Elsevier

Nonlinear Analysis

Volume 212, November 2021, 112427
Nonlinear Analysis

On the Benjamin–Ono equation in the half line

https://doi.org/10.1016/j.na.2021.112427Get rights and content

Abstract

We consider the inhomogeneous Dirichlet initial–boundary value problem for the Benjamin–Ono equation formulated on the half line. We study the global in time existence of solutions to the initial–boundary value problem. This work is a continuation of the ones (Hayashi and Kaikina , 2010, 2012 [15,16]) by Hayashi and Kaikina where the global in time existence and the asymptotic behaviour of solutions for large time were considered.

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 (BO):ut+Huxx+uxu=0,t>0,x>0,u(x,0)=u0(x),u(0,t)=h(t),t>0,where H denotes the Hilbert transform,1 which, as a (Calderón–Zygmund) singular integral operator is defined via Hu(x)PV0u(y)yxdy,uC0(R+).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 Hs(R), s>32, and making use of the conserved quantities he extended globally the result in Hs(R),s2. L. Abdelouhab et al. in [1] the authors proved the global well-posedness of the BO equation in Hs(R), s>32, improving the LWP result of Iorio for the same value of s. In Ponce [31], the author extended the local result for data in H32(R) and the global result for any solution in Hs(R),s32, 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 H1(R), 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 h(t)0, or when (1.1) is endowed with the Neumann condition ux(0,t)=0, it was proved, among other things, in [17], [18], the well-posedness for (1.1) if ψL1,a(R+)H1(R+), where a(0,1). 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 ψ belongs to H1+ε(R+)L1,2(R+) and the boundary condition h(t)H1(R+)L1(R+). Under these conditions, in Theorem 2.1 we prove that there exist a unique global solution u of (1.1) in the space C([0,):H1(R+)L2,1(R+)). 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 uH1(R+) to (1.1). Finally, using the Calderón commutator technique as developed by Ponce and Fonseca [13], we prove that uL2,1(R+).

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 tR. We denote by t1+t2 and {t}|t|t. We write AB if there exist a constant C, such that C does not depend on fundamental quantities on A and B, such that ACB.

We denote the usual direct and inverse Laplace transformation by L and L1, which are the integral operators given by Lϕ(ξ)ϕ̂ξ0exξϕxdx,L1ϕ(x)=12πiiReixξϕ̂ξdξ.The Fourier transform F

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 (LBO):ut+Huxx=0,t>0,x>0,u(x,0)=ψ(x),x>0u(0,t)=h(t),t>0.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 u=Gtψ+B(t)h,where G(t)=G1(t)+G2(t),with G1(t)ψ:=12πiiieK(p)tepxuˆ0(p)dp,G2(t)ψ12π0epxK(t)ψ(p)dp,where K̃(t)ψ(p)1

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 u0, h belong to L1(R+). Then, there exists a unique solution u(x,t) of the initial–boundary value problem (3.1), which has the following integral representation u(x,t)=G(t)ψ+B(t)h, where the operators G(t) and B(t) 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 G(t) and B(t) defined in (3.2), (3.7) respectively.

Lemma 5.1

For ψZ1+ε,2H1+ε(R+)L2(R+,|x|2dx), with ε(0,12), n=0,1, ψ(0)=0, the estimate xnG(t)ψL2(R+)t12(n+12)ψZ1+ε,2(R+) holds.

Proof

Let us recall that G(t)=G1(t)+G2(t), where G1 and G2 are defined in (3.3).

Let ψ=1(0,)ψ be an extension of ψ from R+ to R, since ψ(0)=0. By following  [9, Lemma 2.1], we have that ψZ1+ε(

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 uC([0,T]:H1(R+)). On the other hand, in Proposition (6.2), we show that this solution belongs to C([0,T]:L2,1(R+)).

Proposition 6.1

For any ψZ1+ε,2 and hZ1,1, with ε(0,12), the IBVP (1.1) has a unique global solution uC([0,T]:H1(R+)).

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 .

References (33)

  • CalderónA.P.

    Commutators of singular integral operators

    Proc. Natl. Acad. Sci. USA

    (1965)
  • CaseK.M.

    Benjamin—Ono-related equations and their solutions

    Proc. Natl. Acad. Sci.

    (1979)
  • DanovK.D. et al.

    Nonlinear waves on shallow water in the presence of a horizontal magnetic field

    Fluid Dyn.

    (1983)
  • DawsonL. et al.

    On the decay properties of solutions to a class of Schrödinger equations

    Proc. Amer. Math. Soc.

    (2008)
  • DuoandikoetxeaJ. et al.

    Fourier Analysis, Vol. 29

    (2001)
  • EsquivelL. et al.

    Inhomogeneous Neumann-boundary value problem for one dimensional nonlinear Schrödinger equations via factorization techniques

    J. Math. Phys.

    (2019)
  • Cited by (0)

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