The IVP for a higher dimensional version of the Benjamin-Ono equation in weighted Sobolev spaces
Introduction
This work is concerned with the initial value problem (IVP) for a higher dimensional version of the Benjamin-Ono equation; where , denotes the Riesz transform with respect to the first coordinate defined by the Fourier multiplier operator with symbol , and Δ stands for the Laplace operator in the spatial variables .
When , the Riesz transform coincides with the Hilbert transform, and so we recover the well-known Benjamin-Ono equation, see [23], [19], [26], [15], [12] and the references therein.
When , the (HBO) equation preserves its physical relevance, it describes the dynamics of three-dimensional slightly nonlinear disturbances in boundary-layer shear flows, without the assumption that the scale of the disturbance is smaller along than across the flow, see for instance [1], [21], [27]. Existence and decay rate of Solitary-wave solutions were studied in [18].
Some recent works have been devoted to establish that the IVP associated to (HBO) is locally well-posed (LWP) in the space , and . Here we adopt Kato's notion of well-posedness, which consists of existence, uniqueness, persistence property (i.e., if the data a function space, then the corresponding solution describes a continuous curve in X, ), and continuous dependence of the map data-solution. Regarding the IVP for (HBO), in [14] LWP was deduced for when and for when . In [24], LWP was improved to the range in the case . Up to our knowledge there are no results concerning global well-posedness (GWP) in the current literature. It is worthwhile to mention that local well-posedness issues have been addressed by compactness methods, since one cannot solve the IVP related to (HBO) by a Picard iterative method implemented on its integral formulation for any initial data in the Sobolev space , and . This is a consequence of the results deduced in [14], where it was established that the flow map data-solution for (HBO) is not of class at the origin from to .
Real solutions of (HBO) formally satisfy at least three conservation laws (time invariant quantities) This work is intended to determinate if for a given initial data in the Sobolev space with some additional decay at infinity (for instance polynomial), it is expected that the corresponding solution of (HBO) inherits this behavior. Such matter has been addressed before for the Benjamin-Ono equation in [10], [12], [16], showing that in general polynomial type decay is not preserved by the flow of this model. Here as a consequence of our results, we shall determinate that the same conclusion extends to the (HBO) equation.
Let us now state our results. Our first consequence is motivated from the fact that the weight function is smooth with bounded derivatives when . This property allows us to consider well-posedness issues for a more general class of weights. Proposition 1.1 Let ω be a smooth weight with all its first and second derivatives bounded. Then, the IVP (HBO) is locally well-posed in for all , where and for .
Next, we discuss LWP for the IVP (HBO) in weighted Sobolev spaces and In order to obtain a relation between differentiability and decay in the spaces (1.2), we notice that the linear part of the equation (HBO) commutes with the operators where denotes the Kronecker delta function with if and zero otherwise, thus one has For this reason, it is natural to study well-posedness in weighted Sobolev spaces where the balancing between decay and regularity satisfies the relation, .
Remark For the sake of brevity, from now on we shall state our results for the (HBO) equation only for dimensions two and three. Actually, it will be clear from our arguments that solutions of this model in the spaces (1.1) behave quite different in each of these dimensions. Nevertheless, following our ideas one can extend the ensuing conclusions to arbitrary even and odd dimensions. Theorem 1.1 Consider . Let where and . If with , then the IVP associated to (HBO) is locally well-posed in . If with , then the IVP associated to (HBO) is locally well-posed in .
The proof of Theorem 1.1 is adapted from the arguments used by Fonseca and Ponce in [12] and Fonseca, Linares and Ponce in [11]. Additional difficulties arise from extending these ideas to the (HBO) equation, since here we deal with a several variables model involving Riesz transform operators. Among them, the commutator relation between and a polynomial of a certain higher degree requires to infer weighted estimates for derivatives of negative order. In this regard, as a further consequence of the proof of Theorem 1.1 we deduce. Corollary 1.1 Consider and . Let be a solution of the IVP (HBO) with . Then Theorem 1.2 Assume that . Let u be a solution of the IVP associated to (HBO) such that when and when . If there exist two different times for which Theorem 1.3 Suppose that , and . Let be a solution of the IVP associated to (HBO). If there exist three different times such that
We remark that similar unique continuation properties have been established for the Benjamin-Ono equation in [12] and the dispersion generalized Benjamin-Ono equation in [11]. A difference in the present work is that our proof of Theorem 1.2, Theorem 1.3 incorporates an extra weight in the frequency domain, which allows us to consider less regular solutions of (HBO) to reach these consequences. We also invite the reader to consult the unique continuation principles for the nonlinear Schrödinger equation and the generalized Korteweg-de Vries equation established in [8] and [7] respectively under assumptions on the solutions at two different times.
Remarks When , the conclusion of Theorem 1.1 coincides with the decay rates showed for the Benjamin-Ono equation in [12, Theorem 1]. In this sense our results can be regarded as a generalization of those derived by the Benjamin-Ono equation. As a matter of fact, Theorem 1.1 tells us that an increment in the dimension allows a 1/2 larger decay with respect to the preceding setting. The restrictions on the Sobolev regularity stated in Proposition 1.1 and Theorem 1.1 are imposed from our recent results in [14], which assure that under such considerations the solution satisfies where the Sobolev space is defined as usual with norm . The property (1.4) is essential to establish LWP in . Theorem 1.2 shows that the decay is the largest possible for arbitrary initial datum. In this regard Theorem 1.1 (i) is sharp. In addition, Theorem 1.2 shows that if with and , then the corresponding solution verifies Although, there does not exist a non-trivial solution u corresponding to data with with Theorem 1.3 shows that the decay is the largest possible in the spatial -decay rate. As a result, Theorem 1.2 (ii) is sharp. Apart from this, Theorem 1.3 tells us that there are non-trivial solutions such that and it guarantees that there does not exist a non-trivial solution such that
One may ask wherever the assumption in Theorem 1.3 can be reduced to two different times . In this respect we have the following consequences.
Theorem 1.4 Suppose that , and . Let be a solution of the IVP associated to (HBO). If there exist , , such that and then
Theorem 1.5 Suppose that , and . Let with be a nontrivial solution of the IVP associated to (HBO) such that Let If , then
Remarks Theorem 1.4 tells us that the three times condition in Theorem 1.3 can be reduced to two times provided that Theorem 1.5 asserts that the condition of Theorem 1.3 in general cannot be reduced to two different times. In this sense the result of Theorem 1.4 is optimal. In view of Theorem 1.5, we notice that the number of times involved in Theorem 1.2, Theorem 1.3 is the same required to establish similar unique continuation properties for the Benjamin-Ono equation, see [12, Theorem 2 and Theorem 3]. Therefore, our conclusions on the (HBO) equation are again regarded as a generalization of their equivalents for the Benjamin-Ono model.
Next, we introduce the main ingredient behind the proof of Proposition 1.1 and Theorem 1.1. When dealing with energy estimates, motivated by the structure of the dispersion term in the (HBO) equation, it is reasonable to try to find a commutator relation involving the Riesz transform, in such a way that when applied to a differential operator it redistributes the derivatives lowering the order of the operator. In this direction, we provide a new generalization of Calderón's first commutator estimate [3] in the context of the Riesz transform. Proposition 1.2 Let be the usual Riesz transform in the direction . For any and any multi-index α with , there exists a constant c depending on α and p such that The operator is defined via its Fourier transform as
In the present work, (1.5) is essential to transfer derivatives to some weighted functions. Additionally, the operators defined by (1.6) are useful to symbolize commutator relations between the Riesz transform and polynomials.
We will begin by introducing some preliminary estimates to be used in subsequent sections. In Section 3 we prove Proposition 1.1 and Theorem 1.1, Theorem 1.2, Theorem 1.3, Theorem 1.4, Theorem 1.5 will be deduced in the following Sections 4, 5, 6, 7 and 8 respectively. We conclude the paper with an appendix where we show the commutator estimate stated in Proposition 1.2.
Section snippets
Notation and preliminary estimates
We will employ the standard multi-index notation, , , , and if for all . As usual will denote the standard canonical vector in the k direction.
For any two positive quantities a and b, means that there exists independent of a and b (and in our computations of any parameter involving approximations) such that . Similarly, we define , and states that and . denotes the commutator between the
Proof of Proposition 1.1
In this section, we establish local well-posedness in the space . We require the following result. Lemma 3.1 Let ω be a smooth weight with all its first and second derivatives bounded. Define Then, there exists a constant independent of λ, such that where α is a multi-index of order . Proof The proof is similar to that in [6, Lemma 4.1]. □
Proof of Theorem 1.1
When the decay parameter , the weight satisfies the hypothesis of Theorem 1.1. Thereby, we may assume that .
Let be a solution of (HBO) with initial datum provided by Theorem 2.3. We shall prove that . Once we have established this conclusion, the fact that and the continuous dependence on the initial data follows by the same reasoning in the proof of Proposition 1.1.
We begin by giving a brief sketch of
Proof of Theorem 1.2
We begin by introducing some notation and general considerations independent of the dimension to be applied in the proof of Theorem 1.2. We split defined by (2.1) as In addition, we define and as in (2.3), that is, Without loss of generality we shall assume that , i.e., . The solution of the IVP (HBO) can be represented by
Proof of Theorem 1.3
We first discuss the main ideas leading to the proof of Theorem 1.3. By hypothesis, there exist three different times and such that The equation in (HBO) yields the following identities, and hence If we prove that there exist and such that in view of (6.2) with , it follows that . In this manner,
Proof of Theorem 1.4
Without loss of generality we may assume that Let us treat first the two-dimensional case. Collecting (6.15), (6.20) and (6.3), we have for that whenever . A similar conclusion can be drawn for the three-dimensional case after gathering together (6.35), (6.37) and (6.3) to deduce
Proof of Theorem 1.5
Whenever with , and one has Setting , we can employ (8.1) to replace all the estimates provided in the proof of Theorem 1.3 by their equivalents in the space . This in turn yields for each . On the other hand, when , (8.1) establishes that all the estimates exhibited in the
Acknowledgements
This work was supported by CNPq Brazil. The author wishes to express his gratitude to Prof. F. Linares and A. Mendez for several helpful comments improving this document.
References (27)
- et al.
The IVP for the Benjamin-Ono-Zakharov-Kuznetsov equation in weighted Sobolev spaces
J. Math. Anal. Appl.
(2014) - et al.
On uniqueness properties of solutions of the k-generalized KdV equations
J. Funct. Anal.
(2007) - et al.
The IVP for the Benjamin-Ono equation in weighted Sobolev spaces II
J. Funct. Anal.
(2012) - et al.
The IVP for the dispersion generalized Benjamin-Ono equation in weighted Sobolev spaces
Ann. Inst. Henri Poincaré, Anal. Non Linéaire
(2013) - et al.
The IVP for the Benjamin-Ono equation in weighted Sobolev spaces
J. Funct. Anal.
(2011) On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation
Nonlinear Anal.
(2002)- et al.
Collapse transformation for self-focusing solitary waves in boundary-layer type shear flows
Phys. Lett. A
(1995) - et al.
Multidimensional solitons in shear flows of the boundary-layer type
Sov. Phys. Dokl.
(1992) - et al.
The initial-value problem for the Korteweg-de Vries equation
Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci.
(1975) Commutators of singular integral operators
Proc. Natl. Acad. Sci. USA
(1965)
On commutators of singular integrals and bilinear singular integrals
Trans. Am. Math. Soc.
Au-delà des opérateurs pseudo-différentiels
The sharp Hardy uncertainty principle for Schrödinger evolutions
Duke Math. J.
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