ON THE LOCAL WELL-POSEDNESS OF A BENJAMIN-ONO-BOUSSINESQ SYSTEM

Consider a Benjamin-Ono-Boussinesq system η t + u x + a u x x x + ( u η ) x = 0 , u t + η x + u u x + c η x x x − d u x x t = 0 , where a , c , and d are constants satisfying 0$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"> a = c > 0 , 0$" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"> d > 0 or a 0 , c 0 , 0$" id="E9" xmlns:mml="http://www.w3.org/1998/Math/MathML"> d > 0 . We prove that this system is locally well posed in Sobolev space H s ( ℝ ) × H s + 1 ( ℝ ) , with 1/4$" id="E11" xmlns:mml="http://www.w3.org/1998/Math/MathML"> s > 1 / 4 .


Introduction and main results
We consider the Cauchy problem for a Benjamin-Ono-Boussinesq system    η t + u x + au xxx + (uη) x = 0, t > 0, x ∈ IR u t + η x + uu x + cη xxx − du xxt = 0, t > 0, x ∈ IR η| t=0 = f (x), u| t=0 = g(x), where a, c and d are constants satisfying a = c > 0, d > 0 or a < 0, c < 0, d > 0. (1.2) The system is called a Benjamin-Ono-Boussinesq system because it can be reduced to a pair of equations whose linearization uncouples to a pair of linear Benjamin-Ono equations. Equations of type (1.1) are a class of essential model equations appearing in physics and fluid mechanics. To describe two-dimensional irrotational flows of an inviscid liquid in a uniform rectangular channel, Boussinesq in 1871 derived from the Euler equation the classical Boussinesq system η t + u x + (uη) x = 0, t > 0, x ∈ IR, u t + η x + uu x + 1 3 η xxt = 0, t > 0, x ∈ IR. In [1], Bona, Chen and Saut derived by considering first-order approximations to the Euler equation the following alternative ( a four-parameter Boussinesq system ) to the classical Boussinesq system where the constants obey the relations with θ ∈ [0, 1]. The system (1.1) is one of the four-parameter systems associated with b = 0. When b = 0, Bona, Chen and Saut in [1] determined exactly that the four-parameter systems are linearly well posed if and only if a, c and d satisfy the relation (1.2). The local well-posedness of the nonlinear system (1.1) is considered in [2]. They prove that the system (1.1) associated with (1.2) is locally well-posed in the Sobolev space H s (IR) × H s+1 (IR) with s ≥ 1. In this work we shall give some local well-posedness for the Cauchy problem (1.1) in the Sobolev spaces H s (IR) × H s+1 (IR) with s > 1 4 by using the so-called L p − L q smoothing effect of the Strichartz type. Denote by J the Fourier multiplier with symbol (1 + ξ 2 ) 1/2 , and denote by H the usual Hilbert transform. Our result is

Moreover, for any
In the sequel, we say the pair (p, q) ∈ IR 2 admissible if they satisfy We denote by J the Fourier multiplier with symbol (1 + ξ 2 ) 1/2 , denote by H the usual Hilbert transform and denote by m(D) the Fourier multiplier associated with symbol m(ξ). We also denote the dyadic integers 2 k , k ≥ 0, by λ or µ. Whenever a summation over λ or µ appears it means that we sum over the dyadic integers. The notation A B (resp., A B) means that there exists a harmless positive constant C such that A ≤ CB ( resp., A ≥ CB ). We denote by L p T X (resp., L p I X ) the space of X−valued measurable and p−integrable functions defined on [0, T ] ( resp., I ), equipped with the natural norm. We also use the notation (u 1 , u 2 , · · · , u k ) X = u 1 X + · · · + u k X .
The rest of this paper is organized as follows. In section 2 we prove some Strichartz type estimates for smooth solutions of (1.1). In section 3 we give the proof of the local well-posedness of the Cauchy problem (1.1).

Some estimates
In this section we give some smoothing effects for the equation (1.1). These estimate will be the main ingredient in the proof of local well-posedness of the Cauchy problem (1.1). Consider the following linear system Consider the change of variables where h(D)( resp. h −1 (D) ) is the Fourier multiplier with the symbol h(ξ) ( resp. h −1 (ξ) ). Then .
Using the Cauchy-Schwarz inequality we get Lemma 2.2 Fix T > 0 and σ > 1/2. Let (η, u) be a smooth solution of the system (1.1). Then for every admissible pair (p, q) , Proof. Let (η, u) be a smooth solution of the system (1.1). Then (η λ , u λ ) satisfies the following system Using (2.5) to the system (2.6) and choosing the interval I satisfying |I| ≤ 1 and |I| ≤ 1 λ , we get By the Sobolev embedding, |I| ≤ 1 and |I| ≤ 1/λ, We can choose I k such that their number is bounded by (1 + T )λ. Therefore by (2.8) we obtain we introduce the following estimates which come from Lemma 4.2 and Lemma 4.3 in [4]: and for λ = 1, 2, Notice that the frequencies of order ≤ λ/8 in the Littlewood-Paley decomposition of u do not contribute, therefore Denote by .

Uniqueness
Let (η 1 , u 1 ) and (η 2 , u 2 ) be two solutions of the system (1.1). Let η j = h(D)(v j + w j ) and and integrating by parts easily yield with

Existence
Without loss of generality we assume 1/4 < s < 1. Let (η, u) be a smooth solution of the system (1.1). Setting σ = s + 3/4, δ λ = λ s+1 and I = [0, T ] in Lemma 2.3, we deduce If (p, q) is an admissible pair, then σ + 1/p < σ + 1/4 = s + 1 < 2. Therefore using Theorem 2.1 and (3.6) we get that for every admissible pair (p, q) and every T > 0, Using the Sobolev embedding in the spatial variable together with the Hölder inequality in time variable we can choose an admissible pair (p, q) such that A combination of (3.8) with (3.7) yields Choosing δ λ = λ s+1 , τ = 0 and t = T in (2.34) we deduce Then there exists a positive constant C 0 so small that (3.9) and (3.11) imply that providing H(T ) ≤ C 0 . For every R > 0 we choose a positive constant T R such that, for all T ∈ [0, T R ], Notice that H(0) = 0. A straightforward continuity argument shows that H(T ) ≤ C 0 /2 for all T ≤ T R (η(0), u(0) −1 H s+1 . Using (3.6) we obtain that if (η, u) be a smooth solution of the system (1.1) then it satisfies (3.14) The bounds (3.13) and (3.14) enable us to perform a standard compactness argument. More precisely, consider that the smooth sequence {f n (x), g n (x)} satisfying (f n , g n ) H s+1 ≤ R for some positive constant R, which converges to (f (x), g(x)) in H s (IR) × H s+1 (IR), where we denote bỹ f n = h −1 (D)f n . Let {η n , u n } be the solution of the system (1.1) with data (f n (x), g n (x)) which exists globally in time due to Theorem 3.5 in [2]. We shall prove that {η n , u n } converges and the limit object is a solution of the system (1.1) with data (f (x), g(x)). Indeed, (3.14) implies that {η n , u n } converges in weak -topology of L ∞ ([0, T R ] : H s × H s+1 ) to some limit (η, u). Using (3.5) we deduce that {η n , u n } converges strongly to (η, u) in L ∞ ([0, T R ] : H −1 × L 2 ) and therefore (u n η n ) x and u n (u n ) x converge to (uη) x and uu x , respective, in a distributional sense. This prove that the limit (η, u) satisfies the system (

Continuous dependence on the data
We present a proof of continuous dependence on the data based on Lemma 2.3.