mathematical Physics Here C k is a constant depending only on and coefficients of (6). In particar, the constant C k depends upon u in the coefficients of (6) only in its L ([, ], H k+ (R )) norm. Remark 3. For convenience, we will use the notation Π k in thefollowingtodenotetheproductspaceof(u, V): (u, V) Π k if Π k u C([, ],H k+ (R )) L ([, ],H k+4 (R )) H ([, ],H k (R )), V C([, ],H k+ (R )) L ([, ],H k+ (R )) H ([, ],H k (R )). is a Banach space equipped with the norm ( u (t) k+ + V (t) k+ ) (8) u (s) k+4 + V (s) k+ + V t (s) k + u t (s) k )ds. (9) 3. Existence of Solution for Linearized Problem We will use the continuation method to prove the following existence and uniqueness of the solution to (6). heorem 4. In the initial-value problem (6), letk be any integer and assume that (i) u H k+ (R ) and V H k+ (R ) and f, g L ([, ], H k (R )); (ii) γ δ>and Γ δ>; (iii) all the coefficients ξ (α,a,γ,γ,ε,ε,u ) in (6) are kth order continuously differentiable and are constant outside a bounded domain. hen (6) has a unique solution (u, V) in the space Π k satisfying estimate (7). Remark 5. By the Sobolev imbedding theorem, for k, the solution (u, V) is continuously differentiable in (x, y) up to 4th and nd orders. If k 4and f, g H ([, ], H k (R )), then by (6), we can derive that (u t, V t ) are also continuous. herefore the solution (u, V) in heorem 4 is a classical solution. First we rewrite (6) briefly as follows: Here L (u, V) =u u x +Δu x +αu xx ε V x +γδ u, L (u, V) =a V x ε u x ΓΔV. (3) Consider the following one-parameter family of initial-value problems, denoted as A λ (λ [, ]): u t +λl (u, V) + ( λ) Δ u=f, V t +λl (u, V) ( λ) ΔV =g, u(x,y,)=u (x, y), V (x, y, ) = V (x, y). (3) Obviously, for λ =,theproblema in (3) isthe same initial-value problem in (6). It is readily checked that energy estimate (7) is valid uniformlyfor the solution (u, V) of (3)withtheconstantC k in (7) being independent of the parameter λ [,]. In the following we show that the conclusion on the existence of solution in heorem 4 is true for (3) forall λ [,]. In particar the conclusion in heorem 4 is the case λ=. Let B [, ] be such that, for λ B, heorem 4 is true for (3). o show B = [, ], we need to prove that subset B is not empty, and it is both closed and open. () B is not empty. Actually, B. heproblema is the following two separate initial-value problems for u and V: u t +Δ u=f, u(x,y,)=u (x, y), V t ΔV =g, V (x, y, ) = V (x, y). (33) he existence of the solutions (u, V) for the problem (33) is the standard rest for the Cauchy problem of general parabolic equations; see, for example, [, ]. () B is closed in [, ]. Let λ j B and λ j λ.let(u j, V j ) be the solution of the following initial-value problem: u jt +λ j L (u j, V j )+( λ j )Δ u j =f, V jt +λ j L (u j, V j ) ( λ j )ΔV j =g, u j (x,y,)=u (x, y), V j (x,y,)=v (x, y). (34) By (7), (u j, V j ) is uniformly bounded in Π k.inparticar ( u k+4 + V )ds k+ u t + L (u, V) =f, V t + L (u, V) =g, u(x,y,)=u (x, y), V (x,y,)=v (x, y). (3) C k ( u k+ + V k+ f (s) k + g (s) k )ds) K. (35)

5 ISRN Mathematical Physics 5 Let (u j, V j )=(u j u j, V j V j ) (j =, 3,...) which satisfies u jt +λ j L (u j, V j )+( λ j )Δ u j hen (u j, V j )=(u j u j, V j V j ) (j =, 3,...) satisfies u jt +λ L (u j, V j )+( λ )Δ u j =(λ λ)(l (u j, V j ) Δ u j ), = (λ j λ j )[L (u j, V j ) Δ u j ], V jt +λl (u j, V j ) ( λ) ΔV j (4) V jt +λ j L (u j, V j ) ( λ j )ΔV j (36) =(λ λ)(l (u j, V j ) ΔV j ), = (λ j λ j )[L (u j, V j )+ΔV j ], u j (x,y,)=, V j (x,y,)=. Applying (7)tothesolution(u j, V j ) of (36), we have ( u j (t) k+ + V j (t) ) k+ u + k+4 V C k (λ j λ j ) C k (λ j λ j ) K. )ds k+ ( u j (s) k+4 + V j (s) )ds k+ (37) Since λ j λ,itfollowsthat(u j, V j ) is a Cauchy sequence in Π k and its limit (u, V) is obviously the solution of (3)forλ.hisshowsB is closed in [, ]. (3) B is open in [, ]. Let λ B and λ [,]with λ λ ε. Let (u, V ) be the solution of the following problem: u t +λ L (u, V )+( λ )Δ u =f, V t +λ L (u, V ) ( λ )ΔV =g, u (x,y,)=u (x, y), V (x, y, ) = V (x, y). (38) We can construct a sequence of solutions (u j, V j )(j =, 3,...) as the solution of the following problem: u j (x,y,)=, Since V j (x,y,)=. L (u j, V j ) Δ u j + k L (u j, V j ) ΔV j k M( u j (s) + k+4 V j (s) ), k+ (4) therefore by (7), (u j, V j ) satisfies ( u j (t) k+ + V j (t) ) k+ u + k+4 V )ds k+ C k M (λ λ ) ( u j (s) + k+4 V j (s) )ds. k+ (4) Choose ε such that C k Mε </and (u j, V j ) is a Cauchy sequence with limit (u, V) being the solution of (3) for λ.henceb is open. his concludes the proof of heorem 4. o prepare the study of the nonlinear system (3) inthe next section, we introduce the uniformly local Sobolev space H s (R );seealso[3]. Definition 6. he uniformly local Sobolev space H s (Rn ) is defined by f H s (Rn ) if and only if, for all φ C (Rn ) and φ X (X) φ(x X ), fφ X. H X R n s (R n ) (43) A corresponding norm is defined as f H s (R ), = X R fφ X H s (R ) (44) for a fixed φ C (R ) with φ(x) = in X. u jt +λ L (u j, V j )+( λ )Δ u j =f+(λ λ)(l (u j, V j ) Δ u j ), V jt +λl (u j, V j ) ( λ )ΔV j =g+(λ λ)(l (u j, V j ) ΔV j ), u j (x,y,)=u (x, y), V j (x,y,)=v (x, y). (39) It is readily verified that H s is a Hilbert space. hen we have the following improved version of heorem 4. heorem 7. In the initial-value problem (6), letk be any integer.under thesameassumptions asinheorem 4, except for the requirement on u which is replaced by u C([, ] ;H k+ (R )), (45) then the same conclusion in heorem 4 is true.

6 6 ISRN Mathematical Physics o prove heorem 7, we notice that estimate (7) can be obtained assuming that u C([,];H k+ (R )). Since H k+ (R ) isabanachalgebra,wehaveinstep() (to show B is closed) in the proof of heorem 4 u u (j )x k u u (j ) C k+ u k+ u (j ) k+. (46) Hence (37)follows from(36)for heorem 7. he same argument also applies to the estimate of the right side of (4)inStep(3) (to show B is open) in the proof of heorem 4 and we can obtain (4) correspondingly. where ( u a, Ṽ a ) is the solution for the initial-value problem of the linear system: u at +(U + u ) u ax +Δ u ax +α u axx +γδ u a ε Ṽ ax =, Ṽ at +a Ṽ ax ΓΔṼ a ε u ax =, u a (x,y,)= u (x, y), Ṽ a (x,y,)=ṽ (x, y). (5) From heorems 4 and 7, solution(u a, V a ) Π k of (5) exists and satisfies 4. Existence of Solution for Nonlinear Problem In this section, we prove the existence of solution for the following initial-value problem: u t +uu x +Δu x = αu xx γδ u+ε V x, ( u a (t) k+ + Ṽa (t) k+ ) u a (s) k+4 + Ṽa (s) k+ + u at (s) k + Ṽat (s) k )ds (5) V t +a V x =ΓΔV +ε u x, u(x,y,)=u (x, y), V (x,y,)=v (x, y). (47) C k ( u k+ + Ṽ k+ ). Now we are looking for the solution of (47)intheformof he coefficients γ, Γ, α, a, ε,andε are all positive constants or positive constants outside a bounded domain with possible smooth perturbations in bounded domain. In particar, γ, Γ are uniformly positive: γ δ>, Γ δ>. heorem 8. Let k be a nonnegative integer and assume that (i) all the coefficients γ, Γ, α, a, ε,andε are positive constants with possible smooth perturbations in a bounded domain and γ,γ δ>; (ii) (u, V ) = (U + u,v + Ṽ ) with (U,V ) being constants and ( u, Ṽ ) H k+ (R ) H k+ (R ). hen there is a > such that (47) has a unique solution (u, V) = (U + u, V + Ṽ) C([, ]; H k+ (R ) H k+ (R ) satisfying ( u (t) k+ + Ṽ (t) k+ ) u (s) k+4 + Ṽ (s) k+ + u t (s) k + Ṽt (s) k )ds C k ( u k+ + Ṽ k+ ). (48) For k, such solution is also a classical solution and satisfies (47) in the classical sense. Proof of heorem 8. We use linear iteration to prove heorem 8. First we construct an approximate solution (u a, V a ) of the form (u a, V a ) = (U + u a,v + Ṽ a ), (49) (u, V) = (u a + u, V a + V). (5) Obviously (u, V) is the solution of (47) ifandonlyif( u, V) is the solution of the following problem: u t +(u a + u) u x +Δu x +αu xx +γδ u ε V x +u ax u= f, V t +a V x ΓΔV ε u x = g, with u(x,y,)=, V (x, y, ) =, f = ( u at +u a u ax +Δ u ax +α u axx +γδ u a ε Ṽ ax +u a u ax ), g = (Ṽ at +a Ṽ ax ΓΔṼ a ε u ax ). (53) (54) From (5), ( f, g) L ([, ], H k (R )) and ( f(s) k + g(s) k )ds is controlled by ( u a, Ṽ a ) norms in Π k and consequently controlled by ( u k+ + Ṽ k+ ): ( f (s) + k g (s) k )ds C k ( u k+ + Ṽ k+ ). (55) Denote the nonlinear differential operators on the left side of the equations in (53) asm( u); wecanrewrite(53) briefly as M ( u)( u, V) =( f, g), u(x,y,)=, V (x, y, ) =. (56)

7 ISRN Mathematical Physics 7 ake u (x,y,t) = V (x,y,t) =.Byheorem 7, onecan obtain a sequence of solutions ( u j, V j ),forj =,, 3,..., in the product space Π k by solving the linearized problem Again from (7), we have ( w j (t) + k+ z j (t) ) k+ M ( u j )( u j, V j )=( f, g), u j (x,y,)=, V j (x,y,)=. From (7)ofheorem, ( u j, V j ) satisfy ( u j (t) k+ + u V j (t) k+ ) k+4 + V k+ + V jt (s) + k u jt (s) )ds k C k,j ( f (s) + k g (s) k )ds. (57) (58) + ( w k+4 + z k+ + w jt (s) + k z jt (s) )ds k C k w j (s) u jx (s) ds k C k t [, ] C k K u j k+ t [, ] w j (t) w j (s) ds k+ k+. (6) Choose such that C k K <.henthesequence ( u j, V j ) converges to ( u, V) which is the solution of (53). hen the solution (u,v) of (47)isobtainedby(5). Inequality (48)isobtainedfrom(5), (55), and (58). his completes the proof of heorem 8. References In particar, we notice that the constant C k,j in (58) depends upon u j only in its L ([, ], H k+ (R )) norm, and it is uniform in for small (say, ). So there is aconstantk such that if, for all j, u j (t) K, k+ (59) then the constant C k,j =C k in (58)isuniforminj. ake such that C k ( f (s) + k g (s) k )ds K. (6) hen (58) impliesthat(59) is satisfied for all j. Weconclude that for such,thesequence( u j, V j ) is uniformly bounded in the product space Π k. Let (w j,z j )=( u j+ u j, V j+ V j ). (w j,z j ) satisfies M ( u j )(w j,z j )= (M( u j ) M( u j )) ( u j, V j ) =( w j x u j,), w j (x,y,)=z j (x,y,)=. (6) [] B. A. Malomed, B. F. Feng, and. Kawahara, Stabilized Kuramoto-Sivashinsky system, Physical Review E, vol. 64, no. 4, Article ID 4634, 9 pages,. []B.F.Feng,B.A.Malomed,and.Kawahara, Cylindrical solitary pses in a two-dimensional stabilized Kuramoto- Sivashinsky system, Physica D, vol. 75, no. 3-4, pp. 7 38, 3. [3] D.J.Benney, Longwavesonliquidfilms, Mathematical Physics,vol.45,pp.5 55,966. [4] C.I.ChristovandM.G.Velarde, Dissipativesolitons, Physica D,vol.86,no.-,pp ,995. [5] C. Elphick, G. R. Ierley, O. Regev, and E. A. Spiegel, Interacting localized structures with Galilean invariance, Physical Review A, vol. 44, no., pp., 99. [6]J.A.GearandR.Grimshaw, Weakandstronginteractions between internal solitary waves, Studies in Applied Mathematics,vol.7,no.3,pp.35 58,984. [7]Y.Lenbury,D.Li,andC.Rattanak, Stabilityofsolutionof Kuramoto-Sivashinsky-Korteweg-de Vries system, Computers & Mathematics with Applications,vol.5,no.3-4,pp , 6. [8] X. Carvajal and M. Panthee, Sharp local well-posedness of KdV type equations with dissipative perturbations, [9] M. Cai and D. Li, Global solutions for coupled Kuramoto- Sivashinsky-KdV system, Quarterly of Applied Mathematics, vol. 67, no. 3, pp , 9. [] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, vol.3ofexts in Applied Mathematics, Springer,NewYork,NY,USA,4. [] B. F. Feng, B. A. Malomed, and. Kawahara, Stable periodic waves in coupled Kuramoto-Sivashinsky-Korteweg-de Vries equations, JournalofthePhysicalSocietyofJapan,vol.7,no., pp. 7 77,.

8 8 ISRN Mathematical Physics [] O. A. Ladyzhenskaya, he Boundary Value Problems of Mathematical Physics, vol. 49 of Applied Mathematical Sciences, Springer,NewYork,NY,USA,985. [3]. Kato, he Cauchy problem for quasi-linear symmetric hyperbolic systems, Archive for Rational Mechanics and Analysis,vol.58,no.3,pp.8 5,975.

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