Local well-posedness for dispersion generalized Benjamin-Ono equations in Fourier-Lebesgue spaces

In this paper, we consider the Cauchy problem for the dispersion generalized Benjamin-Ono equation (dgBO)

where $0\leq\alpha\leq 1$. Here $u:{\mathbb{R}}^{2}{\rightarrow}{\mathbb{R}}$ is a real-valued function and $|\partial_{x}|$ is the Fourier multiplier operator with symbol $|\xi|$. These equations arise as mathematical models for the weakly nonlinear propagation of long waves. The case $\alpha=0$ corresponds to the Benjamin-Ono equation and the case $\alpha=1$ corresponds to the Korteweg-de Vries equation. The equation (1.3) is invariant under the following scaling transform:

The scaling critical Sobolev space of (1.3) is $\dot{H}^{-\frac{1+\alpha}{2}}$ in the sense that the homogeneous Sobolev norm is invariant under the scaling transform (1.4).

The low regularity well-posedness of (1.3) in Sobolev spaces $H^{s}(\mathbb{R})$ has been extensively studied in recent years; we will summarize the most recent findings.

$\alpha=0$: (1.3) is known as the Benjamin-Ono equation. Ponce [23] first proved $C^{0}$ local well-posedness ($C^{0}$ LWP) in $H^{s}$ for $s\geq\frac{3}{2}$ using the energy method with enhanced dispersive smoothing. Later, Koch-Tzvetkov [21] refined the energy method and smoothing effect to improve LWP for $s>\frac{5}{4}$. This approach was further refined by Kenig-Keonig [17], which led to LWP for $s>\frac{9}{8}$.

Tao [25] applied a gauge transformation to effectively remove the derivative (or at least the worst terms that involve the derivative) from the nonlinearity and obtained the global well-posedness (GWP) for $s\geq 1$. By a synthesis of Tao’s gauge transformation and $X^{s,b}$ techniques, Ionescu-Kenig [12] improved the range to $s\geq 0$. Recently, Killip-Laurens-Visan [18] obtained a sharp GWP for $s>-\frac{1}{2}$ by using a new gauge transform and delving into the complete integrability of BO equations.

$\alpha=1$: (1.3) is the KdV equation. The first $C^{2}$ local well-posedness by contraction principle was proved by Kenig-Ponce-Vega [15] for $s>\frac{3}{4}$. Bourgain [1] extended this result to $s\geq 0$ by developing $X^{s,b}$ spaces. Then by developing the bilinear estimates in $X^{s,b}$ space, Kenig-Ponce-Vega [16] were able to prove local well-posedness for $s>-\frac{3}{4}$. Christ-Colliander-Tao [2], Guo [8] and Kishimoto [20] discussed the endpoint $s=-\frac{3}{4}$. Recently, Killip-Visan [19] obtained a sharp $C^{0}$ GWP for $s\geq-1$ by using similar strategies mentioned above for BO equations.

$0<\alpha<1$: Kenig-Ponce-Vega [14] have shown that (1.3) is $C^{0}$ locally well-posed provided $s\geq\frac{3}{4}(2-\alpha)$, using the energy method with enhanced smoothing effect. The Sobolev index has been pushed down to $s>1-\alpha$ by Guo [9]. Herr-Ionescu-Kenig-Koch [11] used a para-differential renormalization method to show the range of $s\geq 0$.

Note that there is only $C^{0}$ local well-posedness in $H^{s}$ if $0\leq\alpha<1$. This indicates that only the $H^{s}$ assumption on the initial data is not sufficient to prove local well-posedness of dgBO via Picard iteration, as the ill-posedness result from Molinet-Saut-Tzvetkov [22], by showing the solution mapping fails to be $C^{2}$ smooth from $H^{s}$ to $C\left([0,T];H^{s}\right)$ at the origin for any $s$. The reason is that the dispersive effect of the dispersive group of dgBO is too weak to recover the derivative in the nonlinearity. Hence the $high\times low$ interactions break down the $C^{2}$ smoothness.

Therefore, to study the well-posedness of dgBO when $0\leq\alpha\leq 1$ in the sense that the solution mapping is uniformly continuous, we might choose to abandon $H^{s}$ and prove it still via contraction mapping principle in some other space of initial data. For instance, Herr [10] applied a weighted Sobolev data space to obtain the well-posedness of dgBO equations for $0<\alpha<1$. We also got some inspiration from the work of Grünrock and Vega. Grünrock [5] obtained LWP of the modified KdV equation in Fourier-Lebesgue spaces $\widehat{H}^{s}_{r}$ for $\frac{4}{3}<r\leq 2$ and $s\geq\frac{1}{2}-\frac{1}{2r}$, which was enhanced by Grünrock-Vega [6] to $1<r\leq 2$ and $s\geq\frac{1}{2}-\frac{1}{2r}$. Furthermore, Grünrock in [7] studied the hierarchies of higher order mKdV and KdV equations systematically by using a mixed resolution space ${\widehat{X}}^{s,b}_{r,p}$ where the time parameter $p$ depends on the value of $r$. This initial data space, Fourier-Lebesgue space $\widehat{H}^{s}_{r}$, involves $L^{r}$-type integrability to a spatial weight, in addition to $H^{s}$ regularity, where the norm is defined by

for $s\in\mathbb{R},1\leq r\leq\infty$. From the scaling point of view, the spaces $\widehat{H}^{s}_{r}$ behave like the Bessel potential spaces ${H}^{s}_{r}$ which are embedded in $\widehat{H}^{s}_{r}$ for $1\leq r\leq 2$ by Hausdorff-Young inequality, and like $H^{\sigma}$ if $s-\frac{1}{r}+\frac{1}{2}=\sigma$.

Now, we state our main result:

This paper emphasises the local well-posedness result, which can be obtained by the contraction mapping principle so that the flow map is real analytic. Since the dispersive effect of dgBO is weak when $0\leq\alpha<1$, it previously failed to prove the local well-posedness by Picard iteration in the classical Sobolev space $H^{s}$. However, we observe a stronger local smoothing effect in $\widehat{H}_{r}^{s}$ when $r<2$, hence we may expect something better in this type of space. The answer is yes and the contraction mapping principle works.

We use the notation $X\lesssim Y$ for $X,Y\in\mathbb{R}$ to denote that there exists a constant $C>0$ such that $X\leq CY$. The notation $X\sim Y$ denotes that there exist positive constants $c,C$ such that $cY\leq X\leq CY$. For $a\in{\mathbb{R}}$, $a\pm$ denotes $a\pm\varepsilon$ for any sufficiently small $\varepsilon>0$. We use capitalized variables $\{N,L,N_{1},N_{2},\cdots\}$ to denote dyadic numbers, unless otherwise specified. $\widehat{u}$ denotes the standard Fourier transform $\mathcal{F}_{x}u$, $\mathcal{F}_{t}u$ or $\mathcal{F}_{t,x}u$. Let $\omega(\xi)=-\xi|\xi|^{1+\alpha}$ be the dispersion relation associated with the equation (1.3) and $W_{\alpha}(t)={\mathcal{F}}^{-1}e^{it\omega(\xi)}{\mathcal{F}}$ be the linear propagator.

Let $\psi\in C_{0}^{\infty}({\mathbb{R}})$ be a real-valued, non-negative, even, and radially-decreasing function such that ${\mbox{supp}}\psi\subset[-5/4,5/4]$ and $\psi\equiv 1$ in $[-1,1]$. ${\mathbb{Z}}_{+}={\mathbb{N}}\cup\{0\}$. For a dyadic number $N\in 2^{{\mathbb{Z}}_{+}}$, denote $\chi(\xi):=\psi(\xi)-\psi(2\xi)$ and $\chi_{N}:=\chi(N^{-1}\cdot)$. The Littlewood-Paley projectors for frequency and modulation are defined by

The Fourier-Lebesgue type Bourgain space $\widehat{X}_{r}^{s,b}$ associated to (1.3) is defined by the norm

for $s,b\in\mathbb{R}$ and $1\leq r\leq\infty$, where $\langle\cdot\rangle=(1+|\cdot|^{2})^{\frac{1}{2}}$. When $s=b=0$, we write $\widehat{X}^{s,b}_{r}$ as $\widehat{L}^{r}_{t,x}$ for simplicity. We use the dyadic frequency localization operators $P_{N}$ and $Q_{L}$ to rewrite (2.2) as

By slightly modifying the proof of mKdV equations for the unitary group $\{e^{-t\partial_{x}^{3}}\}$ in [5], it is easy to obtain some linear estimates for the unitary group $\{e^{-t|\partial_{x}|^{1+\alpha}}\}$ in $\widehat{X}^{s,b}_{r}$ spaces.

As a corollary of Lemma 2.1, if $b>1/r$, we have the following embedding

The Duhamel integral of (1.3) is given by

where $\mathcal{N}$ is the nonlinear function of $u$. Let $0<T\leq 1$ and $\psi_{T}(t)=\psi(t/T)$. To apply the contraction principle in $\widehat{X}^{s,b}_{r}$, we shall introduce the restriction norm space $\widehat{X}^{s,b}_{r}(T)$ by

For $u\in\widehat{X}^{s,b}_{r}(T)$ with extension $\tilde{u}\in\widehat{X}^{s,b}_{r}$, an extension of $\Gamma u$ is given by

Thus, it reduces to prove

Furthermore, we prove the following linear estimates.

Therefore, to prove the local well-posedness of dgBO (1.3), we focus on showing nonlinear estimates in $\widehat{X}^{s,b}_{r}$ spaces. Given $b>\frac{1}{r}$ and $b^{\prime}\in(b-1,0]$, it turns to prove

Without loss of generality, let $b=\frac{1}{r}+\varepsilon$ and $b^{\prime}=-\frac{1}{r^{\prime}}+2\varepsilon$. For brevity, in the nonlinear estimate (2.19), we still use $u$ to denote $\tilde{u}$. By duality argument, (2.19) is implied as

For $\xi_{1},\xi_{2}\in{\mathbb{R}}$ and $\omega:{\mathbb{R}}\rightarrow{\mathbb{R}}$, define the resonance function

which plays a crucial role in bilinear estimates of the $X^{s,b}$-type space. See [24] for a perspective discussion. Let $\sigma_{j}$ and $\sigma$ denote the modulations given by

Under the restrictions $\xi=\xi_{1}+\xi_{2}$ and $\tau=\tau_{1}+\tau_{2}$, we have

We apply Littlewood-Paley dyadic decomposition (2.1) to each component of (2.20). Then to prove (2.20), it suffices to prove

where $C(L_{1},L_{2},L)$ and $C(N_{1},N_{2},N)$ are suitable bounds that allow us to sum over all dyadic numbers.

In this section, we prove some dyadic bilinear estimates which are crucial for proving (2.23) in the next section. For compactly supported non-negative functions $f_{1},f_{2}\in L^{r^{\prime}}({\mathbb{R}}\times{\mathbb{R}})$ with

and $f\in L^{r}({\mathbb{R}}\times{\mathbb{R}})$ with

we define

It is convenient to define $L_{\max}$, $L_{\operatorname*{med}}$ and $L_{\min}$ as the maximum, median and minimum values of $L$, $L_{1}$ and $L_{2}$, respectively. Similarly, define $N_{\max}$, $N_{\operatorname*{med}}$ and $N_{\min}$ as the maximum, median and minimum values of $N$, $N_{1}$ and $N_{2}$.

Now, we state our bilinear estimates.

In the sequel, we will consider different interactions. To take advantage of the resonance function $\Omega(\xi_{1},\xi_{2})$, we shall first integrate over $\xi_{1}$ and $\xi_{2}$.