On well-posedness for some dispersive perturbations of Burgers' equation

TL;DR

This paper establishes local and global well-posedness results for dispersive perturbations of Burgers' equation, including the Benjamin-Ono equation, in specific Sobolev spaces, depending on the dispersion parameter.

Contribution

It proves well-posedness for a class of dispersive Burgers' equations, extending known results to low dispersion regimes and identifying conditions for global solutions.

Findings

Local well-posedness in H^s for s > 3/2 - 5α/4

Global well-posedness in H^{α/2} for α > 6/7

Includes the low dispersion Benjamin-Ono equation case

Abstract

We show that the Cauchy problem for a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin-Ono equation $\partial$\_t u -- D^$\alpha$\_x $\partial$\_x u = $\partial$\_x(u^2), 0 < $\alpha$ $\le$ 1, is locally well-posed in H^s (R) when s > 3 /2 -- 5$\alpha$ /4. As a consequence, we obtain global well-posedness in the energy space H^{$\alpha$/2} (R) as soon as $\alpha$ > 6/7 .

Equations

$${\partial }_{t}u-D_x^{\alpha }{\partial }_{x}u={\partial }_x(u^2),0<\alpha \le 1,$$

$${\partial }_{t}u-D_x^{\alpha }{\partial }_{x}u={\partial }_x(u^2),$$

$$\widehat{D_x^{\alpha }\varphi }(\xi )=|\xi {|}^{\alpha }\hat{\varphi }(\xi ).$$

$${\partial }_{t}u-w(D_x){\partial }_{x}u+{\partial }_x(u^2)=0,$$

$$M(u)={\int }_{\mathbb{R}}{u}^2(x,t)dx,$$

$$H(u)=\int_{\mathbb{R}}\big{(}\frac{1}{2}|D^{\frac{\alpha}{2}}u(x,t)|^{2}+\frac{1}{3}u^{3}(x,t)\big{)}\,dx\,.$$

$$u_{\lambda }(x,t)={\lambda }^{\alpha }u(\lambda x,{\lambda }^{\alpha +1}t),$$

$$\Vert u{\Vert }_{L_T^{\infty }{H}_x^s}\le c\Vert u_0{\Vert }_{H_x^s}{e}^{c{\int }_0^T\Vert {\partial }_{x}u{\Vert }_{L_x^{\infty }}dt}.$$

$$\Vert P_{N}u{\Vert }_{L_T^{\infty }{L}_x^2}^2\lesssim \Vert P_N{u}_0{\Vert }_{L_x^2}^2+\underset{t\in ]0,T[}{\sup }\left|{\int }_0^t{\int }_{\mathbb{R}}{P}_N{\partial }_x(u^2)P_{N}udxdt\right|.$$

$${\int }_{\mathbb{R}\times [0,t]}{P}_N{\partial }_x(u^2)P_{N}u={\int }_{\mathbb{R}\times [0,t]}{\partial }_x{P}_N(u_{\gtrsim N}{u}_{\gtrsim N})P_{N}u+{\int }_{\mathbb{R}\times [0,t]}{\partial }_x{P}_N(u_{\ll N}u)P_{N}u$$

$$\Vert u{\Vert }_{X^{s-1,1}}\lesssim \Vert u_0{\Vert }_{H^{s-1}}+\Vert {\partial }_x{J}_x^{s-1}(u^2){\Vert }_{L_{x,T}^2}\lesssim \Vert u_0{\Vert }_{H^s}+\Vert J_x^s(u^2){\Vert }_{L_T^{\infty }{H}_x^s}.$$

$${\partial }_{t}u+L_{\alpha +1}u={\partial }_x(u^2),$$

$$|{\partial }^{\beta }{\omega }_{\alpha +1}(\xi )|\sim |\xi {|}^{\alpha +1-\beta },\beta \in \{0,1,2\},$$

$$|{\partial }^{\beta }{\omega }_{\alpha +1}(\xi )|\lesssim |\xi {|}^{\alpha +1-\beta },\beta \ge 3.$$

$$C([0,T]:H^s(\mathbb{R}))\cap X_T^{s-1,1}\cap L^2(0,T:W^{s-s_{\alpha }+(1-\alpha {)}_{-},\infty }(\mathbb{R})).$$

$$|{\omega }_{\alpha +1}(\xi )|\lesssim |\xi |\text{ for }|\xi |\lesssim 1,$$

$$H_{\alpha +1}(u)={\int }_{\mathbb{R}}(\frac{1}{2}|{\Lambda }^{\alpha /2}u(x,t){|}^2+\frac{1}{3}{u}^3(x,t))dx$$

$$\widehat{{\Lambda }^{\alpha /2}v}(\xi )={\left|\frac{{\omega }_{\alpha +1}(\xi )}{\xi }\right|}^{1/2}\hat{v}(\xi ),$$

$$|{\partial }^{\beta }m({\xi }_1,\cdots ,{\xi }_n)|\lesssim \prod _{i=1}^n|{\xi }_i{|}^{-{\beta }_i},\forall \beta \in {\mathbb{N}}^n.$$

$$J_x^{s}u={\mathcal{F}}_x^{-1}(⟨\xi {⟩}^s{\mathcal{F}}_{x}u)\text{ and }{D}_x^{s}u={\mathcal{F}}_x^{-1}(|\xi {|}^s{\mathcal{F}}_{x}u).$$

$$\eta \in C_0^{\infty }(\mathbb{R}),0\le \eta \le 1,{\eta }_{{|}_{[-1,1]}}=1\text{mbox}and\text{mbox}supp(\eta )\subset [-2,2].$$

$${\varphi }_{2^l}(\xi ):=\varphi (2^{-l}\xi ),{\tilde{\varphi }}_{2^l}(\xi )={\varphi }_{\sim 2^l}(\xi ):=\tilde{\varphi }(2^{-l}\xi ),$$

$${\psi }_{2^l}(\xi ,\tau )={\varphi }_{2^l}(\tau -{\omega }_{\alpha +1}(\xi )).$$

$${\psi }_1(\xi ,\tau ):=\eta (2(\tau -{\omega }_{\alpha +1}(\xi ))).$$

$$\sum _{N>0}{\varphi }_N(\xi )=1\forall \xi \in {\mathbb{R}}^{\ast },\text{mbox}supp({\varphi }_N)\subset \{\frac{N}{2}\le |\xi |\le 2N\},N\in \{2^k:k\in \mathbb{Z}\},$$

$$\sum _{L\ge 1}{\psi }_L(\xi ,\tau )=1\forall (\xi ,\tau )\in {\mathbb{R}}^2,L\in \{2^k:k\in \mathbb{N}\}.$$

$$P_{N}u=\mathcal{F}^{-1}_{x}\big{(}\phi_{N}\mathcal{F}_{x}u\big{)},\quad P_{\sim N}u=\mathcal{F}^{-1}_{x}\big{(}\widetilde{\phi}_{N}\mathcal{F}_{x}u\big{)}\quad Q_{L}u=\mathcal{F}^{-1}\big{(}\psi_{L}\mathcal{F}u\big{)},$$

$$\Vert f{\Vert }_{L_T^p{B}_x}={\Vert \Vert f{\Vert }_B\Vert }_{L^p([0,T])}\text{ and }\Vert f{\Vert }_{L_t^p{B}_x}={\Vert \Vert f{\Vert }_B\Vert }_{L^p(\mathbb{R})}.$$

$$\Vert u{\Vert }_{\overline{M}}=\Vert {\mathcal{F}}_x^{-1}(⟨|\xi {|}^{-1}⟩{\mathcal{F}}_{x}u(\xi )){\Vert }_M.$$

$$\Vert u{\Vert }_{X^{s,b}}=\Vert ⟨\xi {⟩}^s⟨\tau -{\omega }_{\alpha +1}(\xi ){⟩}^b{\mathcal{F}}_{tx}u{\Vert }_{L^2}.$$

Full text

[TABLE]

is locally well-posed in $H^{s}({\mathbb{R}})$ when $s>s_{\alpha}:=\frac{3}{2}-\frac{5\alpha}{4}$. As a consequence, we obtain global well-posedness in the energy space $H^{\frac{\alpha}{2}}({\mathbb{R}})$ as soon as $\frac{\alpha}{2}>s_{\alpha}$, i.e. $\alpha>\frac{6}{7}$.

1. Introduction

This paper is concerned with the initial value problem for a class of dispersive perturbations of Burgers’ equation containing in particular the low dispersion Benjamin-Ono equation

[TABLE]

where $u=u(x,t)$ is a real valued function, $x\in\mathbb{R}$, $t\in\mathbb{R}$, $\alpha>0$ and $D^{\alpha}_{x}$ is the Riesz potential of order $-\alpha$, which is given via Fourier transform by

[TABLE]

The cases $\alpha=2$ and $\alpha=1$ correspond to the well-known Korteweg-de Vries (KdV) and Benjamin-Ono (BO) equations. In the case $\alpha=0$, $\partial_{x}u$ is a transport term, so that there is no dispersion anymore and equation (1.1) corresponds merely to the inviscid Burgers equation.

While the Cauchy problem associated with (1.1) is now very well-understood in the case $\alpha\geq 1$, our objective here is to investigate the case of low dispersion when $0<\alpha<1$, which seems of great physical interest (see for example the introductions in [20, 22] and the references therein). In particular, in the case $\alpha=\frac{1}{2}$, the dispersion is somehow reminiscent of the linear dispersion of finite depth water waves with surface tension. The corresponding Whitham equation with surface tension writes

[TABLE]

where $u=u(x,t)$ is a real valued function, $x\in\mathbb{R}$, $t\in\mathbb{R}$, $\omega(D_{x})$ is the Fourier multiplier of symbol $w(\xi)=\left(\frac{\tanh(\xi)}{\xi}\right)^{\frac{1}{2}}\left(1+\tau\xi^{2}\right)^{\frac{1}{2}}$ and $\tau$ is a positive parameter related to the surface tension. Note that for high frequencies $w(\xi)\sim|\xi|^{\frac{1}{2}}$, which corresponds exactly to equation (1.1) in the $L^{2}$ critical case.

Equation (1.1) is hamiltonian. In particular, the quantities

[TABLE]

and

[TABLE]

are (at least formally) conserved by the flow associated to (1.1). Moreover, equation (1.1) is invariant under the scaling transformation

[TABLE]

for any positive number $\lambda$. A straightforward computation shows that $\|u_{\lambda}\|_{\dot{H}^{s}}=\lambda^{s+\alpha-\frac{1}{2}}\|u_{\lambda}\|_{\dot{H}^{s}}$, and thus the critical index corresponding to (1.1) is $\tilde{s}_{\alpha}=\frac{1}{2}-\alpha$. In particular, equation (1.1) is $L^{2}$-critical for $\alpha=\frac{1}{2}$ and energy critical for $\alpha=\frac{1}{3}$.

Next we recall some important facts about the initial value problem (IVP) associated with (1.1) in $L^{2}$-based Sobolev spaces $H^{s}(\mathbb{R})$222 Recall that the natural space where the quantities (1.3) and (1.4) make sense is $H^{\frac{\alpha}{2}}(\mathbb{R})$, at least when $\alpha\geq\frac{1}{3}$.. For results in weighted Sobolev spaces, we refer to Fonseca, Linares and Ponce [10] and the references therein. It was proved by Molinet, Saut and Tzvetkov [25], that, due to bad high-low frequency interactions in the nonlinearity, the IVP associated with (1.1) cannot be solved by a contraction argument on the corresponding integral equation in any Sobolev space $H^{s}(\mathbb{R})$, $s\in\mathbb{R}$, as soon as $\alpha<2$. Thus, one needs to use compactness arguments based on a priori estimates on the solution and on the difference of two solutions at the required level of regularity.

Standard energy estimates, the Kato-Ponce commutator estimate and Gronwall’s inequality provide the following bound for solutions of (1.1)

[TABLE]

Therefore, one way to obtain a priori estimates in $H^{s}$ is to control $\|\partial_{x}u\|_{L^{1}_{T}L^{\infty}_{x}}$ at the $H^{s}$-level. This can be done easily in $H^{\frac{3}{2}+}(\mathbb{R})$ by using the Sobolev embedding $H^{\frac{1}{2}+}(\mathbb{R})\hookrightarrow L^{\infty}(\mathbb{R})$. In the Bejamin-Ono case $\alpha=1$, Ponce [31] used the smoothing effects (Strichartz estimates, Kato type smoothing estimate and maximal function estimate) associated with the dispersive part of (1.1) to obtain well-posedness in $H^{\frac{3}{2}}(\mathbb{R})$. Later on, Koch and Tzvetkov [21] introduced a refined Strichartz estimate, derived by chopping the time interval in small pieces whose length depends on the spatial frequency of the solution, which allowed them to prove local well-posedness for BO in $H^{\frac{5}{4}+}(\mathbb{R})$. This refined Strichartz estimate was then improved by Kenig and Koenig [17] and the local well-posedness for BO pushed down to $H^{\frac{9}{8}+}(\mathbb{R})$. Recently, Linares, Pilod and Saut [22] extended Kenig and Koenig’s result to (1.1) in the range $0<\alpha<1$ by proving that the corresponding initial value problem is well-posed in $H^{s}(\mathbb{R})$ for $s>\frac{3}{2}-\frac{3\alpha}{8}$. Note that even very few dispersion (when $0<\alpha\ll 1$) allows to enlarge the resolution space, which is not the case anymore when there is no dispersion. Indeed, it is known that the IVP associated with Burgers’ equation is ill-posed in $H^{\frac{3}{2}}(\mathbb{R})$ (c.f. Remark 1.6. in [22]).

Another technique to obtain suitable estimates on the solutions at low regula-rity is the use of a nonlinear gauge transformation which allows to weaken the bad frequency interactions in the nonlinear term. Such transformation was introduced by Tao [34] for the Benjamin-Ono equation and enabled him to prove global well-posedness for BO in $H^{1}(\mathbb{R})$. By using this gauge transformation in the context of Bourgain’s spaces $X^{s,b}$, Burq and Planchon [6], respectively Ionescu and Kenig [15], proved that the IVP associated with BO is well-posed in $H^{\frac{1}{4}+}(\mathbb{R})$, respectively $L^{2}(\mathbb{R})$. We also refer to Molinet and Pilod [26] for another proof of Ionescu and Kenig’s result with stronger uniqueness result (for example unconditional uniqueness in $H^{\frac{1}{4}+}(\mathbb{R})$). In [13], Herr, Ionescu, Kenig and Koch were able to extend Ionescu and Kenig’s result to the whole range $1<\alpha<2$. By using a paradifferential gauge transformation, they proved that the IVP associated to (1.1) is globally well-posed in $L^{2}(\mathbb{R})$ for $1<\alpha<2$.

Recently Molinet and Vento [29] introduced a new method to obtain energy estimates at low regularity for strongly nonresonant dispersive equations. It starts with the classical estimate for the dyadic piece $P_{N}u$ localized in turn of the spatial frequency $N$,

[TABLE]

To control the last term on the right-hand side of the energy estimate (1.5), one performs a paraproduct decomposition

[TABLE]

and put the derivative on the lowest spatial frequencies by “integrating by parts”333Since we work with frequency localized functions, this corresponds actually to use suitable commutator estimates. . The idea is then to perform a dyadic decomposition of each function in term of its modulation variable and to put one of them (the one with the greatest modulation) in the space $X^{s-1,1}$. This allows to recover at least $|\Omega|N^{-1}$ where $\Omega$ is the resonance function. The price to pay is to handle the characteristic function $1_{]0,t[}$ which appears after extending the functions to $\mathbb{R}^{2}$ and is not continuous in $X^{s-1,1}$. On the positive side, the $X^{s-1,1}$ norm of $u$ is relatively simple to control by using the classical linear estimates in Bourgain’s spaces as follows

[TABLE]

Thus, for $s>\frac{1}{2}$, one can easily concludes the bilinear estimate since $H^{s}(\mathbb{R})$ is a Banach algebra. By using this method, Molinet and Vento proved that the IVP associated with (1.1) is locally well-posed in $H^{s}(\mathbb{R})$ for $s\geq 1-\frac{\alpha}{2}$ when $1\leq\alpha\leq 2$. Note that Guo [12] also proved local well-posedness in $H^{s}(\mathbb{R})$ for $s>2-\alpha$ when $1\leq\alpha\leq 2$ without using a gauge transformation. He used instead the short time Bourgain’s spaces in the way of Ionescu, Kenig and Tataru in [16].

Throughout this paper we consider the class of dispersive equations

[TABLE]

where $u=u(x,t)$ is a real-valued function, $x\in{\mathbb{R}}$, $t\in{\mathbb{R}}$, $\alpha>0$ and the linear operator $L_{\alpha+1}$ satisfies the following hypothesis.

Hypothesis 1**.**

We assume that $L_{\alpha+1}$ is the Fourier multiplier operator by $i\omega_{\alpha+1}$ where $\omega_{\alpha+1}$ is a real-valued odd function belonging to $C^{1}({\mathbb{R}})\cap C^{\infty}({\mathbb{R}}^{\ast})$ and satisfying: There exists $\xi_{0}>0$ such that for any $\xi\geq\xi_{0}$, it holds

[TABLE]

and

[TABLE]

Remark 1.1*.*

We easily check that the following operators satisfy Hypothesis 1:

1. (1)

The purely dispersive operator $L_{\alpha+1}=-D_{x}^{\alpha}\partial_{x}$, $\alpha>0$. 2. (2)

The Whitham operator with symbol $\omega(\xi)=\xi\left(\frac{\tanh(\xi)}{\xi}\right)^{\frac{1}{2}}\left(1+\tau\xi^{2}\right)^{\frac{1}{2}}$, $\tau>0$ for $\alpha=1/2$. 3. (3)

The linear Intermediate Long Wave operator $L_{\alpha+1}=\partial_{x}D_{x}\coth(D_{x})$ for $\alpha=1$.

In this article, we show that the initial value problem (IVP) associated with (1.8) is locally well-posed in $H^{s}(\mathbb{R})$ for $s>\frac{3}{2}-\frac{5\alpha}{4}$ when $0<\alpha\leq 1$, which improves Linares, Pilod and Saut’s result in [22].

Theorem 1.2**.**

Assume that $L_{\alpha+1}$ satisfies Hypothesis 1 with $0<\alpha\leq 1$ and let $s>s_{\alpha}=\frac{3}{2}-\frac{5\alpha}{4}$. Then, for any $u_{0}\in H^{s}(\mathbb{R})$, there exist $T=T(\|u_{0}\|_{H^{s}})>0$ and a unique solution $u$ of the IVP associated with (1.8) in the class

[TABLE]

Moreover, for any $0<T^{\prime}<T$, there exists a neighborhood $\mathcal{U}$ of $u_{0}$ in $H^{s}(\mathbb{R})$ such that the flow-map data solution $v_{0}\mapsto v$ is continuous from $\mathcal{U}$ into $C([0,T^{\prime}]:H^{s}(\mathbb{R}))$.

Remark 1.3*.*

In the case $\alpha=1$ and $L_{\alpha+1}=-D_{x}^{\alpha}\partial_{x}$, our result provides a proof of the local well-posedness for BO in $H^{\frac{1}{4}+}(\mathbb{R})$. In other words, we recover Burq and Planchon’s result in [6] without using a gauge transformation.

If we assume moreover that the symbol $\omega_{\alpha+1}$ satisfies

[TABLE]

we easily see that the Hamiltonian

[TABLE]

where $\Lambda^{\alpha/2}$ is the space Fourier multiplier defined by

[TABLE]

as well as (1.3) are conserved by the flow associated to (1.8). Iterating Theorem 1.2, we obtain global well-posedness as soon as $\alpha>\frac{6}{7}$.

Corollary 1.1**.**

Assume that $L_{\alpha+1}$ satisfies Hypothesis 1 and (1.12) with $\frac{6}{7}<\alpha\leq 1$. Then the Cauchy problem associated with (1.8) is globally well-posed in the energy space $H^{\frac{\alpha}{2}}({\mathbb{R}})$.

Remark 1.4*.*

The operators defined in Remark 1.1 also satisfy assumption (1.12).

Remark 1.5*.*

Based on numerical computations by Klein and Saut [20], the global well-posedness of (1.1) was conjectured [20, 22] in the $L^{2}$-subcritical case $\alpha>\frac{1}{2}$. Here, we answer to part of this conjecture when $\alpha>\frac{6}{7}$. Up to our knowledge, this is the first global existence result for $\alpha<1$.

Remark 1.6*.*

It would be interesting to obtain results on the dispersion decay of the solutions associated to small data for (1.1) with low dispersion. Some progress in this direction were recently done by Ifrim and Tataru [14] for the Benjamin-Ono equation.444Note also that the authors give another proof of the well-posedness of the Benjamin-Ono equation in $L^{2}$ without using the $X^{s,b}$ structure but still based on Tao’s renormalization argument together with modified energies.

Remark 1.7*.*

In [23], Linares, Pilod and Saut showed that the solitary waves associated to (1.1) are orbitally stable in the energy space $H^{\frac{\alpha}{2}}(\mathbb{R})$ as soon as $\alpha>\frac{1}{2}$, conditionally to the global well-posedness in $H^{\frac{\alpha}{2}}$ (see Remark 2.1 in [23]). We also refer to Arnesen [2] and Angulo [1] for other proofs of this result. Theorem 2.14 in [23] combined with Theorem 1.2 provides then a complete orbital stability result in the energy space as soon as $\alpha>\frac{6}{7}$.

Now, we discuss the main ingredients in the proof of Theorem 1.2. Since it is not clear wether one can take advantage of a gauge transformation in the case $\alpha<1$ or not, we elect to follow the energy method introduced in [29]. However, we need to add several key ingredients.

Firstly, in order to close the bilinear estimate (1.7) in $H^{s}(\mathbb{R})$ for $s\leq\frac{1}{2}$, we use the norm $\|\cdot\|_{L^{2}_{T}L^{\infty}_{x}}$, which is in turn estimated by using the refined Strichartz estimate as in [21, 17, 22]. Then, we can control the last term on the right-hand side of (1.7) by using the fractional Leibniz rule as $\|J^{s}_{x}(u^{2})\|_{L^{\infty}_{T}H^{s}_{x}}\lesssim\|u\|_{L^{2}_{T}L^{\infty}_{x}}\|J^{s}_{x}u\|_{L^{\infty}_{T}L^{2}_{x}}$.

The norm $\|\cdot\|_{L^{2}_{T}L^{\infty}_{x}}$ is also an important ingredient to close the energy estimate (1.5). This creates a serious technical difficulty. Indeed to handle some commutators with those norms, we need then to use a generalized Coifman-Meyer theorem for multilinear Fourier multipliers $m(\xi_{1},\cdots,\xi_{n})$ satisfying the Marcinkiewicz type condition

[TABLE]

Such a theorem was proved by Muscalu, Pipher, Tao and Thiele [30] in the bilinear case and can be deduced from a result of Bernicot [3] in the multilinear case (see Section 2.3 for more details).

With this theorem in hand, we can estimate the first term of (1.6) corresponding to the high-high frequency interactions by using the norms $\|u\|_{X^{s-1,1}_{T}}$ and $\|J_{x}^{1-\alpha}u\|_{L^{2}_{T}L^{\infty}_{x}}$ as explained above. For the second term, we would like to integrate by parts and use the $\|\cdot\|_{X^{s-1,1}}$-norm as in [29] but the resonance relation $|\Omega|\sim N_{min}N_{max}^{\alpha}$ would not be sufficient to recover the “big” derivative we lost by using this norm. This is one of the main difficulty to work at low dispersion $\alpha<1$. For this reason, we modify the energy by adding a cubic term, constructed so that the contribution of its time derivative coming from the linear part of the equation cancels out the high-low frequency term. It is worth noticing that this modified energy is defined in Fourier variables in the same spirit of the modified energy in the I-method [9]. We also refer to our recent works [27, 28] on the modified Korteweg-de Vries equation both on the line and on the torus for a similar strategy using a modified energy. Note that we gain a factor $N_{min}N_{max}^{\alpha}$ on the additional cubic term. On the other hand, the contribution of its time derivative coming from the nonlinear part of the equation is of order four and contains one more spatial derivative. For $\alpha<1$, it is clear that when this spatial derivative falls on the term with the highest spatial frequencies we should lose $N_{min}^{-1}N_{max}^{1-\alpha}$ which is not acceptable for some high-low frequency interaction terms. The crucial observation here is that there is a fundamental cancellation between two of those terms exhibiting the badest high-low frequency interactions.

Those ingredients are enough to derive a suitable a priori estimate for a solution of (1.8). However, things are more complicated to get an estimate for the difference of two solutions $u_{1}$ and $u_{2}$, since the corresponding equation lacks of symmetry. For this reason, we are only able to derive an energy estimate for the difference $w=u_{1}-u_{2}$ at low regularity $H^{\sigma}$, $\sigma<0$, and with an additional weight on low frequency. This is sufficient for our purposes, since we only need this estimate for the difference of solutions having the same low frequency part in order to prove the uniqueness and the continuity of the flow map (c.f. [15]). However, the bilinear estimate is not straightforward as before when working with negative regularity $H^{\sigma}$, $\sigma<0$. To overcome this last difficulty, we follow the strategy in [29] and work with the sum space $F^{s,\frac{1}{2}}=X^{s-1,1}+X^{s,(\frac{1}{2})_{+}}$ instead of working with $X^{s-1,1}$ only.

Finally, it is worth noticing that even in the particular case of purely power dispersion where scaling invariance occurs, equation (1.8) is $L^{2}$-super critical for $\alpha<1/2$ and thus we will not be able to use a classical scaling argument to prove the local existence result. Roughly speaking, our method consists in cutting the spatial frequencies of the solution into two parts $P_{\leq N_{0}}$ and $P_{>N_{0}}$. We gain some positive factor of the time $T$ (but lose some positive factor of $N_{0}$) when estimating the low frequency part whereas we gain a negative factor of $N_{0}$ when estimating the high frequency part. This will allow us to close our estimates on $]0,T[$ for smooth solution to (1.1) by taking $N_{0}$ big enough and $T>0$ small enough. Finally, the continuity of the solution as well as the continuity with respect to initial data will be proved by using a kind of uniform decay estimate on the high spatial frequencies of the solution.

The paper is organized as follows: in Section 2, we introduce the notation, define the function spaces and state some important preliminary estimates related the generalized Coifman-Meyer theorem. In Section 3, we derive multilinear estimates at the $L^{2}$-level. Those estimates will be used in Sections 4 and 5 to prove estimates for the solution and the difference of two solutions of the equation. Finally, we give the proof of Theorem 1.2 in Section 6.

Acknowledgements

The authors would like to thank Jean-Claude Saut for constant encouragements. They are also grateful to Terence Tao for helpful comments on the generalized Coifman-Meyer theorem in Section 2.3. L.M and S.V were partially supported by the ANR project GEO-DISP. D.P. was partially supported by CNPq/Brasil grant 3035051/2016-7.

2. Notation, function spaces and preliminary estimates

2.1. Notation

For any positive numbers $a$ and $b$, the notation $a\lesssim b$ means that there exists a positive constant $C$ such that $a\leq Cb$, and we denote $a\sim b$ when $a\lesssim b$ and $b\lesssim a$. We also write $a\ll b$ if the estimate $b\lesssim a$ does not hold. If $x\in{\mathbb{R}}$, $x_{+}$, respectively $x_{-}$ will denote a number slightly greater, respectively lesser, than $x$. We also set $\langle x\rangle=(1+x^{2})^{\frac{1}{2}}$.

For $u=u(x,t)\in\mathcal{S}^{\prime}({\mathbb{R}}^{2})$, $\mathcal{F}u=\hat{u}$ will denote its space-time Fourier transform, whereas $\mathcal{F}_{x}u$, respectively $\mathcal{F}_{t}u$ will denote its Fourier transform in space, respectively in time. For $s\in{\mathbb{R}}$, we define the Bessel and Riesz potentials of order $-s$, $J^{s}_{x}$ and $D^{x}_{x}$, by

[TABLE]

Throughout the paper, we fix a smooth cutoff function $\eta$ such that

[TABLE]

We set $\phi(\xi):=\eta(\xi)-\eta(2\xi)$. Let $\widetilde{\phi}\in C_{0}^{\infty}(\mathbb{R})$ be such that $\widetilde{\phi}_{|_{\pm[\frac{1}{2},2]}}\equiv 1$ and $\text{supp}\,(\widetilde{\phi})\subset\pm[\frac{1}{4},4]$. For $l\in\mathbb{Z}$, we define

[TABLE]

and, for $l\in\mathbb{N}^{*}$,

[TABLE]

By convention, we also denote

[TABLE]

Any summations over capitalized variables such as $N$ or $L$ are presumed to be dyadic. Unless stated otherwise, we work with homogeneous dyadic decomposition for the space frequency variables and non-homogeneous decompositions for modulation variables, i.e. these variables range over numbers of the form $\{2^{k}:k\in\mathbb{Z}\}$ and $\{2^{k}:k\in\mathbb{N}\}$ respectively. Then, we have that

[TABLE]

and

[TABLE]

Let us define the Littlewood-Paley multipliers by

[TABLE]

$P_{\geq N}:=\sum_{K\geq N}P_{K}$, $P_{\gtrsim N}:=\sum_{K\geq N}P_{\sim K}$, $P_{\leq N}:=\sum_{K\leq N}P_{K}$, $P_{\lesssim N}:=\sum_{K\leq N}P_{\sim K}$, $Q_{\geq L}:=\sum_{K\geq L}Q_{K}$ and $Q_{\leq L}:=\sum_{K\leq L}Q_{K}$. For the sake of brevity we often write $u_{N}=P_{N}u$, $u_{\leq N}=P_{\leq N}u$, $\cdots$

Finally, if $N_{1}$, $N_{2}$ are two dyadic numbers, we denote $N_{1}\vee N_{2}=\max\{N_{1},N_{2}\}$ and $N_{1}\wedge N_{2}=\min\{N_{1},N_{2}\}$.

2.2. Function spaces

For $1\leq p\leq\infty$, $L^{p}$ denotes the usual Lebesgue space and for $s\in{\mathbb{R}}$, $H^{s}$ is the $L^{2}$-based Sobolev space with norm $\|f\|_{H^{s}}=\|J_{x}^{s}f\|_{L^{2}}$. If $B$ is a space of functions on ${\mathbb{R}}$, $T>0$ and $1\leq p\leq\infty$, we define the spaces $L^{p}_{T}B_{x}$ and $L^{p}_{t}B_{x}$ by the norms

[TABLE]

If $M$ is a normed space of functions, we will denote $\overline{M}$ its subspace associated with the weighted norm:

[TABLE]

For $s,b\in{\mathbb{R}}$ we introduce the Bourgain space $X^{s,b}$ associated with the dispersive Burgers’ equation as the completion of the Schwartz space $\mathcal{S}({\mathbb{R}}^{2})$ under the norm

[TABLE]

We will also work in the sum space $F^{s,b}=X^{s-1,b+\frac{1}{2}}+X^{s,b_{+}}$ endowed with the norm

[TABLE]

For $s\in{\mathbb{R}}$, we define our resolution space $Y^{s}$ by the norm

[TABLE]

We will also need to consider the space $Z^{s}$ equipped with the norm

[TABLE]

Finally, we will use restriction in time versions of these spaces. Let $T>0$ be a positive time and $M$ be a normed space of space-time functions. The restriction space $M_{T}$ will be the space of functions $u:{\mathbb{R}}\times]0,T[\to{\mathbb{R}}$ satisfying

[TABLE]

2.3. Generalized Coifman-Meyer theorem

Definition 2.1**.**

For $n\geq 1$ and $\chi$ a bounded measurable function on ${\mathbb{R}}^{n}$, we define the multilinear Fourier multiplier operator $\Pi^{n}_{\chi}$ on $\mathcal{S}({\mathbb{R}})^{n}$ by

[TABLE]

If $\sigma$ is a permutation of $\{1,\ldots,n\}$, then it is clear that

[TABLE]

where $\chi_{\sigma}(\xi_{1},\ldots,\xi_{n})=\chi(\xi_{\sigma(1)},\ldots,\xi_{\sigma(n)})$. For any $t>0$, we define ${\mathbb{R}}^{n}_{t}={\mathbb{R}}^{n}\times]0,t[$ and for $u_{1},\ldots u_{n+1}\in\mathcal{S}({\mathbb{R}}^{2})$, we set

[TABLE]

When there is no risk of confusion, we will write $G_{t}^{n}=G_{t,\chi}^{n}$ with $\chi\in L^{\infty}({\mathbb{R}}^{n})$.

From Plancherel theorem, it is not too hard to check that

[TABLE]

where $\tilde{\chi}(\xi_{1},\ldots,\xi_{n})=\chi(-\sum_{i=1}^{n}\xi_{i},\xi_{2},\ldots,\xi_{n})$. We deduce from (2.5)-(2.7) that

[TABLE]

for any permutation $\sigma$ of $\{1,\ldots,n+1\}$ with an implicit symbol $\chi_{\sigma}\in L^{\infty}({\mathbb{R}}^{n})$ satisfying $\|\chi_{\sigma}\|_{L^{\infty}}\lesssim\|\chi\|_{L^{\infty}}$.

The classical Coifman-Meyer theorem [8] states that if $\chi$ is smooth away from the origin and satisfies the Hörmander-Milhin condition

[TABLE]

for sufficiently many multi-indices $\beta\in\mathbb{N}^{n}$, then the operator $\Pi_{\chi}^{n}$ is bounded from $L^{p_{1}}(\mathbb{R})\times\cdots\times L^{p_{n}}(\mathbb{R})$ to $L^{p}(\mathbb{R})$ and satisfies

[TABLE]

as long as $1<p_{j}\leq+\infty$, $1\leq p<+\infty$ and $\frac{1}{p}=\frac{1}{p_{1}}+\cdots\frac{1}{p_{n}}$.

In the sequel, we will need the following generalized version of Coifman-Meyer’s theorem.

Theorem 2.2**.**

Let $1<p_{1},\cdots,p_{n}<+\infty$ and $1\leq p<+\infty$ satisfy $\frac{1}{p}=\frac{1}{p_{1}}+\cdots\frac{1}{p_{n}}$. Assume that $f_{1},\ldots,f_{n}\in\mathcal{S}({\mathbb{R}})$ are functions with Fourier variables supported in $\{|\xi|\sim N_{i}\}$ for some dyadic numbers $N_{1},\ldots,N_{n}$.

Assume also that $\chi\in C^{\infty}({\mathbb{R}}^{n})$ satisfies the Marcinkiewicz type condition

[TABLE]

on the support of $\prod_{i=1}^{n}\hat{f}_{i}(\xi_{i})$. Then,

[TABLE]

with an implicit constant that doesn’t depend on $N_{1},\ldots,N_{n}$.

Remark 2.3*.*

Condition (2.9) is too restrictive for our purpose. For instance if $N_{1}\ll N_{2}$ are dyadic numbers and

[TABLE]

then $\chi$ clearly satisfies condition (2.11), but $|\partial_{\xi_{1}}\chi(\xi_{1},\xi_{2})|\sim N_{1}^{-1}\gg N_{2}^{-1}\sim|(\xi_{1},\xi_{2})|^{-1}$, so that $\chi$ does not satisfy (2.9).

Theorem 2.2 was proved by Muscalu, Pipher, Tao and Thiele [30] in the case of bilinear Fourier multipliers555Note that even the extremal case where one the $p_{i}$ is equal to $+\infty$ is proved. (in dimension $2$).

One could certainly prove Theorem 2.2 by extending the arguments in [30] to the multilinear case666Personal communication by Terence Tao.. Instead, we will deduce Theorem 2.2 as a Corollary of Bernicot’s theorem in [3].

Theorem 2.4** ([3], Theorem 1.3).**

Suppose $1<p_{1},\ldots,p_{n}<\infty$, $1\leq p<\infty$ and $1/p=1/p_{1}+\ldots+1/p_{n}$. Assume that $\chi\in C^{\infty}({\mathbb{R}}^{n})$ satisfies

[TABLE]

for some $\lambda_{1},\ldots,\lambda_{n}>0$ and where $d_{\lambda}$ is the metric defined by $d_{\lambda}(\xi,0)=\sum_{i=1}^{n}\lambda_{i}|\xi_{i}|$. Then we have for any smooth functions $f_{1},\ldots,f_{n}\in\mathcal{S}({\mathbb{R}})$

[TABLE]

with an implicit constant that doesn’t depend on $\lambda$.

Proof of Theorem 2.2.

Noticing that

[TABLE]

with $\tilde{\chi}(\xi_{1},\cdots,\xi_{n})=\chi(\xi_{1},\cdots,\xi_{n})\prod_{i=1}^{n}\phi_{\sim N_{i}}(\xi_{i})$, it suffices to show that $\tilde{\chi}$ satisfies (2.13) for suitable $\lambda_{1},\cdots,\lambda_{n}>0$. But setting $\lambda=(\frac{N_{n}}{N_{1}},\ldots,\frac{N_{n}}{N_{n-1}},1)$, this is easily checked since on the one hand

[TABLE]

and on the other hand,

[TABLE]

for $|\xi_{i}|\sim N_{i}$. ∎

Remark 2.5*.*

It is worth noticing that if two symbols $\chi_{1},\chi_{2}$ satisfy (2.11), then this condition also holds for the product function $\chi_{1}\chi_{2}$. This is easily obtained thanks to the Leibniz rule.

Lemma 2.6**.**

Let $0<\alpha\leq 1$. Let $N_{1}\ll N_{2}$ be two dyadic numbers. Then the symbol $\chi$ defined on ${\mathbb{R}}^{2}$ by

[TABLE]

where $\Omega_{2}$ is defined in (3.1), satisfies the Marcinkiewicz condition (2.11) on the set $\{(\xi_{1},\xi_{2})\in{\mathbb{R}}^{2}:|\xi_{1}|\sim N_{1},|\xi_{2}|\sim N_{2}\}$.

Proof.

Let $(\xi_{1},\xi_{2})\in{\mathbb{R}}^{2}$ be such that $|\xi_{1}|\sim N_{1}$ and $|\xi_{2}|\sim N_{2}$. First we estimate $\partial^{\beta}\Omega_{2}(\xi_{1},\xi_{2})$ for $\beta=(\beta_{1},\beta_{2})\in\mathbb{N}^{2}$. From Lemma 3.1 and the mean value theorem we easily get that

[TABLE]

Now classical derivative rules lead to

[TABLE]

where

[TABLE]

Therefore, we deduce from (3.2) as well as (2.15)-(2.16)-(2.17) that

[TABLE]

with

[TABLE]

and

[TABLE]

Noticing that for $\gamma\in C_{\beta}$ we have

[TABLE]

we infer

[TABLE]

Similarly, we get

[TABLE]

We conclude that $A_{\gamma}\geq 0$ and $A_{\gamma}=-B_{\gamma}$, which provides

[TABLE]

∎

2.4. Basic estimates on the sum space $F^{0,\frac{1}{2}}=X^{-1,1}+X^{0,(\frac{1}{2})_{+}}$

By definition of sum space in (2.2), we always have by taking the trivial decompositions $(u_{1},u_{2})=(u,0)$ or $(u_{1},u_{2})=(0,u)$ that

[TABLE]

The next lemma tells us when the reverse holds true.

Lemma 2.7**.**

Let $u\in F^{0,\frac{1}{2}}$ and $L,N$ be two dyadic numbers.

If $1\leq L\lesssim N^{2}$, then

[TABLE]

If $L\gtrsim\langle N\rangle^{2}$, then

[TABLE]

Proof.

It directly follows from the estimate

[TABLE]

∎

3. $L^{2}$-multilinear estimates

3.1. $L^{2}$-bilinear estimates

We follow the strategy in [29] to show $L^{2}$-bilinear estimates related to the dispersive symbol.

Let us define the resonance function of order 2 associated with (1.1) by

[TABLE]

where $\omega_{\alpha+1}$ is the dispersive symbol defined in Hypothesis 1. For $\xi_{1},\xi_{2},\xi_{3}\in{\mathbb{R}}$, it will be convenient to define the quantities $|\xi_{max}|\geq|\xi_{med}|\geq|\xi_{min}|$ to be the maximum, median and minimum of $|\xi_{1}|,|\xi_{2}|$ and $|\xi_{3}|$ respectively.

For the sake of completeness, we recall a few results proved in [29].

Lemma 3.1** ([29], Lemma 2.1).**

Let $\alpha>0$. Let $\xi_{1},\xi_{2}\in{\mathbb{R}}$, and $\xi_{3}=-(\xi_{1}+\xi_{2})$. Then

[TABLE]

Lemma 3.2** ([29], Lemma 2.3).**

Let $L\geq 1$, $1\leq p\leq\infty$ and $s\in{\mathbb{R}}$. The operator $Q_{\leq L}$ is bounded in $L^{p}_{t}H^{s}_{x}$ uniformly in $L\geq 1$.

For any $T>0$, we consider $1_{T}$ the characteristic function of the interval $]0,T[$ and use the decomposition

[TABLE]

for some $R>0$.

Lemma 3.3** ([29], Lemma 2.4).**

For any $R>0$ and $T>0$, it holds

[TABLE]

and

[TABLE]

Lemma 3.4** ([29], Lemma 2.5).**

Let $u\in L^{2}({\mathbb{R}}^{2})$. Then for any $T>0$, $R>0$ and $L\gg R$, it holds

[TABLE]

We are now in a position to prove the main result of this section.

Proposition 3.5**.**

Let $0<\alpha\leq 1$. Assume $0<t\leq 1$ and $u_{i}\in Z^{0}$, $i=1,2,3$ are functions with spatial Fourier support in $\{|\xi|\sim N_{i}\}$ with $N_{i}$ dyadic. Let $\chi\in C^{\infty}({\mathbb{R}}^{2})$ satisfy the Marcinkiewicz condition (2.11).

If $N_{min}\lesssim 1$, then

[TABLE]

If $N_{min}\gg 1$, then

[TABLE]

where $G^{2}_{t,\chi}$ is defined in (2.6).

Proof.

From (2.8) we may always assume $N_{1}\leq N_{2}\leq N_{3}$. Estimate (3.6) is easily obtained thanks to Plancherel identity and Bernstein inequality. Thus it remains to deal with the case $N_{1}\gg 1$. By localization considerations, $G_{t,\chi}^{2}$ vanishes unless $N_{2}\sim N_{3}$. Setting $R=N_{1}^{1+\frac{\alpha}{4}}N_{3}^{\alpha-1}$, we split $G^{2}_{t,\chi}$ as

[TABLE]

where $G^{2}_{\infty}(u,v,w)=\int_{{\mathbb{R}}^{2}}\Pi_{\chi}^{2}(u,v)w$ and $1_{t,R}^{high}$, $1_{t,R}^{low}$ are defined in (3.3).

The contribution of $G_{t}^{2,high}$ is estimated thanks to Lemma 3.3 as well as Hölder inequality by

[TABLE]

To evaluate the contribution of $G_{t}^{2,low}$, we use Lemma 3.1 and we get

[TABLE]

It is worth noticing that since $N_{1}\gg 1$, we have $R\ll N_{1}N_{3}^{\alpha}$. Therefore the contribution of $G_{t,1}^{2,low}$ is easily estimated thanks to Lemma 3.4, Theorem 2.2 and estimate (2.21) by

[TABLE]

where in the last step we used that $0\leq\alpha\leq 1$. Using again Theorem 2.2, Hölder inequality and Lemma 2.7 we estimate the contribution of $G_{t,2}^{2,low}$ by

[TABLE]

On the other hand, observe that an interpolation argument provides

[TABLE]

Since $Q_{\ll L}=I-Q_{\gtrsim L}$, we deduce that

[TABLE]

Combining (3.12)-(3.14) we infer

[TABLE]

Finally, using Lemma 3.2, the contribution of $G_{t,3}^{2,low}$ is estimated in the same way. ∎

3.2. $L^{2}$-trilinear estimates

We first state an elementary estimate.

Proposition 3.6**.**

Let $0<\alpha\leq 1$. Assume $0<t\leq 1$ and $u_{i}\in Z^{0}$ $i=1,2,3,4$ are functions with spatial Fourier support in $\{|\xi|\sim N_{i}\}$ with $N_{i}$ dyadic. Let $\chi\in C^{\infty}({\mathbb{R}}^{3})$ satisfy the Marcinkiewicz condition (2.11).

Then it holds that

[TABLE]

Proof.

We get from (2.12) together with Hölder and Bernstein inequalities that

[TABLE]

We conclude the proof of estimate (3.15) combining

[TABLE]

with (3.13). ∎

Now we define the resonance function of order 3 by

[TABLE]

For $\xi_{1},\xi_{2},\xi_{3},\xi_{4}\in{\mathbb{R}}$, it will be convenient to define the quantities $|\xi_{max}|\geq|\xi_{sub}|\geq|\xi_{thd}|\geq|\xi_{min}|$ to be the maximum, sub-maximum, third-maximum and minimum of $|\xi_{1}|,|\xi_{2}|,|\xi_{3}|$ and $|\xi_{4}|$ respectively.

Lemma 3.7**.**

Let $\alpha>0$. Let $\xi_{1},\xi_{2},\xi_{3}\in{\mathbb{R}}$ and $\xi_{4}=-(\xi_{1}+\xi_{2}+\xi_{3})$. If we assume that $|\xi_{min}|\ll|\xi_{thd}|$ then it holds

[TABLE]

Proof.

Without loss of generality, we may assume $|\xi_{1}|\ll|\xi_{2}|\leq|\xi_{3}|\sim|\xi_{4}|$. Then, estimate (3.17) is a consequence of the identity

[TABLE]

combined with Lemma 3.1. ∎

Proposition 3.8**.**

Let $0<\alpha\leq 1$. Assume $0<t\leq 1$ and $u_{i}\in Z^{0}$, $i=1,2,3,4$ are functions with spatial Fourier support in $\{|\xi|\sim N_{i}\}$ with $N_{i}$ dyadic satisfying $N_{min}\ll N_{thd}$ and $N_{max}\gg 1$. Let $\chi\in C^{\infty}({\mathbb{R}}^{3})$ satisfy the Marcinkiewicz condition (2.11). Then,

[TABLE]

Proof.

From (2.8) it is sufficient to consider the case $N_{1}\ll N_{2}\leq N_{3}\sim N_{4}$. Moreover, we may assume that $N_{2}N_{4}^{\alpha}\gg 1$ and $N_{2}\gg 1$ since otherwise the claim follows from estimate (3.15). We proceed now as in the proof of Proposition 3.5. First we decompose $G_{t}^{3}$ as $G_{t}^{3,high}+G_{t}^{3,low}$ with

[TABLE]

and $R=N_{2}^{1+\frac{\alpha}{4}}N_{4}^{\alpha-1}\ll N_{2}N_{4}^{\alpha}$. The high-part is easily estimated thanks to Lemma 3.3 by

[TABLE]

which is acceptable. To deal with the low-part, we decompose with respect of the modulation variables. Thus

[TABLE]

According to (3.17) the above sum is nontrivial only for $L_{max}\gtrsim N_{2}N_{4}^{\alpha}$. In the case where $L_{max}=L_{1}$, we deduce from (2.12)-(2.21)-(3.13) and Lemma 3.4 that

[TABLE]

In the same way, we get that the sum over $L_{max}=L_{2}$ is controlled by

[TABLE]

Arguing similarly and using (3.14), the sum over $L_{max}=L_{3}$ can be estimated by

[TABLE]

Finally we easily check that the bound in the case $L_{max}=L_{4}$ is obtained similarly. Gathering all these estimates we get the desired result.

∎

4. Estimates for a smooth solution

The aim of this section is to get suitable a priori estimates of a solution of (1.8) in the space $Y^{s}\hookrightarrow Z^{s}$ for $s>s_{\alpha}$.

4.1. Bilinear estimate

Proposition 4.1**.**

Assume that $0<T\leq 1$ and $s\geq 0$. Let $u$ be a smooth solution to (1.8) defined in the time interval $[0,T]$. Then

[TABLE]

Proof.

By using the fractional Leibniz rule (c.f. Kenig, Ponce and Vega [19]), we have for $s\geq 0$

[TABLE]

∎

4.2. Refined Strichartz estimate

Let us first recall the following Strichartz estimate:

[TABLE]

where $U_{\alpha}(t)=e^{tL_{\alpha+1}}$ is the free evolution operator associated to (1.8). This estimate is a direct consequence of Theorem 2.1 in [18] applied with $\phi=(1-\eta)\omega_{\alpha+1}$. From this we get following the proof of Proposition 2.3 in [22] (see also [17]) the refined Strichartz estimate:

Lemma 4.2**.**

Let $0<\alpha\leq 1$. Assume that $0<T\leq 1$ and $\delta\geq 0$. Let $u$ be a solution to

[TABLE]

defined on the time interval $[0,T]$. Then, there exist $0<\kappa_{1},\;\kappa_{2}<\frac{1}{2}$ such that

[TABLE]

and

[TABLE]

for any dyadic number $N\geq 1$.

Proof.

(4.5) is proven in [[22], Proposition 2.3] (see also [17]). To prove (4.6) we modified slightly the procedure (see [27] for a similar modification). Let $N\geq 1$ and let $I=[a,b]\subset{\mathbb{R}}$ be an interval of length $|I|\lesssim T^{\kappa}N^{-\delta}$ for some fixed $\delta>0$ and $0<\kappa<1$. From (4.3) and Hölder’s inequalities, we easily get

[TABLE]

for any $2\leq p\leq 4$ and $u_{0}\in L^{2}({\mathbb{R}})$. By the $TT^{*}$ method and P. Tomas argument, this leads to

[TABLE]

with $\overline{p^{\prime}}=\frac{p^{\prime}}{p^{\prime}-1}$, for any $2\leq p,p^{\prime}\leq 4$ and any $f\in L^{\overline{p^{\prime}}}_{I}L^{1}_{x}$. We need this estimate but on the retarded Duhamel operator $(t,x)\mapsto\int_{a}^{t}U(t-t^{\prime})P_{N}f(t^{\prime},x)dt^{\prime}$. Taking $p=2$ and $p^{\prime}>2$, this can be done by applying Christ-Kiselev Lemma (see [7] and also [33]). We then get

[TABLE]

and Hölder inequalities then yields

[TABLE]

Now, chopping out the interval $[0,T]$ in small intervals of length $T^{\kappa}N^{-\delta}$, we have $[0,T]=\cup_{j\in J}I_{j}$ where $I_{j}=[a_{j},b_{j}]$, $|I_{j}|\sim T^{\kappa}N^{-\delta}$ and $\#J\sim T^{1-\kappa}N^{\delta}$. Since $u_{N}$ satisfies $\partial_{t}u_{N}-L_{\alpha+1}u_{N}=F_{N}$ on each interval $I_{j}$ we have

[TABLE]

and (4.7)-(4.8) yield

[TABLE]

which leads to (4.6) by Bernstein inequalities. ∎

Proposition 4.3**.**

Let $0<\alpha\leq 1$. Assume that $0<T\leq 1$ and $s>s_{\alpha}$. Let $u$ be a smooth solution to (1.8) defined on the time interval $[0,T]$. There exists $0<\kappa<\frac{1}{2}$ such that if $0<T\ll\|u\|_{L^{\infty}_{T}H^{s}_{x}}^{-\frac{1}{\kappa}}$, then

[TABLE]

Proof.

From Bernstein’s inequality, we easily estimate the low frequencies part:

[TABLE]

Taking $\delta=1$ in (4.5), summing over $N\geq 1$ and using the fractional Leibniz rule, we deduce

[TABLE]

Noticing that for $s>s_{\alpha}$ and $0<\alpha\leq 1$, it holds $(s-s_{\alpha})+(1-\alpha)_{-}\geq 0$, we obtain (4.9) by combining the two above estimates and taking $\kappa=\kappa_{1}\vee\kappa_{2}$. ∎

Corollary 4.1**.**

Let $0<\alpha\leq 1$. Assume that $0<T\leq 1$ and $s>s_{\alpha}$. Let $u$ be a smooth solution to (1.8) defined on the time interval $[0,T]$. There exist $0<\kappa<\frac{1}{2}$ and $C_{0}>1$ such that if $0<T\ll\|u\|_{L^{\infty}_{T}H^{s}_{x}}^{-\frac{1}{\kappa}}$, then

[TABLE]

Proof.

We have to extend the function $u$ from $]0,T[$ to ${\mathbb{R}}$. For this we introduce the extension operator $\rho_{T}$ defined by

[TABLE]

where $\eta$ is the smooth cut-off function defined in Section 2.1 and $\mu_{T}$ is the continuous piecewise affine function defined by

[TABLE]

According to classical results on extension operators (see for instance [24]), for any $1/2<b\leq 1$, $f\mapsto\eta f(\mu_{T}(\cdot))$ is linear continuous from $H^{b}([0,T])$ into $H^{b}({\mathbb{R}})$ with a bound that does not depend on $T>0$.

First, the unitarity of the free group $U_{\alpha}(\cdot)$ in $H^{s}({\mathbb{R}})$ easily leads to

[TABLE]

Second, the definition of the $X^{\theta,b}$-norm leads, for $1/2<b\leq 1$ and $\theta\in{\mathbb{R}}$, to

[TABLE]

Finally, for $\theta\in{\mathbb{R}}$,

[TABLE]

whereas (4.3) leads to

[TABLE]

and in the same way

[TABLE]

Noticing that, for $0<\alpha\leq 1$, $s-s_{\alpha}+1-\alpha=s-\frac{1}{2}+\frac{\alpha}{4}\leq s-\frac{1}{4}$, this ensures that

[TABLE]

for any $\varepsilon>0$.

Gathering (4.12)-(4.14), we thus infer that for any $(T,s)\in{\mathbb{R}}_{+}^{*}\times{\mathbb{R}}$, $\rho_{T}$ is a bounded linear operator from $C([0,T];H^{s}({\mathbb{R}}))\cap X^{s-1,1}_{T}\cap L^{2}_{T}W^{(s-s_{\alpha})+(1-\alpha)_{-},\infty}_{x}$ into $Y^{s}$ with a bound that does not depend on $(T,s)$. Therefore (4.1) and (4.9) lead to (4.10). ∎

4.3. Energy estimate

Applying the operator $P_{N}$ with $N>0$ dyadic to equation (1.8), taking the $L^{2}$ scalar product with $P_{N}u$ and integrating on $]0,t[$ we obtain

[TABLE]

Let $N_{0}\geq 2^{9}$ and $N>N_{0}$. Define $\mathcal{J}_{N}$ by

[TABLE]

By localization considerations, we get

[TABLE]

Moreover, from the fundamental theorem of calculus, we easily get

[TABLE]

where we used the bilinear Fourier multiplier notation introduced in Definition 2.1 with

[TABLE]

Inserting this into (4.16) and integrating by parts we deduce $\mathcal{J}_{N}=\mathcal{J}_{N}^{1}+\mathcal{J}_{N}^{2}$ where

[TABLE]

and

[TABLE]

Since $P_{N}P_{\sim N}=P_{N}$, we may rewrite $J_{N}^{1}$ more symmetrically as

[TABLE]

with

[TABLE]

Note that the function $\chi_{1}$ satisfies the condition (2.11). This decomposition of $\mathcal{J}_{N}$ motivates the definition of our modified energy. For $N_{0}>1$, $u\in H^{s}({\mathbb{R}})$, with $s\in{\mathbb{R}}$, and $N>0$ dyadic we define

[TABLE]

where

[TABLE]

$\Omega_{2}(\xi_{1},\xi_{2})$ is the quadratic resonance relation defined in (3.1), and $c$ is a real constant to be fixed later.

We define the modified energy at the $H^{s}$-regularity by using a nonhomogeneous dyadic decomposition in spatial frequency777This means that when summing over $N$, we keep all the low frequencies together and by convention $P_{1}=P_{\leq 1}$.

[TABLE]

Next, we show that if $s>s_{\alpha}$ and $N_{0}>2^{0}$ is large enough then the modified energy $E^{s}(u)$ is equivalent to the $H^{s}$-norm of $u$.

Lemma 4.4** (Coercivity of the modified energy).**

Let $0<\alpha\leq 1$ and let $u\in H^{s}({\mathbb{R}})$ with $s>s_{\alpha}$. Then for any $N_{0}\gg(1+\|u\|_{H^{s_{\alpha}}_{x}})^{\frac{2}{\alpha}}$, it holds

[TABLE]

Proof.

We infer from (4.21) and the triangle inequality that

[TABLE]

Thanks to Young and Bernstein’s inequalities we have

[TABLE]

Finally, we conclude the proof of (4.22) gathering (4.23)-(4.24) and the fact that $\displaystyle\sum_{N\geq 1}\|P_{\sim N}u\|_{H^{s}_{x}}^{2}\sim\displaystyle\sum_{N\geq 1}\langle N\rangle^{2s}\|P_{N}u\|_{L^{2}_{x}}^{2}\,.$ ∎

We now state the main estimate of this subsection.

Proposition 4.5**.**

Let $0<\alpha\leq 1$. Let $s>s^{\prime}>s_{\alpha}$, $0<T\leq 1$ and $u\in Y^{s}_{T}$ be a solution of (1.8) on $[0,T]$. Then for any $N_{0}\gg 1$ we have

[TABLE]

where the implicit constant only depends on $\alpha$.

Proof.

Let $0<t\leq T\leq 1$. First, assume that $N\leq N_{0}=2^{9}$. By using the definition of $\mathcal{E}_{N}$ in (4.20), we have

[TABLE]

which yields after integrating between [math] and $t$ and applying Hölder and Bernstein’s inequalities that

[TABLE]

Thus, we deduce after taking the supreme over $t\in[0,T[$ and summing over $N\leq N_{0}$ that

[TABLE]

where we used that, since $s>0$, $N^{s}\|P_{N}(u^{2})\|_{L^{\infty}_{T}L^{1}_{x}}\lesssim\|u\|_{L^{\infty}_{T}L^{2}_{x}}\|u\|_{L^{\infty}_{T}H^{s}_{x}}$.

Now, for $N\geq N_{0}$, we take the extension $\tilde{u}=\rho_{T}(u)$ defined in (4.11). To simplify the notation we drop the tilde in the sequel. We first notice that

[TABLE]

where $\mathcal{J}_{N}(t)=\mathcal{J}_{N}^{1}(t)+\mathcal{J}_{N}^{2}(t)$ is defined in (4.16), (4.17), (4.18).

Estimate for $\mathcal{J}_{N}^{2}$. We get from Proposition 3.5 that

[TABLE]

Since $s>\frac{1}{2}-\frac{\alpha}{2}$ and $s^{\prime}>s_{\alpha}$, we deduce that

[TABLE]

It remains to estimate $\mathcal{J}_{N}^{1}+c\mathcal{K}_{N}$. Using equation (1.8) we obtain

[TABLE]

Taking $c=1$ this leads to estimate $\mathcal{J}_{N}^{1}+\mathcal{K}_{N}=\mathcal{K}_{N}^{1}+\mathcal{K}_{N}^{2}+\mathcal{K}_{N}^{3}$.

Estimate for $\mathcal{K}_{N}^{1}$. We have

[TABLE]

where

[TABLE]

with $\chi_{1}$ defined in (4.19). From Lemma 2.6, $\chi_{\mathcal{K}^{1}}$ satisfies (2.11). Therefore we get from Proposition 3.6 that

[TABLE]

where in the last step we used that $\frac{1}{2}-\frac{\alpha}{4}-s^{\prime}<0$. We thus infer that

[TABLE]

Estimates for $\mathcal{K}_{N}^{2}+\mathcal{K}_{N}^{3}$. Using, as in Subsection 4.3, that

[TABLE]

where $\tilde{\chi}$ satisfies (2.11), we decompose $\mathcal{K}_{N}^{2}+\mathcal{K}_{N}^{3}$ as $\mathcal{K}_{N}^{31}+\mathcal{K}_{N}^{32}+\mathcal{K}_{N}^{33}$ with

[TABLE]

[TABLE]

and

[TABLE]

Estimate for $\mathcal{K}_{N}^{31}$. We have

[TABLE]

Now a change a variables leads to

[TABLE]

with $\xi=\xi_{1}+\xi_{2}+\xi_{3}$ and

[TABLE]

Let us rewrite $\sigma$ as follows:

[TABLE]

According to Lemma 2.6 and Remark 2.5, it is easy to check that $N_{1}N^{\alpha}\left(\frac{\chi_{1}}{\Omega_{2}}\right)(\xi_{1},\xi_{2}+\xi_{3})$, $\frac{1}{N_{2}}\widetilde{\phi}_{N_{2}}(\xi_{2})\xi_{2}$ and thus $\frac{N_{1}N^{\alpha}}{N_{2}}\left(\frac{\chi_{1}}{\Omega_{2}}\right)(\xi_{1},\xi_{2}+\xi_{3})\widetilde{\phi}_{N_{2}}(\xi_{2})\xi_{2}$ satisfy (2.11). In the same way, $\frac{N_{1}N^{\alpha}}{N_{1}}\left(\frac{\chi_{1}}{\Omega_{2}}\right)(\xi_{1},\xi_{3})\widetilde{\phi}_{N_{1}}(\xi_{1})\xi_{1}$ satisfies (2.11). Now we get from the mean value theorem that for any multi-indice $\beta=(\beta_{1},0,\beta_{3})$, there exists $|\tilde{\xi}_{\beta}|\sim N$ such that

[TABLE]

On the other hand, for any $\beta=(\beta_{1},\beta_{2},\beta_{3})$ with $\beta_{2}\geq 1$, we have

[TABLE]

It thus follows from Lemma 2.6 that $\frac{N_{1}N^{\alpha+1}}{N_{2}}\left[\left(\frac{\chi_{1}}{\Omega_{2}}\right)(\xi_{1},\xi_{1}+\xi_{3})-\left(\frac{\chi_{1}}{\Omega_{2}}\right)(\xi_{1},\xi_{3})\right]$ satisfies (2.11). Therefore we deduce that $\chi_{\mathcal{K}^{31}}:=\frac{N_{1}}{N_{1}\vee N_{2}}N^{\alpha}\sigma$ satisfies (2.11). Rewriting $\mathcal{K}_{N}^{31}$ as

[TABLE]

we get from estimate (3.15) that

[TABLE]

Recalling that $\frac{1}{2}-\frac{\alpha}{4}-s^{\prime}<0$, it follows as in (4.29) that

[TABLE]

Estimate for $\mathcal{K}_{N}^{32}$. We only deal with the first term $\mathcal{K}_{N}^{321}$ of the sum in $\mathcal{K}_{N}^{32}$ since the other is estimated similarly. With the notation of Section 2.3 we obtain

[TABLE]

with

[TABLE]

Noticing that $\chi_{\mathcal{K}^{321}}(\xi_{1},\xi_{2})$ satisfies condition (2.11), estimate (3.15) implies that

[TABLE]

which again, as in (4.29), leads to

[TABLE]

Estimate for $\mathcal{K}_{N}^{33}$. We follow again the same arguments. We only deal with the first term $\mathcal{K}_{N}^{331}$ of the sum in $K_{N}^{33}$ and rewrite it as

[TABLE]

with

[TABLE]

Then, thanks to estimate (3.15), we get

[TABLE]

This leads to

[TABLE]

Combining (4.26)-(4.27)-(4.28)-(4.29)-(4.31)-(4.32)-(4.33), we conclude the proof of Proposition 4.5. ∎

Corollary 4.2**.**

Let $0<\alpha\leq 1$. Let $s>s_{\alpha}$, $0<T\leq 1$ and $u\in Y^{s}_{T}$ be a solution of (1.8) on $[0,T]$. Then for any $N_{0}\gg 1$ we have

[TABLE]

Proof.

According to (4.27), it suffices to bound

[TABLE]

and the result follows from by combining (4.28)-(4.29)-(4.31)-(4.32)-(4.33). ∎

5. Estimates for the difference of two solutions

In this section, we provide the needed estimates for the difference $w$ of two solutions $u,v$ of (1.8). If $w=u-v$ and $z=u+v$, then

[TABLE]

The lack of symmetry in the nonlinear term of (5.1) prevents us to estimate $w$ in $Y^{s}_{T}$, $s>s_{\alpha}$. To overcome this difficulty, we will rather work at a lower regularity level $\sigma<0$ and more precisely with

[TABLE]

Remark 5.1*.*

For $\alpha\in]0,1]$ and $s>s_{\alpha}=\frac{3}{2}-\frac{5}{4}\alpha$, it holds $-\frac{1}{2}+\frac{\alpha}{4}<0$ and $s-2+\frac{3}{2}\alpha>-\frac{1}{2}+\frac{\alpha}{4}$. Therefore, the definition interval in (5.2) is never empty. Moreover, it is worth noticing that $-\sigma<\frac{1}{2}-\frac{\alpha}{4}\leq s_{\alpha}<s$.

Since we are not able to control the $X_{T}^{\sigma-1,1}\cap L^{2}_{T}W^{1-\alpha+(\sigma-s_{\alpha})_{-},\infty}_{x}$ part of $w$ for $\sigma<0$ we need to bound the difference in the sum space $F^{\sigma,\frac{1}{2}}$. Finally, to treat some low-high interactions in the energy estimates, we also need to add a weight on the low space frequencies so that $w$ will take place in $\overline{Z}^{\sigma}$.

5.1. Bilinear estimate

Proposition 5.2**.**

Let $0<\alpha\leq 1$. Assume that $0<T\leq 1$, $s>s_{\alpha}$ and $-\frac{1}{2}+\frac{\alpha}{4}<\sigma<\min(0,s-2+\frac{3}{2}\alpha)$. Let $z\in Y^{s}_{T}$ and let $w\in\overline{Z}^{\sigma}_{T}$ be a solution of (5.1) on $]0,T[$ with $w_{0}\in\overline{H^{\sigma}}$. Then it holds

[TABLE]

Proof.

Let $\tilde{w}=\rho_{T}(w)$ and $\tilde{z}=\rho_{T}(z)$ be the extensions defined in (4.11) and let $\tilde{\tilde{w}}$ satisfying (5.1) with $\partial_{x}(\tilde{z}\tilde{w})$ as second hand member. We will estimate the extension $\breve{w}=\eta\tilde{\tilde{w}}$ of $w$ where $\eta$ is the smooth cut-off function defined in (2.1). To simplify the notation we drop the tilde in the sequel. For $N_{0}\geq 1$ to be chosen later, we rewrite $zw$ as

[TABLE]

Duhamel formula, (2.18), as well as classical Bourgain’s estimate on the linear evolution (cf. [5], [11]) and (2.18) lead to

[TABLE]

Now, using that $0<-\sigma<s$, we easily bound the contribution of the low frequency part $J_{\leq N_{0}}$ by

[TABLE]

The contribution of the high-low interactions $J_{>N_{0}}^{h,l}$ is also easily bounded as follows

[TABLE]

where in the next to the last step we used that $\sigma<0$ yields $\langle N\rangle^{\sigma}\|w_{\lesssim N}\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim\|w\|_{L^{\infty}_{t}H^{\sigma}_{x}}$. To bound the contribution of the low-high interactions $J_{>N_{0}}^{l,h}$ we write

[TABLE]

Now we deal with the (high-high) interactions term

[TABLE]

To estimate the contribution of the sum over $L\gtrsim NN_{1}^{\alpha}$, we take advantage of the $X^{\sigma+1,(-\frac{1}{2})_{+}}$-part of $F^{\sigma+1,-\frac{1}{2}}$. Therefore this term is bounded by

[TABLE]

where we used that $-\frac{\alpha}{2}-\sigma+(s_{\alpha}-s)+(\alpha-1)_{+}\leq-\sigma-(1-\frac{\alpha}{2})<0$ since $0<-\sigma<\frac{1}{2}-\frac{\alpha}{4}$. The contribution of the region $L\ll NN_{1}^{\alpha}$ and $L_{1}\gtrsim NN_{1}^{\alpha}$ is estimated by

[TABLE]

where we used that $-\alpha-s+1-\sigma<s_{\alpha}-s<0$ to sum over $N_{1}$. Finally the contribution of the last region can be bounded thanks to Lemmas 3.2 and 2.7 by

[TABLE]

where we used that $(NN_{1}^{\alpha})^{-\frac{1}{2}}\leq(NN_{1}^{\alpha})^{-1}N_{1}$, since $\alpha\leq 1$. Gathering (5.5)-(5.10) we obtain

[TABLE]

where $c_{1},\,c_{2}\,,c_{3}\geq 1$. This yields the desired result by taking $N_{0}=[2c_{3}(1+\|z\|_{Y^{s}_{T}})]^{2}$ and concludes the proof of Proposition 5.2. ∎

5.2. Refined Strichartz estimate

Proposition 5.3**.**

Let $0<\alpha\leq 1.$ Assume that $0<T\leq 1$, $s>s_{\alpha}$ and $-\frac{1}{2}+\frac{\alpha}{4}<\sigma<\min(0,s-2+\frac{3}{2}\alpha)$. Let $z\in Y^{s}_{T}$ and $w\in\overline{Z}^{\sigma}_{T}$ be a solution of (5.1) on $]0,T[$. Then

[TABLE]

Proof.

The low frequency part is estimated by

[TABLE]

To estimate the high frequency part of the LHS of (5.11), we decompose $zw$ as in (5.4) and we use Lemma 4.2 with $\delta=1$ to get

[TABLE]

where we used that $\sigma<0$ and $s+\sigma>0$. Summing over $N\gg 1$, using that $s>s_{\alpha}\geq\frac{1}{2}-\frac{\alpha}{4}$, (5.11) follows. ∎

Corollary 5.1**.**

Let $0<\alpha\leq 1.$ Assume that $0<T\leq 1$, $s>s_{\alpha}$ and $-\frac{1}{2}+\frac{\alpha}{4}<\sigma<\min(0,s-2+\frac{3}{2}\alpha)$. Let $z\in Y^{s}_{T}$ and $w\in\overline{Z}^{\sigma}_{T}$ be a solution of (5.1) on $]0,T[$ such that $w_{0}\in\overline{H}^{\sigma}$. Then

[TABLE]

Proof.

By the property of the extension $\tilde{w}=\rho_{T}(w)$ defined in (4.11) we have

[TABLE]

and the result follows by gathering this last estimate with (5.3) and (5.11). ∎

5.3. Energy estimate

For $N_{0}\geq 1$, we define the modified energy for the diffe-rence $w$ of two solutions $u$ and $v$ by

[TABLE]

where

[TABLE]

and

[TABLE]

$\Omega_{2}$ is defined in (3.1), $\tilde{\chi}_{1}$, $\tilde{\chi}_{2}$ are symbols satisfying the Marcinkiewicz condition (2.11) and defined later in the proof of Proposition 5.5, and $\widetilde{c_{1}}$, $\widetilde{c_{2}}$ are real constants that will be fixed later in the proof of Proposition 5.5.

We define the modified energy at the $H^{\sigma}$-regularity associated with the difference of two solutions by using a homogeneous dyadic decomposition in spatial frequency

[TABLE]

Lemma 5.4** (Coercivity of the modified energy).**

Let $0<\alpha\leq 1$, $s>s_{\alpha}$, $0<T\leq 1$ and $-\frac{1}{2}+\frac{\alpha}{4}<\sigma<\min(0,s-2+\frac{3}{2}\alpha)$. Let $z\in Y^{s}_{T}$ and $w\in\overline{Z}^{\sigma}_{T}$ be a solution of (5.1). Then for $N_{0}\gg(1+\|z\|_{H^{s}_{x}})^{\frac{2}{\alpha}}$ it holds

[TABLE]

Proof.

We infer from (5.16) and the triangle inequality that, for $N_{0}\gg 1$,

[TABLE]

Thanks to Young and Bernstein’s inequalities we have for $N\geq N_{0}\gg 1$,

[TABLE]

Similarly we bound the contribution of $\widetilde{\mathcal{E}}^{2}_{N}$ for $N\geq N_{0}\gg 1$ by

[TABLE]

Finally, we conclude the proof of (5.17) gathering (5.18)-(5.19)-(5.20) and the fact that $\|w\|_{\overline{H}^{\sigma}_{x}}\sim\displaystyle\sum_{N>0}\langle N\rangle^{2\sigma}\langle N^{-1}\rangle^{2}\|P_{N}w\|_{L^{2}_{x}}^{2}$. ∎

Proposition 5.5**.**

Let $0<\alpha\leq 1$. Let $s>s_{\alpha}$, $0<T\leq 1$ and $-\frac{1}{2}+\frac{\alpha}{4}<\sigma<\min(0,s-2+\frac{3}{2}\alpha)$. Let $u,v\in Y^{s}_{T}$ two solutions of (1.8) such that $w=u-v\in\overline{Z}^{\sigma}_{T}$. Then, setting $z=u+v$, it holds

[TABLE]

where $\gamma=s-2+\frac{3}{2}\alpha-\sigma>0$.

Proof.

We argue as in the proof of Proposition 4.5. To deal with the low frequencies $N\leq N_{0}$, we use equation (5.1) to deduce

[TABLE]

for any $t\in(0,T]$. Integrating this on $(0,t)$ it follows after a dyadic decomposition of $P_{N}(zw)$ that

[TABLE]

On the one hand, we infer

[TABLE]

by using that $\langle N^{-1}\rangle\langle N\rangle^{\sigma}\|w_{\lesssim N}\|_{L^{\infty}_{T}L^{2}_{x}}\lesssim\|w\|_{L^{\infty}_{T}\overline{H}^{\sigma}_{x}}$. On the other hand, recalling that $0<-\sigma<s$, we get

[TABLE]

Therefore, we deduce by summing over $N\leq N_{0}$ that

[TABLE]

We consider now the case $N>N_{0}$. We take the extensions $\tilde{w}=\rho_{T}(w)$ and $\tilde{z}=\rho_{T}(z)$ defined in (4.11), and we drop the tilde in the sequel. Arguing as in the proof of Proposition 4.5, we get

[TABLE]

with

[TABLE]

and

[TABLE]

Proceeding as in the Section 4.3, we split $\widetilde{\mathcal{J}}_{N}$ as $\widetilde{\mathcal{J}}_{N}^{1}+\widetilde{\mathcal{J}}_{N}^{2}+\widetilde{\mathcal{J}}_{N}^{3}$ with

[TABLE]

where $\widetilde{\chi_{1}}=-\frac{1}{2}\langle N^{-1}\rangle^{2}\chi_{1}$ and \widetilde{\chi_{2}}(\xi_{1},\xi_{2})=\langle N^{-1}\rangle^{2}\Bigl{(}\frac{\langle N\rangle}{N}\Bigr{)}^{2\sigma}\phi_{N}^{2}(\xi_{1}+\xi_{2}).

Estimate for $\widetilde{\mathcal{J}}_{N}^{3}$. We infer from Proposition 3.5 that

[TABLE]

where in the last step we used that $-s-\sigma+(1-\alpha)_{+}<-(s-s_{\alpha})_{+}<0$ to sum over $N_{1}$. Therefore we get

[TABLE]

Estimate for $-\widetilde{\mathcal{J}}_{N}^{1}+\widetilde{c_{1}}\widetilde{\mathcal{K}}_{N}$. We deduce using equation (5.1) that

[TABLE]

We choose $\widetilde{c_{1}}=-1$ so that the first term on the right-hand side cancels out with $-\widetilde{\mathcal{J}}_{N}^{1}$ and it suffices to estimate $\widetilde{\mathcal{K}}_{N}^{1}+\widetilde{\mathcal{K}}_{N}^{2}+\widetilde{\mathcal{K}}_{N}^{3}$.

Estimate for $\widetilde{\mathcal{K}}_{N}^{1}$. The contribution of $\widetilde{\mathcal{K}}_{N}^{1}$ may be treated exactly as $K_{N}^{1}$ in the proof of Proposition 4.5. We obtain

[TABLE]

Estimate for $\widetilde{\mathcal{K}}_{N}^{2}+\widetilde{\mathcal{K}}_{N}^{3}$. We decompose $P_{\sim N}(zw)$ into dyadic pieces as follows:

[TABLE]

As in the proof of Proposition 4.5, this leads to estimate $\sum_{j=1}^{4}\widetilde{\mathcal{K}}_{N}^{3j}$ where $\widetilde{\mathcal{K}}_{N}^{3j}$ denotes the contribution to $\widetilde{\mathcal{K}}_{N}^{2}+\widetilde{\mathcal{K}}_{N}^{3}$ of the jth term in the RHS of (5.24).

Estimate for $\widetilde{\mathcal{K}}_{N}^{31}$ and $\widetilde{\mathcal{K}}_{N}^{32}$. Since in these terms, both occurrences of $w$ are localized at frequency $\sim N$, they may be estimated as $K_{N}^{31}$ and $K_{N}^{32}$ in the proof of Proposition 4.5. We infer that

[TABLE]

Estimate for $\widetilde{\mathcal{K}}_{N}^{33}$. It suffices to consider the contribution $\widetilde{\mathcal{K}}_{N}^{331}$ of $\widetilde{\mathcal{K}}_{N}^{2}$ to $\widetilde{\mathcal{K}}_{N}^{33}$ since the contribution to $\widetilde{\mathcal{K}}_{N}^{3}$ can be estimated in exactly the same way.

[TABLE]

where

[TABLE]

satisfies (2.11). Estimate (3.15) gives

[TABLE]

Since $\frac{1}{2}-\frac{\alpha}{4}-s<0$ and $\frac{1}{2}-\frac{\alpha}{4}-\sigma>0$, this yields

[TABLE]

Estimate for $\widetilde{\mathcal{K}}_{N}^{34}$. Again, we only estimate the contribution of

[TABLE]

It follows from estimate (3.15) that

[TABLE]

where in the last step we used that $s+\sigma>0$ and $\frac{1}{2}-\frac{\alpha}{4}-s<0$. We conclude that

[TABLE]

Estimate for $-\widetilde{\mathcal{J}}_{N}^{2}+\widetilde{c_{2}}\widetilde{\mathcal{L}}_{N}$. Using equation (5.1) we rewrite $\widetilde{\mathcal{L}}_{N}$ as

[TABLE]

where we used that $z=u+v$ solves

[TABLE]

Taking $\widetilde{c_{2}}=-1$ it remains to estimate $\sum_{j=1}^{4}\widetilde{\mathcal{L}}_{N}^{j}$.

Estimate for $\widetilde{\mathcal{L}}_{N}^{1}$. We may rewrite this term as

[TABLE]

with

[TABLE]

The contribution $\widetilde{\mathcal{L}}_{N}^{11}$ of the region where $N_{2}\vee N_{3}\lesssim N$ is estimated thanks to (3.15) by

[TABLE]

where we used that $\frac{1}{2}-\frac{\alpha}{4}-s<0$ and $\frac{1}{2}-\frac{\alpha}{4}-\sigma>0$. For the other contribution $\widetilde{\mathcal{L}}_{N}^{12}$, we must have $N_{2}\sim N_{3}$ and by virtue of (3.15) again, we deduce that

[TABLE]

where we used that $s+\sigma>0$ and $-s+1-\frac{3}{4}\alpha\leq s_{\alpha}-s$ for $\alpha\leq 1$. Therefore we infer that

[TABLE]

Estimate for $\widetilde{\mathcal{L}}_{N}^{2}$. We need to bound

[TABLE]

where

[TABLE]

We may always assume $N_{2}\leq N_{3}$. The contribution $\widetilde{\mathcal{L}}_{N}^{21}$ of the sum over $N_{2}\sim N_{3}\gg N$ is estimated thanks to Proposition 3.8 by

[TABLE]

where in the first step we used that $\sigma<s-2+\frac{3}{2}\alpha$ and in the last step we used that $\sigma\geq-\frac{1}{2}+\frac{\alpha}{4}>-\frac{1}{2}_{-}$. We also used the weight $\langle N_{1}^{-1}\rangle$ of $\overline{Z}^{\sigma}$ to sum over $N_{1}\leq 1$. This leads to

[TABLE]

Similarly, we bound the contribution $\widetilde{\mathcal{L}}_{N}^{22}$ of the sum over $N_{1}\ll N_{2}$ and $N_{3}\sim N$ by

[TABLE]

where in the last step we used that $\sigma>-\frac{1}{2}$. We also used the weight $\langle N_{1}^{-1}\rangle$ of $\overline{Z}^{\sigma}$ to sum over $N_{1}\leq 1$. Setting $\gamma=s-2+\frac{3}{2}\alpha-\sigma>0$, this leads to

[TABLE]

To deal with the last region $N_{3}\sim N$ and $N_{2}\lesssim N_{1}$, we use estimate (3.15) to get

[TABLE]

where in the last step we used that $\frac{1}{2}-\frac{\alpha}{4}-\sigma>0$ since $\sigma<0$ and that $-\frac{1}{2}-\frac{\alpha}{4}-\sigma<0$ since $\sigma>\frac{\alpha}{4}-\frac{1}{2}$. It follows that

[TABLE]

and we deduce gathering (5.29)-(5.30)-(5.31) that

[TABLE]

Estimate for $\widetilde{\mathcal{L}}_{N}^{3}+\widetilde{\mathcal{L}}_{N}^{4}$. Performing a dyadic decomposition for $P_{\sim N}(z^{2})$ and $P_{\sim N}(zw)$, we get from (4.30) and (5.24) that

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Estimate for $\widetilde{\mathcal{L}}_{N}^{41}$. Arguing as for the term $K_{N}^{31}$ in the proof of Proposition 4.5, we obtain

[TABLE]

The contribution $\widetilde{\mathcal{L}}_{N}^{411}$ of the sum over $N_{2}\lesssim N_{1}$ is bounded thanks to Proposition 3.6 by

[TABLE]

Using Proposition 3.8, the other contribution $\widetilde{\mathcal{L}}_{N}^{412}$ is estimated by

[TABLE]

since $s>\frac{1}{2}-\frac{\alpha}{4}$, $\sigma-s+2<\frac{3}{2}\alpha$ and where we also used the weight $\langle N_{1}^{-1}\rangle$ of $\overline{Z}^{\sigma}$ to sum over $N_{1}\leq 1$. Combining estimates (5.34)-(5.35) we infer that

[TABLE]

Estimate for $\widetilde{\mathcal{L}}_{N}^{42}$. Noticing that

[TABLE]

it is clear that we may follow the same lines as the estimate for $\widetilde{\mathcal{L}}_{N}^{41}$ to prove

[TABLE]

Estimate for $\widetilde{\mathcal{L}}_{N}^{43}$, $\widetilde{\mathcal{L}}_{N}^{44}$ and $\widetilde{\mathcal{L}}_{N}^{45}$. It is not too hard to check that $\widetilde{\mathcal{L}}_{N}^{43}$ and $\widetilde{\mathcal{L}}_{N}^{45}$ may be estimated as $\widetilde{\mathcal{L}}_{N}^{21}$ above, whereas we can deal with $\widetilde{\mathcal{L}}_{N}^{44}$ by following the bounds on $\widetilde{\mathcal{L}}_{N}^{22}$ and $\widetilde{\mathcal{L}}_{N}^{23}$. Thus we get

[TABLE]

This concludes the proof of Proposition 5.5.

∎

6. Proof of Theorem 1.2

Let us fix $0<\alpha\leq 1$.

6.1. Lipschitz bound and uniqueness

Let $s>s_{\alpha}$, $0<T\leq 1$ and assume that $u\in Y^{s}_{T}$ and $v\in Y^{s}_{T}$ are two solutions to (1.8) on $]0,T[$ associated with initial data $u_{0},v_{0}\in H^{s}({\mathbb{R}})$ such that $u_{0}-v_{0}\in\overline{L}^{2}({\mathbb{R}})$. We fix $-\frac{1}{2}+\frac{\alpha}{4}<\sigma<\min(0,s-2+\frac{3}{2}\alpha)$ and set $w=u-v$. It is clear that $w(0)=w_{0}\in\overline{H}^{\sigma}$ and the continuous embedding from $Y^{s}_{T}$ into $Z^{\sigma}_{T}$ ensures that $w\in Z^{\sigma}_{T}$. Now, from Duhamel formula we have

[TABLE]

and thus ,

[TABLE]

Moreover, classical linear estimates in the context of Bourgain’s space (cf. [5], [11]) lead to

[TABLE]

These estimates combined with (5.14) and the fact that $w\in Z^{\sigma}_{T}$, ensure that $w\in\overline{Z}^{\sigma}_{T}$.

Combining Corollary 5.1, Lemma 5.4 and Proposition 5.5, we obtain that, for any $N_{0}\gg(1+\|z\|_{L^{\infty}_{T}H^{s}_{x}})^{\frac{2}{\alpha}}$,

[TABLE]

where $\gamma=s-2+\frac{3}{2}\alpha-\sigma>0$. Taking $N_{0}\gg(1+\|u\|_{Y^{s}_{T}}^{2}+\|v\|_{Y^{s}_{T}}^{2})^{\frac{3}{\delta}}$ with

[TABLE]

This forces

[TABLE]

for 0<T^{\prime}\lesssim\min\Bigl{\{}(1+\|u\|_{Y^{s}_{T}}^{2}+\|v\|_{Y^{s}_{T}}^{2})^{-\frac{9}{2\delta}},T\Bigr{\}}.

Therefore, taking $u_{0}-v_{0}=0$, we obtain that $u\equiv v$ on $]0,T^{\prime}[$. Noticing, that equation (1.1) ensures that $u_{t},v_{t}\in L^{\infty}(0,T;H^{s-2}(\mathbb{R}))$ and thus $u,v\in C([0,T];L^{2}({\mathbb{R}}))$, it follows that $v(T^{\prime})=u(T^{\prime})$. Repeating this argument a finite number of times we extend the uniqueness result on $]0,T[$.

6.2. A priori estimates on smooth solutions

According to [32] (see also [4] to get the continuity of the flow-map) for any $u_{0}\in H^{\theta}({\mathbb{R}})$, with $\theta\geq 3$, there exists a positive time $T=T(\|u_{0}\|_{H^{3}})$ and a unique solution $u\in C([0,T];H^{\theta}({\mathbb{R}}))$ to (1.8) emanating from $u_{0}$. Moreover, for any fixed $R>0$, the map $u_{0}\mapsto u$ is continuous from the ball of $H^{\theta}({\mathbb{R}})$ of radius $R$ centered at the origin into $C([0,T];H^{\theta}({\mathbb{R}}))$.

Let $u_{0}\in H^{\infty}({\mathbb{R}})$. From the above result $u_{0}$ gives rise to a solution $u\in C([0,T^{*}[;H^{\infty}({\mathbb{R}}))$ to (1.8) with $T^{*}\geq T(\|u_{0}\|_{H^{3}})$ and

[TABLE]

Let $0<T<T^{*}$. Since $u\in C([0,T];H^{\infty}({\mathbb{R}}))$ is a solution to (1.8), we must have $u_{t}\in L^{\infty}(0,T;H^{\infty}({\mathbb{R}}))$ and thus it is easy to check that $u\in Y^{\theta}_{T}$ for any $\theta\in{\mathbb{R}}$ and

[TABLE]

In the sequel, $\kappa>0$ and $C_{0}>1$ are the constants appearing in Corollary 4.1.

We claim that there exist $A_{0}>0$, $0<\beta_{0}\ll 1$ such that $T^{*}\geq A_{0}(1+\|u_{0}\|_{H^{s^{\prime}}})^{-\frac{1}{\beta}_{0}}$ and, for any $s_{\alpha}<s^{\prime}\leq 3$,

[TABLE]

Indeed, fixing $s_{\alpha}<s^{\prime}\leq 3$, it follows from (6.3) that

[TABLE]

is a non empty interval of ${\mathbb{R}}_{+}^{*}$. Let us set $T_{0}=\sup\Lambda_{s^{\prime}}$. We proceed by contradiction, assuming that $T_{0}<A_{0}(1+\|u_{0}\|_{H^{s^{\prime}}})^{-\frac{1}{\beta}_{0}}$ since otherwise we are done. Note that by continuity

[TABLE]

According to Corollary 4.1, Lemma 4.4 and Proposition 4.5, there exist $C_{1},C_{2}\gg 1$ and $0<\varepsilon_{0}\ll 1$ such that for any $s>s_{\alpha}$, $N_{0}\geq C_{1}(1+\|u_{0}\|_{H^{s_{\alpha}}})^{\frac{1}{\alpha}}$ and any $0<T<\min\{\varepsilon_{0}\|u_{0}\|_{H^{s^{\prime}}}^{-\frac{1}{\kappa}},T_{0}\}$, it holds

[TABLE]

We take $A_{0}\leq\varepsilon_{0}$ and $\beta_{0}\leq\kappa$ so that $\min\{\varepsilon_{0}\|u_{0}\|_{H^{s^{\prime}}}^{-\frac{1}{\kappa}},T_{0}\}=T_{0}$ and thus, by continuity, (6.5) is satisfied with $T=T_{0}$. Now, applying (6.5) with $s=3$, $N_{0}=[8C_{2}(1+\|u_{0}\|_{H^{s^{\prime}}})^{2}]^{{\frac{1}{\delta}}+}$, where $\delta=\min\{\alpha,{s^{\prime}}-s_{\alpha}\}$, and $T=\min\{T_{0},(8C_{2}N_{0}^{\frac{3}{2}})^{-1}\}$, we get

[TABLE]

Therefore, taking $A_{0}\leq\varepsilon_{0}$ and $\beta_{0}\leq\kappa$ small enough so that

[TABLE]

we obtain that (6.6) is satisfied with $T=T_{0}$. In view of (6.2), this forces $T^{*}>T_{0}$. Now taking $s=s^{\prime}$ and proceeding in the same way we get

[TABLE]

But since $T^{*}>T_{0}$, by continuity this ensures that $\|u\|_{Y^{s^{\prime}}_{T}}^{2}\leq 2^{4}C_{0}^{2}\|u_{0}\|_{H^{s^{\prime}}}^{2}$ for some $T_{0}<T<T^{*}$ which contradicts the definition of $T_{0}$ . This concludes the proof of (6.4).

Note that Lemma 4.4 and Corollary 4.2 then ensure that for any $N_{0}\geq C_{1}(1+\|u_{0}\|_{H^{s_{\alpha}}})^{\frac{1}{\alpha}}$, it holds

[TABLE]

where $T>0$ is defined as in (6.4).

6.3. Local existence in $H^{s}({\mathbb{R}})$, $s>s_{\alpha}$.

Now let us fix $s>s_{\alpha}$ and $u_{0}\in H^{s}({\mathbb{R}})$. We set $u_{0,n}=P_{\leq n}u_{0}$ and we denote by $u_{n}\in C([0,T_{n}^{\star});H^{\infty}({\mathbb{R}}))$ the solutions to (1.8) emanating from $u_{0,n}$. Setting

[TABLE]

it follows from (6.4) that for any $n\in\mathbb{N}^{*}$, $T_{n}^{\star}\geq T$ and

[TABLE]

Let $-\frac{1}{2}+\frac{\alpha}{4}<\sigma<\min(0,s-2+\frac{3}{2}\alpha)$. For $n\geq m\geq 1$, clearly $u_{0,n}-u_{0,m}\in\overline{L}^{2}({\mathbb{R}})$ and thus (6.1) ensures that

[TABLE]

where $0<T^{{}^{\prime\prime}}=T^{{}^{\prime\prime}}(\|u_{0}\|_{H^{s^{\prime}}})\leq T$. This last inequality combined with (6.7) ensure that

[TABLE]

for any $N_{0}\geq C_{1}(1+\|u_{0}\|_{H^{s_{\alpha}}})^{\frac{1}{\alpha}}$. This proves that $\{u_{n}\}$ is a Cauchy sequence in $C([0,T^{{}^{\prime\prime}}];H^{s}({\mathbb{R}}))$ and thus converges to some $u$ in this space. It is then not hard to check that $u\in Y^{s}_{T^{{}^{\prime\prime}}}$ and is a solution to (1.8) emanating from $u_{0}$. By the uniqueness result, this is the only one. Repeating this argument a finite number of times we obtain that actually $\{u_{n}\}$ converges to $u$ in $C([0,T];H^{s}({\mathbb{R}}))$ with $T$ defined in (6.8).

6.4. Continuity of the solution-map

Finally, to prove the continuity with respect to initial data, we take a sequence $\{u_{0}^{j}\}\subset B_{H^{s}}(0,2\|u_{0}\|_{H^{s}})$ that converges to $u_{0}$ in $H^{s}({\mathbb{R}})$. We denote by respectively $u^{j}$ and $u^{j}_{n}$ the associated solutions to (1.8) emanating from respectively $u_{0}^{j}$ and $P_{\leq n}u_{0}^{j}$. Noticing that

[TABLE]

we infer from (6.11) that

[TABLE]

with $T^{{}^{\prime\prime}}=T^{{}^{\prime\prime}}(\|u_{0}\|_{H^{s}})>0$. From

[TABLE]

and the continuity with respect to initial data in $H^{3}({\mathbb{R}})$ (note that $P_{\leq n}u_{0}$ and $P_{\leq n}u_{0}$ belong to $H^{\infty}({\mathbb{R}})$), it follows that $u^{j}\to u$ in $C([0,T^{{}^{\prime\prime}}];H^{s}({\mathbb{R}}))$. Iterating this process a finite number of times we obtain that $u^{j}\to u$ in $C([0,T];H^{s}({\mathbb{R}}))$ with $T$ defined in (6.8) which completes the proof of Theorem 1.2.