Smoothing properties for a Hirota-Satsuma systems

We study local existence and smoothing properties for the initial value problem associated to HirotaSatsuma systems that describes an interaction of two long waves with different dispersion relations.


Introduction
This paper is concerned with gain in regularity of solutions of the Hirota-Satsuma system where x ∈ R, t ∈ R and u = u(x, t), v = v(x, t) are real unknown functions.a and b are real constants with b > 0. In equation (1.1), 2 b v v x acts as a force term on the Korteweg-de Vries(KdV) wave system with the linear dispersion relation ω = a κ 3 .This system was introduced by Hirota and Satsuma [19] to describe and interaction of two long waves with different dispersion relations.If there is no effect of one of the long waves on the other, the latter obeys the ordinary KdV equation.They showed that for all values of a an b this system possesses three conservation laws.Indeed (1.5) They further showed that for all values of b, but only a = 1 2 , the system possesses two further conservation laws (1.6) (1.7) The system (1.1)- (1.4) has been studied by several authors, see [8,19,20] and the references there.In 1986, N. Hayashi et al. [16] showed that for the nonlinear Schrödinger equation (NLS): i u t + u xx = λ |u| p−1 u, (x, t) ∈ R × R with initial condition u(x, 0) = u 0 (x), x ∈ R and a certain assumption on λ and p, all solutions of finite energy are smooth for t = 0 provided the initial functions in H 1 (R)(or on L 2 (R)) decay sufficiently rapidly as |x| → ∞.The main tool is the operator J defined by Ju = e i x 2 /4 t (2 i t) ∂ x (e − i x 2 /4 t u) = (x + 2 i t ∂ x )u which has the remarkable property that it commutes with the operator L defined by L = (i ∂ t + ∂ 2 x ), namely LJ − JL = [L, J] = 0.For the Korteweg-de Vries type equation (KdV), Saut and Temam [29] remarked that a solution u cannot gain or lose regularity.They showed that if u(x, 0) = u 0 (x) ∈ H s (R) for s ≥ 2, then u( • , t) ∈ H s (R) for all t > 0. The same result was obtained independently by Bona and Scott [4] though a different method.For the KdV equation on the line, Kato [22] motivated by work of Cohen [9] showed that if u(x, 0) = u 0 (x) ∈ L 2 b ≡ H 2 (R) ∩ L 2 (e bx dx)(b > 0) then the solution u(x, t) of the KdV equation becomes C ∞ for all t > 0. A main ingredient in the proof was the fact that formally the semi-group S(t) = e − ∂ 3 x in L 2 b (R) is equivalent to S b (t) = e − t (∂x−b) 3 in L 2 (R) when t > 0. One would be inclined to believe that this was a special property of the KdV equation.This is not however the case.The effect is due to the dispersive nature of the linear part of the equation.Kruzkov and Faminskii [26] proved that u(x, 0) = u 0 (x) ∈ L 2 (R) such that x α u 0 (x) ∈ L 2 ((0, +∞)) the weak solution of the KdV equation has l-continuous space derivatives for all t > 0 if l < 2 α.The proof of this result is based on the asymptotic behavior of the Airy function and its derivatives, and on the smoothing effect of the KdV equation which was found in [22,26].While the proof of Kato appears to depend on special a priori estimates, some of this mystery has been resolved by the result of local gain of finite regularity for various others linear and nonlinear dispersive equation due to Constantin and Saut [13], Ginibre and Velo [15] and others.However, all of them require growth conditions on the nonlinear term.In 1992, W. Craig et al. [12] proved for fully nonlinear KdV equation u t + f (u xxx , u xx , u x , u, x, t) = 0 and certain additional assumption over f that C ∞ solutions u(x, t) are obtained for all t > 0 if the initial data u 0 (x) decays faster than polynomially on R + = {x ∈ R : x > 0} and has certain initial Sobolev regularity.Following with this idea, in 2001, O. Vera and G. Perla Menzala [33,34] proved that the solutions of the initial value problem (P ) are locally smooth due to the dispersive of the coupled system of equations of Korteweg -de Vries type (P ) where u = u(x, t), v = v(x, t) are real-valued functions of the variables x and t and a 1 , a 2 , a 3 , b 1 , b 2 are real constants with b 1 > 0 and b 2 > 0. The original coupled system is where u = u(x, t), v = v(x, t) are real-valued functions of the variables x and t and a 1 , a 2 , a 3 , b 1 , b 2 are real constants with b 1 > 0 and b 2 > 0. The power p is an integer larger than or equal to one.The system ( P ) has the structure of a pair of Korteweg -de Vries equations coupled through both dispersive and nonlinear effects.In the case p = 1, system ( P ) was derived by Gear and Grimshaw in 1984 [14] as a model to describe the strong interaction of weakly nonlinear, long waves.Mathematical results on the system ( P ) were given by J. Bona et al. [3].They proved that ( P ) is globally well posed in The system ( P ) has been intensively studied by several authors.See [2,3] and the references therein.We have the following conservation laws The time-invariance of the functionals φ 1 and φ 2 expresses the property that the mass of each mode separately is conserved during interaction, while that of φ 3 is a expression of the conservation of energy EJQTDE, 2011 No. 61, p. 2 for the system of two models taken as a whole.Solutions of ( P ) satisfy an additional conservation law which is revealed by the time-invariance of the functional The functional φ 4 is a Hamiltonian for the system ( P ) and if b 2 a 2 3 < 1, φ 4 will be seen to provide an a priori estimate for the solutions (u, v) of ( P ) in the space H 1 (R) × H 1 (R).Furthermore, the linearization of ( P ) about the rest state can be reduced to two, linear Korteweg -de Vries equations by a process of diagonalization.Using this remark and the smoothing properties (in both the temporal and spatial variables) for the linear Korteweg -de Vries derived by Kato [22,23], Kenig, Ponce and Vega [24,25] it will be shown that ( P ) is locally well-posed in H s (R) × H s (R) for any s ≥ 1 whenever √ b 2 a 3 = 1.Indeed, all this appears in the following Theorem: Consider the system ( P ) together with these initial conditions.Let p ≥ 1, p be an integer and This result was improved by J. Marshall et al. [1] They proved that the system ( P )(with p = 1), is globally well-posed in This kind of dispersive problem exhibits the interesting phenomenon of dispersive smoothing, that is, If the initial data belong to a certain Sobolev space and has a good behavior as |x| → +∞, then the solutions in any time t = 0 are smoother than the initial data.Our aim in this paper, is to study gain in regularity for the equation (1.1)- (1.4).Specifically, we prove conditions on (1.1)-(1.4)for which initial data (u 0 , v 0 ) possessing sufficient decay at infinity and minimal amount of regularity will lead to a unique solution (u(t), where T is the existence time of the solution.This paper is organized as follows: Section 2 outlines briefly the notation and terminology to be used subsequently.Section 3 we prove the main inequality.Section 4 we prove an important a priori estimate.Section 5 we prove a basic-local-in-time existence and uniqueness theorem.Section 6 we develop a series of estimates for solutions of equations (1.1)-(1.4) in weighted Sobolev norms.These provide a starting point for the a priori gain of regularity.In section 7 we prove the following theorem: Theorem 1.2 (Main Theorem) Let T > 0, a < 0 and (u, v) be a solution of (1.1)- (1.4) for all 0 ≤ l ≤ L − 1 and all σ > 0 where the weight classes will be defined in Section 2.
Example 2.4 Let Definition 2.5 Let N be a positive integer.By H N (W σ, i, k ) we denote the Sobolev space on R with a weight; that is, with the norm for any ξ ∈ W σ i k and 0 < t < T.Even though the norm depends on ξ, all such choices lead to equivalent norms.
Lemma 2.7 (See [7]).For ξ ∈ W σ i 0 and σ ≥ 0, i ≥ 0, there exists a constant C > 0 such that, for Lemma 2.8 (The Gagliardo-Nirenberg inequality).Let q, r be any real numbers satisfying 1 ≤ q, r ≤ ∞ and let j and m be a nonnegative integers such that j ≤ m.Then where q for all a in the interval j m ≤ a ≤ 1, and M is a positive constant depending only m, j, q, r and a. Definition 2.9 By L 2 ([0, T ] : H N (W σ i k )) we denote the space of functions v(x, t) with the norm(N positive integer) EJQTDE, 2011 No. 61, p. 4 Remark 2.10 The usual Sobolev space is H N (R) = H N (W 0 0 0 ) without a weight.

Remark 2.11
We shall derive the a priori estimates assuming that the solution is C ∞ , bounded as x → − ∞, and rapidly decreasing as x → + ∞, together with all of its derivatives.
Considering the above notation, the Hirota-Satsuma system can be written as where x ∈ R, t ∈ R and u = u(x, t), v = v(x, t) are real unknown functions.b and a are real constants with b > 0.
Throughout this paper C is a generic constant, not necessarily the same at each occasion(it will change from line to line), which depend in an increasing way on the indicated quantities.In this part we only consider the case t > 0. The case t < 0 can be treated analogously.

Main Inequality
Lemma 3.1 Let (u, v) be a solution to (2.8)-(2.9)with enough Sobolev regularity(for instance, where Proof.Differentiating (2.8) α-times(for α ≥ 0) over x ∈ R leads to Each term is calculated separately, integrating by parts Using Leibniz's Formula, we have Hence replacing in (3.8) and performing straightforward calculus we have Differentiating (2.9) α-times of (for α ≥ 0) over x ∈ R leads to Multiply this equation by 2 ξ v α and integrate over x ∈ R we have Performing straightforward calculations as above we obtain 1 6 Adding (3.9) and (3.12) we have We take β = α in (3.14) we obtain the lemma follows.
Lemma 3.2 For µ 1 , µ 2 ∈ W σ i k an arbitrary weight functions and a < 0, there exist ξ 1 , ξ 2 ∈ W σ i+1 k respectively such that Indeed, we have The expression R α in the inequality of Lemma 3.1 is a sum of terms of the form where 1 ≤ ν 1 ≤ ν 2 ≤ α and Proof.The result follows using (3.6).

An a priori estimate
We show now a fundamental a priori estimate used for a basic local-in-time existence theorem.We construct a mapping Z : with the following property: Given where s and C 0 > 0 are constants.This property tells us that Z : To guarantee this property, we will appeal to an a priori estimate which is the main object of this section.
Differentiating (2.8) and (2.9) respectively two times leads to The equations (4.3), (4.4) are linearized equations by substituting a new variable θ and φ in each coefficient: Equations (4.5) and (4.6) are linear equations at each iteration which can be solved in any interval of time in which the coefficients are defined.These equations have the form We consider the following Lemma to help us to set up the iteration scheme.
and a < 0. Then there exists a unique solution of (4.7), (4.8)where The solution is defined in any time interval in which the coefficients are defined.
Proof.From equations (4.7)-(4.8)we have where Let T > 0 be arbitrary and M > 0 a constant.Define L = 2 ξ (∂ t − A ∧∂ 5 − B 1, 2 ∧∂ 3 ).Then in (4.9) we have LW = 2 ξ C (0) .We consider the bilinear form We have Each term is treated separately integrating by parts.The first two terms we have The other terms are calculated the same form Multiplying (4.10) for e − M t , and integrate in time t for t ∈ [0, T ] and U = (w, z) ∈ D.
EJQTDE, 2011 No. 61, p. 9 w(x, T ) = 0 and z(x, T ) = 0}.The same form for L * the formal adjoint of L we show that From (4.11) we have that L * is one to one.Therefore L * W, L * W is an inner product on D * .Denote by X the completion of D * with respect to this inner product.By the Riesz representation Theorem, there exists a unique solution V ∈ X, such that for any where we used that ξ C (0) ∈ X.
Remark 4.1 To obtain higher regularity of the solution, we repeat the proof with higher derivatives included in the inner product.It is a standard approximation procedure to obtain a result for general initial data.
Proof.We begin by applying ∂ to (4.5), our equation become Each term in (4.14) is treated separately.The first two terms yield EJQTDE, 2011 No. 61, p. 11 The other terms in (4.14) are treated the similar form, using integrating by parts.

Uniqueness and Existence of a Local Solution
In this section, we study the uniqueness and the existence of local strong solutions in the Sobolev space H N (R) for N ≥ 3 for problem (2.8), (2.9).To establish the existence of strong solutions for (2.8), (2.9) we use the a priori estimate together with an approximation procedure.
We suppose that there is at least one local strong solution of (2.8), (2.9) in the interval [0, T ].Then there is at most one strong solution ), (2.9) with initial data u(x, 0) = u 0 (x) and v(x, 0) = v 0 (x).
EJQTDE, 2011 No. 61, p. 16 hence in (5.3) we have where using that a < 0 we obtain (5.4) The difference (v − v ′ ) satisfies (5.5) Multiplying (5.5) by 2 ξ (v − v ′ ), integrating over x ∈ R and performing the similar calculations our equation becomes No. 61, p. 17 where Adding (5.4) with (5.7) and using straightforward calculus it follows that (ξ ∈ W 0 i 0 ) for some positive constant c.Using that u(x, 0) − u ′ (x, 0) ≡ 0 and v(x, 0) − v ′ (x, 0) ≡ 0, and Gronwall's inequality it follows that We conclude that u ≡ u ′ and v ≡ v ′ .This proves the uniqueness of the solution.We construct the mapping where the initial condition is given by u (0) = u 0 (x), v (0) = v 0 (x) and the first approximation is given by u is in a place of v in the equations (4.5), (4.6) which are the solution of the equations (4.5), (4.6).By Lemma 4.1, (u (n) , v (n) ) exists and is unique in C((0, +∞) : A choice of C 0 and the use of the a priori estimate in Section 4 show that Z : Theorem 5.2 (Local solution).Let a < 0 and N an integer ≥ 3.
be the maximum time such that: Integrating (5.12) over [0, t] we have that for 0 ≤ t ≤ T (n) 0 and j = 0, 1.
does not approach 0. On the contrary, assume that T as n → +∞, we have which is a contradiction.Consequently T (n) 0 → 0. Choosing T = T (c, c ′ ) sufficiently small, and T not depending on n, one concludes that for 0 ≤ t ≤ T. This show that T (n) 0 ≥ T. Hence, from (5.13) we imply that there exist subsequences Claim.u = ∧w and v = ∧z are solutions.
In the linearized equation (5.8) we have ∧w EJQTDE, 2011 No. 61, p. 19 Let (u j , v j ) be the solution to (2.8)-(2.9)with u j (x, 0) = u j 0 (x) and v j (x, 0) = v j 0 (x).According to the above argument, there exist T which is independent of n but depending on sup j u j 0 and sup j v j 0 such that u j , v j exists on [0, T ] and a subsequence As a consequence of Theorem 5.1 and 5.2 and its proof, one obtains the following result.

Persistence Theorem
As a starting point for the a priori gain of regularity results that will be discussed in the next section, we need to develop some estimates for solutions of (2.8), (2.9) in weighted Sobolev norms.The existence of these weighted estimates is often called the persistence of a property of the initial data (u 0 , v 0 ).We show that if (u The time interval of that persistence is at least as long as the interval guaranteed by the existence Theorem 5.2. Theorem 6.1 (Persistence).Let i ≥ 1, L ≥ 3 be non negative integers and 0 < T < +∞.Assume that (u 0 (x), v 0 (x)) = (u(x, 0), v(x, 0)) ∈ H 3 (R) × H 3 (R) and a < 0.
where σ is arbitrary, We derive formally some a priori estimate for the solution where the bound, involves only the norms of (u, v) in L ∞ ([0, T ] : ) and the norm of u 0 (x), v 0 (x) in H 3 (W 0 i 0 ).We do this by approximating (u(x, t), v(x, t)) by smooth solutions, and weight functions by smooth bounded functions.By Theorem 5.2, we have EJQTDE, 2011 No. 61, p. 22 In particular To obtain (6.1)-(6.2) and (6.3) there are two ways of approximations.We approximate general solutions by smooth solutions, and we approximate general weight functions by bounded weight functions.The first of these procedure has already been discussed, so we shall concentrate on the second.Given a smooth weight function µ 1 (x) ∈ W σ, i−1, 0 with σ > 0, we take a sequence µ δ1 (x) of smooth bounded weight functions approximating µ 1 (x) from below, uniformly on any half line (−∞, c).Similarly, given a smooth weight function µ 2 (x) ∈ W σ, i−1, 0 with σ > 0, we take a sequence µ δ2 (x) of smooth bounded weight functions approximating µ 2 (x) from below, uniformly on any half line (−∞, c).Define the weight functions for the αth induction step as x −∞ µ δ2 (y, t) dy (6.4) then the ξ δ1 and ξ δ2 are bounded weight functions which approximate a desired weight functions ξ 1 , ξ 2 ∈ W 0 i 0 respectively from below, uniformly on a compact set.
For α = 0 (simple case), multiplying (2.8) by 2 ξ δ u and integrating over x ∈ R we have Each term is treated separately.In the first term we have For the others terms, using integration by parts, we have 12 Replacing in (6.5), we obtain We multiply (2.9) by 2 ξ δ v and integrating over Each term is treated separately.Performing straightforward calculations as above we obtain Adding (6.6) and (6.8) we have We apply Gronwall's lemma to conclude for 0 ≤ t ≤ T and C not depending on δ > 0, the weighted estimate remains true for δ → 0. Now, we assume that the result is true for (α − 1) and we prove that it is true for α.To prove this, we start from the main inequality (3.1) where Using (2.7) and the Gagliardo-Nirenberg in the first term of the right side we obtain According to (3.17), R R α dx contains a term of R ξ δ u ν1 u ν1 u α dx (6.11) the other terms are estimate the same form.If ν 2 ≤ α − 2, using integrating by parts and the Hölder inequality by the induction hypothesis we have is bounded.If ν 1 = ν 2 = α − 1, then by (3.18) we have α = 3 and The other terms.Using those estimate and straightforward calculus Applying the Gronwall's argument, we obtain for 0 ≤ t ≤ T, where C 0 and C 1 are independent δ such that letting the parameter δ → 0 the desired estimates (6.2)-(6.3)are obtained.
EJQTDE, 2011 No. 61, p. 25 In this section we state and prove our main theorem, which states that if the initial data (u 0 , v 0 ) decays faster than polynomially on R = {x ∈ R : x > 0} and possesses certain initial Sobolev regularity, then the solution (u(x, t), v(x, t)) ∈ C ∞ (R) × C ∞ (R) for all t > 0. For the main theorem, we take 4 ≤ α ≤ L + 2. For α ≤ L + 2, we take Lemma 7.1 (Estimate of Error Terms).If 4 ≤ α ≤ L + 2 and the weight functions are chosen as in (7.1), then where C depends only on the norms of u, v in Proof.We must estimate R, θ 1 and θ 2 .We begin with a term of the form ξ u ν1 u ν2 u α (7.4) (the other terms are calculated the similar form) assuming that ν 1 ≤ α − 2. By the induction hypothesis, Then in the term of the form ξu ν1 u ν2 u α we estimate u ν1 using (7.5).We estimate u ν2 and u α using the weight L 2 bounds and the same with ν p replaced by α.It is sufficient to check the powers of t, and the powers x as x → +∞ and the exponentials x as x → −∞.For x > 1.In the term (7.4), the factor ξ contributes according to (7.1)-(7.2) Claim M ≥ 0 is large enough, that the extra power of t can be bounded by a constant. Proof.

M
Claim T ≤ 0 so that the extra power x T can be bounded as x → +∞.Proof.which is bounded.For x < 1 the estimate is similar except for the exponential weight.This completes the estimate of R. Now we estimate the terms θ 1 u 2 α and θ 2 v 2 α where θ 1 and θ 2 are given in the fundamental inequality, follow that θ 1 and θ 2 involves derivatives of u and v only up to order 1 and hence θ 1 u 2 α and θ 2 v 2 α is a sum of terms of the same type we have already encountered in R, so that its integral can be bounded in the same manner.This completes the proof of Lemma 7.1.
Remark 7.3 If the assumption (7.8) holds for all L ≥ 2, the solution is infinitely differentiable in the x-variable.From the equations (2.8), (2.9) itself the solution is C ∞ in both of its variables.