Analyticity and Smoothing Effect for the Coupled System of Equations of Korteweg-de Vries Type with a Single Point Singularity

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1 Acta Appl Math () : 75 DOI.7/s Analyticity and Smoothing Effect for the Coupled System of Equations of Korteweg-de Vries Type with a Single Point Singularity Margareth S. Alves Bianca M.R. Calsavara JaimeE.MuñozRivera Mauricio Sepúlveda Octavio Vera Villagrán Received: Decemer 8 / Accepted: 6 Septemer / Pulished online: 8 Septemer Springer Science+Business Media B.V. Astract Using Bourgain spaces and the generator of dilation P = t t +x x, which almost commutes with the linear Korteweg-de Vries operator, we show that a solution of the initial value prolem associated for the coupled system of equations of Korteweg-de Vries type which appears as a model to descrie the strong interaction of wealy nonlinear long waves, has an analyticity in time and a smoothing effect up to real analyticity if the initial data only have a single point singularity at x =. Keywords Evolution equations Bourgain space Smoothing effect Mathematics Suject Classification () 5Q5 M.S. Alves Departamento de Matemática, Universidade Federal de Viçosa-UFV, 657- Viçosa, MG, Brazil malves@ufv.r B.M.R. Calsavara Universidade Estadual de Campinas-Unicamp, Limeira, SP, Brazil iancamrc@yahoo.com J.E. Muñoz Rivera National Laoratory for Scientific Computation, Rua Getulio Vargas, Quitandinha-Petrópolis, Rio de Janeiro, RJ, Brazil rivera@lncc.r M. Sepúlveda ( ) CI²MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 6-C, Concepción, Chile mauricio@ing-mat.udec.cl O.V. Villagrán Departamento de Matemática, Universidad del Bío-Bío, Collao, Casilla 5-C, Concepción, Chile overa@uioio.cl

2 76 M.S. Alves et al. Introduction In this wor, we consider the following nonlinear coupled system of equations of Kortewegde Vries type u t + u xxx + a v xxx + uu x + a vv x + a (uv) x =, x,t R, (.) v t + v xxx + a u xxx + vv x + a uu x + a (uv) x =, (.) u(x, ) = u (x), v(x, ) = v (x) (.) where u = u(x, t), v = v(x,t) are real-valued functions of the variales x and t and a, a, a,, are real constants with > and >. This system has the structure of a pair of Korteweg-de Vries equations coupled through oth dispersive and nonlinear effects. It was derived y Gear and Grimshaw [9] as a model to descrie the strong interaction of wealy nonlinear, long waves. Mathematical results on the system (.) (.) were given y J. Bona et al. [5]. They proved that the system is gloally well posed in H s (R) H s (R) for any s provided that a <. This system has een intensively studied y several authors (see [, 5, 8] and the references therein). It have the following conservation laws E (u) = udx, E (v) = vdx, E (u, v) = ( u + v )dx. R R The time-invariance of the functionals E and E expresses the property that the mass of each mode separately is conserved during interaction, while that of E is an expression of the conservation of energy for the system of two models taen as a whole. Solutions of (.) (.) satisfy an additional conservation law which is revealed y the time-invariance of the functional E 4 = ( u x + vx + u a u x v x R a u v a u v a uv v R ) dx. The functional E 4 is a Hamiltonian for the system (.) (.)andif a <, E 4 will e seen to provide an a priori estimate for the solutions (u, v) of (.) (.) in the space H (R) H (R). Furthermore, the linearization of (.) (.) aout the rest state can e reduced to two, linear Korteweg-de Vries equations y a process of diagonalization (see Sect. ). Using this remar and the smoothing properties (in oth the temporal and spatial variales) for the linear Korteweg-de Vries derived y Kato [ ], Kenig, Ponce and Vega [6, 7] it will e shown that the system is locally well-posed in H s (R) H s (R) for any s whenever a. Later on, this result was improved y J.M. Ash et al. [], showing that the system (.) (.) is gloally well-posed in L (R) L (R) with a. In 4, F. Linares and M. Panthee [8] improved this result showing that the system (.) (.) is locally well-posed in H s (R) H s (R) for s> /4 and gloally well-posed in H s (R) H s (R) for s> / under some conditions on the coefficients, indeed for a = and =. In, O. Vera [9], following the idea of W. Craig et al. [7] shownthatc solutions (u(,t),v(,t)) to (.) (.) are otained for t> if the initial data (u (x), v (x)) elong to a suitale Soolev space satisfying reasonale conditions as x. In, K. Kato and T. Ogawa [] shown that a solution of the Korteweg-de Vries equation has a smoothing effect up to real analyticity if the initial data only has a single point singularity at x =. It is shown that for H s (R) (s > /4) data satisfying the condition A (x x ) u(x, ) < +, H s (R)

3 Analyticity and Smoothing Effect for the Coupled System of Equations 77 the solution is analytic in oth space and time variales. This result generalized the result given y W. Craig et al. [7]. On the other hand, ecause (.) (.) is a coupled system of Korteweg-de Vries equations, it is natural to as whether it has a smoothing effect up to real analyticity if the initial data only has a single point singularity at x = similar to those results otained y K. Kato and T. Ogawa [] for the Korteweg-de Vries equation. In this paper, we give an answer to this question which will improve those results otained in O. Vera [9] (seealso[]). For this reason, first we will decouple the dispersive term in the system (.) (.) and we will consider the associated linear system. Our main tool to otain the main result is to use the generator of dilation P = t t + x x, which almost commutes with the linear Korteweg-de Vries operator L = t + x. A typical example of initial data satisfying the assumption of Theorem. is the Dirac delta measure, since (x x ) δ(x) = ( ) δ(x). The other example of the data is the Cauchy principal value p.v. ( ). Linear x comination of those distriutions with analytic H s (R) data satisfying the assumption is also possile. In this sense, the Dirac delta measure adding to the soliton initial data can e taen as an initial datum. This paper is organized as follow: In Sect. we have the reduction of the prolem, notation, terminology to e used susequently and results that will e used several times. In Sect. we prove a theorem of existence and well-posedness of the solutions. In Sect. 4 we prove the following theorem: Theorem. Suppose that the initial data (u,v ) H s (R) H s (R), s> /4 and A,A > such that A (x x ) u H < + and A s (x x ) v H (R) < +. (.4) s (R) Then for some (/, 7/), there exist T = T( u H s (R), v H s (R)) and a unique solution (u, v) C(( T,T): H s (R)) X s C(( T,T): H s (R)) X s to the coupled system (.) (.) in a certain time ( T,T) and the solution (u, v) is time locally well-posed, i.e., the solution continuously depends on the initial data. Moreover the solution (u, v) is analytic at each point (x, t) R ( T,T), t. As a consequence of this theorem is the following Gevrey regularity result Corollary. Let s> /4, (/, 7/). Suppose that the initial data (u,v ) H s (R) H s (R), and A,A > such that (.4) is verified. Then there exists a unique solution (u, v) C(( T,T),H s (R)) X s C(( T,T),Hs (R)) X s to the coupled system of Korteweg-de Vries equations (.) (.) for a certain ( T,T) and for any t ( T,T), t, the pair (u, v) are analytic functions in the variale space and for x R, u(x, ) and v(x, ) is Gevrey as function of the time variale. Remar. In Theorem. and Corollary., the assumption on the initial data implies analyticity and Gevrey regularity except at the origin respectively. In this sense, those results are stating that the singularity at the origin immediately disappears after t> or t<, up to analyticity. Remar. The crucial part for otaining a full regularity is to gain the L (R ) regularity of the solutions (u,v ) from the negative order Soolev space. This part is considered in

4 78 M.S. Alves et al. Proposition 4. in Sect. 4. We utilize a three steps recurrence argument for treating the nonlinearity appearing in the right hand side of t x u = Pu + x xu + tb (u, u) + tb (v, v) + tb (u, v), (.5) t x v = Pv + x xv + tc (u, u) + tc (v, v) + tc (u, v). (.6) Then step y step, we otain the pointwise analytic estimates sup t m x l u H t [t ɛ,t +ɛ] (x ɛ,x +ɛ) cam+l (m + l)!, l,m=,,,..., sup t [t ɛ,t +ɛ] m t l x v H (x ɛ,x +ɛ) cam+l (m + l)!, l,m=,,,... Since initially we do not now whether the solution elongs to even L (R ) we should mention that the local well-posedness is essentially important for our argument and therefore it merely satisfies the coupled system equations in the sense of distriution. Reduction of the Prolem and Preliminary Results If a = in the system (.) (.), there is no coupling in the dispersive terms. Let us assume that a. We are interested into decouple the dispersive terms in the system (.) (.). For this, let a. We consider the associated linear system W t + AW xxx =, W(x,) = W (x) where W = [ ] [ u, A= v a a / / The eigenvalues α + and α of A are given y ( α ± = + ± ) + 4 a ]. which are distinct since >, > anda. Our assumption a guarantees that α ±. Thus we can write the system (.) (.) in a matrix form as in [8]. After we mae the change of scale u(x, t) = ũ(α / + x,t) and v(x,t) = ṽ(α / x,t), (.) otaining the coupled system of equations ũ t + α + ũ xxx + aũ ũ x + ṽ ṽ x + c(ũ ṽ) x =, x,t R, (.) ṽ t + α ṽ xxx + ã ũ ũ x + ṽ ṽ x + c(ũ ṽ) x =, (.) ũ(x, ) = ũ (x), ṽ(x,) = ṽ (x) (.4)

5 Analyticity and Smoothing Effect for the Coupled System of Equations 79 where and a,, c and ã,, c are constant. ũ(x, t) = u(α / x,t) and ṽ(x,t) = v(α / + x,t), (.5) Remar. Notice that the nonlinear terms involving the functions ũ and ṽ are not evaluated at the same point. Therefore, those terms are not local anymore. Hence we should e careful in maing the estimates (see for instance [8]). We remar additionally, that than to the property of homogeneity of the generator of dilation P,wehaveũ = P u = P ũ = ũ. Thus, for the sae of simplicity, while there is no amiguity we will drop and using the notation u, v, u,v in the system (.) (.4). For s, R define the following Bourgain spaces X s and Xs to e the completion of the Schwartz space S(R ) with respect to the norms ( ) / u X s := ( + τ ξ ) ( + ξ ) s û(ξ, τ) dξdτ R R and u X s,wherex s ={u S (R ) : u X s < } (see [6]). Let F x and F x,t e the Fourier transforms in the x and (x, t) variales respectively. The Riesz operator D x is defined y D x = F ξ ξ F x. The fractional derivative is defined y D x s := F ξ ξ s F x = F ξ ( + ξ ) s/ F x, D x,t s := F ξ,τ ξ + τ s F x,t. For, = ( + ) /, we have the following notation (i) H (R:H r (R)) = D t D x r L x,t (R ) (ii) H s (R) ={u S (R) : D x s u L (R)} (iii) H s (R) = D x s L (R). Remar. With the aove notation we have (a) u H s x (R) = ξ s û L (R) () u L t (R:H r x (R)) = ξ r û L (R ) (c) D x s u L (R) = u H s (R) (d) D t D x r u L x,t (R ) = u H t (R:H r x (R)) (e) D x,t s u L t (R:H r x (R)) = ξ r ξ + τ s û(ξ, τ) L (R ). Let L = t + x ; we introduce the generator of dilation P = t t + x x for the linear part of the coupled system (.) (.4)andthelocalized dilation operator P = t t + x x. By employing a localization argument, we loo the operator P as a vector field P = t t + x x near a fixed point (x,t ) R ( T,T), t. Since P is a directional derivative towards (x,t), we introduce another operator L = t x which plays the role of a non-tangential vector field to P.SinceP and L are linearly independent, the space and time derivative can e covered y those operators. The main reason why we choose L is ecause the corresponding variale coefficients operator L = t x can e treated via (.5) (.6) and a cut-off procedure enales us to handle the right hand side of those equations.

6 8 M.S. Alves et al. Remar. For L and P we have the following properties: (a) [L,P] LP PL= L; () LP = (P + ) L; (c) (P + ) x = x (P + ) ; (d) (P + ) x = x P ; (e) P P = PP + P x x. Notation. The summation = + + is simply areviated y = + +.,, Let P u = u,then t (P u) + x (P u) = LP u = (P + ) Lu = (P + ) ( t u + x u) [ a = (P + ) x(u ) + ] x(v ) + c x (uv) = a (P + ) x (u ) (P + ) x (v ) c(p + ) x (uv) = a x(p + ) (u ) x(p + ) (v ) c x (P + ) (uv). Noting that (P + ) u = j= ( j) j P j u. Hence Therefore B (u, u) = a x(p + ) (u ) = a = + +!!! x (u u ), B (v, v) = x(p + ) (v ) = B (u, v) = c x(p + ) (uv) = c t (P u) + x (P u) = + + = + + = a = + +!!! x (u u ) c = + +!!! x (u v )!!! x (v v ),!!! = + + x (u v ).!!! x (v v ) = B (u, u) + B (v, v) + B (u, v). (.6) Performing similar calculations as aove we otain t (P v) + x (P v) = ã = + +!!! x (u u ) = + +!!! x (v v )

7 Analyticity and Smoothing Effect for the Coupled System of Equations 8 c = + +!!! x (u v ) = C (u, u) + C (v, v) + C (u, v). The aove nonlinear terms maintain the ilinear structure lie that of the original coupled system of equations of KdV type, since Leiniz s rule can e applicale to operations of P. Now each u and v satisfies the following system of equations t u + x u = B (u, u) + B (v, v) + B (u, v) B, (.7) t v + x v = C (u, u) + C (v, v) + C (u, v) C, (.8) u (x, ) = (x x ) u (x) u (x), v (x, ) = (x x ) v (x) v (x). (.9) In order to otain a well-posedness result for the system (.7) (.9) we use Duhamel s principle and we study the following system of integral equations equivalent to the system (.7) (.9) t ψ(t)u = ψ(t)v(t)u ψ(t) V(t t )ψ T (t )B (t )dt, t ψ(t)v = ψ(t)v(t)v ψ(t) V(t t )ψ T (t )C (t )dt where V(t)= e t x is the unitary groups associated with the linear prolems and ψ(t) (R), ψ is a cut-off function such that C {, if t < ψ(t)=, if t > and ψ T (t) = ψ(t/t). The following results are going to e used several times in the rest of this paper. Lemma. ([4, 5]) Let s R, a,a (, /), (/, ) and δ<. Then for any =,,,...we have ψ δ φ X s a cδ (a a )/4( a ) φ X s, a ψ δ V(t)φ X s cδ / φ H s (R), t X ψ δ V(t t )F (t )dt cδ / F X s. s Lemma. ([]) Let s> /4,, (/, 7/) with <. Then for any,l =,,,...we have x (u v l ) X s c v X s v l X s. Lemma. ([]) Let s<, (/, 7/) and ψ = ψ(x,t) e a smooth cut-off function such that the support of ψ is in B () and ψ = on B (). We set ψ ɛ = ψ((x x )/ɛ, (t

8 8 M.S. Alves et al. t )/ɛ). Then for f X s, we have ψ ɛ f X s cɛ s 5 ψ ɛ X s + where the constant c is independent of ɛ and f. f s+ X, Lemma.4 ([]) Let P e the generator of the dilation and D x,t e an operator defined y F ξ,τ τ + ξ F x,t. We fix an aritrary point (x,t ) R ( T,T), t. Then. Suppose that (, ], r (, ] and g X r with supp g B ɛ(x,t ) and t x g, P g X r. If ɛ>is sufficiently small, then we have ) D x,t g L (R:H r (R)) c ( g X + t r x g X r + P g X (.) r where the constant c = c(x,t ; ɛ).. If g H μ (R ) with suppg B ɛ (x,t ) and t x g, P g H μ (R ). Then for small ɛ, we have D x,t μ g L (R ) c ( g H μ (R ) + t x g ) H μ (R ) + P g H μ (R ) (.) where the constant c = c(x,t ; ɛ). Lemma.5 ([]) Let s,r n/ with n/ s + r and suppose f H s (R n ) and g H r (R n ). Then for any σ<s+ r n/, we have fg H σ (R n ) and where ɛ = s + r n/ σ. fg H σ (R n ) c(ɛ) f H s (R n ) g H r (R n ), Corollary. ([]) For / << and /4 <s<, we have ψf X s c f X s, where ψ C (R ) and c is independent of f. Lemma.6 ([]) Let ψ(x) e a smooth cut-off function in C ((, )) with ψ(x)= on (, ). We set ψ ɛ = ψ(x/ɛ) for <ɛ<. Then for r, and f H r, we have { cɛ δ f H ψ ɛ f H r (R) r (R) if / r, cɛ /+r f H r (R) if r< / where δ> is an aritrary small constant and c is independent of ɛ. Proposition. Let s> /4,, (/, 7/) with <. Then for any, l =,,,...we have x (ũ ṽ l ) X s X s ṽ l X s, (.) x (ũ ṽ l ) X s X s ṽ l X s, (.) where ũ, ṽ, ũ and ṽ are defined y the change of scales (.) and (.5), from u and v.

9 Analyticity and Smoothing Effect for the Coupled System of Equations 8 Proof The proof of (.) (.) follows the same argument used in [4, 5] to prove x (u v l ) X s c u X s v l X s. It is not difficult to prove the same inequality since the only changes coming out are from the contriutions given y the constants α + and α (see for example [8, Remar.]). Throughout this paper c is a generic constant, not necessarily the same at each occasion (it will change from line to line), which depends in an increasing way on the indicated quantities. Existence of the Solutions In this section, we first solve the following (slightly general) system of equations t u + x u = B (u, u) + B (v, v) + B (u, v) B, (.) t v + x v = C (u, u) + C (v, v) + C (u, v) C, (.) u (x, ) = (x x ) u (x) u (x), v (x, ) = (x x ) v (x) v (x) (.) where B and C are as aove. Definition. Let f = (f,f,...,f ) denotes the infinity series of distriutions and define } A A (X {f s ) = (f,f,...,f ), f i X s (i =,,...)such that f AA (X s ) < + where f AA (X s ) A f X s. Similarly, for u ={u,u,...,u,...} and v ={v,v,...,v,...} we set u AA (H s (R)) respectively. A u H s (R) and v AA (H s (R)) A v H s (R) Remar. The corresponding of the solution to the coupled system of Korteweg-de Vries equations is accompanied y the following estimate (for =,,,...) P u X s ca, and P v X s ca. Theorem. Let s> /4, (/, 7/). Suppose that u, v H s (R)(=,,,...) and satisfy u AA (X s ) = A u H s (R) < + and v AA (X s ) = A v H s (R) < +.

10 84 M.S. Alves et al. Then there exist T = T( u H s (R), v H s (R)) and a unique solution u = (u,u,...) and v = (v,v,...)of the system (.) (.) with u,v C(( T,T): H s (R)) X s and A u X s (R) < + and A v X s (R) < +. Moreover, the map (u,v ) (u(t), v(t)) is Lipschitz continuous, i.e., u(t) u(t) AA (X s ) + u(t) u(t) C(( T,T):H s (R)) c(t ) u u AA (H s (R)), v(t) v(t) AA (X s ) + v(t) v(t) C(( T,T):H s (R)) c(t ) v v AA (H s (R)), for two initial conditions (u, v ) and (u,v ) in (.), and its respective solutions (u, v) and (u,v) of (.) (.). Proof For given (u,v ) A A (H s (R)) A A (H s (R)) and >/, let us define, } H R,R = {(u, v) A A (X s ) A A (X s ) : u AA (X s ) R, v AA (X s ) R where R = c u AA (H s (R)) and R = c v AA (H s (R)).ThenH R,R is a complete metric space with norm (u, v) HR,R = u AA (X s ) + v AA (X s ). Without loss of generality, we may assume R > andr >. For (u, v) H R,R,letus define the maps, t u (u, v) = ψ(t)v(t)u ψ(t) V(t t )ψ T (t )B (t )dt, t v (u, v) = ψ(t)v(t)v ψ(t) V(t t )ψ T (t )C (t )dt. We prove that maps H R,R into H R,R and it is a contraction. In fact, using Lemma., Lemma., Proposition. and the fact that u and v are not evaluated at the same point (see Remar. and proof of Proposition.)wehave u (u, v) X s = ψ(t)v(t)u X s + ψ(t) t V(t t )ψ T (t )B (t )dt X s c u H s (R) + c T μ B X s c u H s (R) + c T μ a = + +!!! u X s u X s + c T μ + c T μ c = + + = + +!!! v X s v X s!!! u X s v X s.

11 Analyticity and Smoothing Effect for the Coupled System of Equations 85 Applying a sum over we have A u (u, v) X s c + c T μ + c T μ c A u H s (R) + c T μ a A A = + + = + + c u AA (H s (R)) + c T μ a + c T μ + c T μ c A = + +!!! u X s u X s!!! v X s v X s!!! = A A =! = = A! = A!! v X s A! u X s u = X s v A A = = X s! u X s! v X s A! v X s = A! u X s = c u AA (H s (R)) + c T μ a ea u A A (X s ) + c T μ ea v A A (X s ) + c T μ ce A u AA (X s ) v AA (X s ). Hence, choosing d = max{a/,/,c} we have u (u, v) AA (X s ) c u AA (H s (R)) ] + c T μ de A [ u AA (X s) + v AA (X s) + u AA (X s ) v AA (X s ) c u AA (H s (R)) + ] c dt μ e A [ u AA (X s) + v AA (X s). (.4) In a similar way, choosing d = max{ã/, /, c} we have Choosing T μ v (u, v) AA (X s ) c v AA (H s (R)) + ] c dt μ e A [ u AA (X s) + v AA (X s). (.5) that is, ( u, v ) H R,R. max{c,c }(R +R ), we otain from (.4) and(.5), the following estimates u (u, v) AA (X s ) R and v (u, v) AA (X s ) R,

12 86 M.S. Alves et al. Now, we will show that u v : (u, v) ( u (u, v), v (u, v)) is a contraction. Let (u, v),(u,v) H R,R, then as aove we get for d = max{a/,/,c,ã/, /, c} Choosing T μ u (u, v) u (u,v) AA (X s ) ] c dt μ e A (R + R ) [ u u AA (X s ) + v v AA (X s ). 6max{c,c }(R +R ),weotain u (u, v) u (u,v) AA (X s ) 4 In a similar way [ u u AA (X s ) + v v AA (X s ) ]. v (u, v) v (u,v) AA (X s ) ] [ u u AA (X 4 s ) + v v AA (X s ). Therefore, the map u v is a contraction and we otain a unique fixed point (u, v) which solves the initial value prolem (.) (.) fort<t μ. The remainder of the proof follows a standard argument. Corollary. Let s> /4, (/, 7/). Suppose that (x x ) u,(x x ) v H s (R), with =,,,...and satisfy A u H s (R) < + and A v H s (R) < +. Then there exist T = T( u H s (R), v H s (R)) and a unique solution (u, v) of the coupled system equations KdV type (.) (.) with u, v C(( T,T): H s (R)) X s and A P u X s (R) < +, A P v X s (R) < +. Moreover, the map (u,v ) (u(t), v(t)) is Lipschitz continuous in the following sense: u(t) u(t) X s + u(t) u(t) C(( T,T):H s (R)) c(t ) v(t) v(t) X s + v(t) v(t) C(( T,T):H s (R)) c(t ) A (x x) (u u ) H s (R), A (x x) (v v ) H s (R). 4 The Main Result In this section we prove analyticity of the solution otained in the previous section. We treat the solution u P u and v P v as if it satisfies the coupled system of (.) (.)inthe classical sense. This can e justified y a proper approximation procedure. The following results are going to e used in this section. Let (x,t ) e aritrarily taen in R ( T,T),

13 Analyticity and Smoothing Effect for the Coupled System of Equations 87 t. By ψ(x,t) we denote a smooth cut-off function in C (B ()) and ψ ɛ = ψ((x x )/ɛ, (t t )/ɛ). Let ψ e a smooth cut-off function around the freezing point (x,t ) with supp ψ C (B ɛ(x,t )). Proposition 4. For the cut-off function ψ defined aove, there exist positive constant c and A such that for =,,,... ψp u L x,t (R ) ca (), ψp v L x,t (R ) ca (). Proof Using (.) with r = s, we otain D x,t ψp u L t (R:Hx s (R)) c ( ψu X s + t x (ψu ) X s + P (ψu ) X s ). (4.) Each term in (4.) is estimated separately. For first term in the right hand side we use Lemma.. Indeed, ψu X s ψu X s c ψ s + X u X s c(ψ)a, =,,,... (4.) The third term is estimated again using Corollary. P (ψu ) X s l= c(ψ) c = c! l(l )! P l ψp l u X s l= l= l=! l(l )! P l u X s! l(l )! P +l u X s A +l ( + l)! ca, (4.) for =,,,...Forthesecondterm,weuse(.) and reduce the third derivative in space to the dilation operator P. On the other hand, from the generator of dilation Pu = t t u + x x u we remar that t t u = Pu x xu. (4.4) Multiplying (.)yψt,wehave ψt t u + ψt x u = ψtb. (4.5)

14 88 M.S. Alves et al. Replacing (4.4) in(4.5) we otain ψt x u = ψpu + ψx xu + ψtb. Hence ψt x u X s = ψpu X s + ψx xu X s + ψtb X s Usingtheassumptionintheorem,wehave = F + F + F. (4.6) Similarly, we otain F = ψpu X s c ψ X s P + u X s c P + u X s ca + ( + )! ca 4. (4.7) F = ψx xu X s x(ψxu ) X s + x(ψx)u X s x(ψxv ) X s + c x (ψx) X s u X s Using Lemmas.,., and Proposition.,wehave ψx X s u X s + c x (ψx) X s u X s ( ) c ψx X s + x (ψx) X s A 5 ca 6. (4.8) F = ψtb X s c ψ X s B + B + B X s ( ) c B X s + B X s + B X s c = + +!!! u X s u X s + c + c = + +!!! u X s v c = + +!!! A 7!A 7!+c + c c c = + + A 7 = =!!! A! A 7 + A 8 ( A 7 ) + A 8! A 7 = = X s 9!A! = + + = + +!!! v X s v X s!!! A 8!A 8! A =! 8 + = = = ( A 8 )! + = + + = + +! A! 9 A 9 A A

15 Analyticity and Smoothing Effect for the Coupled System of Equations 89 ce /A 7 A 7 +ce/a 8 A 8 +c c ( e /A 7 + e /A 8 ) A +c = + + = + +!! 9 A A 9 A A, =,,,... (4.9) Hence, replacing (4.7), (4.8) and(4.9) in(4.6) we otain that there exist positive constants c, A 9, A and A, such that ψt x u X s ca +c = + +! 9 A A, =,,,... (4.) On the other hand, using x (ψf ) = ψ x f + x ( xψf ) x ( x ψf ) + x ψf we have t x (ψu ) X s tψ x u X s + x (t xψu ) X s + x (t x ψu ) X s Using Lemma. and Proposition., we otain + t x ψu X s. (4.) x (t xψu ) X s x (t x ψu ) X s c t x ψ X s u X s ca, (4.) x (t x ψu ) X s x (t x ψu ) X s c t x ψ X s u X s t x ψu X s ca, (4.) c D x,t / t x ψ X s + u X s c u X s ca. (4.4) Hence, replacing (4.), (4.), (4.) and(4.4) in(4.) we otain that there exist constants c and A 4 such that t x (ψu ) X s ca 4 +c = + +! 9 A A, =,,,... (4.5) Therefore, replacing (4.), (4.) and(4.5) in(4.) we otain that there exist constants c and A 5 such that D x,t ψu L t (R:Hx s (R)) ca 5 +c = + +! 9 A A In a similar way, we otain that there exist constants c and A 6 such that, =,,,... (4.6)

16 9 M.S. Alves et al. D x,t ψv L t (R:Hx s (R)) ca 6 +c Adding (4.6) and(4.7)wehave = + +! 9 A A, =,,,... (4.7) D x,t ψu L t (R:H s x ca 5 +ca 6 +c c(a 5 + A 6 )+c ca 7 +c (R)) + D x,t ψv L t (R:Hx s (R)) = + + = + + = + +!!! A 9 A We estimate now the last term in the right hand side of (4.8) = + +! A 9 A Replacing (4.9) in(4.8) we otain = m m= j= A A A A A 9 A 9 A (m j)! (m j) A j 9 A m m m= j= m m= j= m= j= A m (m j)!. (4.8) ( A9 ) j ( A [ ( A ) j ( ) ] 9 4 m + 4 A j! ( A 9 4 ) j j! m m= j= ( A 9 4 ) j j! + A m m= j= + A m= j= ) m ( 4 ) m A m! m! m ( 4 ) m A m! e A 9 /4 A +e4/a A ca. (4.9) D x,t ψu L t (R:H s x (R)) + D x,t ψv L t (R:Hx s (R)) ca 7 +ca 9 () ca 7 () + ca 9 () ca () (4.) and the result follows.

17 Analyticity and Smoothing Effect for the Coupled System of Equations 9 Remar 4.. For simplicity, we only illustrate the conclusion for the case s / δ with = /+ δ/ (for small δ>) and the case s = /4 + δ and = 7/ δ/. If s = / δ with = / + δ/, the initial data can involve Dirac s delta measure δ and the latter is the critical case of the local well-posedness.. The following inequality is simple to verify in oth cases, ψu L x,t (R ) D x (ψu ) L t (R:Hx s (R)) c D x,t (ψu ) L t (R:Hx s (R)). Proposition 4. Under the same assumptions as in Proposition 4., there exist positive constants c and A such that for =,,,... ψp u H 7/ (R ) ca () and ψp v H 7/ (R ) ca (). Proof We apply Lemma.4 to ψu ψp u with = andr =. D x,t ψp u L (R:L x (R)) ( ) c ψu L (R:L x (R)) + t x (ψu ) L (R:L x (R)) + P (ψu ) L (R:L x (R)). (4.) Therefore, if we wish to estimate the second term in the right hand side of (4.) with the aid of (.7) ψt x u = ψpu + ψx xu + tψb it is necessary to estimate ψu L t (R:Hx (R)) which is not yet otained. Hence, we start from the lower regularity setting, i.e., apply (.) in Lemma.4 to ψu with μ = /. Let ψ e a smaller size of smooth cut-off function with ψ ψ and ψ = around (x,t ). Applying (.)aψu = ψp u with μ = / wehave D x,t ψ P u H 5/ (R ) c D x,t ψ P u L (R ) c ( ψ u H 5/ (R ) + t x (ψ u ) H 5/ (R ) + P (ψ u ) H 5/ (R )). (4.) The first term on the right hand side of (4.) has already een estimated. For the third term we have P (ψ u ) H 5/ (R ) P (ψ u ) L x,t (R ) = c l= l=! l!( l)! P l ψ P l u L x,t (R )! l!( l)! P l ψ L x,t (R ) P l u L x,t (R ) l=! l!( l)! P +l u L x,t (R )

18 9 M.S. Alves et al. c l= A +l ca ca (). (4.) For the second term on the right side hand we use the same idea of the aove remar, using the dilation operator P. Indeed, t x (ψ u ) H 5/ (R ) ψ t x u H 5/ (R ) + x (t xψ u ) H 5/ (R ) The last three term are ounded y the following: c + x (t x ψ u ) H 5/ (R ) + t x ψ u H 5/ (R ). (4.4) x l ψ L x,t (R ) ψu L x,t (R ) ca ca (). (4.5) l= On the other hand, using ψ t x u H 5/ (R ) ψ Pu L (R:L x (R)) + xψ x u H 5/ (R ) + tψ B H 5/ (R ) Thus = F + F + F. (4.6) F c ψ L x,t (R ) ψp + u L x,t (R ) c ψp+ v L x,t (R ) ca + 4 ( + )! ca 5 ca 5 (), F xψ x v L (R:Hx (R)) x (xψ v ) L (R:Hx (R)) + x(xψ )ψv L (R:Hx xψ v L x,t (R ) + x(xψ ) L x,t (R ) ψv L x,t (R ) ( ) xψ L x,t (R ) + x (xψ ) L x,t (R ) ψv L x,t (R ) ca 6 ca 6 (). Using Lemma.5 (case σ = 5/, s = 5, r = 5/) and replacing B y (.6), we have F = tψ B H 5/ (R ) c ψ H 5 (R ) ψ B x H 5/ (R ) c a + c + c c c a = + + = + + = + + = + +!!! ψu ψu H / (R )!!! ψv ψv H / (R )!!! ψu ψv H / (R )!!! ψu L (R ) ψu L (R ) (R))

19 Analyticity and Smoothing Effect for the Coupled System of Equations 9 + c + c c c ( + c a ( a = c + c = + + = + +!!! ψv L (R ) ψv L (R )!!! ψu L (R ) ψv L (R ) = + +! A = + + A 7 = = = + + A! 9 A )! A 7 + A! 9 A ) = + + A 8 = = a c e/a 7 A 7 ( + )!+c e/a 8 A 8 ( + )! + c c = + + A! 9 A Replacing (4.5), (4.7) and(4.4) in(4.6) we otain ψ t x u H 5/ (R ) c A +c c Replacing (4.5) and(4.8) in(4.4) t x (ψ u ) H 5/ (R ) c A +c c = + + = + + +! A 8! A 8. (4.7) A! 9 A Now replacing (4.) and(4.9) in(4.) we otain A! 9 A D x,t ψu H 5/ (R ) c 4 A +c c for =,,,... In particular ψu H / (R ) c 5 A 4 +c c, =,,,... (4.8), =,,,... (4.9) = + + = + + A! 9 A, A! 9 A,

20 94 M.S. Alves et al. for =,,,... Using similar argument as aove for D x,t ψp u H / (R ) with μ = / in(.) and replacing the support of the cut-off function ψ ɛ we otain ψu H / (R ) c 5 A 4 +c c for =,,,... In a similar way we have ψv H / (R ) c 5 A 5 +c c = + + = + + A! 9 A A! 9 A, (4.), (4.) for =,,,... Adding (4.) with (4.) and performing straightforward calculations as (4.9) we otain ψu H / (R ) + ψv H / (R ) CA (), =,,,... Remar 4. To otain the estimate for ψp u H 7/ (R ) and ψp v H 7/ (R ) we repeat the aove method with μ = 7/. Proposition 4. Suppose that ψu H 7/ (R ) ca () and ψv H 7/ (R ) ca () for =,,,... Then we have sup (t / x )P u H (x ɛ,x +ɛ) c A +l [( + l)!], t [t ɛ,t +ɛ] sup (t / x )P v H (x ɛ,x +ɛ) c A +l 4 [( + l)!] t [t ɛ,t +ɛ] for,l =,,,... where ɛ> is so small that ψ near I = (x ɛ,x + ɛ) (t ɛ,t + ɛ). Proof Let I t = (t ɛ,t + ɛ) and I x = (x ɛ,x + ɛ), thenwehavei = I x I t.for any fixed t I x,letl = t / x. We show that for some positive constants c and A the following inequality holds L l P u H x (I x ) ca+l [( + l)!],, l =,,,... (4.) Induction over l. By the trace theorem, we have L l P u H x (I x ) t l/ l x P u(t) H x (I x ) (t + ɛ) l/ x l P u H / (I x I t ) (t + ɛ) l/ P u H 7/ (I x I t ) (t + ɛ) l/ ψp u H 7/ (R ) (t + ɛ) l/ c A

21 Analyticity and Smoothing Effect for the Coupled System of Equations 95 (t + ɛ) l/ c A +l ( + l) (t + ɛ) l/ c A +l [( + l)!], where we tae c = (t + ɛ) l/ c and A = max{,a }. Hence, in the case l =,,, it is easytoshowthat(4.) follows directly from the assumption. Now, we assume that (4.)istrueforl. Applying P to the (.) equation we have t (P u) + x (P u) = LP u = (P + ) Lu = (P + ) ( t u + x [ u) a = (P + ) x(u ) + ] x(v ) + c x (uv) = a (P + ) x (u ) (P + ) x (v ) c(p + ) x (uv) = a x(p + ) (u ) x(p + ) (v ) c x (P + ) (uv) such that t t (P u) + t x (P u) = a t x(p + ) (u ) t x(p + ) (v ) ct x (P + ) (uv). (4.) Moreover, P = t t + x x.then t t (P u) = P + u x x(p u). (4.4) Replacing (4.4) in(4.) we otain L P u = t x (P u) = P + u + x x(p u) a t x(p + ) (u ) t x(p + ) (v ) ct x (P + ) (uv). Hence, applying L l we have L l+ P u H x (I x ) = Ll L P u H x (I x ) Ll P + u H x (I x ) + Ll x x (P u) H x (I x ) + a tll x (P + ) (u ) H x (I x ) + tll x (P + ) (v ) H x (I x ) + c tl l x (P + ) (uv) H x (I x ) = F + F + F + F 4 + F 5.

22 96 M.S. Alves et al. Using the induction assumption, we otain F c A +l+ 4 ( + l + )!. (4.5) We estimate the term L l (x x ) for l. Let r = l, then we estimate L r (x x ) for r. r x (x x) = r ( ) r x r (x) x ( x). (4.6) Replacing in (4.6), we otain { if = r, x r (x) = if r x r (x x) = r x r ( x ) + x x r ( x) = r x r + x x( x r ) = (l ) x (l ) + x x ( x (l ) ), thus L l (x x ) = x x L l + (l )L l for l. For F we have F x x L l P u H x (I x ) + (l ) Ll P u H x (I x ) xt / L l P u H x (I x ) + (l ) Ll P u H x (I x ) c(t ɛ)( x +ɛ + ) L l P u H x (I x ) + (l ) Ll P u H x (I x ) (t ɛ) / ( x + ɛ + )c A +l 4 ( + l )!+c A +l 4 (l )( + l )! c A +l+ 4 ( + l + )! (4.7) where we tae A 4 larger than (t ɛ) / ( x +ɛ + ) and. Using that L = t / x,and then tl l x = tt (l )/ x (l ) x = tt / t (l )/ x (l ) = t / L l, we have F = a t / L l (P + ) (u ) H x (I x ) a (t + ɛ) / (l )! l = + +!l!!!! Using the induction assumption l =l +l c L l P u H x (I x ) Ll P u H x (I x ). F a (t + ɛ) / (l + )! l!! l =l +l = + + c c (l + )! A +l 4 l!! a (t + ɛ) / c c (l + )!A+l 4 (l )!! l =l +l = + +!

23 Analyticity and Smoothing Effect for the Coupled System of Equations 97 Using that l =l +l (l + )! (l + )! (l )! l!! l!! (l + )!. = + +! (l + )! (l + )! l!! l!! (l )! (l + )! e (l + )! we otain F (t + ɛ) / c c e (l + )!A +l 4 c A +l+ 4 ( + l + )! (4.8) wherewetaea 4 larger than (t ɛ) / c c e, and. In a similar way F 4 c A +l+ 5 ( + l + )! (4.9) wherewetaea 5 larger than (t ɛ) / c 4 c e, and. Finally, in a similar way F 5 c 6A +l+ 6 ( + l + )! (4.4) where we tae A 6 larger than (t ɛ) / c 6 c 5 e, and. Therefore, from (4.5), (4.7), (4.8), (4.9)and(4.4) we otain L l+ P u H x (I x ) c 7A +l+ 7 ( + l + )!. In a similar way, we otain L l+ P v H x (I x ) c 7A +l+ 7 ( + l + )! and the result follows. Proposition 4.4 Suppose that there exists positive constants c,c and A 4,A 5 such that sup x l P u H x (x ɛ,x +ɛ) c A +l 4 [( + l)!], t [t ɛ,t +ɛ] sup t [t ɛ,t +ɛ] for,l =,,,... Then we have respectively x l P v H x (x ɛ,x +ɛ) c A +l 5 [( + l)!] sup t m x l u Hx (x ɛ,x +ɛ) c A m+l 6 [(m + l)!], t [t ɛ,t +ɛ] sup t [t ɛ,t +ɛ] t m x l v Hx (x ɛ,x +ɛ) c 4A m+l 7 [(m + l)!] for m, l =,,,... where c,c 4 and A 6,A 7 only depend on c,c and A 4, A 5, respectively and ɛ, (x,t ). Proof Using the idea of Proposition 4.,wefixt I x. First we show that for some positive constants c, A 6 and B 6 (x x ) m x l P v H x (I x ) c A +m+l 6 B6 m ( + m + l)!,,m,l=,,,.... (4.4)

24 98 M.S. Alves et al. We use induction. Suppose that (4.4) istrueform. (x x ) m+ l x P v H x (I x ) = (x x )(x x ) m l x P v H x (I x ) ( x +ɛ + ) (x x + I) m x l+ P v H x (I x ) m ( ) m c( x,ɛ) (x x ) j x l+ P v j H x (I x ) c m j= ( m j j= ) c A +l+j+ 6 B j 6 ( + l + j + )! c A +l+m+ 6 B m 6 ( + l + m + )! m j= e A 6B 6 c A +l+m+ 6 B6 m ( + l + m + )!, (A 6 B 6 ) (m j) (m j)! m! ( + l + j + )! j! ( + l + m + )! where we tae B 6 so large that B 6 max{ x +ɛ +, }. We show that for some positive constants c 4, A 7 such that (t t ) m x l u Hx (I x ) c 4A l+m 7 (l + m)!, l,m=,,,... Using that t t = (P x x), we otain (t t ) m l x u H x (I x ) = m (P x x ) m l x u H x (I x ) m m! j m=j +j!j! (x x) j P j x l u Hx (I x ) m m! j m=j +j!j! (x x) j x l (P l)j u H x (I x ) m m! j m=j +j +j!j!j! lj (x x ) j x l P j u H x (I x ), where we replacing j into j + j. Now, using the induction hypothesis we have (with B 7 A 6 B 6 ) (t t ) m l x u H x (I x ) m m! j m=j +j +j!j!j! lj c B j +j +l 7 (j + j + l)! m c B m+l 7 (m + l)! B j 7 m=j +j +j Oserving that l j (j +j +l)! (m+l)!, we otain in (4.4) m! (j j!j!j! lj + j + l)!. (4.4) (m + l)!

25 Analyticity and Smoothing Effect for the Coupled System of Equations 99 (t t ) m l x u H x (I x ) m c ( + B 7 )m B l+m 7 (l + m)! c 4 A l+m 7 (l + m)! where we tae A 7 = max{b 7, B 7 ( + B 7 )}. We show that for some positive constants c 4, A 8 and B 8 we have Induction in m. (t t ) j m t l x u H x (I x ) c 4A j+m+l 8 B 8 (j + m + l)!, j,l,m=,,,... (4.4) (t t ) j m+ t l x u H x (I x ) t(t t I) m m t l x u H x (I x ) Using the induction hypothesis = t t t (t t I) j t m x l u Hx (I x ) j ( ) j (t ɛ) (t t ) j+ t m x l u Hx (I x ). j = (t t ) j t m+ x l u Hx (I x ) j ( ) j (t ɛ) c 4 A j +l+m+ 8 B8 m (j + l + m + )! j = j = c 4 (t ɛ) A j +l+m+ 8 B8 m (j + l + m + )! j A (j j ( ) 8 j (j + m + l + )!(j j )! (j j )! (j + m + l + )! j = j = c 4 (t ɛ) e A 8 A j+l+m+ 8 B8 m (j + l + m + )! c 4 A j+l+m+ 8 B8 m (j + l + m + )! wherewetaeb 8 larger than (t ɛ) e A 8. Finally, we choose j = in(4.4) andtae c = c 4 and A 5 = A 8 B 8. The result of analyticity follows. Proof of the Main Theorem The theorem is a direct consequence of Propositions 4., 4., 4. and 4.4. We note that the result of Theorem. is otained for the scaled solutions (ũ, ṽ) of (.) (.4). However, the result of analyticity remains also valid for the solution (u, v) of the coupled system (.) (.). Acnowledgements M.S. has een supported y Fondecyt (Project #7694), FONDAP and BASAL projects CMM, Universidad de Chile, and CI MA, Universidad de Concepción. This wor was done while the fourth author was visiting the National Laoratory for Scientific Computation (LNCC/MCT) Petrópolis- Brazil and the Universidade Federal de Viçosa-Brazil. This research was partially supported y PROSUL Project. j References. Alves, M., Villagrán, O. Vera: Smoothing properties for a coupled system of nonlinear evolution dispersive equations. Indag. Math. (), 85 7 (9)

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