On the Cauchy problem for higher dimensional Benjamin-Ono and Zakharov-Kuznetsov equations

TL;DR

This paper studies a family of dispersive equations connecting higher dimensional Benjamin-Ono and Zakharov-Kuznetsov equations, establishing new well-posedness results through advanced mathematical techniques.

Contribution

It introduces new well-posedness results for fractional Zakharov-Kuznetsov equations using transversality and frequency localization methods.

Findings

Proved well-posedness for a family of dispersive equations.

Established techniques applicable to fractional higher-dimensional equations.

Enhanced understanding of dispersive equation behavior in higher dimensions.

Abstract

A family of dispersive equations is considered which links a higher dimensional Benjamin-Ono equation and the Zakharov-Kuznetsov equation. For these fractional Zakharov-Kuznetsov equations new well-posedness results are proved using transversality and localization of time to small frequency dependent time intervals.

Equations

$$\left\{\begin{array}[]{cl}\partial_{t}u+\partial_{x_{1}}(-\Delta)^{a/2}u&=u\partial_{x_{1}}u,\quad(t,x)\in\mathbb{R}\times\mathbb{K}^{n},\quad 1\leq a\leq 2\\ u(0)&=u_{0}\in H^{s}(\mathbb{K}^{n}),\end{array}\right.$$

$$(P_{N}f)\widehat{(}\xi )=\{\begin{array}{l}\Phi (\xi )\hat{f}(\xi ),N=1\\ {\chi }_N(\xi )\hat{f}(\xi ),\text{else,}\end{array}$$

$$\widehat{S_a(t)u_0}(\xi )=e^{-it{\xi }_1|\xi {|}^a}{\hat{u}}_0(\xi )$$

$${\partial }_{x_1}(P_{N}u{P}_{K}u)(K\ll N)$$

$$\Vert P_N{S}_a(t)u_0{P}_K{S}_a(t)v_0{\Vert }_{L_{t,x}^2(\mathbb{R}\times {\mathbb{R}}^n)}\lesssim {\left(\frac{K^{n-1}}{N^a}\right)}^{1/2}\Vert P_N{u}_0{\Vert }_{L^2}\Vert P_K{v}_0{\Vert }_{L^2}$$

$$\begin{array}{rl}& \Vert {\partial }_{x_1}(P_N{S}_a(t)u_0{P}_K{S}_a(t)v_0){\Vert }_{L^1([0,T];L^2({\mathbb{R}}^n))}\\ & \lesssim N|T{|}^{1/2}\Vert P_N{S}_a(t)u_0{P}_K{S}_a(t)v_0{\Vert }_{L^2([0,T];L_x^2({\mathbb{R}}^n)}\\ & \lesssim |T{|}^{1/2}N{\left(\frac{K^{n-1}}{N^a}\right)}^{1/2}\Vert P_N{u}_0{\Vert }_{L^2({\mathbb{R}}^n)}\Vert P_K{v}_0{\Vert }_{L_x^2({\mathbb{R}}^n)}\end{array}$$

$$\begin{array}{rl}\Vert S_a(t)f{\Vert }_{L_t^q(\mathbb{R},L_x^p({\mathbb{R}}^n))}& \lesssim \Vert f{\Vert }_{{\dot{H}}^s({\mathbb{R}}^n)}\\ \Vert S_a(t)f{\Vert }_{L_t^q([0,T],L_x^p({\mathbb{R}}^n))}& {\lesssim }_T\Vert f{\Vert }_{H^s({\mathbb{R}}^n)}\end{array}$$

$$\phi (\xi )=\sum _{i=1}^n\chi ({\xi }_i)$$

$${\partial }_{t}u+\sum _{i=1}^n{\partial }_{x_i}|D_{x_i}{|}^{a}u=\sum _{i=1}^n{\partial }_{x_i}(u^2)/2,(t,x)\in \mathbb{R}\times {\mathbb{K}}^n,$$

$$\left\{\begin{array}[]{cl}\partial_{t}u-\partial_{x_{1}}D_{x_{1}}^{a}u+\partial_{x_{1}}\partial_{x_{2}}^{2}u&=u\partial_{x_{1}}u,\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{2},\quad 1\leq a\leq 2\\ u(0)&=u_{0}\in H^{s}(\mathbb{R}^{n}),\end{array}\right.$$

$$E(u)={\int }_{{\mathbb{R}}^n}|D^{a/2}u{|}^2-\frac{1}{3}{u}^3(t,x)dx$$

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

$$\Vert f{\Vert }_{H^{s,0}({\mathbb{T}}^2)}^2=\sum _{(\xi ,\eta )\in {\mathbb{Z}}^2}(1+|\xi {|}^2{)}^s|\hat{f}(\xi ,\eta ){|}^2$$

$$\left|\int \psi (|\xi |)e^{i(t{\xi }_1|\xi {|}^a+x.\xi )}d\xi \right|\le C|t{|}^{-1}$$

$$\begin{array}{rl}I(x,t)& ={\int }_0^{\infty }dr\underset{\rho (r)}{\underbrace{r^{n-1}\psi (r)}}{\int }_{{\mathbb{S}}^{n-1}}d\sigma (\omega )e^{it(r^{a+1}{\omega }_1+x_{1}r{\omega }_1+\dots +x_{n}r\omega )}\\ & ={\int }_0^{\infty }\rho (r)\hat{\sigma }(y_{x,t}(r))dr,\end{array}$$

$$|\hat{\sigma }(y)|\lesssim (1+|y|{)}^{-\frac{n-1}{2}}$$

$${\int }_{1/2}^2{1}_{\{|t{r}^a+x_1|\le 2\}}(r)\rho (r)dr={\int }_{r^{\prime }\sim 1}{1}_{\{|t{r}^{\prime }+x_1|\le 2\}}\rho (r^{\prime })d{r}^{\prime }\le C|t{|}^{-1}$$

$$\begin{array}{rl}{\int }_{E_2}\rho (r)|\hat{\sigma }(y_{x,t}(r))|dr& \le {\int }_{|t{r}^a+x_1|\ge 2}\rho (r)|t{r}^{a+1}+x_{1}r{|}^{-\frac{n-1}{2}}dr\\ & \le C{\int }_{|tr+x_1|\ge 2}|tr+x_1{|}^{-\frac{n-1}{2}}dr=C|t{|}^{-\frac{n-1}{2}}{\int }_{|r+x_1/t|\ge 2/|t|}|r+x_1/t{|}^{-\frac{n-1}{2}}dr\end{array}$$

$$\hat{\sigma }(y)=c\frac{e^{i\Vert y\Vert }}{\Vert y\Vert }+c\frac{e^{-i\Vert y\Vert }}{\Vert y\Vert }+{\mathcal{E}}_{x,t}(y),$$

$$F^1$$

$$F^2$$

$$\begin{array}{rl}{f}^{\prime }(r)& =2t^2(a+1)r(r^a-r_{-})(r^a-r_{+}),\\ r_{\pm }& =-\frac{(a+2)x_1}{2(a+1)t}\pm \sqrt{{\left(\frac{a+2}{a+1}\right)}^2{\left(\frac{x_1}{t}\right)}^2-\frac{x_1^2}{(a+1)t^2}-\frac{\Vert x^{\prime }{\Vert }^2}{(a+1)t^2}}\end{array}$$

$$\int \frac{e^{i\varphi (r)}}{\varphi (r)}\rho (r)dr\sim \int \frac{d}{dr}[e^{i\varphi (r)}]\frac{\rho (r)}{f^{\prime }(r)}dr$$

$$\Vert S_a(t)P_{1}f{\Vert }_{L^p({\mathbb{R}}^n)}\lesssim \Vert {\tilde{P}}_{1}f{\Vert }_{L^{p^{\prime }}({\mathbb{R}}^n)}(2\le p\le \infty )$$

$$\Vert S_a(t)P_{1}f{\Vert }_{L_t^q(\mathbb{R},L_x^p({\mathbb{R}}^n))}\lesssim \Vert {\tilde{P}}_{1}f{\Vert }_{L^2({\mathbb{R}}^n)}$$

$$\Vert S_a(t)P_{N}f{\Vert }_{L_t^q(\mathbb{R},L_x^p({\mathbb{R}}^n))}\lesssim N^s\Vert {\tilde{P}}_{N}f{\Vert }_{L^2({\mathbb{R}}^n)},s=n\left(\frac{1}{2}-\frac{1}{p}\right)-\frac{a+1}{q}$$

$$\int {\int }_{{\mathbb{R}}^2\times [0,T]}{P}_{N_1}{S}_a(t)u_0{P}_{N_2}{S}_a(t)v_0{P}_{N_3}{S}_a(t)w_{0}dxdydt,N\sim N_1\sim N_2\sim N_3,$$

$$\Vert e^{it\phi (\nabla /i)}{u}_1{e}^{it\phi (\nabla /i)}{u}_2{\Vert }_{L_{t,x}^2(\mathbb{R}\times {\mathbb{R}}^n)}{\lesssim }_n\frac{r^{\frac{n-1}{2}}}{N^{1/2}}\Vert u_1{\Vert }_{L^2({\mathbb{R}}^n)}\Vert u_2{\Vert }_{L^2({\mathbb{R}}^n)}$$

$${\partial }_1{\phi }_a(\xi )=|\xi {|}^a+a{\xi }_1^2|\xi {|}^{a-2},{\partial }_2{\phi }_a(\xi )=a{\xi }_1{\xi }_2|\xi {|}^{a-2}$$

$$\Vert P_{N}S(t)u_0{P}_{K}S(t)v_0{\Vert }_{L_{t,x}^2}^2\lesssim \sum _L\Vert R_{L}S(t)u_0{P}_{K}S(t)v_0{\Vert }_{L_{t,x}^2}^2$$

$$\text{eqref}eq:bilinearDecomposition\lesssim \sum _L\left(\frac{K}{N}\right)\Vert R_L{u}_0{\Vert }_{L^2}^2\Vert P_K{v}_0{\Vert }_{L^2}^2={\left(\frac{K}{N^a}\right)}^{1/2}\Vert P_N{u}_0{\Vert }_{L^2}^2\Vert P_K{v}_0{\Vert }_{L^2}^2,$$

Full text

Financial support by the German Science Foundation (IRTG 2235) is gratefully acknowledged.

1. Introduction

In this note well-posedness of the higher-dimensional fractional Zakharov-Kuznetsov equations

[TABLE]

is discussed, where $n\geq 2$ and $\mathbb{K}\in\{\mathbb{R},\mathbb{T}\}$.

In the one-dimensional case (1) becomes the Benjamin-Ono equation (cf. [2], see e.g. [32] for a recent survey) for $a=1$ and the Korteweg-De Vries equation (see [19] for the sharp global well-posednesss result) for $a=2$. In the one-dimensional case the equations are best understood and extensively studied.

In higher dimensions (1) yields a generalization of the Benjamin-Ono equation for $a=1$ (cf. [25, 26, 29]) and for $a=2$ the Zakharov-Kuznetsov equation (cf. [36]) is recovered.

By local well-posedness we mean that for any $u_{0}\in H^{s}$ there is $T=T(\|u_{0}\|_{H^{s}})$ such that $S_{T}^{\infty}:H^{\infty}\rightarrow C([0,T],H^{\infty})$ extends uniquely to a continuous mapping $S_{T}^{s}:H^{s}\rightarrow C([0,T],H^{s})$.

The energy method [3] yields well-posedness for $s>\frac{n+2}{2}$, but neglects the dispersive properties. These are clearly stronger in Euclidean space than in the fully periodic case. We discuss solutions in Euclidean space first, for which we can show stronger well-posedness results consequently.

Already in the one-dimensional case it is well-known that the data-to-solution mapping for dispersion coefficients $1\leq a<2$ is not uniformly continuous (cf. [21, 16, 27]).

Also, in two-dimensions it was proved for $a=1$ in [25] that the data-to-solution mapping is not $C^{2}$. Local well-posedness was proved for $a=1$ provided that $s>5/3$ in [25] using shorttime linear Strichartz estimates (see also [20]).

Here, we improve the local well-posedness for $n=2$ and interpolate between $a=1$ and $a=2$ to recover in the limiting case the currently best local well-posedness for the Zakharov-Kuznetsov equation $s>1/2$ (cf. [11, 28]) in two dimensions and $s>1$ in three dimensions [28, 30]. The results in higher dimensions seem to be new for $1<a\leq 2$.

Here, we use transversality and localization of time to small frequency dependent time intervals (cf. [17, 33]) to prove the following theorem:

Theorem 1.1**.**

Let $\mathbb{K}=\mathbb{R}$, $1\leq a<2$ and $s>\frac{n+3}{2}-a$. Then (1) is locally well-posed.

We sketch the method of proof. Let $N\in 2^{\mathbb{N}_{0}}$ denote a dyadic number and $P_{N}$ the inhomogeneous Littlewood-Paley projector, i.e.,

[TABLE]

where $\Phi,\chi_{N}\in C^{\infty}_{c}$, $supp(\Phi)\subseteq B(0,2)$, $supp(\chi_{N})\subseteq B(0,2N)\backslash B(0,N/2)$ and $\Phi+\sum_{N}\chi_{N}\equiv 1$.

Further, let $S_{a}(t)$ denote the linear propagator of (1), that is

[TABLE]

The most problematic interaction happens in case a low frequency interacts with a high frequency because the derivative nonlinearity

[TABLE]

possibly requires one to recover a whole derivative. The derivative loss is partially ameliorated by the following bilinear Strichartz estimate:

Proposition 1.2**.**

Let $n\geq 2,\;K,N\in 2^{\mathbb{N}_{0}},\;K\ll N$. Then, we find the following estimate to hold:

[TABLE]

This proposition is an easy consequence of general transversality considerations (cf. [5]).

Apparently, this is still insufficient to recover the derivative loss for $1\leq a<2$. To overcome the gap we additionally localize time in a frequency dependent way (cf. [17]).

In the following we motivate at which frequency dependent time localization we can treat the most problematic $High\times Low\rightarrow High$-interaction utilizing (2). For $K\ll N$ one finds

[TABLE]

This suggests that for $T(N)=N^{2-a}$ this peculiar interaction can be estimated for $s>(n-1)/2$, which will be carried out in Section 5. In the one-dimensional case this had been done for dispersion generalized Benjamin-Ono equations (cf. [12, 13]).

This argument will be sufficient to handle $High\times Low\rightarrow High$-interactions and $High\times High\rightarrow High$-interactions for $n=2$. For $High\times High\rightarrow High$-interactions at $n\geq 3$ linear Strichartz estimates (cf. [25]) are used:

Proposition 1.3**.**

Let $n\geq 3$, $1\leq a\leq 2$ and $2\leq p,q\leq\infty$, $p\neq\infty$. Then, we find the following estimate to hold

[TABLE]

provided that $\frac{2}{q}+\frac{2}{p}=1$ and $s=n\left(\frac{1}{2}-\frac{1}{p}\right)-\frac{a+1}{q}$.

Since localization in time erases the dependence on the initial data, one still has to carry out energy estimates, which will give a worse regularity threshold to close the argument, namely $s>\frac{n+3}{2}-a$. This will be done in Section 6.

We will use a variant of the function spaces from [17, 9] to prove a priori estimates in the first step, next, $L^{2}$-Lipschitz dependence for initial data of higher regularity is discussed and finally, continuous dependence is proved by the Bona-Smith argument (cf. [3]).

The strategy of the proof closely follows the arguments from [33] where the argument was applied to periodic solutions.

The approach from [33] does not apply directly to periodic solutions because this would require the dispersion relation to split

[TABLE]

This is true for another possible generalization of the (fractional) Benjamin-Ono equation

[TABLE]

where $\mathbb{K}\in\{\mathbb{R};\mathbb{T}\}$.

Here, for $n=2$, $a=2$ we recover a Cauchy problem which is equivalent to the Zakharov-Kuznetsov equation (cf. [1, 11]).

Another Benjamin-Ono-Zakharov-Kuznetsov equation was considered in [31]:

[TABLE]

Here, only dispersion in the $x_{1}$-component was decreased. Local and global well-posedness results for (6) were also proved via frequency dependent time localization.

Lastly, we remark that the local well-posedness result from Theorem 1.1 gives global well-posedness in the energy space $H^{a/2}(\mathbb{R}^{2})$ for sufficiently large $a$ in the two-dimensional case due to conservation of energy

[TABLE]

Another conserved quantity is the mass

[TABLE]

but a well-posedness result in $L^{2}$ seems to be far beyond the methods of this paper.

Thus, iteration of Theorem 1.1 for $s=a/2$ yields:

Corollary 1.4**.**

Let $n=2,\mathbb{K}=\mathbb{R}$ and $a>5/3$. Then, (1) is globally well-posed for $s=a/2$.

We turn to a discussion of the fully periodic case. In the two-dimensional case the anisotropic Sobolev space $H^{s,0}(\mathbb{T}^{2})$ is also considered for the Zakharov-Kuznetsov equation, where

[TABLE]

In previous works ([24, 8]) local well-posedness has only been considered in isotropic Sobolev spaces, but since this is a larger space we also consider well-posednesss for these initial data. We prove the following theorem:

Theorem 1.5**.**

Let $\mathbb{K}=\mathbb{T}$, $n=2$ and $1\leq a\leq 2$ or $n\geq 2$, $a=2$ and $s>(n+1)/2$. Then (1) is locally well-posed in $H^{s}(\mathbb{T}^{n})$ and for $n=2$, $a=2$ (1) is locally well-posed in $H^{s,0}(\mathbb{T}^{2})$.

In case $n=2$ this improves the results from [24, 8] where local well-posedness was proved in $H^{s}(\mathbb{T}^{2})$ for $a\in\{1,2\}$ provided that $s>5/3$ for $a=2$ and $s>7/4$ for $a=1$. In these works shorttime linear Strichartz estimates were used. In the present work this result is modestly improved by transversality considerations and corresponding results are proven in higher dimensions. However, the covered regularities are still far from the energy space. To make further progress one probably needs a better comprehension of the resonance set which appears to be more delicate than for the Kadomtsev Petviashvili-equations (cf. [4, 37]).

The strategy of proof is the same as for solutions on Euclidean space: In suitable function spaces we will prove a priori estimates for solutions, Lipschitz continuous dependence for differences of solutions in $L^{2}$ for higher regular initial data and finally continuous dependence in $H^{s}$ by the Bona-Smith approximation. The conclusion of the proof is similar to the Euclidean case.

Key ingredient will be bilinear convolution estimates for the space-time Fourier transform of functions which will be localized in frequency and modulation. These will be derived in Subsection 8.2. Here, the transversality considerations from Euclidean space will again come into play. However, we always have to localize time reciprocally to the highest involved frequency so that transversality becomes observable. Therefore, we can not lower the regularity at which our method of proof yields local well-posedness as the dispersion coefficients increase compared to the Euclidean case.

After deriving these bilinear convolution estimates the argument follows the real line case. Thus, for the sake of clarity of presentation the details after the derivation of the bilinear convolution estimates will only be presented for the two-dimensional Zakharov-Kuznetsov equation.

2. Linear Strichartz estimates

This section is devoted to the proof of Proposition 1.3 which was carried out for $a=1$ in [25]. The required modifications are easy, but the proof is contained for the sake of completeness. We start with a dispersive estimate:

Proposition 2.1**.**

Let $a\geq 1$, $n\geq 3$ and $\psi:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a smooth radial function supported in $B_{n}(0,2)\backslash B_{n}(0,1/2)$. Then, we find the following estimate to hold:

[TABLE]

with $C$ only depending on $n$, $\psi$ and $a$.

Proof.

We rewrite the integral in spherical coordinates to find

[TABLE]

where $y_{x,t}(r)=(tr^{a+1}+x_{1}r,x_{2}r,\ldots,x_{n}r)$.

Recall the decay

[TABLE]

This is already enough to prove the claim for $n\geq 4$.

Indeed, partition $supp(f)=E_{1}\cup E_{2}$, where $E_{1}=\{r\in supp(\rho)||tr^{a+1}+x_{1}r|\leq 1\}$ and $|E_{1}|\lesssim|t|^{-1}$. To see this note that $|tr^{a+1}+x_{1}r|\leq 1$ implies $|tr^{a}+x_{1}|\leq 2$ and, by change of variables,

[TABLE]

where $C$ depends on $\psi$, $n$ and $a$.

Similarly, $E_{2}\subseteq\{r\in supp(\rho)||tr^{a}+x_{1}|\geq 2\}$ and consequently,

[TABLE]

and after linear change of variables we estimate by $C|t|^{-1}$.

We turn to $n=3$. Here, we make use of the asymptotic expansion

[TABLE]

where $|\mathcal{E}_{x,t}(y)|\lesssim\|y\|^{-2}\quad(\|y\|\gg 1)$.

Set $\phi(r)=\sqrt{f(r)}$, where $f(r)=(tr^{a+1}+x_{1}r)^{2}+r^{2}\|x^{\prime}\|^{2}$ and

[TABLE]

Below, we see that $|F^{1}|\lesssim|t|^{-1}$, which means that this contribution is controlled by $|\hat{\sigma}|\lesssim 1$.

Moreover, the contribution of $\mathcal{E}_{x,t}$ when integrating over $F^{2}$ is controlled by the higher dimensional argument due to $F^{2}\subseteq E^{2}$ and sufficient decay to run the above argument.

A computation yields

[TABLE]

We can suppose that $\frac{x_{1}}{t}\sim 1$ and $\frac{\|x^{\prime}\|^{2}}{t^{2}}\ll 1$, since otherwise $|f(r)|\gtrsim|t|$, so that the roots are real and separated.

In fact, $|r_{\pm}|\sim 1$ and $|r_{+}-r_{-}|\sim 1$. Moreover, whenever $f^{\prime}$ vanishes, then $|f^{\prime\prime}|$ is still bounded away from zero and thus, $|F^{1}|\lesssim|t|^{-1}$.

For the contribution of $e^{i\|y\|}/\|y\|$ over $F^{2}$ note that we can write

[TABLE]

Next, the domain of integration is divided into a finite union of intervals, where $\rho/f^{\prime}$ is monotone. On each such interval integration by parts yields the desired result. ∎

Remark 2.2**.**

The dispersive estimate follows also from [22, Proposition 4.7., p. 260].

From the dispersive estimate Strichartz estimates are derived by standard arguments.

Proof of Proposition 1.3.

For $n\geq 3$ the dispersive estimate and conservation of mass give by interpolation

[TABLE]

and combination with the $TT^{*}$-argument (cf. [35, 10, 18]) proves Strichartz estimates

[TABLE]

provided that $\frac{2}{q}+\frac{2}{p}=1$, $p\neq\infty$. A scaling argument gives for $p,q$ like above

[TABLE]

and (4) follows from Littlewood-Paley theory. ∎

3. Bilinear Strichartz estimates

Purpose of this section is to prove bilinear Strichartz estimates as stated in Proposition 1.2. Whereat the proof is straight-forward in case of separated frequencies, it requires more care to treat the $High\times High\times High$-interaction

[TABLE]

where we shall see that it is still amenable to a bilinear Strichartz estimate.

Both cases follow from the following more general well-known transversality estimate:

Proposition 3.1**.**

*Let $U_{i}$ be open sets in $\mathbb{R}^{n}$, $\varphi_{i}\in C^{1}(U_{i},\mathbb{R})$ and let $u_{i}$ have Fourier support in balls of radius $r$ which are contained in $U_{i}$ for $i=1,2$. Moreover, suppose that $|\nabla\varphi(\xi_{1})-\nabla\varphi(\xi_{2})|\geq N>0$, whenever $\xi_{1}\in U_{1}$, $\xi_{2}\in U_{2}$.

Then, we find the following estimate to hold:*

[TABLE]

In order to apply Proposition 3.1 we have to analyze the group velocity $v_{a}(\xi)=-\nabla\varphi_{a}(\xi)$, where $\varphi_{a}(\xi)=\xi_{1}|\xi|^{a}$.

We have

[TABLE]

Proof of Proposition 1.2.

First, divide $B_{2N}\backslash B_{N/2}$ into finitely overlapping balls of radius $K$, which we denote by the family $(R_{L})$. Then, from almost orthogonality

[TABLE]

To estimate the terms from the sum we use Proposition 3.1. From (10) we find $|\partial_{1}\varphi(\xi)|\geq(N/2)^{a}$ for $|\xi|\geq N/2$ and $|\partial_{2}\varphi_{a}(\xi)|\leq(1+a)(2K)^{a}$ and (9) implies

[TABLE]

which completes the proof. ∎

Next, we turn to the case of three comparable frequencies in the plane as depicted in (8). We prove the following proposition:

Proposition 3.2**.**

Let $N\gg 1$ and suppose that $\xi_{i}\in\mathbb{R}^{2},N/8\leq|\xi_{i}|\leq 8N$ for $i=1,2,3$ and $\xi_{1}+\xi_{2}+\xi_{3}=0$. Then, there are $i,j\in\{1,2,3\}$ with

[TABLE]

Proof.

A key observation is that for $|\xi_{2}|\leq c|\xi|$ or $|\xi_{1}|\leq c|\xi|$, where $c$ is a small constant, a Taylor expansion of $|\xi|$ around the large component reveals

[TABLE]

This means that as soon as one component dominates the other one, the propagation into $x_{1}$-direction is essentially governed by the group velocity associated to a (fractional) one-dimensional Benjamin-Ono equation, which has been considered in [33].

To deal with different sizes of the components for $\xi\in\mathbb{R}^{2}$ we introduce the notation $\xi\in(A,B)$, where $A,B\in\{Low,Medium,High\}$ and $\xi\in(X,Y)$, where $X,Y\in\{+,-\}$ to indicate $\xi_{1}\geq 0,\xi_{2}\leq 0$. E.g. $\xi\in(High(+),Medium(-))$ means $|\xi_{1}|\geq\frac{c|\xi|}{2}$, $|\xi_{2}|\in[c^{3}|\xi|,\frac{c|\xi|}{2}]$, $\xi_{1}\geq 0,\xi_{2}\leq 0$ or $\xi\in(Low,High(-))$ means $|\xi_{1}|\leq c^{3}|\xi|$, $|\xi_{2}|\geq\frac{c|\xi|}{2}$, $\xi_{2}\leq 0$.

Here, $c$ is a small dimensional constant chosen so that the error terms in the above Taylor expansion can be neglected in the following considerations.

We sort the frequencies according to the above system.

Suppose that the components of any frequency are all at least of medium size, so that no component of the three frequencies is low.

Then, by (10) $|\partial_{2}\varphi_{a}(\xi)|\geq c^{5}|\xi|^{a}$ for $i=1,2,3$. Next, observe that for $\xi_{i}\in(+,+)$ or $\xi_{i}\in(-,-)$ we have $\partial_{2}\varphi(\xi_{i})\geq c^{5}|\xi|^{a}$ and in case of mixed signs $\xi_{i}\in(+,-)$ or $\xi_{i}\in(-,+)$ we have $\partial_{2}\varphi_{a}(\xi_{i})\leq-c^{5}|\xi|^{a}$, and the estimate $|\partial_{2}\varphi_{a}(\xi_{i})-\partial_{2}\varphi_{a}(\xi_{j})|\geq c^{5}|\xi|^{a}$ is immediate.

Next, we turn to the case where all components have size greater than $c^{3}|\xi|$ and all frequencies are of equal signs (the case of mixed signs will be analogous).

Say $\xi_{1}\in(High(+),Medium(+))$, $\xi_{2}\in(High(+),High(+))$, $\xi_{3}\in(High(-),High(-))$.

Write $\xi_{21}=\alpha\xi_{11}$, $\xi_{22}=\beta\xi_{12}$, where $\alpha,\beta\in[c^{5},c^{-5}]$ and it follows

[TABLE]

Next, we suppose that there is one low component involved, say $\xi_{1}\in(Low,High)$. Suppose that there is a frequency $\xi_{j}\in(High,High)$. Then, we find $|\partial_{2}\varphi_{a}(\xi_{1})|=O(c^{3}|\xi|^{a})$ and $|\partial_{2}\varphi_{a}(\xi_{j})|\gtrsim c^{2}|\xi|^{a}$, hence $|\partial_{2}\varphi_{a}(\xi_{1})-\partial_{2}\varphi_{a}(\xi_{j})|\gtrsim c^{2}|\xi|^{a}$ which yields the desired transversality.

With $|\xi_{12}|\sim|\xi|$ there is another frequency, say $\xi_{2}$ with $|\xi_{22}|\sim|\xi|$ and by the above consideration suppose next that $\xi_{2}\in(Low,High)$ or $\xi_{2}\in(Medium,High)$.

Either way, $|\xi_{31}|\leq|\xi_{11}|+|\xi_{12}|\leq c|\xi_{11}|$ and we can expand $\partial_{1}\varphi(\xi_{i})$ in the second component of the frequencies to find that the analysis reduces to the one-dimensional fractional Benjamin-Ono equation and hence, there are $\xi_{i}$ and $\xi_{j}$ with

[TABLE]

The same argument applies in case $\xi_{1}\in(High,Low)$. In case there is $\xi_{j}\in(High,High)$ the difference satisfies $|\partial_{2}\varphi_{a}(\xi_{1})-\partial_{2}\varphi_{a}(\xi_{j})|\gtrsim c^{2}|\xi|^{a}$ and in case there is no $\xi_{j}\in(High,High)$ we can expand in the first frequency component to reduce the analysis to the one-dimensional fractional Benjamin-Ono equation according to which there are $\xi_{i},\xi_{j}$ such that $|\partial_{1}\varphi_{a}(\xi_{i})-\partial_{1}\varphi_{a}(\xi_{j})|\gtrsim|\xi|^{a}$.

The proof is complete. ∎

4. Function spaces

In this section we discuss the shorttime function spaces which are used to prove the local well-posedness results. The iteration scheme is the same for solutions in Euclidean space and for fully periodic solutions. However, in Euclidean space we do not have to use Fourier transform in time which allows for a simplification of the construction compared to the periodic case.

Shorttime $L^{2}$-valued $U^{p}$-/$V^{p}$-spaces will be utilized like in [9, 33]. Here, we will be very brief and instead refer to these works for a presentation of the basic function space properties. The notation will be the same like in the aforementioned works. For a careful exposition see [14, 15]. The $V^{p}$-spaces are the usual function spaces containing functions of bounded $p$-variation and the $U^{p}$-spaces are atomic spaces which are the respective predual spaces. Roughly, $U^{2}$ serves as a substitute for $H^{1/2}$, which does not embed into $L^{\infty}$, but any $U^{p}$-function is bounded.

The $U^{p}$-/$V^{p}$-spaces are adapted to free solutions in the usual way:

[TABLE]

Motivated by (3) we choose $T(N)=N^{a-2}$ as frequency dependent time localization.

Below we shall only deal with the case $1\leq a<2$, since for $a=2$ the localization to small frequency dependent time intervals is no longer necessary and the analysis comes down to the Fourier restriction analysis without localization in time from [11].

Letting $\chi_{I}$ denote a sharp cut-off to a time interval $I$ the shorttime $U^{2}$-space into which the solution to (1) will be placed is given by

[TABLE]

The corresponding space for the nonlinearity is defined by

[TABLE]

and the energy space is

[TABLE]

The shorttime norm of a smooth solution to (1) is propagated as follows:

[TABLE]

(cf. [33, Lemma 3.8, p. 12]).

Moreover, since $U^{p}_{a}$-atoms are piecewise free solutions estimates for free solutions extend to $U^{p}_{a}$-functions.

Proposition 4.1**.**

Let $n\geq 3$, $1\leq a\leq 2$, $N\in 2^{\mathbb{N}_{0}}$ and $I$ be an interval. Suppose that $2/q+2/p=1$, $2\leq q,p<\infty$. Then, we find the following estimate to hold:

[TABLE]

where $s=n\left(\frac{1}{2}-\frac{1}{p}\right)-\frac{a+1}{q}$.

This also remains valid for bilinear estimates.

Proposition 4.2**.**

Let $1\leq a<2$, $N_{1}\gg N_{2}$ and $I$ be an interval with $|I|=N_{1}^{a-2}$. Then, we find the following estimates to hold:

[TABLE]

Proof.

(13) is immediate from atomic decompositions (cf. [14, Proposition 2.19, p. 929]) and (14) follows from an interpolation argument (cf. [14, Proposition 2.20, p. 930]). ∎

5. Nonlinear estimates

This section is devoted to the propagation of the nonlinearity in the shorttime function spaces.

Proposition 5.1**.**

Let $1\leq a\leq 2$, $n\geq 2$, $s>(n-1)/2$. Then, we find the following estimates to hold:

[TABLE]

Proof.

After using Littlewood-Paley theory we are reduced to the analysis of $High\times Low\rightarrow High$-, $High\times High\rightarrow High$- and $High\times High\rightarrow Low$-interaction. Carrying out the summation in the shorttime function spaces gives (15) and (16).

Suppose that $N_{3}\sim N_{1}\gg N_{2}$. Then, we compute

[TABLE]

Suppose that $N_{1}\sim N_{2}\sim N_{3}$ and $n=2$. Using duality we have

[TABLE]

Now, we use Proposition 3.2 to apply a bilinear Strichartz estimate on two factors, say $w$ and $u$, to find

[TABLE]

which is sufficient.

For $n\geq 3$ we use two $L^{4}_{t,x}$-Strichartz estimates instead:

[TABLE]

which is again sufficient.

Finally, suppose that $N_{3}\ll N_{1}\sim N_{2}$ . Here, we have to add localization in time which amounts to a factor $(N_{1}/N_{3})^{2-a}$. Again we use duality to write

[TABLE]

and again carrying out the summation is straight-forward for $s>(n-1)/2$. ∎

6. Energy estimates

First, we turn to the energy estimate which will yield a priori estimates provided that $s>s_{a}$:

Proposition 6.1**.**

Let $n\geq 2$, $1\leq a<2$ and let $u$ be a smooth solution to (1). Then, we find the following estimate to hold

[TABLE]

provided that $s>s_{a}$.

Proof.

The fundamental theorem of calculus yields

[TABLE]

The time integral we treat with Littlewood-Paley decompositions and analyze the possible interactions separately.

Suppose that $N_{1}\sim N_{3}\gg N_{2}$. Then integration by parts and a commutator estimate yields after localization in time to intervals of size $N_{1}^{2-a}$

[TABLE]

In case $N_{1}\lesssim N_{2}\sim N_{3}$ there is no point to integrate by parts, but apart from that the estimate is concluded along the lines of the above argument. ∎

Next, we proof the energy estimates which will yield Lipschitz continuity in $L^{2}$ for initial data in $H^{s}$, $s>s_{a}$ and continuity of the data-to-solution mapping after invoking the Bona-Smith approximation.

Proposition 6.2**.**

Let $n\geq 2$, $1\leq a<2$ and $u_{1},u_{2}$ be two smooth solutions to (1) and denote $v=u_{1}-u_{2}$. Then, we find the following estimate to hold

[TABLE]

provided that $s>s_{a}$.

Proof.

Performing the same reductions like above we have to estimate

[TABLE]

for $N_{1}\sim N_{3}\gg N_{2}$, $N_{1}\lesssim N_{2}\sim N_{3}$ and $N_{3}\lesssim N_{1}\sim N_{2}$.

The first case can be dealt with like in the corresponding estimate for solutions because we can still integrate by parts.

The second case does not require integration by parts and thus can be estimated like above. Finally, for the case $N_{1}\lesssim N_{2}\sim N_{3}$ we estimate

[TABLE]

This yields (19) after summation.

To prove (20) one writes

[TABLE]

The first term has the same symmetries like the term we encountered when proving a priori estimates for solutions. For the second term the only new estimate one has to carry out (due to impossibility to integrate by parts) is

[TABLE]

which follows by the above means. ∎

7. Proof of Theorem 1.1

We shall be brief because the concluding arguments are already standard (cf. [17]). Below fix $s>s_{a}$.

By rescaling we are reduced to consider sufficiently small initial data. Firstly, we only consider initial data $u_{0}\in H^{\infty}(\mathbb{R}^{n})$. The energy method yields existence of solutions in $C([0,T^{*}],H^{s}(\mathbb{R}^{n}))$ for $s>n/2+1$, where $\lim_{T\to T^{*}}\|u(t)\|_{H^{2s}}=\infty$.

In a first step, we prove a priori estimates from

[TABLE]

for solutions to (1) by a bootstrap argument for $s>\frac{n+3}{2}-a$.

The above set of estimates yields

[TABLE]

Next, we invoke continuity of $E^{s}(T)$ and

[TABLE]

For details see e.g. [23].

Consequently, the above set of estimates yields

[TABLE]

provided that $\|u_{0}\|_{H^{s}}$ is chosen sufficiently small.

For $s^{\prime}>s$ we have

[TABLE]

Together with (21) this implies

[TABLE]

This a priori estimate for higher regularities together with the blow-up alternative shows that $T^{*}\geq 1$ provided that $\|u_{0}\|_{H^{s}}$ is chosen sufficiently small.

Next, we argue that the set of estimates

[TABLE]

yield an a priori estimate for $v$ in $L^{2}$ in dependence of $\|u_{i}\|_{H^{s}}$ for $s>\frac{n+3}{2}-a$.

Finally, the set of estimates

[TABLE]

allows us to conclude continuous dependence on the initial data by the classical Bona-Smith approximation (cf. [3, 17]).

For this purpose, let $u_{2}$ be the solution associated to $P_{\leq N}u_{0}$ and $u_{1}$ be the solution associated to $u_{0}$.

Due to the difference of initial data consisting only of high frequencies, the gain from estimating $\|v\|_{F^{0}}$ compensates the loss from estimating

[TABLE]

The data-to-solution mapping $H^{s}\rightarrow C([0,T],H^{s})\cap F^{s}(T)$, which can also be constructed by the above means, is continuous, but not uniformly continuous because the approximation depends on the distribution of the Sobolev energy along the high frequencies, i.e., $\|P_{\geq N}u_{0}\|_{H^{s}}$.

8. Periodic solutions to fractional Zakharov-Kuznetsov equations

Below, the above considerations regarding shorttime nonlinear and energy estimates are extended to the fully periodic case. Firstly, the function spaces are introduced.

8.1. Function spaces in the periodic case

Here, shorttime $X^{s,b}$-spaces adapted to periodic solutions are used (cf. [37]) to overcome the derivative loss. We will be brief because the function spaces are defined completely analogous to [37] with the basic function space properties remaining valid.

The dispersion relation for the two-dimensional Zakharov-Kuznetsov equation we denote by

[TABLE]

For $k\in\mathbb{N}$ let $I^{x}_{k}=\{\xi\in\mathbb{R}||\xi|\in[2^{k-1},2^{k})\}$ denote dyadic ranges on the real line and $I_{k}=\{(\xi,\eta)\in\mathbb{R}^{2}||(\xi,\eta)|\in[2^{k-1},2^{k})\}$. By $P_{k}$ and $P_{k,x}$ we denote the corresponding frequency projectors, i.e.,

[TABLE]

Most of the time it will be fine to work with sharp cutoffs though in Subsection 8.4 we adapt to smooth cutoffs, which will be denoted by $\tilde{P}_{k,x}$ or $\tilde{P}_{k}$, respectively.

For a time $T_{0}\in(0,1]$, let $k_{0}\geq 0$ be the greatest integer $k$ such that $2^{k}<1/T_{0}$. For $k\in\mathbb{N}\cup\{0\}$ define the dyadic $X^{s,b}$-type normed spaces

[TABLE]

Recall the basic properties ([37, Remark 2.1, p. 259]): For $f_{k}\in X_{k}$ we find the following estimate to hold:

[TABLE]

If $k,l\in\mathbb{N}\cap\{0\}$, $l\geq k_{0}$ and $f_{k}\in X_{k}$, then

[TABLE]

Consequently, for $f\in X_{k}$ we find for $l\geq k_{0}$, $t_{0}\in\mathbb{R}$, $\gamma\in\mathcal{S}(\mathbb{R})$

[TABLE]

The $X_{k}$-spaces relate to the space-time Fourier transform of the original functions after frequency localization. Let

[TABLE]

Next, define

[TABLE]

and for $T\in(0,T_{0}]$, let

[TABLE]

The spaces $F^{s}_{x}(T)$, $N_{x}^{s}(T)$, $E^{s}_{x}(T)$ are assembled by Littlewood-Paley theory. Let $C=C([-T,T],H^{\infty}_{0}(\mathbb{T}^{2}))$ and define

[TABLE]

The isotropic pendant spaces $F_{k},N_{k},F_{k}(T),N_{k}(T),F^{s}(T),N^{s}(T),E^{s}(T)$ are defined mutatis mutandi, replacing the anisotropic frequency projector $P_{k,x}$ with $P_{k}$. The multiplier properties (cf. [37, p. 260]) hold independent of the dispersion relation.

8.2. Bilinear estimates

Next, we derive bilinear convolution estimates for space-time Fourier transforms of functions localized in frequency and modulation.

Due to the lack of dispersion, Strichartz estimates like in Euclidean space are not available. Still the supports of the involved functions are estimated by properties of the resonance function. First, we focus on the two-dimensional Zakharov-Kuznetsov dispersion

[TABLE]

For $k,j\in\mathbb{N}_{0}$, let

[TABLE]

First, we give an estimate in the anisotropic case:

Lemma 8.1**.**

*Let $k_{i},j_{i}\in\mathbb{N}$, $f_{i}:\mathbb{Z}^{2}\times\mathbb{R}\rightarrow\mathbb{R}_{+},\;f_{i}\in L^{2},supp(f_{i})\subseteq D^{x}_{k_{i},\leq j_{i}}$ for $i\in\{1,2,3\}$.

Then, we find the following estimate to hold:*

[TABLE]

Proof.

Set $f_{i}^{\#}(\xi,\eta,\tau)=f_{i}(\xi,\eta,\tau+\omega(\xi,\eta))$ for $i\in\{1,2,3\}$ such that $\|f_{i}^{\#}\|_{L^{2}}=\|f_{i}\|_{L^{2}}$ and

[TABLE]

where $\text{supp}(f_{i}^{\#})\subseteq\{(\xi,\eta,\tau)||\xi|\sim 2^{k_{i}},\;|\tau|\leq 2^{j_{i}}\}$.

Observe that $|\partial^{2}\Omega/\partial\eta_{2}^{2}|\sim|\xi_{1}|\sim 2^{k_{1}}$. Consequently, an application of the Cauchy-Schwarz inequality in $\eta_{2}$ yields

[TABLE]

where the penultimate estimate follows from Cauchy-Schwarz in $\tau_{1}$, $\eta_{1}$ and $\xi_{1}$ and the last line follows from Cauchy-Schwarz in $\tau_{2}$ and $\xi_{2}$. ∎

Next, we turn to the isotropic case:

Lemma 8.2**.**

Let $k_{i},j_{i}\in\mathbb{N}$, $f_{i}:\mathbb{Z}^{2}\times\mathbb{R}\rightarrow\mathbb{R}_{+}$, $f_{i}\in L^{2}$, $supp(f_{i})\subseteq D_{k_{i},\leq j_{i}}$ for $i\in\{1,2,3\}$.

1. (a)

Let $|k_{1}-k_{3}|\leq 5$, $k_{2}\leq k_{1}-10$. Then, we find the following estimate to hold:

[TABLE] 2. (b)

Let $|k_{1}-k_{2}|\leq 5,|k_{2}-k_{3}|\leq 5$ and $j_{i}\geq k_{i}$. Then, we find the following estimate to hold:

[TABLE] 3. (c)

The estimate

[TABLE]

holds true.

Proof.

(a): For the representation (24) we find

[TABLE]

and an application of the Cauchy-Schwarz inequality in $\xi_{2}$ yields

[TABLE]

where the penultimate estimate follows from applications of Cauchy-Schwarz inequality in $\tau_{1},\xi_{1}$ and $\eta_{1}$ and the last line from applications in $\eta_{2}$ and $\tau_{2}$.

(b): In case there are $x_{1}$-frequencies comparable to $2^{k_{1}}$ the estimate (23) is sufficient as the claim follows due to $j_{i}\geq k_{i}$.

Hence, we suppose next that the $x_{1}$-frequencies are much smaller than $2^{k_{1}}$ and the $x_{2}$-frequencies are comparable to $2^{k_{1}}$. Then, either $\eta_{1}$ or $\eta_{2}$ has the same sign like $\eta_{1}+\eta_{2}$. Suppose that it is $\eta_{2}$. Then, we estimate

[TABLE]

where in the first estimate we used $\partial\Omega/\partial\xi_{2}=\eta_{1}(\eta_{1}+2\eta_{2})$, $|\partial\Omega/\partial\xi_{2}|\sim 2^{2n_{1}}$ and the Cauchy-Schwarz inequality in $\xi_{2}$ and in the second line Cauchy-Schwarz in $\xi_{1},\eta_{1},\tau_{1}$ and finally in $\tau_{2}$ and $\eta_{2}$.

(c): (27) follows from applications of Cauchy-Schwarz inequality without using the resonance function. ∎

The above estimates in the isotropic case extend to dispersion generalizations by virtue of the transversality considerations from Section 3. Let

[TABLE]

The considerations from Section 3 can be utilized in the following way:

Consider $u_{i}\in L^{2}(\mathbb{T}^{2}\times\mathbb{R})$, real-valued, with $f_{i}=\mathcal{F}_{t,x}[u_{i}]$, $supp(f_{i})\subseteq D^{a}_{k_{i},\leq j_{i}}$ and moreover,

[TABLE]

Suppose that $|\partial_{\xi}\varphi_{a}(\xi_{1},\eta_{1})-\partial_{\xi}\varphi_{a}(\xi_{2},\eta_{2})|\gtrsim V$. In case another partial derivative dominates the conclusion follows likewise.

It follows that

[TABLE]

Next, use Cauchy-Schwarz inequality in $\xi_{2}$ which will give a factor $1+(2^{j_{3}}/V)^{1/2}$ and following along the above lines we find

[TABLE]

We record the following isotropic estimates for dispersion relations of two dimensional generalized Benjamin-Ono equations.

Lemma 8.3**.**

*Let $k_{i},j_{i}\in\mathbb{N}$, $i\in\{1,2,3\}$, $f_{i}:\mathbb{Z}^{2}\times\mathbb{R}\rightarrow\mathbb{R}_{+}$, $f_{i}\in L^{2}$, $supp(f_{i})\subseteq D^{a}_{k_{i},\leq j_{i}}$ for $i\in\{1,2,3\}$.

1. (a)

If $|k_{1}-k_{3}|\leq 5$, $k_{2}\leq k_{1}-10$ and $j_{3}\geq k_{3}$, then

[TABLE] 2. (b)

If $|k_{1}-k_{2}|\leq 5$, $|k_{2}-k_{3}|\leq 5$ and $j_{i}\geq k_{i}$, then

[TABLE]

Proof.

The proofs of (a) and (b) are consequences of (29). For (a) observe that $|\nabla\varphi_{a}(\xi_{2})-\nabla\varphi_{a}(\xi_{3})|\sim 2^{ak_{1}}$ and for (b) use the symmetry in $f_{1}$, $f_{2}$ and $f_{3}$ and Proposition 3.2. ∎

Recall that in higher dimensions we have the following transversality in case of separated frequencies: Let

[TABLE]

Suppose that $k_{1}\leq k_{2}-10$, $\xi_{i}\in A_{k_{i}}$ for $i=1,2$. Then

[TABLE]

For the Zakharov-Kuznetsov dispersion, i.e., $a=2$ we can also prove a certain transversality for frequencies of comparable size:

Lemma 8.4**.**

*Let $\xi_{i}\in\mathbb{Z}^{n}$, $i=1,2,3$ and $n\geq 3$.

Assume $\xi_{1}+\xi_{2}+\xi_{3}=0$ and $|\xi_{i}|\sim 2^{k}$ and $1\leq|\xi_{i1}|\ll 2^{k}$ for $i=1,2,3$. Then there are $i,j\in\{1,2,3\}$ such that*

[TABLE]

Proof.

There must be one coordinate of size $2^{k}$, say $|\xi_{12}|\sim 2^{k}$. By this choice we use the symmetry of $\varphi_{2}$ in $\xi_{2},\ldots,\xi_{n}$.

By convolution constraint there is another frequency with second component of size $2^{k}$, that is $|\xi_{22}|\sim 2^{k}$.

Suppose that $|\xi_{32}|\ll 2^{k}$.

In this case by convolution constraint we find $|\xi_{i1}|\gtrsim|\xi_{31}|$ for an $i\in\{1,2\}$ and easily

[TABLE]

Next, suppose that $|\xi_{32}|\sim 2^{k}$. Here, we argue with the signs of the involved frequencies.

Suppose that $sgn(\xi_{12})=sgn(\xi_{22})=1$, $sgn(\xi_{32})=-1$.

In case $sgn(\xi_{11})\neq sgn(\xi_{21})$ we have

[TABLE]

Thus, suppose that $sgn(\xi_{11})=sgn(\xi_{21})=1$ or $sgn(\xi_{11})=sgn(\xi_{21})=-1$.

Then,

[TABLE]

because all terms have the same sign in the above sum. ∎

This argument is insufficient to prove transversality for the higher dimensional fractional Zakharov-Kuznetsov dispersion relations in case of comparable frequencies.

Thus, we have no non-trivial estimates for the $High\times High\rightarrow High$-interaction for fractional Zakharov-Kuznetsov equations in higher dimensions. It might be possible though to prove an $L^{4}_{t,x}$-Strichartz estimate by decoupling (cf. [6, 7]) adapting the argument from [34].

8.3. Nonlinear estimates

Proposition 8.5**.**

Let $T\in(0,T_{0}]$. We find the following estimates to hold

[TABLE]

provided that $1<s\leq s^{\prime}$. Moreover, estimates (35) and (36) also hold true, when replacing $N_{x}$ and $F_{x}$ by $N$ and $F$, respectively.

Remark 8.6**.**

The argument below yields nonlinear estimates up to $H^{1/2}(\mathbb{T}^{2})$. The regularity threshold $s>3/2$ comes from carrying out energy estimates.

Proof.

We prove the estimates in case of anisotropic frequency localization first. Choose $\tilde{u},\tilde{v}\in C(\mathbb{R},H^{3,0})$ such that

[TABLE]

for $k\in\mathbb{N}$. Set $u_{k}=P_{k,x}\tilde{u}$ and $v_{k}=P_{k,x}\tilde{v}$. Then it suffices to consider the interactions $High\times Low\rightarrow High$:

[TABLE]

$High\times High\rightarrow High$:

[TABLE]

$High\times High\rightarrow Low$:

[TABLE]

In fact, the above estimates imply in case of $High\times Low\rightarrow High$- and $High\times High\rightarrow High$-interaction

[TABLE]

and regarding $High\times High\rightarrow Low$-interaction

[TABLE]

Then the claim follows from the definition of the function spaces by summing over the frequencies.

We start with $High\times Low\rightarrow High$-interaction. By the definition of $N_{k,x}$ and $F_{k,x}$-spaces it suffices to show the estimate

[TABLE]

Here,

[TABLE]

To prove (40) we use duality to write

[TABLE]

where estimate (23) was applied in the first step and the conclusion is due to $j_{i}\geq k$. Plugging (41) into (40) yields (37).

The $High\times High\rightarrow High$-interaction is handled along the same lines.

In the case of $High\times High\rightarrow Low$-interaction we add further localization in time to length of $2^{-k_{1}}$ to estimate the resulting functions in $F_{k,x}$-spaces. Let $\gamma:\mathbb{R}\rightarrow[0,1]$ such that $\sum_{n\in\mathbb{Z}}\gamma^{2}(x-n)=1\quad\forall x\in\mathbb{R}$ and suppose that $k_{1}\geq k_{2}$.

Then the lhs of (39) is dominated by

[TABLE]

Thus, like above from the properties of the shorttime function spaces it suffices to prove

[TABLE]

where $f_{i}:\mathbb{Z}^{2}\times\mathbb{R}\rightarrow\mathbb{R}_{+}$ supported in $D_{k_{i},\leq j_{i}},j_{i}\geq k_{1}$, $i=1,2$.

The sum over $j$ we split into $k\leq j\leq 2k_{1}$ and $j>k_{1}$. For the first part we use duality and estimate (23) like above to find

[TABLE]

In the second case we apply duality and estimate (23) in another way to find

[TABLE]

which is more than enough.

We turn to the estimates in the case of isotropic frequency localization. Again, we have to analyze the interactions from above. The pendant of (37) reduces to

[TABLE]

where $supp(f_{i})\subseteq D_{k_{i},j_{i}}$, $j_{i}\geq k$.

To prove the above display use duality and apply estimate (25) to find

[TABLE]

For the $High\times High\rightarrow High$-interaction we split the sum over the output modulation variable into $n\leq j\leq 2n$ and $j\geq 2n$ to find

[TABLE]

after applying duality and estimate (26).

For the high modulation output apply duality and estimate (27) to find

[TABLE]

For $High\times High\rightarrow Low$-interaction we argue similarly: Taking into account the additional time localization it suffices to prove

[TABLE]

where $supp(f_{i})\subseteq D_{k_{i},\leq j_{i}}$, $j_{i}\geq j$ for $i=1,2$.

Again, the sum over $j$ is split into $k\leq j\leq 2k_{1}$, $j\geq 2k_{1}$. In the first case, we use duality and apply (25) to find

[TABLE]

In the second case, estimate (27) yields

[TABLE]

The proof is complete. ∎

8.4. Energy estimates

Purpose of this section is to propagate the energy norm of solutions and differences of solutions in terms of shorttime norms. We prove the following proposition:

Proposition 8.7**.**

Let $T\in(0,1]$, $s>3/2$ and $u\in C([-T,T],H^{\infty}_{0}(\mathbb{T}^{2}))$ be a smooth solution to (1) for $a=2$, $n=2$. Then we find the following estimate to hold:

[TABLE]

For two solutions to (1) $u_{i}\in C([-T,T],H^{\infty}_{0})$ with initial data $\phi_{i}$, $i=1,2$, the function $v=u_{1}-u_{2}$ satisfies the estimate

[TABLE]

and

[TABLE]

The above estimates also remain valid after replacing $E^{s}$, $F^{s}$ with $E^{s}_{x}$, $F^{s}_{x}$, respectively.

The proof will be carried out by estimating the energy transfer in the following way: Suppose that $u$ is a smooth solution to

[TABLE]

Then we find for the evolution of the $L^{2}$-norm of the frequencies

[TABLE]

The key estimates are carried out in the following lemma:

Lemma 8.8**.**

Let $T>0$, $u_{i}\in F_{k_{i}}(T)$, $i=1,2,3$. We find the following estimate to hold:

[TABLE]

Suppose that $k_{1}<k-10$. Then we find the following estimate to hold:

[TABLE]

Furthermore, estimates (49) and (50) hold true after replacing $\tilde{P}_{k}$ with $\tilde{P}_{k,x}$ and $F_{k_{i}}(T)$ with $F_{k_{i},x}(T)$.

Proof.

We start with the proof of the isotropic estimates. By symmetry we can assume that $k_{1}\leq k_{2}\leq k_{3}$. Let $\tilde{u}_{i}\in F_{k_{i}}$ with $\|\tilde{u}_{i}\|_{F_{k_{i}}}\leq 2\|u_{i}\|_{F_{k_{i}}(T)}$, $i=1,2,3$ from the definitions.

The $\tilde{u}_{i}$ will be denoted by $u_{i}$ to lighten the notation. In order to estimate the functions in the shorttime function spaces time has to be localized according to the highest frequency. Let $\gamma:\mathbb{R}\rightarrow[0,1]$ be a smooth function supported in $[-1,1]$ with

[TABLE]

The lhs of (49) is dominated by

[TABLE]

where

[TABLE]

In (51) read $\eta_{j_{i}}=\eta_{\leq j_{i}}$; it is sufficient to derive bounds for this modulation variable decomposition according to (8.1).

Apparently, $|A|\leq 10$, $|B|\leq C_{0}T2^{k_{3}}$. The main contribution of $B$ is handled first.

Denote

[TABLE]

We do not distinguish between different values of $n$ because the following estimates are independent of $n$.

In case $k_{1}\leq k_{2}-10$ an application of (25) yields

[TABLE]

because $|k_{2}-k_{3}|\leq 5$ and $j_{3}\geq k_{3}$.

In case $|k_{1}-k_{2}|\leq 5$, $|k_{2}-k_{3}|\leq 5$ an application of (26) gives

[TABLE]

For the boundary terms note that sharp cutoffs in time are almost bounded in $X_{k}$, that is for an interval $I\subseteq\mathbb{R}$, $k\in\mathbb{N}_{0}$, $f_{k}\in X_{k}$ and $f_{k}^{I}=\mathcal{F}(1_{I}(t)\mathcal{F}^{-1}(f_{k}))$ (cf. [37, p. 267])

[TABLE]

An application of Cauchy-Schwarz yields

[TABLE]

which yields the claim.

For the proof of (50) we integrate by parts (cf. [17]) to find

[TABLE]

where $\psi_{i}$, $i=1,2$ are bounded and regular multipliers. The resulting expressions can be handled by (25).

In the anisotropic case we use similar arguments, but use (23) instead to conclude (49) and the commutator estimate for (50) is actually easier because there are no derivatives in $x_{2}$-direction involved. ∎

We are ready to prove Proposition 8.7.

Proof of Proposition 8.7.

Following the remark after Proposition 8.7 we find for a solution to (1)

[TABLE]

For the integral we consider the following interactions: $High\times Low\rightarrow High$:

[TABLE]

$High\times High\rightarrow High$:

[TABLE]

$High\times High\rightarrow Low$:

[TABLE]

$High\times Low\rightarrow High$-interaction is estimated by (50) to

[TABLE]

and summing over $k_{1}\leq k-10$ and square summing over $k$ gives (45).

In case of $High\times High\rightarrow High$-interaction estimate (49) is used to obtain

[TABLE]

and square summing over $k$ gives (45).

$High\times High\rightarrow Low$-interaction is handled similarly; again, there is no point in rearranging the derivative.

To prove (50) we write

[TABLE]

and estimate $High\times High\rightarrow High$-interaction and $High\times High\rightarrow Low$-interaction like above to obtain (50). In case of $High\times Low\rightarrow High$-interaction one finds two different terms:

[TABLE]

and

[TABLE]

(55) is estimated along the above lines because we can integrate by parts to arrange the derivative on the smallest frequency.

For (56) we use estimate (49) instead to find

[TABLE]

and square summing in $k$ and summing over $k_{1}\leq k-10$ gives (46).

To prove (47) the solution to the difference equation is rewritten as

[TABLE]

When estimating $\|v\|_{E^{s}(T)}$ for $s>3/2$ the contribution of $\partial_{x_{1}}(v^{2})$ can be handled like in the proof of (45), which gives

[TABLE]

The contribution of $\partial_{x_{1}}(vu_{2})$ can be treated like in the proof of (45) except for the interaction

[TABLE]

because here we can not integrate by parts like above. Instead estimate (49) and square summing in $k$ and summation in $k_{1}\leq k-10$ gives

[TABLE]

In the anisotropic case the same strategy applies after deploying the energy estimate for anisotropic frequency localization. ∎

8.5. Proof of Theorem 1.5

Proof of Theorem 1.5.

Fix $s>3/2$. Here, instead of rescaling to small initial values which are considered for large times like in the proof of Theorem 1.1 we consider arbitrary initial data for small times as in [37].

We only demonstrate the proof of a priori estimates for smooth initial values. The additionally required arguments to construct the data-to-solution mapping are like in the proof of Theorem 1.1. For $0<T\leq T_{0}$ we find for a smooth solution

[TABLE]

This implies

[TABLE]

and since

[TABLE]

as in [37, Lemma 6.3, p. 278]. After choosing $T_{0}=T_{0}(\|u_{0}\|_{H^{s}})$ one proves a priori estimates by a bootstrap argument. ∎