On the Benjamin–Ono equation in the half line
Introduction
When studying dispersive problems on the half line, boundary terms provide a perturbation to the behaviour of their solutions and the analysis for these problems can be treated by using techniques of complex analysis, namely, methods of analytic continuation. By following this philosophy, and by exploiting the Calderón commutator technique, in this work we shall study the initial–boundary value problem (IBV problem) for the Generalised Dispersive Benjamin–Ono equation on the half line where denotes the Hilbert transform,1 which, as a (Calderón–Zygmund) singular integral operator is defined via For a comprehensive study of this operator on the half-line we refer to the interested reader to [32]. Between the family of dispersive equations, the Benjamin–Ono equation is a model describing long internal gravity waves in a stratified fluid with infinite depth (see Benjamin [2] and Ono [30], for instance) and turns out to be important in other physical phenomena as well (we refer the reader to Danov and Ruderman [6], Ishimori [21] and Matsuno Kaup [27] and references therein). In the case of the whole line, for the Benjamin–Ono model are well known very noticeable properties: it defines a Hamiltonian system, can be solved by an analogue of the inverse scattering method (see Ablowitz and Fokas [12]), admits (multi-)soliton solutions, and satisfies infinitely many conserved quantities (see Case [5]). It is worth to mention that, regarding the IVP associated to the BO equation (in the case of the whole line), local and global results have been obtained by various authors. Iorio [22] showed local well-posedness for data in , , and making use of the conserved quantities he extended globally the result in . L. Abdelouhab et al. in [1] the authors proved the global well-posedness of the BO equation in , , improving the LWP result of Iorio for the same value of . In Ponce [31], the author extended the local result for data in and the global result for any solution in , and further improvements were done by Molinet, Saut, and Tzvetkov,2 Koch and Tzvetkov,3 Jinibre and Velo [15], Tao who showed in [33] that the IVP associated to the BO equation is globally well-posed in , Ionescu and Kenig [20], Molinet and Riboud [28], [29] and [26], just to mention a few.
In the case of the half-line, the BO equation and other non-linear evolution problems for pseudo-differential operators on the half-line have been considered by Esquivel, Hayashi and Kaikina [10], [11], Hayashi and Kaikina [16], [17], [18] and Kaikina [23], [24], [25]. In the homogeneous case of (1.1), with , or when (1.1) is endowed with the Neumann condition , it was proved, among other things, in [17], [18], the well-posedness for (1.1) if , where . As far as we know, the case of nonhomogeneous boundary condition for the initial–boundary value problem (1.1) was not studied previously. The main problem addressed in this paper is to study the global in time existence of solutions to (1.1) in the case where the initial data belongs to and the boundary condition . Under these conditions, in Theorem 2.1 we prove that there exist a unique global solution of (1.1) in the space . In this result we observe the influence of the boundary data on the behaviour of solutions.
The novelty of the present work is that we combine two different approaches between the real and the complex analysis. First, we start our work by applying the analytic continuation method by Hayashi and Kaikina (in the aforementioned references) related to the Riemann–Hilbert problem. Indeed, the construction of the Green operator is based on the introduction of a suitable necessary condition at the singularity points of the symbol, the integral representation for the sectionally analytic function, and the theory of singular integrodifferential equations with Hilbert kernels and with discontinuous coefficients, (see [14], [24] and Section 3 for details).
Later on, via the contraction principle, in Theorem 2.1 we deduce of global existence of a solutions to (1.1). Finally, using the Calderón commutator technique as developed by Ponce and Fonseca [13], we prove that .
This paper is organised as follows. In Section 2 we present the notation used in our work and our main result in the form of Theorem 2.1. For the benefit of the reader, in Section 3 we explain the techniques that we follow in the proof of our main theorem. The linear problem associated to (1.1) will be analysed in Section 4 and some technical lemmata will be established in Section 5. We end our work with the proof of Theorem 2.1 in Section 6.
Section snippets
Notation and main result
We will introduce the necessary notations and the function spaces used in the formulation of our main result, and in our further analysis. Let . We denote by and . We write if there exist a constant , such that does not depend on fundamental quantities on and , such that .
We denote the usual direct and inverse Laplace transformation by and , which are the integral operators given by The Fourier transform
Sketch of the proof
For the convenience of the reader we briefly explain our strategy. First of all, we consider the linear Benjamin–Ono equation with inhomogeneous boundary condition In Lemma 4.1 we construct the Green function and the boundary operator for Eq. (3.1), indeed we prove that the solution of this equation can be represented as where with where
Linear problem
We consider the linearised version of the problem (1.1), that is the initial–boundary value problem in (3.1). We prove the following lemma.
Lemma 4.1 Suppose that the initial and boundary data , belong to . Then, there exists a unique solution of the initial–boundary value problem (3.1), which has the following integral representation , where the operators and were given in (3.2).
In order to prove Lemma 4.1 we recall some basic results related with the analytic
Preliminaries
In this section we present some essential lemmas for our further analysis. Firstly, we prove the main properties of the operators and defined in (3.2), (3.7) respectively.
Lemma 5.1 For , with , , , the estimate holds.
Proof Let us recall that , where and are defined in (3.3). Let be an extension of from to , since . By following [9, Lemma 2.1], we have that
Proof of Theorem 2.1
The proof of Theorem 2.1 will be exposed in two propositions. In the first of them, Proposition 6.1, we prove there exist, under certain conditions of the initial and boundary conditions, a solution in the functional space . On the other hand, in Proposition (6.2), we show that this solution belongs to .
Proposition 6.1 For any and , with , the IBVP (1.1) has a unique global solution .
Proof By Lemma 4.1 we rewrite the initial–boundary value
Acknowledgements
D. C. was supported by the Foundation –Flanders, Belgium Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations, L. E. was supported by Gran Sasso Science Institute, Italy .
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