Global well-posedness for low regularity data in the 2d modified Zakharov-Kuznetsov equation

TL;DR

This paper proves global well-posedness for the 2D modified Zakharov-Kuznetsov equation with low regularity initial data using the $I$-method, extending previous results to broader function spaces and initial conditions.

Contribution

It establishes global well-posedness for $H^s$ solutions with $s > 3/4$ in the defocusing case and under mass constraints in the focusing case, improving prior results.

Findings

Global well-posedness for $s > 3/4$ in the defocusing case.

Global well-posedness under mass constraints in the focusing case.

Extension of previous results by Linares and Pastor.

Abstract

We consider the modified Zakharov-Kuznetsov (mZK) equation in two space dimensions in both focusing and defocusing cases. Using the $I$-method, we prove the global well-posedness of the $H^s$ solutions for $s>\frac{3}{4}$ for any data in the defocusing case and under the assumption that the mass of the initial data is less than the mass of the ground state solution of $\Delta \varphi - \varphi + \varphi^3 = 0$ in the focusing case. This improves the global well-posedness result of Linares and Pastor [20].

Equations

$$\{\begin{array}{l}{v}_t+{\partial }_x(\Delta v)+\sigma {\partial }_x(v^3)=0\text{, }(x,y)\in {\mathbb{R}}^2,t>0\\ v(x,y,0)=v_0(x,y),\end{array}$$

$$v_t+{\partial }_x(\Delta v+v^2)=0,$$

$$\mathcal{M}[v(t)]={\int }_{{\mathbb{R}}^2}{v}^2(x,y,t)dxdy\equiv \mathcal{M}[v_0]$$

$$\mathcal{E}[v(t)]=\frac{1}{2}{\int }_{{\mathbb{R}}^2}|\nabla v(x,y,t){|}^{2}dxdy-\frac{\sigma }{4}{\int }_{{\mathbb{R}}^2}{v}^4(x,y,t)dxdy\equiv \mathcal{E}[v_0].$$

$$v_{\lambda }(x,y,t):=\lambda v\left(\lambda x,\lambda y,{\lambda }^{3}t\right).$$

$$\Delta \phi -\phi +{\phi }^3=0,(x,y)\in {\mathbb{R}}^2.$$

$$v(x,y,t)={\phi }_c(x-ct,y)$$

$${\phi }_c(x,y)=\sqrt{c}\phi (\sqrt{c}x,\sqrt{c}y)$$

$$\Vert f{\Vert }_{L^4({\mathbb{R}}^2)}^4\le \frac{2}{\Vert \phi {\Vert }_{L^2({\mathbb{R}}^2)}^2}\Vert f{\Vert }_{L^2({\mathbb{R}}^2)}^2\Vert \nabla f{\Vert }_{L^2({\mathbb{R}}^2)}^2.$$

$$\Vert \nabla v(t){\Vert }_{L^2({\mathbb{R}}^2)}^2\le \frac{2}{\left(1-\frac{\Vert v_0{\Vert }_{L^2({\mathbb{R}}^2)}^2}{\Vert \phi {\Vert }_{L^2({\mathbb{R}}^2)}^2}\right)}\mathcal{E}[v_0].$$

$$\Vert v_0{\Vert }_{L^2({\mathbb{R}}^2)}<\Vert \phi {\Vert }_{L^2({\mathbb{R}}^2)}$$

$$\underset{t\in [0,T]}{\sup }\Vert v(t){\Vert }_{H^s({\mathbb{R}}^2)}\lesssim {\left(1+T\right)}^{\frac{1-s}{4s-3}+}.$$

$$\{\begin{array}{l}{\partial }_{t}u+({\partial }_x^3+{\partial }_y^3)u+\sigma a({\partial }_x+{\partial }_y)(u^3)=0\\ u(x,y,0)=u_0,\end{array}$$

$$\Vert u_0{\Vert }_{L^2({\mathbb{R}}^2)}<\sqrt{2ab}\Vert \phi {\Vert }_{L^2({\mathbb{R}}^2)},$$

$$\underset{t\in [0,T]}{\sup }\Vert u(t){\Vert }_{H^s({\mathbb{R}}^2)}\lesssim {\left(1+T\right)}^{\frac{1-s}{4s-3}+}.$$

$${\mathcal{F}}_{x,y,t}(f)(\xi ,\eta ,\tau )={\iiint }_{{\mathbb{R}}^3}f(x,y,t)e^{-i(x\xi +y\eta +t\tau )}dxdydt.$$

$${\mathcal{F}}_{x,y}(f)(\xi ,\eta ,t)={\iint }_{{\mathbb{R}}^2}f(x,y,t)e^{-i(x\xi +y\eta )}dxdy.$$

$$\Vert u{\Vert }_{L_x^p{L}_y^q{L}_t^r}={\left({\int }_{\mathbb{R}}{\left({\int }_{\mathbb{R}}{\left({\int }_{\mathbb{R}}|u(x,y,t){|}^{r}dt\right)}^{\frac{q}{r}}dy\right)}^{\frac{p}{q}}dx\right)}^{\frac{1}{p}}$$

$$\Vert u{\Vert }_{H^s({\mathbb{R}}^2)}=\Vert J^{s}u{\Vert }_{L^2({\mathbb{R}}^2)}$$

$$\Vert u{\Vert }_{{\dot{H}}^s({\mathbb{R}}^2)}=\Vert D^{s}u{\Vert }_{L^2({\mathbb{R}}^2)}.$$

$$\Vert u{\Vert }_{X_{s,b}}=\Vert ⟨\zeta {⟩}^s⟨\tau -({\xi }^3+{\eta }^3){⟩}^b\mathcal{F}(u)(\xi ,\eta ,\tau ){\Vert }_{L_{\xi ,\eta ,\tau }^2}<\infty .$$

$$\Vert u{\Vert }_{X_{s,b}^T}=\inf \left\{\Vert v{\Vert }_{X_{s,b}}:v\in X_{s,b}\text{ and }v(t)=u(t)\text{ on }[0,T]\right\}.$$

$$\{\begin{array}{l}{x}^{\prime }=ax+by\\ y^{\prime }=ax-by,\end{array}$$

$$\begin{array}{r}{\partial }_{x}v(x,y)=a({\partial }_{x^{\prime }}+{\partial }_{y^{\prime }})u(x^{\prime },y^{\prime })\\ {\partial }_{y}v(x,y)=b({\partial }_{x^{\prime }}-{\partial }_{y^{\prime }})u(x^{\prime },y^{\prime }).\end{array}$$

$${\left|\nabla v\right|}^2=v_x^2+v_y^2$$

$$\Vert \nabla v{\Vert }_{L^2({\mathbb{R}}^2)}^2=\frac{a^2+b^2}{|2ab|}\Vert \nabla u{\Vert }_{L^2({\mathbb{R}}^2)}^2-\frac{2(a^2-b^2)}{|2ab|}{\int }_{{\mathbb{R}}^2}{u}_{x^{\prime }}{u}_{y^{\prime }}d{x}^{\prime }d{y}^{\prime }.$$

$$\frac{2}{a^2+b^2}|2ab|\mathcal{E}[v]$$

$$E[u(t)]=\frac{|2ab|}{a^2+b^2}\mathcal{E}[v(t)].$$

$$E[u(t)]=\frac{1}{2}{\int }_{{\mathbb{R}}^2}|\nabla u(x,y,t){|}^{2}dxdy-\frac{1}{2}{\int }_{{\mathbb{R}}^2}(u_x{u}_y)(x,y,t)dxdy-\frac{\sigma a}{4}{\int }_{{\mathbb{R}}^2}{u}^4(x,y,t)dxdy,$$

$$u_t=({\partial }_x+{\partial }_y)E^{\prime }(u).$$

Full text

1. Introduction

We consider the two-dimensional initial value problem (IVP)

[TABLE]

where $v$ is a real-valued function, $\Delta=\partial_{x}^{2}+\partial_{y}^{2}$ is the Laplacian operator in 2d and $\sigma=\pm 1$ denotes the focusing (plus sign) and defocusing (minus sign) cases of the equation. This equation is a modification (thus, the name) of the standard Zakharov-Kuznetsov (ZK) equation

[TABLE]

introduced in 3d by Zakharov and Kuznetsov [26] to model the propagation of nonlinear ion-acoustic waves in magnetized plasma.

The mZK equation has been extensively studied in recent years, we recall relevant well-posedness results in 2d. Biagioni and Linares [4] studied the local well-posedness of solutions to mZK with data in $H^{1}(\mathbb{R}^{2})$. Linares and Pastor [19] proved the local well-posedness in $H^{s}(\mathbb{R}^{2})$ for $s>\frac{3}{4}$ and they also showed the ill-posedness (non-uniform data to solution map) for $s\leq 0$, so one can not expect well-posedness in the critical space $L^{2}(\mathbb{R}^{2})$. Ribaud and Vento [22] proved local well-posedness in $H^{s}(\mathbb{R}^{2})$ for $s>\frac{1}{4}$, which is currently the best result known for the local well-posedness for the 2d mZK equation. Linares and Pastor [20] proved the global well-posedness in $H^{s}(\mathbb{R}^{2})$ for $s>\frac{53}{63}$ with an additional assumption on the size of the initial data (related to the ground state) in the focusing case. In this paper, we use the I-method and obtain the global well-posedness in $H^{s}(\mathbb{R}^{2})$ space for $s>\frac{3}{4}$, thus, improving the result of Linares and Pastor [20].

During their lifespan, solutions $u(x,y,t)$ to the mZK equation (1) conserve the mass $\mathcal{M}$ and energy $\mathcal{E}$:

[TABLE]

and

[TABLE]

Note that if $v$ is a solution to (8), then so is $v_{\lambda}$, its rescaled version

[TABLE]

The equation (1) is referred to as the $L^{2}$-critical (or mass-critical), since its $L^{2}$ norm is invariant under this rescaling. As with most focusing $L^{2}$-critical equations, solutions may blow up in finite time, which is also the case for this equation, see [12], and thus, in order to consider the global well-posedness we have to put a restriction on the size of the initial data. For that we recall the notion of the ground state or traveling waves. Let $\varphi$ be the only radial and positive solution of

[TABLE]

Then

[TABLE]

is a solution of the focusing mZK equation which travels only in the $x$-direction. Here, $\varphi_{c}$ is the dilation of $\varphi$ given by

[TABLE]

and solves the equation $\Delta\varphi_{c}-c\varphi_{c}+\varphi_{c}^{3}=0$.

The existence of solutions of the equation (4) in 2d was considered by Berestycki, Gallouët and Kavian [1], see Strauss [23], Berestycki and Lions [2], Berestycki, Lions and Peletier [3] for the existence in other dimensions. Regarding the uniqueness, Kwong [18] showed that radial and positive solutions are unique.

The function $\varphi$, also known as the ground state, is related to the following sharp Gagliardo-Nirenberg inequality (see Weinstein [25])

[TABLE]

Recall that from the definition of the energy (3) in the defocusing case $\sigma=-1$, we immediately have $\|{\nabla v(t)}\|^{2}_{L^{2}(\mathbb{R}^{2})}\leq 2\mathcal{E}[v_{0}]$. On the other hand, in the focusing case $\sigma=1$, if we assume that $\|v_{0}\|_{L^{2}(\mathbb{R}^{2})}<\|\varphi\|_{L^{2}(\mathbb{R}^{2})}$, then the sharp Gagliardo-Nirenberg inequality (5) yields

[TABLE]

Therefore, the solution of (1) is global in $H^{1}(\mathbb{R}^{2})$ for all $v_{0}\in H^{1}(\mathbb{R}^{2})$ when $\sigma=-1$, and for all $v_{0}\in H^{1}(\mathbb{R}^{2})$ such that $\|v_{0}\|_{L^{2}(\mathbb{R}^{2})}<\|\varphi\|_{L^{2}(\mathbb{R}^{2})}$ when $\sigma=1$.

In this paper we are interested in global well-posedness question in $H^{s}(\mathbb{R}^{2})$ with $s<1$, and in particular, in the application of the almost conservation method to the setting of Zakharov-Kuznetsov model. We use the $I$-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao [8] (see also [9]). Briefly recalling the approach, we note that since the energy is not well-defined for initial data in $H^{s}(\mathbb{R}^{2})$ with $s<1$, a smoothing operator $I$ is introduced, so that the energy of the smoothened solution (or, the modified energy) is finite. Even though the modified energy of the solution is not conserved in time, it can be shown to be slowly growing (or, almost conserved). By controlling the growth of this modified energy, we can iterate a local existence result finitely many times to obtain the existence of the solution for any time $T>0$.

One of the main ingredients of proving the type of the local existence theorem we need is to establish a trilinear estimate in the suitable Bourgain spaces $X_{s,b}$ associated to the linear part of the equation. To prove the trilinear estimate, we use a smoothing effect proved by Faminskii [10], which is the 2d upgrade of the smoothing attained by Kenig, Ponce and Vega [17] for the Airy equation. To complete the proof, we need an $L^{4}$-based maximal in time estimate, which can be easily obtained by following Kenig, Ponce and Vega’s approach [17] for the KdV equation, if the dispersion relation associated to the underlying Bourgain spaces is symmetric in both spatial variables (see, for example, Grünrock [14]). Therefore, following Grünrock and Herr [15], we symmetrize the mZK equation and work with the symmetrized version instead. We prove the trilinear estimate in the Bourgain spaces associated with the linear part of the symmetrized mZK equation. Since the symmetrization changes the conserved quantities associated with the IVP, we also prove that the $H^{1}(\mathbb{R}^{2})$ norm of the smoothened solution is bounded by its energy. Finally, we obtain a polynomial growth of the $H^{s}(\mathbb{R}^{2})$ norm of the solution.

In this paper, we consider both focusing and defocusing cases of the mZK equation. In the focusing case, $\sigma=1$, we prove global well-posedness in $H^{s}(\mathbb{R}^{2})$, $s>\frac{3}{4}$, under the assumption that $L^{2}$ norm of the initial data is less than the $L^{2}$ norm of the ground state solution. In the defocusing case, $\sigma=-1$, we prove the same result without any restriction on the size of initial data. As discussed before, this is exactly the situation in $H^{1}(\mathbb{R}^{2})$. Our first result is the following

Theorem 1.1**.**

The initial value problem (1) is globally well-posed in $H^{s}(\mathbb{R}^{2})$, $s>\frac{3}{4}$, for any $v_{0}\in H^{s}(\mathbb{R}^{2})$ when $\sigma=-1$, and for $v_{0}\in H^{s}(\mathbb{R}^{2})$ such that

[TABLE]

when $\sigma=1$; here, $\varphi$ is the ground state solution of (4).

Furthermore, for any time $T>0$, the solution $v(t)$ satisfies the following polynomial bound

[TABLE]

To prove the above result we deal with the symmetrized version of the mZK equation. For that we make a linear change of variables $x\mapsto ax+by$ and $y\mapsto ax-by$ with $a=2^{-\frac{2}{3}}$ and $b=3^{\frac{1}{2}}2^{-\frac{2}{3}}$, following Grünrock and Herr [15], to obtain

[TABLE]

where $\sigma$ still denotes the sign; see details on this symmetrization and properties of the new equation in Section 2.1. Since this change of variables (12) is essentially a rotation, we study the equation (8) instead of (1) without changing the well-posedness theory. Our second result is the following

Theorem 1.2**.**

The initial value problem (8) is globally well-posed in $H^{s}(\mathbb{R}^{2})$, $s>\frac{3}{4}$, for any $u_{0}\in H^{s}(\mathbb{R}^{2})$ when $\sigma=-1$, and for $u_{0}\in H^{s}(\mathbb{R}^{2})$ such that

[TABLE]

when $\sigma=1$.

Furthermore, for any time $T>0$, the solution $u(t)$ of (8) satisfies the following polynomial bound

[TABLE]

Remark 1.3**.**

Since the Jacobian of the above change of variables is $\left\lvert 2ab\right\rvert$, undoing the change, we see that the threshold condition (9) is equivalent to (6), and (10) implies (7). Therefore, Theorem 10 and Theorem 1.1 are equivalent, including the polynomial growth bounds (10) and (7).

This paper is organized as follows. In the next section, we introduce some notations and preliminaries that will be used throughout the paper. In Section 3 we prove a trilinear estimate required for a variant of the usual local existence theorem. Section 4 contains a refinement of bilinear Strichartz estimate when frequencies are separated. In Section 5, we introduce the modified energy and prove the almost conservation law. Section 6 is about the local existence theorem of the modified solution. Finally, in Section 7, we prove the global result stated in Theorem 10, which in turn implies Theorem 1.1.

1.1. Acknowledgements

L.G.F. was partially supported by CNPq, CAPES and FAPEMIG (Brazil). S.R. was partially supported by the NSF CAREER grant DMS-1151618 and NSF grant DMS-1815873. D.B. had partial graduate research and travel support to work on this project from grant DMS-1151618 (PI: Roudenko).

2. Notations, symmetrization and linear estimates

Throughout this paper, we shall denote the two dimensional spatial variable pair by $(x,y)$ and its dual Fourier variable by $\zeta=(\xi,\eta)$. For any integer $j$, we denote $\zeta_{j}=(\xi_{j},\eta_{j})$. As usual, we denote the time variable by $t$ and its Fourier dual variable by $\tau$.

We use $\mathcal{F}_{x,y,t}$ (or $\mathcal{F}$) to denote the Fourier transform both in space and time variables

[TABLE]

When the Fourier transform is computed in one or two variables out of $x$, $y$ and $t$, we write the variable(s) as a subscript of $\mathcal{F}$. In this way, for a function $f\in\mathcal{S}$, the Schwartz class, for example, $\mathcal{F}_{x,y}$ is defined by

[TABLE]

$\mathcal{F}_{x,y,t}$ and $\mathcal{F}_{x,y}$ are also denoted by $\widehat{(.)}$ and $\widetilde{(.)}$, respectively.

Given $f(\xi,\eta,\tau)$ in the Schwartz class, $\mathcal{F}^{-1}_{\xi,\eta,\tau}(f)(x,y,t)$ denotes the inverse Fourier transform of $f(\xi,\eta,\tau)$ in both space and time variables and the inverse Fourier transform is written in $(x,y,t)$ variables. Similar to $\mathcal{F}$, when the inverse Fourier transform is computed in one or two variables out of $\xi$, $\eta$ and $\tau$, we denote the variables as a subscript of $\mathcal{F}^{-1}$.

By $\|.\|_{L^{p}}$ or $\|.\|_{p}$ we denote the $L^{p}(\mathbb{R})$ norm. We use a subscript to denote the variable with respect to which norm is computed. The mixed Lebesgue norm is defined by

[TABLE]

with obvious modifications when $p$, $q$ or $r$ is $\infty$. We abbreviate $L_{x}^{p}L_{y}^{p}L_{t}^{q}$ and $L_{x}^{p}L_{y}^{p}L_{t}^{p}$ by $L_{x,y}^{p}L_{t}^{q}$ and $L_{x,y,t}^{p}$, respectively.

By $D^{\alpha}$ and $J^{\alpha}$, we define the Fourier multipliers with symbols $|\zeta|^{\alpha}$ and $\langle\zeta\rangle^{\alpha}$ respectively, where $\langle\zeta\rangle=\sqrt{1+|\zeta|^{2}}$. Also, $D_{x}^{\alpha}$ and $D_{y}^{\alpha}$ denote the Fourier multipliers with symbols $|\xi|^{\alpha}$ and $|\eta|^{\alpha}$, respectively. $J_{y}^{\alpha}$ and $J^{\alpha}_{y}$ denote the Fourier multipliers with symbols $\left\langle\xi\right\rangle^{\alpha}$ and $\left\langle\eta\right\rangle^{\alpha}$, respectively. In this notation, the norm in the Sobolev spaces $H^{s}(\mathbb{R}^{2})$ and $\dot{H}^{s}(\mathbb{R}^{2})$ are defined by

[TABLE]

and

[TABLE]

Let $s,\ b\in\mathbb{R}$. The Bourgain space $X_{s,b}$ is defined as the space of all tempered distributions $u$ on $\mathbb{R}^{2}\times\mathbb{R}$ such that

[TABLE]

Also, for $T>0$, we define the localized $X_{s,b}^{T}$ norm by

[TABLE]

These spaces were first used by Bourgain [5, 6] to study the nonlinear Schrödinger equation and the KdV equation, respectively.

Given $A,B\geq 0$, we write $A\lesssim B$, if for some universal constant $K>2$ we have $A\leq KB$. We write $A\sim B$, if both $A\lesssim B$ and $B\lesssim A$ hold. We write $A<<B$, if there is a universal constant $K>2$ such that $KA<B$. For arbitrarily small $\varepsilon>0$, we use $a+$ and $a-$ to denote $a+\varepsilon$ and $a-\varepsilon$, respectively. By $a++$ and $a--$ we denote $a+2\varepsilon$ and $a-2\varepsilon$, respectively.

2.1. Symmetrization Details

To symmetrize the initial mZK equation (1), we perform the linear change of variables with $a=2^{-\frac{2}{3}}$ and $b=3^{\frac{1}{2}}2^{-\frac{2}{3}}$

[TABLE]

and denote $u(x^{\prime},y^{\prime})=v(x,y)$ and $u_{0}(x^{\prime},y^{\prime})=v_{0}(x,y)$. Then

[TABLE]

Thus, re-defining $x^{\prime}$ by $x$ and $y^{\prime}$ by $y$, the IVP (1) becomes (8) Note that, from (13), we can write

[TABLE]

Thus, using the change of variables (12), we get

[TABLE]

Writing the energy $\mathcal{E}[v]$ in terms of $u$, we get,

[TABLE]

Observe that $(a^{2}+b^{2})=2^{\frac{2}{3}}$ and $\frac{2(a^{2}-b^{2})}{a^{2}+b^{2}}=-1$. We next define the energy of $u$ by

[TABLE]

Then

[TABLE]

where $a=(a^{2}+b^{2})^{-1}=2^{-\frac{2}{3}}$.

Remark 2.1**.**

In view of (16), the symmetrized ZK equation (8) can be written in the Hamiltonian form

[TABLE]

Since $E[u(t)]$ is conserved in time, so is $\mathcal{E}[v(t)]$. From (2) we also have mass conservation for $u(t)$ for all time $t>0$

[TABLE]

2.2. Linear estimates

We denote the unitary group associated to the linear part of symmetrized equation (8) by

[TABLE]

that is, for any $u_{0}\in H^{s}(\mathbb{R}^{2})$ the propagator $U(t)u_{0}$ is the solution to the linear problem

[TABLE]

Next, we recall some linear estimates in the mixed Lebesgue spaces as well as in the $X_{s,b}$ spaces. Following the proof of Theorem 2.2 of Faminskii [10] and using the fact that $(\xi^{3}+\eta^{3})$ is monotonic in both $\xi$ and $\eta$ variables, we get the smoothing estimates

[TABLE]

and

[TABLE]

From Theorem 3.1.(ii) of Kenig, Ponce and Vega [16] (see also estimate (7) of Grünrock and Herr [15]), we have the Strichartz type estimate

[TABLE]

where $\frac{2}{p}+\frac{2}{q}=1$, $p>2$. Taking $p=q=4$, we have,

[TABLE]

Also, taking $p=5$, $q=10/3$ in (20), we deduce

[TABLE]

From (21), using Sobolev embedding in $x$ and $y$ variables, and applying Lemma 2.3 of Ginibre, Tsutsumi and Velo [13], we have

[TABLE]

Interpolating (22) with the trivial estimate $\|u\|_{L^{2}_{x,y,t}}\leq\|u\|_{X_{0,0}}$, we obtain

[TABLE]

We recall the classical inequality (see Ginibre, Tsutsumi and Velo [13] or Tao’s book [24, Corollary 2.10])

[TABLE]

Applying the Sobolev embedding $H^{1+}(\mathbb{R}^{2})\hookrightarrow L^{\infty}(\mathbb{R}^{2})$ to (24), we have

[TABLE]

Interpolation between (22) and (25) yields

[TABLE]

where $\alpha(p)=(1+)\left(\frac{p-5}{p}\right)$ and $p\in(5,\infty)$.

3. A Trilinear Estimate

In this section, we prove a trilinear estimate, which will be the key ingredient in the proof of the local well-posedness theory of the modified solution $I_{N}u$ (see Theorem 6.4 below). First, we state the $L^{4}$-based maximal in time estimate from Grünrock [14].

Lemma 3.1** (Proposition 1 in [14]).**

Let $U(t)u_{0}$ be the free solution to the linear symmetrized ZK equation (18). Then

[TABLE]

An immediate consequence of the previous result is the following lemma.

Lemma 3.2**.**

Let $U(t)u_{0}$ be the free solution to the linear symmetrized ZK equation (18). Then

[TABLE]

Proof.

Since $\xi\eta\lesssim|\xi|^{2}+|\eta|^{2}$, we have $|\xi\eta|^{\frac{1}{4}}\lesssim|\zeta|^{\frac{1}{2}}$. Hence, (27) implies

[TABLE]

On the other hand, taking $L_{x}^{4}L_{t}^{\infty}$ in the both sides of the following Sobolev inequality in $y$ variable (see [24, Page 336])

[TABLE]

we get

[TABLE]

Now, using Minkowski’s inequality, we can interchange the $L_{t}^{\infty}$ and $L^{4}_{y}$ norms in the right-hand side of (30) to get

[TABLE]

Then, from (31) and (29), we have

[TABLE]

Since $\left\lvert\eta\right\rvert\leq\left\lvert\zeta\right\rvert$, (28) follows. ∎

Now, we prove the following trilinear estimate.

Lemma 3.3**.**

For any $\frac{3}{4}<s<1$, the following inequality holds

[TABLE]

Proof.

Since $|\xi|+|\eta|\lesssim\sqrt{|\xi|^{2}+|\eta|^{2}}$, it is enough to show that

[TABLE]

We proceed as in the work of the second author [11]. First note that it is enough to show

[TABLE]

Next, from the definition,

[TABLE]

where $\ast$ denotes the set $\{\sum_{i=1}^{3}(\xi_{i},\eta_{i},\tau_{i})=(\xi,\eta,\tau)\}$. We decompose the domain of integration according to the relative sizes of spacial frequencies. By symmetry, we assume $|\zeta_{1}|\geq|\zeta_{2}|\geq|\zeta_{3}|$. We consider the following 3 regions

[TABLE]

In region A, we have

[TABLE]

Hence, via Plancherel’s theorem and Holder’s inequality,

[TABLE]

where we have used (26) with $p=6$ and $\alpha(6)=\frac{1}{6}+<3/4<s$.

In region B, it is easy to see that $|\zeta|\langle\zeta\rangle^{s}\lesssim|\zeta_{1}|\langle\zeta_{1}\rangle^{s}$. Moreover, using Lemma 2.3 in [13], from the smoothing estimate (19) and the $L_{x}^{4}$-maximal function estimate (28), we get

[TABLE]

and

[TABLE]

Applying Hölder’s inequality, (33) and (34), we get

[TABLE]

In region C, we have

[TABLE]

Thus,

[TABLE]

Hence, using Hölder’s inequality, we have

[TABLE]

where we have used the $L^{5}$ Strichartz estimate (22) and the inequality (26) with $p=\frac{20}{3}$ (note that $\alpha(\frac{20}{3})+\frac{1}{2}=(\frac{1}{4}+)+\frac{1}{2}=\frac{3}{4}+$). ∎

4. A refinement of bilinear Strichartz estimate

In this section, we prove a refinement of the bilinear Strichartz estimate when supports of frequencies are separated. Originally, Bourgain [7] introduced such a refinement in the context of the nonlinear Schrödinger equation in two space dimensions. The second authot [11] also derived an improvement of the bilinear Strichartz estimate associated with the KdV equation for frequencies that are separated. It should be mentioned that Molinet and Pilod [21] proved a similar refinement for the unitary group associated with the linear part of the ZK equation (1) using a dyadic decomposition. Here, we follow a different approach and prove a refinement for the linear part associated with the symmetrized mZK equation (8).

Lemma 4.1**.**

Let $u$ and $v$ be supported on frequencies $\zeta_{1}$ and $\zeta_{2}$, respectively, with $\zeta_{1}\sim N_{1}$, $\zeta_{2}\sim N_{2}$ and $N_{1}<<N_{2}$. Then, we have

[TABLE]

Proof.

Let $u=U(t)u_{0}$ and $v=U(t)v_{0}$. It suffices to show that

[TABLE]

Recall that $\zeta=(\xi,\eta)$ and $\zeta_{i}=(\xi_{i},\eta_{i})$ for $i=1,2$. We make change of variables $\zeta_{2}:=\zeta-\zeta_{1},\zeta_{1}:=\zeta_{1}$ to write

[TABLE]

Now, we make the change of variables $\varphi(\xi_{1},\eta_{1},\xi_{2},\eta_{2})=(a,b,c,d)$, where

[TABLE]

Then, using $|\zeta_{i}|^{2}=\xi_{i}^{2}+\eta_{i}^{2}$ for $i=1,2$, the Jacobian of the change of variables $\varphi$ is given by

[TABLE]

Now, since $N_{1}<<N_{2}$, we have

[TABLE]

Thus,

[TABLE]

Using Plancherel’s theorem in time and space variables, Cauchy-Schwarz inequality and (36), we have

[TABLE]

Since $c=\xi_{1}-\eta_{1}$, we have $\left\lvert c\right\rvert\leq|\xi_{1}|+|\eta_{1}|\lesssim|\zeta_{1}|\lesssim N_{1}$. From (36), we obtain,

[TABLE]

Now, changing the variables $a,b,c,d$ back to $\xi_{1},\xi_{2},\eta_{1},\eta_{2}$ and using (36) again, we deduce,

[TABLE]

which concludes the proof of the lemma. ∎

Corollary 4.2**.**

Let $u$ and $v$ be as in the hypothesis of Lemma 4.1. Then we have the following bilinear refinement of Strichartz estimate

[TABLE]

Proof.

Replacing $u$ by $D^{\frac{1}{2}}u$ and $v$ by $D^{-1}v$ in (35), we have

[TABLE]

∎

5. Modified energy and almost conservation law

In this section, we define the modified energy $E^{1}$ for a solution to the IVP (8) and prove an almost conservation law. We denote the hyperplane $\Gamma_{n}$ of $(\mathbb{R}^{2})^{n}$ by

[TABLE]

equipped with the measure

[TABLE]

for any measurable function $f:(\mathbb{R}^{2})^{n}\rightarrow\mathbb{C}$. For a natural number $n$, we define $S_{n}$ as the group of all permutations on the set $\{1,\dots,n\}$. A measurable function $M_{n}$ defined on $\Gamma_{n}$ is called a symmetric $n$-multiplier if for any permutation $\pi\in S_{n}$, we have

[TABLE]

First, we derive a formula for the time-derivative of a general $n$-linear functional.

Proposition 5.1**.**

Let $u$ solve the IVP (8) and $M_{n}$ be a symmetric $n$-multiplier.

Let $\Lambda_{n}$ be an $n$-linear functional defined by

[TABLE]

Then, recalling the notation $\zeta_{i}=(\xi_{i},\eta_{i})$, we have

[TABLE]

where

[TABLE]

Proof.

Applying Fourier transform to (8) in spatial variable, we get

[TABLE]

Differentiating (38) with respect to $t$ and applying (41), we get

[TABLE]

Now applying a permutation of indexes, interchanging $j$ with $n$ (or the variables $\zeta_{j}$) in the integration in the last term above, we rewrite the last term and obtain

[TABLE]

where $\alpha_{n}$ is defined in (40). ∎

5.1. Modified energy functional

For a large positive real number $N$ and $0<s<1$, define a Fourier multiplier operator $I_{N}:H^{s}(\mathbb{R}^{2})\rightarrow H^{1}(\mathbb{R}^{2})$ by

[TABLE]

where $m_{N}$ is smooth, radially symmetric, non-increasing function of $|\zeta|$ such that

[TABLE]

Note that for any $f\in X_{s_{0},b_{0}}$, we have

[TABLE]

For simplicity, we drop the subscript $N$ of $m_{N}$ and only write $m$. Further, for any $j\in\mathbb{N}$, we define

[TABLE]

Moreover, we sometimes abuse the notation to define $m$ as a function of a scalar variable $r\in[0,\infty)$ by $m(r)=m(\zeta)$, where $r=|\zeta|$.

We define, for all time $t>0$, the modified energy $E^{1}[u]$ of $u$ by

[TABLE]

Proposition 5.2**.**

Let $s>\frac{3}{4}$, $N>>1$ and $u\in H^{s}(\mathbb{R}^{2})$ be a solution to (8) on $[0,\delta]$. Then we have the following growth of the modified energy

[TABLE]

Proof.

First, using Proposition 5.1, we write $\frac{d}{dt}E^{1}[u](t)$ as a sum of $n$-linear functionals. By definition, on $\Gamma_{2}$, we have, $\zeta_{1}=-\zeta_{2}$ and $\eta_{2}=-\eta_{1}$. Hence, using Plancherel Theorem, we can write

[TABLE]

and

[TABLE]

Finally, one can see that

[TABLE]

Therefore, the energy (16) can be written as

[TABLE]

Also, using (44) and the definition (43), we obtain

[TABLE]

Moreover, since $\alpha_{2}=0$ on $\Gamma_{2}$ and $\zeta_{2}+\zeta_{3}+\zeta_{4}=-\zeta_{1}$ on $\Gamma_{4}$, differentiating the first term of (45) in time and using (39), we have

[TABLE]

Similarly, differentiating the second term of (45), we get

[TABLE]

where we have used $\Lambda_{4}(m_{1}\dots m_{4}(\xi_{j}^{3}+\eta_{j}^{3}))=\Lambda_{4}(m_{1}\dots m_{4}(\xi_{1}^{3}+\eta_{1}^{3}))$ for all $j=1,\dots,4$, by permuting the variables.

Collecting all the $\Lambda_{4}$ terms appearing in $\frac{d}{dt}E^{1}[u](t)$, we get

[TABLE]

Since $\sigma=\pm 1$, collecting all the $\Lambda_{6}$ terms, we get

[TABLE]

Hence, the derivative of the modified energy $E^{1}[u](t)$ is

[TABLE]

The Fundamental Theorem of Calculus yields

[TABLE]

Now, we integrate (46) in time variable from [math] to $\delta$ and take the absolute value to get

[TABLE]

To estimate the first term on the right-hand side, we decompose the function $u$ into dyadic constituents and work with a typical term in the infinite sum.

For example, we decompose $\widehat{I_{N}u}(\zeta_{1},t)$ as

[TABLE]

where $\mbox{supp}\,(\widehat{u}_{l_{1}})\subset[2^{l_{1}},2^{l_{1}+1}]$ for each $l_{1}\in\mathbb{N}$. We do a similar decomposition for other functions as well and index the projections by $l_{2},l_{3},l_{4}$. Then, the first term on the right-hand side of (47) can be written as

[TABLE]

where

[TABLE]

and $\mbox{supp}\,(\widehat{u}_{l_{i}})\subset[N_{l_{i}},2N_{l_{i}}]$, $N_{l_{i}}=2^{l_{i}}$ for each $l_{i}\in\mathbb{N},\ i=1,2,3,4$.

Our aim is to show that there exists $\varepsilon>0$ such that

[TABLE]

so that, after applying the infinite sum over $l_{1},\dots,l_{4}$, we get $N^{-1+}$ on the right-hand side, since $\sum\limits_{l_{i}\in\mathbb{N}}2^{-\varepsilon l_{i}}<\infty$, for $i=1,\dots,4$.

Define $N_{max}=\max\left\{N_{l_{i}}:i=1,2,3,4\right\}$. Since $N_{max}^{\varepsilon}\geq N_{l_{1}}^{\frac{\varepsilon}{4}}N_{l_{2}}^{\frac{\varepsilon}{4}}N_{l_{3}}^{\frac{\varepsilon}{4}}N_{l_{4}}^{\frac{\varepsilon}{4}}$, it suffices to show that

[TABLE]

Here, for brevity we write $u_{i}$ instead of $u_{l_{i}}$ for $i=1,2,3,4$. Thus, a typical term in the sum (48) is given by

[TABLE]

Without loss of generality, we assume that the Fourier transform of all the functions is non-negative and that $N_{2}\geq N_{3}\geq N_{4}$ by the symmetry of the term

[TABLE]

in $\zeta_{2},\zeta_{3},\zeta_{4}$ variables. Moreover, we can assume that $N_{2}\gtrsim N$, otherwise, the multiplier is zero, since $m_{1}=\dots=m_{4}=1$. Also, the condition $\sum\limits_{i=1}^{4}\zeta_{i}=0$ implies $N_{1}\lesssim N_{2}$.

Now, we consider the following nested subcases. Recall that our largest frequency is $N_{2}$. Based on where the second largest frequency $N_{3}$ is located compared to $N$, we get the following cases

1. (1)

$N_{3}<<N$. In this subcase, we have $N_{2}\gtrsim N>>N_{3}\geq N_{4}$. We also have that $N_{1}\sim N_{2}$. 2. (2)

$N\lesssim N_{3}$. In this case, based on whether $N_{3}$ is comparable to $N_{2}$ or not, we consider the subcases

1. (a)

$N_{3}<<N_{2}$. This is the case where $N_{4}\leq N_{3},\ N\lesssim N_{3}<<N_{2}$. Here, we also have $N_{1}\sim N_{2}$. 2. (b)

$N_{3}\sim N_{2}$. This subcase can be broken into two further subcases based on the comparison between the lowest frequency $N_{4}$ and $N$

1. (i)

$N_{4}<<N$. Based on the comparison between $N_{1}$ and $N_{2}$, we have two further subcases

1. (A)

$N_{1}<<N_{2}$ 2. (B)

$N_{1}\sim N_{2}$ 2. (ii)

$N\lesssim N_{4}$. Again, comparing $N_{4}$ and $N_{3}$, we have the further subcases

1. (A)

$N_{4}<<N_{3}$ 2. (B)

$N_{4}\sim N_{3}$

Now, we bound the multiplier in every terminal subcase in a pointwise manner.

Case 1. $N_{2}\gtrsim N>>N_{3}\geq N_{4}$. This implies $N_{1}\sim N_{2}$. By the Mean Value Theorem,

[TABLE]

which we substitute in (49). Using the bilinear Strichartz estimate (37), we get

[TABLE]

Case 2.(a). $N_{2}>>N_{3}\gtrsim N$ and $N_{3}\geq N_{4}$. In this case, we still have $N_{1}\sim N_{2}$.

Here, we use the pointwise bound

[TABLE]

Since $m$ is a non-decreasing function and $N_{1}\sim N_{2}$, we have $m(\zeta_{1})\lesssim m(\zeta_{2})$. Then, the bound on the multiplier becomes

[TABLE]

Hence, using the bilinear Strichartz estimate (37), we have,

[TABLE]

where we have used that for any $p>0$ with $p+s>1$, the function $m(\zeta)\left\lvert\zeta\right\rvert^{p}$ is increasing and $m(\zeta)\left\langle\zeta\right\rangle^{p}$ is bounded below.

Case 2.(b).i.A. In this case, we have $N_{4}<<N\lesssim N_{3}\sim N_{2}$ and $N_{1}<<N_{2}$.

Here, we pair $I_{N}u_{1}$ with $I_{N}u_{2}$ and $I_{N}u_{3}$ with $I_{N}u_{4}$. In the following subcases, we use the following crude bound for the multiplier

[TABLE]

Thus,

[TABLE]

where in the last step we have used $N_{max}\sim N_{2}$, $m(\zeta)\left\lvert\zeta\right\rvert^{p}$ is non-decreasing and $m(\zeta)\left\langle\zeta\right\rangle^{p}>1$ for $s+p\geq 1$, hence, $m(N_{3})N_{3}^{\frac{1}{2}}\gtrsim N^{\frac{1}{2}},\ m(N_{2})N_{2}\gtrsim N$ and $m(N_{4})\left\langle N_{4}\right\rangle\gtrsim 1$.

Case 2.(b).i.B. In this case, we have $N_{4}<<N\lesssim N_{3}\sim N_{2}\sim N_{1}$.

Here, we use the same bound for the multiplier as the previous case. The only frequencies that are separated are $N_{4}$ and $N_{2}$ (or, $N_{3}$ since $N_{2}\sim N_{3}$). So, we use inequality (37) on the pair $I_{N}u_{2}I_{N}u_{4}$ and inequality (23) on the other pair to get

[TABLE]

In the last step, we have used $m(N_{i})N_{i}^{\frac{1}{2}-}\gtrsim N^{\frac{1}{2}-}$, since $N_{i}\gtrsim N$ for $i=2,3$, $m(N_{4})\left\langle N_{4}\right\rangle^{\frac{1}{2}}\gtrsim 1$ and $N_{max}\sim N_{2}$.

Case 2.(b).ii.A. In this case, we have $N\lesssim N_{4}<<N_{3}\sim N_{2}$.

Here, we again use the same bound for the multiplier as in Case 2.(b).i.A. The only frequencies that are separated are $N_{4}$ and $N_{2}$ (or, $N_{3}$ since $N_{2}\sim N_{3}$). So, we use the bilinear refinement of Strichartz on the pair $I_{N}u_{2}I_{N}u_{4}$ and $L^{4}$-Strichartz estimate for the other pair.

[TABLE]

In the last step, we have used $m(N_{i})N_{i}^{p}\gtrsim N^{p}$ for any $p>0$ with $p+s>1$, since $N_{i}\gtrsim N$ for $i=2,3,4$ and $N_{max}\sim N_{2}$.

Case 2.(b).ii.B. In this case, we have $N\lesssim N_{4}\sim N_{3}\sim N_{2}$.

Here, we cannot use the bilinear refinement (37), since no two frequencies are separated. Therefore, we use $L^{4}$ Strichartz estimate (23) to control the term. We use the same bound for the multiplier as in Case 2.(b).i.A to get

[TABLE]

In the last step, we have used $m(N_{i})N_{i}^{p}\gtrsim N^{p}$ for any $p>0$ with $p+s>1$ since $N_{i}\gtrsim N$ for $i=2,3,4$ and $N_{max}\sim N_{2}$.

Remark 5.3**.**

Note that the cases 2.(b).i.B and 2.(b).ii.B provide the worst growth ($N^{-1+}$), which we use in (68) to determine the lower bound on the Sobolev index, i.e., $\frac{3}{4}$, for which the solution is globally well-posed. Thus, improving the growth in these cases will improve the global well-posedness result.

Now, we turn to the second term on the right-hand side of (47). Again, we perform a dyadic decomposition. Following the previous discussion, we define

[TABLE]

We arrange the frequencies $N_{1},\dots,N_{6}$ in descending order and call them $N_{1}^{*},\dots,N_{6}^{*}$, respectively (e.g., $N_{1}^{*}$ is the largest frequency, etc.). Also, we define $u^{*}_{i}=u_{i}(\zeta_{i}^{*},t)$, where $\zeta_{i}^{*}$ is the frequency variable associated with $N_{i}^{*}$.

For any $1\leq i\leq 6$, we have

[TABLE]

Thus,

[TABLE]

If $N_{1}^{*}<<N$, $m(\zeta_{1})=m(\zeta_{2})=m(\zeta_{3})=m(\zeta_{4}+\zeta_{5}+\zeta_{6})=1$. Then, defining $\nu=(1,1)$, we write

[TABLE]

Now, permuting the variables $\zeta_{1},\dots,\zeta_{6}$, we have

[TABLE]

where $S_{6}$ is the symmetric group on the set $\{1,\dots,6\}$.

Thus, we assume that $N\lesssim N_{1}^{*}$. Also, we cannot have $N_{1}^{*}>>N_{2}^{*}\geq N_{3}^{*}$, since the sum of frequencies is zero. Therefore, we have $N_{1}^{*}\sim N_{2}^{*}$. We shall break the frequency interactions into the following two cases, based on a comparison between $N_{2}^{*}$ and $N_{3}^{*}$,

1. (1)

$N_{1}^{*}\sim N_{2}^{*}\sim N_{3}^{*}\gtrsim N$ 2. (2)

$N_{1}^{*}\sim N_{2}^{*}>>N_{3}^{*}$ and $N_{1}^{*}\sim N_{2}^{*}\gtrsim N$.

Case 1. Using $m(N_{1}^{*})^{2-}(N_{1}^{*})^{2-}\gtrsim N^{2-}$ together with inequalities (22) and (26), we get

[TABLE]

where we have used (26) with $\alpha(\frac{15}{2})=\frac{1}{3}+$ and the inequality (42). Here, $(\frac{1}{3}+)+1-s<\frac{7}{12}+<1$ for any $s\in(3/4,1)$. Thus, we have

[TABLE]

Case 2. $N_{1}^{*}\sim N_{2}^{*}>>N_{3}^{*}$ and $N_{2}^{*}\gtrsim N$. We use the following bound

[TABLE]

and estimates (37) and (22) to get

[TABLE]

Here, we also used

[TABLE]

and

[TABLE]

due to (42), since $(\frac{1}{2}+)+1-s<\frac{3}{4}+$ for $s\in(\frac{3}{4},1)$. This completes the proof of Proposition 5.1. ∎

6. A variant of the local existence theorem

Applying the operator $I_{N}$ on IVP (8), we get the modified IVP

[TABLE]

The Duhamel’s formula for the IVP (50) is

[TABLE]

To work on $X_{s,b}$ spaces, we consider the following integral equation instead

[TABLE]

where $\psi,\psi_{\delta}$ are as in Lemma 6.1 below. It is clear that if $u$ is a solution to this equation, then $u|_{[0,\delta]}$ is a solution to (51). We will use the following two Lemmas (from Grünrock and Herr [15] and Colliander, Keel, Staffilani, Takaoka and Tao [9]) to establish a local existence result for $I_{N}u$ in $H^{1}(\mathbb{R}^{2})$ space.

Lemma 6.1** (Lemma 2.2 in [15]).**

Let $\psi\in C_{0}^{\infty}([-2,2])$ be even, $0\leq\psi\leq 1$, $\psi(t)=1$ for $|t|\leq 1$. Let $\psi_{\delta}(t)=\psi(\frac{t}{\delta})$ for $\delta>0$. For all $s,b\in\mathbb{R}$,

[TABLE]

Also, for $-\frac{1}{2}<b^{\prime}\leq 0\leq b\leq(b^{\prime}+1)$ and $0<\delta<1$,

[TABLE]

Lemma 6.2** (Lemma 12.1 in [9]).**

Let $s_{0}>0$, $n\geq 1$ and $Z,X_{1},\dots,X_{n}$ be translation-invariant Banach spaces. If $T$ is a translation-invariant n-linear operator such that

[TABLE]

for all $u_{1},\dots,u_{n}$ and for all $0\leq s\leq s_{0}$, then,

[TABLE]

for all $N\geq 1$, for all $0\leq s\leq s_{0}$, and for all $u_{1},\dots,u_{n}$ with the implicit constant independent of N.

Corollary 6.3**.**

For all $\frac{3}{4}<s<1$ and $N>>1$, we have

[TABLE]

Proof.

Using Lemma 6.2, it is enough to show that

[TABLE]

Next, note that for any $b\in\mathbb{R}$ and $\frac{3}{4}<s<1$ we have

[TABLE]

Indeed, recalling the definition of $m$ and decomposing the domain of integration in two parts, we can write

[TABLE]

where

[TABLE]

and

[TABLE]

When $\left\lvert\zeta\right\rvert\leq 2$, we have $m_{1}(\zeta)\leq 1$ and

[TABLE]

which implies

[TABLE]

On the other hand, when $\left\lvert\zeta\right\rvert\geq 2$, we have

[TABLE]

and hence,

[TABLE]

Thus, we also have

[TABLE]

and therefore,

[TABLE]

Similarly, we can show that

[TABLE]

since when $\left\lvert\zeta\right\rvert\leq 1$, we have $\left\langle\zeta\right\rangle^{s}\lesssim\left\langle\zeta\right\rangle=\left\langle\zeta\right\rangle m_{1}(\zeta)$ and when $\left\lvert\zeta\right\rvert\geq 1$, we have $\left\langle\zeta\right\rangle^{s}\lesssim\left\lvert\zeta\right\rvert^{s}=\frac{\left\lvert\zeta\right\rvert}{\left\lvert\zeta\right\rvert^{1-s}}\lesssim\frac{\left\langle\zeta\right\rangle}{\left\lvert\zeta\right\rvert^{1-s}}\lesssim\left\langle\zeta\right\rangle m_{1}(\zeta)$. Thus, since $I_{1}$ commutes with $(\partial_{x}+\partial_{y})$, (56) is equivalent to (32), completing the proof. ∎

Now, we prove a local existence result for the modified IVP (50). As a consequence, we also obtain a bound on the $X^{\delta}_{1,\frac{1}{2}+}$ norm of the modified solution $I_{N}u$, uniformly in the existence time, in terms of the initial data.

Theorem 6.4**.**

Let $\frac{3}{4}<s<1$ and $u_{0}\in H^{s}(\mathbb{R}^{2})$. Then there exists $\delta=\delta(\|I_{N}u_{0}\|_{H^{1}(\mathbb{R}^{2})})>0$ such that the modified IVP (50) has a solution $I_{N}u\in C([0,\delta];H^{1}(\mathbb{R}^{2}))$ with

[TABLE]

Proof.

Applying $X_{1,\frac{1}{2}+}^{\delta}$ norm on both sides of (52) and applying estimates (53) and (54) with $b=\frac{1}{2}+$ and $b^{\prime}=-\frac{1}{2}++$, we obtain

[TABLE]

where $\varepsilon=(1-b+b^{\prime})>0$. By definition of the localized norm (11), we have

[TABLE]

where the function $\phi(x,y,t)=u(x,y,t)$ on $\mathbb{R}^{2}\times[0,\delta]$ and

[TABLE]

Using the trilinear estimate (55) for $\phi$ and the relation (58), we get

[TABLE]

for some $0<\delta<1$ and $\varepsilon>0$. Setting $r=2\|I_{N}u_{0}\|_{H^{1}}$ and taking $\delta^{\epsilon}\leq\min\{\frac{1}{16\,r^{2}},1\}$ (and recalling that $\|I_{N}u(t)\|_{X_{1,\frac{1}{2}+}^{\delta}}$ is continuous in $\delta$), the operator $I_{N}$ defined in (52) maps the ball $B_{r}$ in $X_{1,\frac{1}{2}+}^{\delta}$ centered at the origin and of radius $r$ into itself and is a contraction by similar argument. Thus,

[TABLE]

completing the proof. ∎

Remark 6.5**.**

We can obtain a more precise bound on $\delta$ by observing that the trilinear estimate (32) holds for $b^{\prime}=0$ (see the proof of Lemma 32). Thus, the estimate (55) can be modified to

[TABLE]

Moreover, the inhomogeneous linear estimate (54) holds for $b^{\prime}=0$. Thus, the proof of the above theorem works for $\varepsilon=1-b+b^{\prime}=\frac{1}{2}-$, and hence, we get

[TABLE]

Next we prove an a priori bound on the $\dot{H}^{1}$ norm of the initial data of the modified IVP (50) in terms of its energy. In the focusing case, $\sigma=1$, the size of the $L^{2}$ norm of the initial data of (8) has to be bounded from above by the $L^{2}$ norm of the ground state.

Lemma 6.6**.**

Let $I_{N}u\in C([0,\delta];H^{1}(\mathbb{R}^{2}))$ be the solution of the modified IVP (50) given by Theorem 6.4. If $\sigma=-1$, we have,

[TABLE]

If $\sigma=+1$, the same conclusion (60) holds if

[TABLE]

Proof.

This lemma is straightforward in the defocusing case, and follows from the Gagliardo-Nirenberg inequality in the focusing case. For convenience of the reader, we include the proof. From (16) the energy of $I_{N}u$ at time $t=0$ is

[TABLE]

When $\sigma=-1$ the last term is positive, thus,

[TABLE]

The basic inequality $\alpha\beta\leq\frac{1}{2}(\alpha^{2}+\beta^{2})$ applied to the middle term, yields

[TABLE]

or

[TABLE]

Now we turn to the focusing case $\sigma=+1$. It is possible to rewrite the sharp Gagliardo-Nirenberg inequality (5) for the symmetrized version with $u$ (the sharp constant will change accordingly), or to go back to the original variable $v$ and use (5) as is. Here we use the second approach. Thus, using the change of variables (12), we rewrite the multiplier operator as

[TABLE]

Computing the inverse Fourier transform with the change of variables $p=a(\xi+\eta)$ and $q=b(\xi-\eta)$, we get

[TABLE]

where

[TABLE]

In particular,

[TABLE]

Thus, we have

[TABLE]

Note that since $I_{N}u_{0}\in H^{1}(\mathbb{R}^{2})$, so is $\widetilde{I}_{N}v_{0}$. Applying the sharp Gagliardo-Nirenberg inequality (5) to $f(x,y)=\widetilde{I}_{N}v_{0}(x,y)$, from (3) we get

[TABLE]

The condition (61) implies

[TABLE]

and we conclude that

[TABLE]

Next, using the equations (14) , (15) and (62), we can return to $u_{0}$. Indeed,

[TABLE]

and thus,

[TABLE]

Recalling that $\dfrac{2|a^{2}-b^{2}|}{a^{2}+b^{2}}=1$ and splitting the middle term above, we conclude that

[TABLE]

completing the proof. ∎

Now, we state a local existence result for the modified rescaled solution $I_{N}u_{\lambda}$, which will be used in the proof of the main theorem (10). Under the assumption that the modified energy of the rescaled solution is uniformly bounded from above, we conclude that that the time of existence of $I_{N}u_{\lambda}$ is a constant depending only on $\|u_{0}\|_{L^{2}(\mathbb{R}^{2})}$ (in particular, independent of the scaling factor $\lambda$). To arrive at this conclusion, the assumption (61) is required in the focusing case.

For $\lambda>0$ define the rescaled solution

[TABLE]

Lemma 6.7**.**

Assume that $E[I_{N}u_{0,\lambda}]<1$. Further assume that either $\sigma=-1$, or $\sigma=+1$ with (61) holds.

Then, there exists $\delta=\delta\left(\|u_{0}\|_{2}\right)>0$ such that $I_{N}u_{\lambda}\in C\left([0,\delta];H^{1}\left(\mathbb{R}^{2}\right)\right)$ with

[TABLE]

If $\sigma=+1$, the same conclusion (60) holds if

[TABLE]

Proof.

If we assume that $E[I_{N}u_{0,\lambda}]<1$ and either $\sigma=-1$ or $\sigma=+1$ and (61) holds, then, in view of (60), we have

[TABLE]

Moreover, using $m\leq 1$ and the fact that the IVP (8) is $L^{2}$-critical, we have

[TABLE]

Thus, we have

[TABLE]

Since $u_{\lambda}$ solves the IVP (8), from (59), the time of existence of $I_{N}u_{\lambda}$ given by Theorem 6.4 depends only on $\|u_{0}\|_{L^{2}}$, that is, $\delta=\delta(\|u_{0}\|_{L^{2}})>0$. Finally, using (57) and (64), we have (63). ∎

7. Proof of main theorem

As we mentioned in the introduction, it suffices to prove Theorem 10 as it is equivalent to Theorem 1.1.

Proof of Theorem 10: Let $u_{0}\in H^{s}$, where $\frac{3}{4}<s<1$. Given any $T>0$, we will show that the solution $u$ to (8) exists for time $[0,T]$, which is equivalent to showing that $I_{N}u_{\lambda}\in H^{1}(\mathbb{R}^{2})$ for time $\lambda^{3}T$. We will do this by iterating Lemma 6.7.

Note that the $\dot{H}^{s}$ norm of the rescaled solution is $\|u_{0,\lambda}\|_{\dot{H}^{s}(\mathbb{R}^{2})}=\lambda^{-s}\|u_{0}\|_{\dot{H}^{s}(\mathbb{R}^{2})}$, and thus, from (42), we deduce

[TABLE]

From (16), using a simple bound $-\int_{\mathbb{R}^{2}}u_{x}u_{y}dxdy\leq\frac{1}{2}\int_{\mathbb{R}^{2}}|\nabla u|^{2}dxdy$, Gagliardo-Nirenberg inequality (5) and the identity (65), we get

[TABLE]

where we used $\|I_{N}u_{0,\lambda}\|_{L^{2}(\mathbb{R}^{2})}^{2}\leq\|u_{0}\|_{L^{2}(\mathbb{R}^{2})}^{2}$.

Take $\lambda$ such that

[TABLE]

or,

[TABLE]

This implies

[TABLE]

We apply Lemma 6.7 and Proposition 5.2 to $u_{\lambda}$ and conclude that there exists a $\delta=\delta(\|u_{0}\|_{L^{2}(\mathbb{R}^{2})})$ such that

[TABLE]

Choosing $N$ large, we have $E[I_{N}u_{\lambda}(\delta)]<1$.

Now, since $\|I_{N}u_{\lambda}(\delta)\|_{L^{2}(\mathbb{R}^{2})}\leq\|u_{\lambda}(\delta)\|_{L^{2}(\mathbb{R}^{2})}=\|u_{0}\|_{L^{2}(\mathbb{R}^{2})}$, we can apply Lemma 6.7 again with $t=\delta$ as the starting time, followed by Proposition 5.2. In other words, starting at $t=\delta$, the solution $I_{N}u_{\lambda}$ exists for an additional time $\delta=\delta\left(\|u_{0}\|_{L^{2}(\mathbb{R}^{2})}\right)>0$ with

[TABLE]

Note that in the defocusing case (i.e. $\sigma=+1$), the requirement (61) to apply Lemma 6.7 again is trivially satisfied due the mass-conservation law (17). Indeed, under the assumption (61), as long as the solution exists, we have

[TABLE]

We repeat this process $M$ times, as long as $E[I_{N}u_{\lambda}(M\delta)]<1$ and additionally for $\sigma=+1$, as long as $\|u(M\delta)\|_{L^{2}(\mathbb{R}^{2})}<\sqrt{2ab}\|\varphi\|_{L^{2}(\mathbb{R}^{2})}$, which holds again by (67). Thus, the iterative application of Lemma 6.7 is valid as long as

[TABLE]

To show that the solution $I_{N}u_{\lambda}$ exists for time $\lambda^{3}T$, we need that

[TABLE]

where we have used (68) and the fact that $\delta$ depends only on $\|u_{0}\|_{L^{2}(\mathbb{R}^{2})}$. Using the relation (66) between $N$ and $\lambda$ , we have

[TABLE]

Now, we need the power of $N$ to be positive so that $T$ can be taken as large as we want. Therefore,

[TABLE]

Finally, we derive a polynomial bound for the $H^{s}$ norm of the solution to IVP (8). Note that in the previous argument we can select time $T\sim N^{\frac{4s-3}{s}-}$. By definition,

[TABLE]

The first term on the right hand side can be bounded by $\|u_{0}\|_{L^{2}}^{2}$, since $m\leq 1$ and mass conservation (17) holds. To bound the second term on the right-hand side of (69), first, we note that $E(I_{N}u_{\lambda})(\lambda^{3}T)\lesssim 1$. Moreover, when $\sigma=+1$, for any time $t\in[0,T]$, we have $\|u_{\lambda}(t)\|_{L^{2}(\mathbb{R}^{2})}\lesssim\sqrt{2ab}\|\varphi\|_{L^{2}(\mathbb{R}^{2})}$ from (67). Then using relations (60) and (66), we get

[TABLE]

From (69), we have

[TABLE]

which concludes the proof of Theorem 10. ∎