Global well-posedness and limit behavior for a higher-order Benjamin-Ono equation
In this paper, we prove that the Cauchy problem associated to the following higher-order Benjamin-Ono equation
is globally well-posed in the energy space . Moreover, we study the limit behavior when the small positive parameter tends to zero and show that, under a condition on the coefficients , , and , the solution to (0.1) converges to the corresponding solution of the Benjamin-Ono equation.
Key words and phrases:Initial value problem, Benjamin-Ono equation, gauge transformation
2010 Mathematics Subject Classification:Primary 35Q53, 35A01; Secondary 76B55
LMPT, Université François Rabelais Tours, Fédération Denis Poisson-CNRS,
Parc Grandmont, 37200 Tours, France.
UFRJ, Instituto de Matemática, Universidade Federal do Rio de Janeiro,
Caixa Postal 68530, CEP: 21945-970, Rio de Janeiro, RJ, Brazil.
Considered here is the following higher-order Benjamin-Ono equation
where , , is a real-valued function, , and are positive constants, is a small positive parameter and is the Hilbert transform, defined on the line by
The equation above corresponds to a second order approximation of the unidirectional evolution of weakly nonlinear dispersive internal long waves at the interface of a two-layer system of fluids, the lower one being infinitely deep. It was derived by Craig, Guyenne and Kalisch (see equation (5.38) in [CGK]), using a Hamiltonian perturbation theory. Here, represents the dislocation of the interface around its position of equilibrium, the coefficients , , and are respectively given by
where represents the depth of the upper layer when the fluid is at rest, is the density of the upper fluid and is the density of the lower fluid. Moreover, the system is assumed to be in a stable configuration, which is to say that , so that the coefficients , , and are positive.
It is worth noting that the equation obtained at the first order approximation of the above physical model is the well-known Benjamin-Ono equation
are conserved by the flow associated to (1.1).
The initial value problem (IVP) associated to the Benjamin-Ono equation on the line has been extensively studied in the recent years and has been proved to be globally well-posed in by Ionescu and Kenig in [IK] (see [MP] for another proof and [abfs, BP, Io, KK, KT, Po1, Ta] for former results). The IVP associated to (1.1) presents the same mathematical difficulties as for the Benjamin-Ono equation. Indeed, it has been shown in [Pi] that the flow map data-solution cannot be in any -based Sobolev space , , by using the same counter-example as for the Benjamin-Ono equation in [MST]. On the other hand, the Cauchy problem associated to (1.1) was proved in [LPP] to be locally well-posed in , for (and also in weighted Sobolev spaces , for , ). However, there are no conserved quantities at the level and thus it is not known wether these local solutions extend globally in time or not. Therefore, as commented in [LPP], the question of the local well-posedness in , which would directly imply global well-posedness by using (1.6) and (1.7), arises naturally.
The first aim of this paper is to give a positive answer to this issue. The result states as follows.
Fix and let be given. Then, for all and all , there exists a unique solution to equation (1.1) in the space
where is a spatial primitive of defined in Section 3.
Moreover, and the flow map data-solution is continuous from into .
Note that above denotes the space of all real-valued functions with the usual norm, while and are Bourgain spaces defined in Subsection 2.2.
Since it follows from the result of ill-posedness in [Pi] that the Cauchy problem associated to (1.1) cannot be solved by using a fixed point theorem on the integral equation, we use a compactness argument based on the smooth solutions obtained in [LPP]. To derive a priori estimates at the level, we introduce a gauge transformation which weakens the high-low frequency interactions in the nonlinearity of (1.1), as it was done by Tao in [Ta] for the Benjamin-Ono equation. Note that the same kind of gauge transformation was already introduced in [LPP] to obtain the solutions in . However, to lower the regularity till , we will need to combine this transformation with the use of Bourgain’s spaces (as it was already done in [BP, IK, MP] for BO). More precisely, we need to work in a Besov version of Bourgain’s spaces (introcuded in [Tat] in the context of waves maps). Indeed, on one hand we have to work in Bourgain’ spaces of conormal regularity to establish the main bilinear estimate (see Proposition 4.2 below). On the other hand, to control some remaining terms appearing in the transformation, we need the full Kato smoothing effect for functions that are localized in space frequencies (see Proposition 4.4). The rest of the proof follows closely the one in [MP] for the Benjamin-Ono equation (see also [Mo]).
In the second part of this article, we investigate the limit behavior of the solutions to (1.1), obtained in Theorem 1.1, as tends to zero. First, it is interesting to observe that a direct argument based on compactness methods (see for example [Mo2] in the case of the Benjamin-Ono-Burgers equation) does not seem to work. Indeed, the leading terms in the energy , which is to say and , have opposite signs, so that (1.6) and (1.7) do not provide a priori bounds, uniformly in , on . Therefore, the problem of studying the limit of , as goes to zero, turns out to be far from trivial.
Nevertheless, we are able to prove the convergence of solutions of (1.1) toward a solution of the Benjamin-Ono equation in the special case where the ratio of the densities is equal to .
Assume that . Let and for any denote by the solution to (1.1) emanating from . Then for any it holds
where is the solution to the Benjamin-Ono equation emanating from .
In the case where , the spatial primitive chosen to perform the gauge transformation for equation (1.1) corresponds to the one chosen for the Benjamin-Ono equation. Then, we can show that the Cauchy problem associated to (1.1) is uniformly in well-posed in , which will in a classical way (see for example [GW]) lead to Theorem 1.2. The main difficulty here arises from the fact that the dispersive linear terms and compete together as in the Benjamin equation (see the introduction in [ABR]). Therefore, we are only allowed to use the dispersive smoothing effects associated to (1.1) in some well behaved regions in spatial frequency and we need to refine the bilinear estimates obtained in the proof of Theorem 1.1 in the other regions.
It would be interesting to derive a class of higher-order equation for internal long waves from the first order one derived by Bona, Lannes and Saut in [BLS]. Among those equations, which would be formally equivalent to (1.1), one might find some with better behaved linear parts, which would avoid to deal with those technical difficulties.
Finally, we observe that the techniques introduced here would likely lead to similar results for the following intermediate long wave equation
where is the Fourier multiplier , is a real-valued solution, and and are positive constants, and which was also derived in [CGK]. Note that the same ill-posedness results as for equation (1.1) also hold for this equation (see [Pi]).
The paper is organized as follows: in the next section, we introduce the notations, define the functions spaces and recall some classical estimates. Sections 3 and 4 are devoted the key nonlinear estimates, which are used in Section 5 to prove Theorem 1.1. Finally, in Section 6, we prove Theorem 1.2.
2. Notations, function spaces and preliminary estimates
For any positive numbers and , the notation means that there exists a positive constant such that . We also denote when and . Moreover, if , , respectively , will denote a number slightly greater, respectively lesser, than .
For , will denote its space-time Fourier transform, whereas , respectively , will denote its Fourier transform in space, respectively in time. For , we define the Bessel and Riesz potentials of order , and , by
Throughout the paper, we fix a smooth cutoff function such that
Then if is a positive number, denote the Fourier multiplier whose symbol is given by and is defined by . For , we define
By convention, we also denote
Any summations over capitalized variables such as , or are presumed to be dyadic with , or , i.e., these variables range over numbers of the form . Then, we have that
Let us define the Littlewood-Paley multipliers by
and . Moreover, we also define the operators , , and by
Let and denote the projection on respectively the positive and the negative Fourier frequencies. Then
and we also denote , , , and . Observe that , , , , and are bounded (uniformly in ) operators on for , while are only bounded on for . We also note that
Finally, we denote by the free group associated with the linearized part of equation (1.1), which is to say,
2.2. Function spaces
For , is the usual Lebesgue space with the norm , and for , the real-valued Sobolev spaces and denote the spaces of all real-valued functions with the usual norms
If is a function defined for and in the time interval , with , if is one of the spaces defined above, and , we will define the mixed space-time spaces , , by the norms
Moreover, if , and denotes one of the mixed space-time spaces defined above, we define its dyadic version as
In the special case , the space will be simply denoted by .
For , , we introduce the Bourgain spaces related to the linear part of (1.1) as the completion of the Schwartz space under the norm
where . We will also use a dyadic version of those spaces introduced in [Tat] in the context of wave maps. For , , , will denote the completion of the Schwartz space under the norm
Moreover, we define a localized (in time) version of these spaces. Let be a positive time and or . Then, if , we have that
When , we will denote , , and .
Finally we list some useful properties of the Bourgain spaces defined above.
Fix , and . Then it holds that
for satisfying . In other words, the injections
2.3. Linear estimates
First, we recall some linear estimates in Bourgain’s spaces which will be needed later (see for instance [Tat]).
Lemma 2.2 (Homogeneous linear estimate).
Let and . Then
Lemma 2.3 (Non-homogeneous linear estimate).
Let and . Then, it holds that
Next, we derive local and global smoothing effects associated to the group , for the KdV scaling, in the context of Bourgain’s spaces. We begin with the Strichartz estimates.
For all , and , we have that
First, we observe, arguing as in Lemma 2.1 in [LPP], that is a solution to the linear equation
if and only if
is a solution to
Next, we use Lemma 3.3 in [Gi] to rewrite estimate (2.16) in the context of Bourgain’s spaces. We get that
Therefore, we deduce by using Stein’s theorem to interpolate estimate (2.17) with Plancherel’s identity , that
Next, we turn to the local Kato type smoothing effect.
Let and and . Then, it holds that
Since , we obtain applying estimate (4.3) in Theorem 4.1 of [KPV3] that
Moreover, by applying the Fourier inverse formula, it follows that
Therefore, Minkowski’s inequality, estimate (2.21), Plancherel’s identity and the Cauchy-Schwarz inequality imply that
which leads to estimate (2.19) since
On the other hand, if , we deduce from the Sobolev embedding , whenever , that
Therefore, we deduce arguing as above that
Finally, we derive the maximal function estimate.
Let , , and be such that . Then, we have that
The -maximal function for the KdV group derived in Theorem 2.7 of [KPV2] implies that
if . Then, a scaling argument and estimate (2.25) yield
since and .
whenever and , satisfying . This implies estimate arguing as in (2.22) that
for any and , which leads to (2.24) by squaring and summing over . ∎
2.4. Fractional Leibniz’s rules
First we state the classical fractional Leibniz rule estimate derived by Kenig, Ponce and Vega (See Theorems A.8 and A.12 in [KPV2]).
Let , with and with . Then,
Moreover, for , the value is allowed.
The next estimate is a frequency localized version of estimate (2.29), proved in [Mo], in the same spirit as Lemma 3.2 in [Ta].
Let and . Then,
with , and , and .
We also state an estimate to handle the multiplication by a term on the form , where is a real-valued function, in fractional Sobolev spaces.
Let and . Consider and two real-valued functions such that and belong to . Then, it holds that
The proof follows the lines of Lemma 2.7 in [MP] (see also [Mo, Mo1]). A version of Lemma 2.9 could also be stated for .
3. The gauge transformation
The gauge transform we will use is the one introduced by Tao in [Ta]. First we define an antiderivative of . We determine on the time axis by solving the ODE
Then we extend on the whole plan by setting
Clearly, it holds
and, according to the choice of on the time axis, it satisfies the equation
Now, we perform the following nonlinear transformation
First, using the identity , we compute
Then using (3.1) and the identity it follows that