The IVP for a higher dimensional version of the Benjamin-Ono equation in weighted Sobolev spaces

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Abstract

We study the initial value problem associated to a higher dimensional version of the Benjamin-Ono equation. Our purpose is to establish local well-posedness results in weighted Sobolev spaces and to determinate according to them some sharp unique continuation properties of the solution flow. In consequence, optimal decay rate for this model is determined. A key ingredient is the deduction of a new commutator estimate involving Riesz transforms.

Introduction

This work is concerned with the initial value problem (IVP) for a higher dimensional version of the Benjamin-Ono equation;{tuR1Δu+ux1u=0,xRd,tR,u(x,0)=u0, where d2, R1=x1(Δ)1/2 denotes the Riesz transform with respect to the first coordinate defined by the Fourier multiplier operator with symbol iξ1|ξ|1, and Δ stands for the Laplace operator in the spatial variables xRd.

When d=1, 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 d=2, 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 Hs(Rd), sR and d2. Here we adopt Kato's notion of well-posedness, which consists of existence, uniqueness, persistence property (i.e., if the data u0X a function space, then the corresponding solution u() describes a continuous curve in X, uC([0,T];X),T>0), and continuous dependence of the map data-solution. Regarding the IVP for (HBO), in [14] LWP was deduced for s>5/3 when d=2 and for s>(d+1)/2 when d3. In [24], LWP was improved to the range s>3/2 in the case d=2. 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 Hs(Rd), d2 and sR. This is a consequence of the results deduced in [14], where it was established that the flow map data-solution u0u for (HBO) is not of class C2 at the origin from Hs(Rd) to Hs(Rd) d2.

Real solutions of (HBO) formally satisfy at least three conservation laws (time invariant quantities)I(u)=u(x,t)dx,M(u)=u2(x,t)dx,H(u)=|(Δ)1/4u(x,t)|213u3(x,t)dx. This work is intended to determinate if for a given initial data in the Sobolev space Hs(Rd) 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 xr=(1+|x|2)r/2 is smooth with bounded derivatives when r[0,1]. 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 Hs(Rd)L2(ω2dx) for all s>sd, where s2=5/3 and sd=d/2+1/2 for d3.

The proof of Proposition 1.1 is similar in spirit to that in [6] for a two-dimension model. A remarkable difference is that our recent results in [14] enable us to prove Proposition 1.1 in Sobolev spaces of lower regularity compared with those obtained by implementing a parabolic regularization argument.

Next, we discuss LWP for the IVP (HBO) in weighted Sobolev spacesZs,r(Rd)=Hs(Rd)L2(|x|2rdx),s,rR andZ˙s,r(Rd)={fHs(Rd)L2(|x|2rdx):fˆ(0)=0},s,rR. 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) L=tR1Δ commutes with the operatorsΓl=xl+tδ1,l(Δ)1/2+txlR1,l=1,,d, where δ1,l denotes the Kronecker delta function with δ1,l=1 if l=1 and zero otherwise, thus one has[L,Γl]=LΓlΓlL=0. For this reason, it is natural to study well-posedness in weighted Sobolev spaces Zs,r(Rd) where the balancing between decay and regularity satisfies the relation, rs.

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 d=2,3. Let s>sd where s2=5/3 and s3=2.

  • (i)

    If r[0,d/2+2) with rs, then the IVP associated to (HBO) is locally well-posed in Zs,r(Rd).

  • (ii)

    If r[0,d/2+3) with rs, then the IVP associated to (HBO) is locally well-posed in Z˙s,r(Rd).

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 R1 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 d=2,3 and r0[0,d/2). Let uC([0,T];Z˙s,r(Rd)) be a solution of the IVP (HBO) with (d/2+2)rs. Then||1uC([0,T];L2(|x|2r0dx)).

In the above display the operator ||1 is defined by the Fourier multiplier |ξ|1=(ξ12++ξd2)1/2. Next we state some continuation principles for the (HBO) equation.

Theorem 1.2

Assume that d=2,3. Let u be a solution of the IVP associated to (HBO) such that uC([0,T];Z2+,2(R2)) when d=2 and uC([0,T];Z3,3(R3)) when d=3. If there exist two different times t1,t2[0,T] for whichu(,tj)Zd/2+2,d/2+2(Rd),j=1,2thenuˆ0(0)=0.

In Theorem 1.2, uZ2+,2(R2) means that uHs+(R2)L2(|x|4dx), where there exists a positive number ϵ1 such that uHs+ϵ(R2).

Theorem 1.3

Suppose that d=2,3, r2=3 and r3=4. Let uC([0,T];Z˙rd,rd(Rd)) be a solution of the IVP associated to (HBO). If there exist three different times t1,t2,t3[0,T] such thatu(,tj)Zd/2+3,d/2+3(Rd),j=1,2,3thenu(x,t)=0.

It is worth pointing out that the deduction of Theorem 1.2, Theorem 1.3 is more involved in the odd dimension case, where the decay rates d/2+2 and d/2+3 are not integer numbers. Roughly speaking, transferring decay to regularity in the frequency domain, on this setting one has to deal with an extra 1/2-fractional derivative to achieve these conclusions.

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

  • (i)

    When d=1, 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.

  • (ii)

    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 u(x,t) satisfiesuL1([0,T);W1,(Rd)), where the Sobolev space W1,(Rd) is defined as usual with norm fW1,:=fL+fL. The property (1.4) is essential to establish LWP in Zs,r(Rd).

  • (iii)

    Theorem 1.2 shows that the decay r=(d/2+2) 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 u0Zs,r(Rd) with d/2+2rs and u0ˆ(0)0, then the corresponding solution u=u(x,t) verifies|x|(d/2+2)uL([0,T];L2(Rd)),T>0. Although, there does not exist a non-trivial solution u corresponding to data u0 with u0ˆ(0)0 with|x|d/2+2uL([0,T];L2(Rd)), for some T>0.

  • (iv)

    Theorem 1.3 shows that the decay r=(d/2+3) is the largest possible in the spatial L2-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 u=u(x,t) such that|x|(d/2+3)uL([0,T];L2(Rd)),T>0 and it guarantees that there does not exist a non-trivial solution such that|x|d/2+3uL([0,T];L2(Rd)),for some T>0.

One may ask wherever the assumption in Theorem 1.3 can be reduced to two different times t1<t2. In this respect we have the following consequences.

Theorem 1.4

Suppose that d=2,3, r2=3 and r3=4. Let uC([0,T];Z˙rd,rd(Rd)) be a solution of the IVP associated to (HBO). If there exist t1,t2[0,T], t1t2, such thatu(,tj)Zd/2+3,d/2+3(Rd),j=1,2, andx1u(x,t1)dx=0orx1u(x,t2)dx=0, thenu0.

Theorem 1.5

Suppose that d=2,3, r2=3 and r3=4. Let uC([0,T];Z˙s,rd(Rd)) with sd/2+4 be a nontrivial solution of the IVP associated to (HBO) such thatu0Z˙d/2+3,d/2+3(Rd)andx1u0(x)dx0. Lett:=4u0L22x1u0(x)dx. If t(0,T], thenu(t)Z˙d/2+3,d/2+3(Rd).

Remarks

  • (i)

    Theorem 1.4 tells us that the three times condition in Theorem 1.3 can be reduced to two times t1t2 provided thatx1u(x,t1)dx=0 or x1u(x,t2)dx=0.

  • (ii)

    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.

  • (iii)

    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 Rl be the usual Riesz transform in the direction l=1,,d. For any 1<p< and any multi-index α with |α|1, there exists a constant c depending on α and p such thatRl(aαf)aRlαf1|β|<|α|1β!βaDRlβαfLpcα,p|β|=|α|βaLfLp. The operator DRlβ is defined via its Fourier transform asDRlβgˆ(ξ)=i|β|ξβ(iξl|ξ|)gˆ(ξ).

In Proposition 1.2 the convention for the empty summation (such as 1|β|<1) is defined as zero. Consequently, when |α|=1 we find[Rl,a]αfLpaLfLp, where[Rl,a]αf=Rl(aαf)aRlαf. Estimates of the form (1.5) are of interest on their own in harmonic analysis, see [17] for similar results and several applications dealing with homogeneous differential operators. The result of Proposition 1.2 may be of independent interest. Indeed, we believe that it could certainly be used to derive other properties for the (HBO) equation.

In the present work, (1.5) is essential to transfer derivatives to some weighted functions. Additionally, the operators DRlβ 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, α=(α1,,αd)Nd, α=x1α1xdαd, |α|=j=1dαj, α!=α1!αd! and αβ if αjβj for all j=1,,d. As usual ekRd will denote the standard canonical vector in the k direction.

For any two positive quantities a and b, ab means that there exists C>0 independent of a and b (and in our computations of any parameter involving approximations) such that aCb. Similarly, we define ab, and ab states that ab and ba. [A,B] denotes the commutator between the

Proof of Proposition 1.1

In this section, we establish local well-posedness in the space Hs(Rd)L2(ω2dx). We require the following result.

Lemma 3.1

Let ω be a smooth weight with all its first and second derivatives bounded. Defineωλ(x)=ω(x)eλ|x|2,xRd,λ(0,1). Then, there exists a constant c>0 independent of λ, such thatαωλc, where α is a multi-index of order 1|α|2.

Proof

The proof is similar to that in [6, Lemma 4.1]. 

Now we proceed to prove Proposition 1.1. Given u0Hs(Rd)L2(ω2dx), from Theorem 2.3, there exist T=T(u0Hs)

Proof of Theorem 1.1

When the decay parameter r[0,1], the weight xr satisfies the hypothesis of Theorem 1.1. Thereby, we may assume that 1<rs.

Let uC([0,T];Hs(Rd)) be a solution of (HBO) with initial datum u0Zs,r(Rd) provided by Theorem 2.3. We shall prove that uL([0,T];L2(|x|2rdx)). Once we have established this conclusion, the fact that uC([0,T];L2(|x|2rdx)) 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 F3k defined by (2.1) asF3,1k(t,ξ,f)=ξk3(itξ1|ξ|)eitξ1|ξ|f(ξ), and F3,2k(t,ξ,f)=F3k(t,ξ,f)F3,1k(t,ξ,f). In addition, we define F˜3,1k and F˜3,2k as in (2.3), that is,F˜3,lk(t,ξ,f)=eitξ1|ξ|F3,lk(t,ξ,f),l=1,2. Without loss of generality we shall assume that t1=0<t2, i.e., u0Zd/2+2,d/2+2(Rd). 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 t1,t2 and t3 such thatu(,tj)Zd/2+3,d/2+3(Rd),j=1,2,3. The equation in (HBO) yields the following identities,ddtxlu(x,t)dx=δ1,l2u(t)L22=δ1,l2u0L22,l=1,,d and hencexlu(x,t)dx=xlu0(x)dx+δ1,l2u0L22t,l=1,,d. If we prove that there exist t˜1(t1,t2) and t˜2(t2,t3) such thatx1u(x,t˜j)dx=0, for all j=1,2, in view of (6.2) with l=1, it follows that u0. In this manner,

Proof of Theorem 1.4

Without loss of generality we may assume thatt1=0 and x1u0(x)dx=0. Let us treat first the two-dimensional case. Collecting (6.15), (6.20) and (6.3), we have for t20 thatξk4uˆ(,t2)L2(R2) implies ξk4uˆ(,t2)L2(ξ8dξ), this holds if and only if 0=0t2x1u(x,τ)dxdτ=120t2τu(τ)L22dτ=t224u0L22, whenever k=1,2. A similar conclusion can be drawn for the three-dimensional case after gathering together (6.35), (6.37) and (6.3) to deduceξ14uˆ(,t2)H1/2(R3) implies ξ2ξ14uˆ(,t2)H1/2(R3),

Proof of Theorem 1.5

Whenever uC([0,T];Z˙s,rd(Rd)) with r2=3, r3=4 and sd/2+4 one hasux1uL([0,T];Zd/2+3,d/2+3(Rd)). Setting d=2, we can employ (8.1) to replace all the L2(ξ8dξ) estimates provided in the proof of Theorem 1.3 by their equivalents in the space L2(R2). This in turn yieldsξk4uˆ(,t)L2(R2), if and only if 0=0tx1u(x,τ)dxdτ=0tx1u0(x)dx+τ2u0L22dτ=0, if and only if t(x1u0(x)dx+t4u0L22)=0, for each k=1,2. On the other hand, when d=3, (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.

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