The Zakharov–Kuznetsov equation in high dimensions: small initial data of critical regularity

Abstract

The Zakharov–Kuznetsov equation in spatial dimension \(d\ge 5\) is considered. The Cauchy problem is shown to be globally well-posed for small initial data in critical spaces, and it is proved that solutions scatter to free solutions as \(t \rightarrow \pm \infty \). The proof is based on i) novel endpoint non-isotropic Strichartz estimates which are derived from the \((d-1)\)-dimensional Schrödinger equation, ii) transversal bilinear restriction estimates, and iii) an interpolation argument in critical function spaces. Under an additional radiality assumption, a similar result is obtained in dimension \(d=4\).

Introduction

This paper is concerned with the Zakharov–Kuznetsov equation

$$\begin{aligned} \begin{array}{l@{\quad }l} \partial _t u+\partial _{x_1}\Delta u=\partial _{x_1}u^2 &{}\text { in}{\mathbb {R}}\times {\mathbb {R}}^d\\ u(0,\cdot )=u_0 &{}\text { on } {\mathbb {R}}^d \end{array} \end{aligned}$$
(1.1)

where \(d\ge 2\), \(u=u(t,x)\), \((t,x) =(t,x_1,\ldots , x_d) \in {\mathbb {R}}\times {\mathbb {R}}^d\), u is real-valued, and \(\Delta \) denotes the Laplacian with respect to x.

The Zakharov–Kuznetsov equation was introduced in [13] as a model for propagation of ion-sound waves in magnetic fields. The Zakharov–Kuznetsov equation can be seen as a multidimensional extension of the well-known Korteweg–de Vries (KdV) equation. In contrast to the KdV equation, the Zakharov–Kuznetsov equation is not completely integrable, but possesses two invariants,

$$\begin{aligned} M(u) := \int _{{\mathbb {R}}^d} u^2 dx, \quad E(u) := \int _{{\mathbb {R}}^d} \frac{1}{2} |\nabla u|^2 + \frac{1}{3} u^3 dx. \end{aligned}$$

In the following, \(H^s({\mathbb {R}}^d)\) denotes the standard \(L^2\)-based inhomogeneous Sobolev space and \(B^s_{2,1}({\mathbb {R}}^d)\) is the Besov refinement, and the dotted versions their homogeneous counterparts, see below for definitions. The scale-invariant regularity threshold for (1.1) is \(s_c=\frac{d-4}{2}\).

Before we state our main results, let us briefly summarize the progress which has been made regarding the well-posedness problem associated to (1.1). In the two-dimensional case, Faminskiĭ [3] established global well-posedness in the energy space \(H^1({\mathbb {R}}^2)\). Later, Linares and Pastor [9] proved local well-posedness in \(H^s({\mathbb {R}}^2)\) for \(s>3/4\); before Grünrock and Herr [4] and Molinet and Pilod [11] showed local well-posedness for \(s > 1/2\). Recently, the second author [7] proved local well-posedness for \(s>-1/4\). In dimension \(d=3\), Linares and Saut [10] obtained local well-posedness in \(H^s({\mathbb {R}}^3)\) for \(s > 9/8\). Ribaud and Vento [12] proved local well-posedness for \(s > 1\) and in \(B_2^{1,1}({\mathbb {R}}^3)\). The global well-posedness in \(H^s({\mathbb {R}}^3)\) for \(s>1\) was obtained by Molinet and Pilod in [11]. Recently, in dimensions \(d \ge 3\), local well-posedness in \(H^s({\mathbb {R}}^d)\) in the full subcritical range \(s>s_c\) was proved in [5], which implies global well-posedness in \(H^1({\mathbb {R}}^d)\) if \(3 \le d \le 5\) and in \(L^2({\mathbb {R}}^3)\). We refer the reader to these papers for a more thorough account on the Zakharov–Kuznetsov equation and more references.

In the present paper, we address the problem of global well-posedness and scattering for small initial data in critical spaces. By well-posedness we mean existence of a (mild) solution, uniqueness of solutions (in some subspace) and (locally Lipschitz) continuous dependence of solutions on the initial data. We say that a global solution \(u \in C({\mathbb {R}},H^s({\mathbb {R}}^d))\) of (1.1) scatters as \(t \rightarrow \pm \infty \), if there exist \(u_\pm \in H^s({\mathbb {R}}^d)\) such that

$$\begin{aligned} \Vert u(t)-e^{tS}u_\pm \Vert _{H^s({\mathbb {R}}^d)}\rightarrow 0 \quad \left( t \rightarrow \pm \infty \right) . \end{aligned}$$

Here, \(e^{tS}\) denotes the unitary group generated by the skew-adjoint linear operator \(S = -\partial _{x_1} \Delta \), so that \(e^{tS}u_\pm \) solves the linear homogeneous equation.

Our first main result covers small data in dimension \(d=5\).

Theorem 1.1

For \(d =5\), the Cauchy problem (1.1) is globally well-posed for small initial data in \(B^{s_c}_{2,1}({\mathbb {R}}^5)\), and solutions scatter as \(t\rightarrow \pm \infty \). The same result holds in \(\dot{B}^{s_c}_{2,1}({\mathbb {R}}^d)\).

In dimensions \(d\ge 6\), we can extend this to Sobolev regularity.

Theorem 1.2

For \(d \ge 6\), the Cauchy problem (1.1) is globally well-posed for small initial data in \(H^{s_c}({\mathbb {R}}^d)\), and solutions scatter as \(t\rightarrow \pm \infty \). The same result holds in \(\dot{H}^{s_c}({\mathbb {R}}^d)\).

Note that in \(d=6\) this result includes the energy space \(\dot{H}^1({\mathbb {R}}^6)\).

If we restrict to initial data which is radial in the last \((d-1)\) variables (see below for definitions), we obtain small data global well-posedness and scattering in the critical Sobolev spaces for any dimension \(d\ge 4\).

Theorem 1.3

For \(d \ge 4\), the Cauchy problem (1.1) is globally well-posed for small data in \(H^{s_c}_{\mathrm {rad}}({\mathbb {R}}^d)\), and solutions scatter as \(t\rightarrow \pm \infty \). The same result holds for radial data in \(\dot{H}^{s_c}_{\mathrm {rad}}({\mathbb {R}}^d)\).

As the proof shows, the radiality assumption can be weakened to an angular regularity assumption, but we do not pursue this. One of the most interesting special cases here is \(d=4\), when \(s_c=0\); hence, the result covers the radial \(L^2\) space.

The main idea of this paper is to combine a new set of non-isotropic Strichartz estimates with the bilinear transversal estimate and an interpolation argument in critical function spaces.

The paper is structured as follows: In Sect. 1.1 we introduce notation. In Sect. 2 we derive Strichartz-type estimates which are based on the well-known Strichartz estimates for the \((d-1)\)-dimensional Schrödinger equation and allow us to treat the case \(d=5\). In Sect. 3 we combine this with the bilinear transversal estimate and an interpolation argument, which leads to a proof of Theorem 1.2. Finally, in Sect. 4 we discuss an variation of these ideas under the additional radiality assumption and a proof of Theorem 1.3.

Notation

We write \( x'=(x_2,\ldots ,x_d)\), \(D_{x_j}=-i \partial _j\), \(D=(-\Delta )^\frac{1}{2}\), \(|\nabla _{x'}|^s=\mathcal {F}_{x'}^{-1} |\xi '|^s \mathcal {F}_{x'}\), and \(\langle \nabla _{x'}\rangle ^s=\mathcal {F}_{x'}^{-1} \langle \xi '\rangle ^s \mathcal {F}_{x'}\). Here and in the sequel, we denote the Fourier transform of u in time, space, and the first spatial variable, by \(\mathcal {F}_t u\), \(\mathcal {F}_{x} u\), \(\mathcal {F}_{x_1} u\), and \(\mathcal {F}_{x'}\), respectively. \(\mathcal {F}_{t, x} u = \widehat{u}\) denotes the Fourier transform of u in time and space. Choose a nonnegative bump function \(\psi \in C_c^\infty ({\mathbb {R}})\) supported in the interval (1/2, 2) with the property that \(\sum _{N \in 2^{\mathbb Z}} \psi (r/N)=1\) for \(r>0\), and set \(\psi _N=\psi (\cdot /N)\). For N, \(\lambda , M \in 2^{{\mathbb Z}}\), we define (spatial) frequency projections \(P_N\), \(Q_{\lambda }\), and \(R_M\) as the Fourier multipliers with symbols \(\psi _{N}(|\xi | )\), \(\psi _{\lambda }(|\xi _1| )\), and \( \psi _{M}(|\xi '| )\), respectively, where \((\tau ,\xi )=(\tau ,\xi _1,\xi ')=(\tau ,\xi _1,\ldots ,\xi _d) \in {\mathbb {R}}\times {\mathbb {R}}^d\) are temporal and spatial frequencies. In addition, we define

$$\begin{aligned} P_{\le 1}=\sum _{1\ge N \in 2^{\mathbb Z}}P_N, \quad Q_{\le 1}:= \sum _{1\ge \lambda \in 2^{\mathbb Z}} Q_\lambda , \text { and } R_{\le 1}:= \sum _{1\ge M \in 2^{\mathbb Z}} R_M. \end{aligned}$$

As usual, the Sobolev space \(H^s({\mathbb {R}}^d)\) is defined as the completion of the Schwartz functions with respect to the norm

$$\begin{aligned} \Vert f\Vert _{H^s({\mathbb {R}}^d)}=\left( \int _{{\mathbb {R}}^d} {\langle {\xi } \rangle }^{2s} |\widehat{f}(\xi )|^2 d\xi \right) ^{\frac{1}{2}}, \end{aligned}$$

and the (smaller) Besov space \(B^s_{2,1}({\mathbb {R}}^d)\) as the completion of the Schwartz functions \(\mathcal {S}({\mathbb {R}}^{d})\) with respect to the norm

$$\begin{aligned} \Vert f\Vert _{B^s_{2,1}({\mathbb {R}}^d)}=\Vert P_{\le 1}f\Vert _{L^2} +\sum _{N\in 2^{{\mathbb N}} }N^s\Vert P_Nf\Vert _{L^2}. \end{aligned}$$

Similarly, for \(s\ge 0\), the homogeneous Sobolev space \(\dot{H}^s({\mathbb {R}}^d)\) is defined as the completion of the Schwartz functions with respect to the norm

$$\begin{aligned} \Vert f\Vert _{\dot{H}^s({\mathbb {R}}^d)}=\Big (\int _{{\mathbb {R}}^d} |\xi |^{2s} |\widehat{f}(\xi )|^2 d\xi \Big )^{\frac{1}{2}}, \end{aligned}$$

and the homogeneous Besov space \(\dot{B}^s_{2,1}({\mathbb {R}}^d)\) as the completion of the Schwartz functions with respect to the norm

$$\begin{aligned} \Vert f\Vert _{\dot{B}^s_{2,1}({\mathbb {R}}^d)}=\sum _{N\in 2^{{\mathbb Z}} }N^s\Vert P_Nf\Vert _{L^2}. \end{aligned}$$

The radial subspaces \(H_{\mathrm {rad}}^s({\mathbb {R}}^d)\) and \(\dot{H}^s_{\mathrm {rad}}({\mathbb {R}}^d)\) are defined by the requirement that \(f(x_1,x') = f (x_1, y')\) if \( |x'|=|y'|\), i.e., for fixed \(x_1\), the functions are radial in \(x'\).

Finally, the Duhamel operator is denoted by

$$\begin{aligned} \mathcal {I}(F)(t) := \int _0^t e^{(t-t')S} F(t') d t'. \end{aligned}$$

Strichartz estimates and the proof of Theorem 1.1

For \(d \ge 2\), we say (qr) is \((d-1)\)-admissible if

$$\begin{aligned} 2 \le q,r \le \infty , \; 2/q = (d-1)(1/2-1/r), \; (d, q, r) \not = (3,2,\infty ). \end{aligned}$$

Theorem 2.1

Let \(d \ge 2\) and \((q_1, r_1)\), \((q_2,r_2)\) be \((d-1)\)-admissible. Then, we have

$$\begin{aligned}&\Vert D_{x_1}^{\frac{1}{q}} e^{t S} f \Vert _{L_t^{q_1} L_{x'}^{r_1} {{L}_{x_1}^2}} \lesssim \Vert f \Vert _{L_x^2}. \end{aligned}$$
(2.1)
$$\begin{aligned}&\Vert D_{x_1}^{\frac{1}{q_1}+\frac{1}{q_2}}\mathcal {I} F\Vert _{L_t^{q_1} L_{x'}^{r_1} {{L}_{x_1}^2}} \lesssim \Vert F\Vert _{L_t^{q_2'} L_{x'}^{r_2'} {{L}_{x_1}^2}}, \end{aligned}$$
(2.2)

where \(1/q_2'=1-1/q_2\) and \(1/r_2'=1-1/r_2\).

Proof

Let \(\Delta _{x'} = \sum _{j=2}^d \partial _{x_j}^2\). For fixed \(\xi _1 \in {\mathbb {R}}\), define \(V_{\xi _1}(t) f(x') := (e^{-i t \xi _1 \Delta _{x'}} f)(x')\). Since \(V_{\xi _1}(t/\xi _1) = e^{it \Delta _{x'}}\), for \(f \in \mathcal {S}({\mathbb {R}}^{d-1})\) and \(F \in \mathcal {S}({\mathbb {R}}\times {\mathbb {R}}^{d-1})\), the Strichartz estimates of Schrödinger equations in \({\mathbb {R}}^{d-1}\) imply

$$\begin{aligned} \Vert |\xi _1|^{\frac{1}{q_1}} V_{\xi _1}(t) f \Vert _{L_t^{q_1} L_{x'}^{r_1}} \lesssim {}&\Vert f \Vert _{L_{x'}^2}, \end{aligned}$$
(2.3)
$$\begin{aligned} \Bigl \Vert \int _0^t |\xi _1|^{\frac{1}{q_1}+\frac{1}{q_2}} V_{\xi _1}(t-t') F(t') d t' \Bigr \Vert _{L_t^{q_1} L_{x'}^{r_1}} \lesssim {}&\Vert F\Vert _{L_t^{q_2'} L_{x'}^{r_2'}}, \end{aligned}$$
(2.4)

see [6, Theorem 1.2] for details. We deduce from Plancherel’s theorem, Minkowski’s inequality and (2.3) that

$$\begin{aligned} \Vert D_{x_1}^{\frac{1}{q}} e^{t S} f \Vert _{L_t^q L_{x'}^r L_{x_1}^2} = \left( \int _{{\mathbb {R}}} \Vert |\xi _1|^{\frac{1}{q}} V_{\xi _1} \mathcal {F}_{x_1} f \Vert _{L_t^q L_{x'}^r}^{2} d \xi _1 \right) ^{\frac{1}{2}} \lesssim \Vert f \Vert _{L_x^2}, \end{aligned}$$

which is (2.1). Similarly, by (2.4),

$$\begin{aligned}&\Vert D_{x_1}^{\frac{1}{q_1}+\frac{1}{q_2}} \mathcal {I} (F)\Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \\&\quad = \left( \int _{{\mathbb {R}}} \Bigl \Vert \int _0^t |\xi _1|^{\frac{1}{q_1}+\frac{1}{q_2}} V_{\xi _1}(t-t') \mathcal {F}_{x_1}(F)(t') d t' \Bigr \Vert _{L_t^{q_1} L_{x'}^{r_1}}^2 d \xi _1 \right) ^{\frac{1}{2}}\\&\quad \lesssim \Vert F\Vert _{L_t^{q_2'} L_{x'}^{r_2'} L_{x_1}^2}, \end{aligned}$$

which is (2.2). \(\square \)

Now, we can complete the proof of Theorem 1.1. Recall that \(d=5\) implies \(s_c = 1/2\).

Definition 2.2

We define

$$\begin{aligned} \Vert u \Vert _{Z^{\frac{1}{2}}} :={}&\Vert P_{\le 1}u \Vert _{L_t^{\infty } L_x^2} + \Vert P_{\le 1} D_{x_1}^{\frac{1}{2}} u\Vert _{L_t^2 L_{x'}^4 L_{x_1}^2} \\ {}&+ \sum _{N \in 2^{{\mathbb N}}}N^{\frac{1}{2}} \left( \Vert P_{N}u \Vert _{L_t^{\infty } L_x^2} + \Vert P_{N} D_{x_1}^{\frac{1}{2}} u\Vert _{L_t^2 L_{x'}^4 L_{x_1}^2} \right) , \\ \Vert u \Vert _{\dot{Z}^{\frac{1}{2}}} :={}&\sum _{N \in 2^{{\mathbb Z}}}N^{\frac{1}{2}} \left( \Vert P_{N}u \Vert _{L_t^{\infty } L_x^2} + \Vert P_{N} D_{x_1}^{\frac{1}{2}} u\Vert _{L_t^2 L_{x'}^4 L_{x_1}^2} \right) , \end{aligned}$$

and the corresponding Banach spaces.

By the standard argument involving the contraction mapping principle, it suffices to prove the following:

Proposition 2.3

Let \(d=5\). Then, we have

$$\begin{aligned} \Vert \mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert _{Z^{\frac{1}{2}}} \lesssim \Vert u_1 \Vert _{Z^{\frac{1}{2}}} \Vert u_2 \Vert _{Z^{\frac{1}{2}}}, \quad \Vert \mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert _{\dot{Z}^{\frac{1}{2}}} \lesssim \Vert u_1 \Vert _{\dot{Z}^{\frac{1}{2}}} \Vert u_2 \Vert _{\dot{Z}^{\frac{1}{2}}}. \end{aligned}$$

Proof

Let \(N_{\max } = \max (N_1,N_2,N_3)\) and \(N_{\min } = \min (N_1,N_2, N_3)\). Theorem 2.1 gives

$$\begin{aligned}&\Vert P_{N_3}\mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert |_{L_t^{\infty } L_x^2} + \Vert P_{N_3}D_{x_1}^{\frac{1}{2}} \mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert _{L_t^2 L_{x'}^4 L_{x_1}^2}\\&\quad \lesssim \Vert P_{N_3} D_{x_1}^{\frac{1}{2}} \left( u_1 \, u_2\right) \Vert _{L_t^2 L_{x'}^{\frac{4}{3}} L_{x_1}^2}. \end{aligned}$$

We observe that \(P_{N_3}=P_{N_3}Q_{\le N_3}\). Since the kernel of the operator \(P_{N_3}\) is uniformly bounded in \(L^1_x\), we obtain

$$\begin{aligned} \Vert P_{N_3} D_{x_1}^{\frac{1}{2}} (u_1 \, u_2) \Vert _{L_t^2 L_{x'}^{\frac{4}{3}} L_{x_1}^2} \lesssim \Vert Q_{\le N_3} D_{x_1}^{\frac{1}{2}} (u_1 \, u_2) \Vert _{L_t^2 L_{x'}^{\frac{4}{3}} L_{x_1}^2}. \end{aligned}$$

Now, by freezing \((t,x')\), we can use Plancherel’s theorem and Bernstein’s inequality in \(x_1\) to compute

$$\begin{aligned} \Vert Q_{\le N_3} D_{x_1}^{\frac{1}{2}} (u_{N_1} \, u_{N_2}) \Vert _{L_{x_1}^2}\lesssim N_{\min }^{\frac{1}{2}}(\Vert D_{x_1}^{\frac{1}{2}} u_{N_1}\Vert _{L_{x_1}^2}\Vert u_{N_2} \Vert _{L_{x_1}^2}+\Vert u_{N_1} \Vert _{L_{x_1}^2}\Vert D_{x_1}^{\frac{1}{2}} u_{N_2}\Vert _{L_{x_1}^2}) \end{aligned}$$

Finally, Hölder’s inequality with respect to \((t,x')\) implies

$$\begin{aligned}&\Vert P_{N_3} D_{x_1}^{\frac{1}{2}} (u_{N_1} \, u_{N_2}) \Vert _{L_t^2 L_{x'}^{\frac{4}{3}} L_{x_1}^2} \\&\quad \lesssim N_{\min }^{\frac{1}{2}}\Vert D_{x_1}^{\frac{1}{2}} u_{N_1} \Vert _{L_t^2 L_{x'}^4 L_{x_1}^2} \Vert u_{N_2} \Vert _{L_t^{\infty } L_x^2} + N_{\min }^{\frac{1}{2}} \Vert u_{N_1} \Vert _{L_t^{\infty } L_x^2} \Vert D_{x_1}^{\frac{1}{2}} u_{N_2} \Vert _{L_t^2 L_{x'}^4 L_{x_1}^2}. \end{aligned}$$

This can be summed up both in the homogeneous and in the inhomogeneous version.

\(\square \)

This argument also implies the scattering claim, since it implies that the Duhamel integral converges to a free solutions as \(t\rightarrow \pm \infty \). We omit the details of this standard argument.

Transversal estimates and the proof of Theorem 1.2

Lemma 3.1

Let \(d\ge 2\) and \(f_{N_1,\lambda _1} = Q_{\lambda _1} P_{N_1} f\), \(g_{N_2,\lambda _2} = Q_{\lambda _2} P_{N_2} g\). For all \(\lambda _j,N_j\in 2^{\mathbb Z}\) such that

$$\begin{aligned} |\nabla \varphi (\xi )-\nabla \varphi (\eta )| > rsim \max \{\lambda _1,\lambda _2\} N_{\max }, \end{aligned}$$

for all \(\xi \in {\text {supp}}\widehat{f}_{N_1,\lambda _1}\), \(\eta \in {\text {supp}}\widehat{g}_{N_2,\lambda _2}\), it holds that

$$\begin{aligned} \Vert P_{N_3} (e^{tS}f_{N_1,\lambda _1} \, e^{tS}g_{N_2,\lambda _2}) \Vert _{L_t^2 L_x^2} \lesssim \left( \frac{N_{\min }^{d-1}}{ \max \{\lambda _1,\lambda _2\} N_{\max }}\right) ^{\frac{1}{2}} \Vert f_{N_1,\lambda _1}\Vert _{L^2} \Vert g_{N_2,\lambda _2} \Vert _{L^2}. \end{aligned}$$
(3.1)

This is an instance of the well-known bilinear transversal estimate, e.g., a special case of [1, Lemma 2.6], where a proof can be found.

Next, we recall the definitions of \(U^p\) and \(V^p\) spaces, which have been introduced in [8] the dispersive PDE context. We refer the reader to [1] and the references therein for further details. For \(1\le p <\infty \), we call a function \(a: {\mathbb {R}}\rightarrow L^2({\mathbb {R}}^d)\) a \(p-\)atom, if there exists a finite partition \(\mathcal {J}=\{(-\infty ,t_1),[t_2,t_3), \ldots , [t_K,\infty )\}\) of the real line such that

$$\begin{aligned} a(t)=\sum _{J \in \mathcal {J}}\mathbf {1}(t)f_J, \quad \sum _{J \in \mathcal {J}}\Vert f_J\Vert _{L^2}^p\le 1. \end{aligned}$$

Now, \(U^p\) is defined as the space of all \(u: {\mathbb {R}}\rightarrow L^2({\mathbb {R}}^d)\), such that there exists an atomic decomposition \(u=\sum _{j=1 }^\infty c_j a_j\), where \((c_j)\in \ell ^1({\mathbb N})\) and the \(a_j\)’s are \(p-\)atoms. Then, \(\Vert u\Vert _{U^p}=\inf \sum _{j=1}^\infty |c_j|\) is a norm (the infimum is taken with respect to all possible atomic decompositions), so that \(U^p\) is a Banach space. Further, let \(V^p\) denote the space of all right-continuous functions \(v: {\mathbb {R}}\rightarrow L^2({\mathbb {R}}^d)\), such that

$$\begin{aligned} \Vert v\Vert _{V^p}=\Vert v\Vert _{L^\infty _t L^2_x}+\sup \Big (\sum _{j \in {\mathbb Z}}\Vert v(t_j)-v(t_{j-1})\Vert _{L^2_x}^p\Big )^{\frac{1}{p}}<\infty , \end{aligned}$$

where the supremum is taken over all increasing sequences \((t_j)\). Now, we define the atomic space \(U_{S}^p = e^{\cdot \, S} U^p\) with norm \(\Vert u \Vert _{U_S^p} = \Vert e^{- \, \cdot \, S} u\Vert _{U^p}\), and \(V_{S}^p = e^{\cdot \, S} V^p\) with norm \(\Vert u \Vert _{V_S^p} = \Vert e^{- \, \cdot \, S} u\Vert _{V^p}\).

There is the embedding \(V_{S}^p\subset U_{S}^q\) if \(p<q\), see [8, Lemma 6.4]. Due to the atomic structure of \(U_{S}^q\) and the Strichartz estimate (2.1), we have

$$\begin{aligned} \Vert D_{x_1}^{\frac{1}{q}} u\Vert _{L_t^{q} L_{x'}^{r} L_{x_1}^2} \lesssim \Vert u \Vert _{U^{q}_S} \end{aligned}$$
(3.2)

for \((d-1)-\)admissible pairs, and \(\Vert u \Vert _{U^{q}_S}\) may be replaced by \(\Vert u \Vert _{V^{2}_S}\) for non-endpoint pairs, i.e., when \(q>2\).

Let \(\lambda _{\max } := \max (\lambda _1,\lambda _2,\lambda _3)\) and \(\lambda _{\min } := \min (\lambda _1,\lambda _2,\lambda _3)\). We use the shorthand notation \(u_N := P_{N} u\), \(u_{N, \lambda } := Q_{\lambda } P_{N} u \), etc.

Proposition 3.2

Let \(d\ge 6\) and the pair (qr) be \((d-1)\)-admissible with \(2<q<\frac{2(d-3)}{d-5}\), and let \(\varepsilon >0\). Suppose

$$\begin{aligned} |\nabla \varphi (\xi )-\nabla \varphi (\eta )| > rsim \max \{\lambda _1,\lambda _2\} N_{\max }, \end{aligned}$$

for all \(\xi \in {\text {supp}}_{\xi } \widehat{u}_{N_1,\lambda _1}\), \(\eta \in {\text {supp}}_{\xi } \widehat{v}_{N_2,\lambda _2}\). Then, for all \(\lambda _j,N_j \in 2^{\mathbb Z}\),

$$\begin{aligned} \begin{aligned}&\Vert P_{N_3} Q_{\lambda _3} ( u_{N_1,\lambda _1} v_{N_2,\lambda _2}) \Vert _{L_t^{q'} L_{x'}^{r'} L_{x_1}^2} \\&\quad \lesssim \lambda _{\max }^{-\frac{1}{2}+ \frac{2 \varepsilon }{d-3}} \lambda _{\min }^{-\frac{1}{2}+\frac{1}{d-3}+\frac{1}{q}-\frac{2 \varepsilon }{d-3}} N_{\max }^{-\frac{1}{2} + \frac{1}{d-3}} N_{\min }^{\frac{d-3}{2} - \frac{2}{d-3}} \Vert u_{N_1,\lambda _1} \Vert _{V^2_S} \Vert v_{N_2,\lambda _2}\Vert _{V^2_S}. \end{aligned} \end{aligned}$$
(3.3)

Proof

By symmetry, we may assume that \(\lambda _1 \sim \lambda _{\max }\). For a sufficiently small \(\varepsilon >0\), we define the \((d-1)\)-admissible pairs \((q_1,r_1)\) and \((q_2,r_2)\) by

$$\begin{aligned} \left( \frac{1}{q_1}, \ \frac{1}{r_1} \right)= & {} \left( \frac{1}{2} -\varepsilon , \frac{1}{2}-\frac{1- 2\varepsilon }{d-1} \right) , \ \left( \frac{1}{q_2}, \ \frac{1}{r_2}\right) \\= & {} \left( \frac{d-3}{4}- \frac{d-3}{2 q} + \varepsilon , \ \frac{d-3}{q(d-1)} + \frac{1- 2 \varepsilon }{d-1} \right) . \end{aligned}$$

In addition, letting

$$\begin{aligned} \frac{1}{\alpha } = \frac{1}{q_1} + \frac{1}{q_2} , \qquad \frac{1}{\beta } = \frac{1}{r_1} + \frac{1}{r_2}, \end{aligned}$$

by using (3.2), we have

$$\begin{aligned} \Vert P_{N_3}Q_{\lambda _3}( u_{N_1,\lambda _1} v_{N_2,\lambda _2}) \Vert _{L_t^{\alpha } L_{x'}^{\beta } L_{x_1}^2}&\lesssim \lambda _{\min }^{\frac{1}{2}} \Vert u_{N_1,\lambda _1} \Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \Vert v_{N_2,\lambda _2}\Vert _{L_t^{q_2} L_{x'}^{r_2} L_{x_1}^2}\nonumber \\&\lesssim \lambda _{\min }^{\frac{1}{2}-\frac{1}{q_2}} \lambda _{\max }^{-\frac{1}{q_1}} \Vert u_{N_1,\lambda _1} \Vert _{U^{q_1}_S} \Vert v_{N_2,\lambda _2}\Vert _{U_S^{q_2}}. \end{aligned}$$
(3.4)

Lemma 3.1 immediately extends from free solutions to \(2-\)atoms. Therefore, the atomic structure of \(U^2\) implies

$$\begin{aligned} \Vert P_{N_3}Q_{\lambda _3}\left( u_{N_1,\lambda _1} v_{N_2,\lambda _2}\right) \Vert _{L_t^2 L_x^2} \lesssim \left( \frac{N_{\min }^{d-1}}{ \lambda _{\max } N_{\max }}\right) ^{\frac{1}{2}} \Vert u_{N_1,\lambda _1} \Vert _{U^{2}_S} \Vert v_{N_2,\lambda _2}\Vert _{U_S^{2}}. \end{aligned}$$
(3.5)

For \(\theta = \frac{2}{d-3}\), it is observed that

$$\begin{aligned} \frac{\theta }{\alpha } + \frac{1- \theta }{2} = 1 - \frac{1}{q} \Bigl ( =: \frac{1}{q'} \Bigr ), \quad \frac{\theta }{\beta } + \frac{1- \theta }{2} = 1 - \frac{1}{r} \Bigl (=: \frac{1}{r'} \Bigr ). \end{aligned}$$
Fig. 1
figure1

Choice of parameters in the interpolation argument

Now, we interpolate (3.4) and (3.5) to obtain (3.3). More precisely, we follow the argument in [1, p. 1203]: For brevity, we set \(u:=u_{N_1,\lambda _1} \), \(v:=v_{N_2,\lambda _2} \). Then, [8, Lemma 6.4] implies that there exist decompositions \(u=\sum _{k=1}^\infty u_{k}\), such that \(\widehat{u_{k}}\subset {\text {supp}}\widehat{u}\), and for any \(q \ge 2\) we have \( \Vert u_{k}\Vert _{U^{q}_S}\lesssim 2^{k(\frac{2}{q}-1)}\Vert u\Vert _{V^2_S}, \) and the analogous decomposition for v. Then, by convexity, we obtain

$$\begin{aligned} \Vert P_{N_3} Q_{\lambda _3} ( u v) \Vert _{L_t^{q'} L_{x'}^{r'} L_{x_1}^2} \lesssim {}&\sum _{k,k'\in {\mathbb N}}\Vert P_{N_3} Q_{\lambda _3} \left( u_k v_{k'}\right) \Vert _{L_t^{q'} L_{x'}^{r'} L_{x_1}^2} \\ \lesssim {}&\sum _{k,k'\in {\mathbb N}}\Vert P_{N_3} Q_{\lambda _3} \left( u_k v_{k'}\right) \Vert _{L_t^{\alpha } L_{x'}^{\beta } L_{x_1}^2}^\theta \Vert P_{N_3} Q_{\lambda _3} \left( u_k v_{k'}\right) \Vert _{L^2_{t,x}}^{1-\theta }. \end{aligned}$$

Estimates (3.4) and (3.5) further imply

$$\begin{aligned} \Vert P_{N_3} Q_{\lambda _3} ( u v) \Vert _{L_t^{q'} L_{x'}^{r'} L_{x_1}^2} \lesssim \Big (\sum _{k,k'\in {\mathbb N}} 2^{k\theta (\frac{2}{q_1}-1)}2^{k'\theta (\frac{2}{q_2}-1)}\Big ) \frac{\lambda _{\min }^{\frac{\theta }{2}-\frac{\theta }{q_2}} N_{\min }^{\frac{d-1}{2}(1-\theta )}}{ \lambda _{\max }^{\frac{\theta }{q_1}+\frac{1-\theta }{2}} N_{\max }^{\frac{1-\theta }{2}}} \Vert u\Vert _{V^2_S}\Vert v\Vert _{V^2_S} . \end{aligned}$$

Since \(q_1,q_2>2\), the sums converge, and the proof of (3.3) is complete. \(\square \)

Define

$$\begin{aligned} \mathcal {R}_{\lambda _{\max },N_{\max }} = \{ (\xi _1, \xi ') \in {\mathbb {R}}\times {\mathbb {R}}^{d-1} \, | \, |\xi _1|\ll \lambda _{\max }, \ |\xi '| \ll N_{\max } \}. \end{aligned}$$

Lemma 3.3

Assume that there exist \(\gamma _1\), \(\gamma _2\), \(\gamma _3 \in {\mathbb {R}}^{d}\) such that \(\gamma _1 + \gamma _2 -\gamma _3 \in \mathcal {R}_{4 \lambda _{\max }, 4N_{\max }}\) and

$$\begin{aligned} {\text {supp}}_{\xi } \widehat{u}_{N_i, \lambda _i} \subset \mathcal {R}_{\lambda _{\max },N_{\max }} + \gamma _i := \{\xi \in {\mathbb {R}}^d \, | \, \xi - \gamma _i \in \mathcal {R}_{\lambda _{\max },N_{\max }} \} . \end{aligned}$$
(3.6)

Then, we have either

$$\begin{aligned} |\nabla \varphi (\xi ) - \nabla \varphi (\eta )| > rsim \lambda _{\max }N_{\max }, \end{aligned}$$
(3.7)

for all \(\xi \in {\text {supp}}_{\xi } \widehat{u}_{N_1, \lambda _1}\), \(\eta \in {\text {supp}}_{\xi } \widehat{u}_{N_2, \lambda _2}\), or

$$\begin{aligned} |\nabla \varphi (\eta )-\nabla \varphi (\zeta )| > rsim \lambda _{\max }N_{\max }, \end{aligned}$$
(3.8)

for all \(\eta \in {\text {supp}}_{\xi } \widehat{u}_{N_2, \lambda _2}\), \(\zeta \in {\text {supp}}_{\xi } \widehat{u}_{N_3, \lambda _3}\).

Proof

Firstly, we consider the case \(\max (|\xi '|, |\eta '|, |\zeta '|) \ll N_{\max }\). We deduce from \(\partial _1 \varphi (\xi )= 3 \xi _1^2 + |\xi '|^2\) that

$$\begin{aligned}&|\partial _1 \varphi (\xi ) - \partial _1 \varphi (\eta )| + | \partial _1 \varphi (\eta ) - \partial _1 \varphi (\xi + \eta )|\\&\quad \ge 3|\xi _1^2 - \eta _1^2| + 3|\xi _1\left( \xi _1+2\eta _1\right) | - |\xi '|^2-|\eta '|^2-|\xi '+\eta '|^2 > rsim N_{\max }^2, \end{aligned}$$

which implies the claim since \(|\nabla ^2 \varphi (\xi )| \lesssim |\xi |\).

Next, we assume \(\max (|\xi '|, |\eta '|, |\zeta '|) \sim N_{\max }\). For all \(\xi \in {\text {supp}}_{\xi } \widehat{u}_{N_1, \lambda _1}\), \(\eta \in {\text {supp}}_{\xi } \widehat{u}_{N_2, \lambda _2}\), \(\xi + \eta \in {\text {supp}}_{\xi } \widehat{u}_{N_3, \lambda _3}\), we will show

$$\begin{aligned} \sum _{j=2}^d \left( |\partial _j \varphi (\xi ) - \partial _j \varphi (\eta )| + |\partial _j \varphi (\eta )-\partial _j \varphi (\xi + \eta )| \right) > rsim \lambda _{\max }N_{\max }. \end{aligned}$$
(3.9)

We may assume \(|\xi '| \sim N_{\max }\), \(\lambda _1 \sim \lambda _{\max }\). For \(2 \le j \le d\), it is observed that \(\partial _j \varphi (\xi ) = 2 \xi _1 \xi _j\). Then, for (3.9), it suffices to show

$$\begin{aligned} |\xi _1 \xi ' - \eta _1 \eta '| + |\eta _1 \eta ' - \left( \xi _1+\eta _1\right) \left( \xi ' + \eta '\right) | > rsim \lambda _1 N_1. \end{aligned}$$
(3.10)

Since \(|\xi '| \sim N_{\max }\), \(\lambda _1 \sim \lambda _{\max }\), if either \(\lambda _{\min } \ll \lambda _{\max }\) or \(\min (|\eta '|, |\xi '+\eta '|) \ll N_{\max }\) holds, we easily verify (3.10). Then, we assume \(\lambda _1 \sim \lambda _2 \sim \lambda _3\) and \(|\xi '| \sim |\eta '| \sim |\xi '+\eta '|\). We observe

$$\begin{aligned}&|\eta _1 \eta ' - \left( \xi _1+\eta _1\right) \left( \xi ' + \eta '\right) |\\&\quad = \bigl |\eta _1 \eta ' -\left( \xi _1 + \eta _1\right) \left( \xi ' - \frac{\eta _1}{\xi _1} \eta ' + \frac{\eta _1}{\xi _1} \eta ' + \eta ' \right) \bigr |\\&\quad \ge \bigl | \eta _1 \eta ' - (\xi _1 + \eta _1) \left( 1 + \frac{\eta _1}{\xi _1}\right) \eta ' \bigr | - \bigl | \left( 1+\frac{\eta _1}{\xi _1}\right) \left( \xi _1 \xi ' - \eta _1 \eta '\right) \bigr |\\&\quad = \bigl | 1+\frac{\eta _1}{\xi _1}+ \frac{\xi _1}{\eta _1} \, \bigr | |\eta _1 \eta '| - \bigl | \left( 1+\frac{\eta _1}{\xi _1}\right) \left( \xi _1 \xi ' - \eta _1 \eta '\right) \bigr |. \end{aligned}$$

Since \(|\alpha + \alpha ^{-1}| \ge 2\) for any \(\alpha \in {\mathbb {R}}\), this completes the proof of (3.10).

From (3.9), without loss of generality, we can assume that there exist \(\xi _0 \in {\text {supp}}_{\xi } \widehat{u}_{N_1, \lambda _1}\), \(\eta _0 \in {\text {supp}}_{\xi } \widehat{u}_{N_2, \lambda _2}\) such that

$$\begin{aligned} \sum _{j=2}^d |\partial _j \varphi \left( \xi _0\right) - \partial _j \varphi \left( \eta _0\right) | > rsim \lambda _{\max }N_{\max }. \end{aligned}$$
(3.11)

For \(2 \le j, k \le d\) and all \(\xi \in {\text {supp}}_{\xi } \widehat{u}_{N_1, \lambda _1}\), \(\eta \in {\text {supp}}_{\xi } \widehat{u}_{N_2, \lambda _2}\), since \(|\partial _1 \partial _j \varphi (\xi )| + |\partial _1 \partial _j \varphi (\eta )| \lesssim N_{\max }\) and \(|\partial _k \partial _j \varphi (\xi )| +|\partial _k \partial _j \varphi (\eta )| \lesssim \lambda _{\max }\), we get

$$\begin{aligned} | \partial _j \varphi (\xi ) - \partial _j \varphi \left( \xi _0\right) | + | \partial _j \varphi (\eta ) - \partial _j \varphi \left( \eta _0\right) | \ll \lambda _{\max } N_{\max }, \end{aligned}$$

for all \(\xi \in {\text {supp}}_{\xi } \widehat{u}_{N_1, \lambda _1}\), \(\eta \in {\text {supp}}_{\xi } \widehat{u}_{N_2, \lambda _2}\). This estimate and (3.11) yield the claim (3.7). \(\square \)

Now, we define the solution spaces as \(Y^s:=C({\mathbb {R}}; H^s({\mathbb {R}}^d)) \cap {\langle {\nabla _x} \rangle }^{-s} V^2_S\) and \( \dot{Y}^s:=C({\mathbb {R}}; \dot{H}^s({\mathbb {R}}^d)) \cap {|\nabla _x|}^{-s} V^2_S\), with norms

$$\begin{aligned} \Vert u \Vert _{Y^s} :=&\left( \sum _{N \in 2^{{\mathbb Z}}} \langle N\rangle ^{2 s} \Vert P_N u \Vert _{V^2_S}^2 \right) ^{1/2} , \\ \Vert u \Vert _{\dot{Y}^s} :=&\left( \sum _{N \in 2^{{\mathbb Z}}} N^{2 s} \Vert P_N u \Vert _{V^2_S}^2 \right) ^{1/2}, \end{aligned}$$

respectively.

Proposition 3.4

Let \(d \ge 6\). Then, we have

$$\begin{aligned} \Vert \mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert _{Y^{s_c}} \lesssim \Vert u_1 \Vert _{Y^{s_c}} \Vert u_2 \Vert _{Y^{s_c}}, \quad \Vert \mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert _{\dot{Y}^{s_c}} \lesssim \Vert u_1 \Vert _{\dot{Y}^{s_c}} \Vert u_2 \Vert _{\dot{Y}^{s_c}}. \end{aligned}$$

Proof

We show first that there exists \(\varepsilon >0\) such that for any \(N_1,N_2,N_3\in 2^{\mathbb Z}\) we have

$$\begin{aligned} \Bigl | \iint P_{N_1} u_1 P_{N_2} u_2 \partial _{x_1} P_{N_3} u_3 dx dt \Bigr | \lesssim N_{\min }^{s_c + \varepsilon } N_{\max }^{- \varepsilon } \prod _{i=1}^3 \Vert P_{N_i}u_{i} \Vert _{V^2_S}. \end{aligned}$$
(3.12)

As before, we use the shorthand notation \(u_{N_j} := P_{N_j} u_j\), \(u_{N_j, \lambda _j} := Q_{\lambda _j} P_{N_j} u_j \), etc. Obviously, (3.12) is implied by

$$\begin{aligned} \sum _{\lambda _1, \lambda _2, \lambda _{3} \in 2^{{\mathbb Z}}}\lambda _3 \Bigl | \iint u_{N_1,\lambda _1} u_{N_2,\lambda _2} u_{N_3,\lambda _3} dx dt \Bigr | \lesssim N_{\min }^{s_c + \varepsilon } N_{\max }^{- \varepsilon } \prod _{i=1}^3 \Vert u_{N_i} \Vert _{V^2_S}. \end{aligned}$$
(3.13)

Now we show (3.13). After harmless decompositions, we may assume that there exist \(\gamma _1\), \(\gamma _2\), \(\gamma _3 \in {\mathbb {R}}^{d}\) such that \(\gamma _1 + \gamma _2 -\gamma _3 \in \mathcal {R}_{4 \lambda _{\max }, 4N_{\max }}\) and (3.6). Lemma 3.3 provides either \(|\nabla \varphi (\xi )-\nabla \varphi (\eta )| > rsim \lambda _{\max }N_{\max }\) for all \(\xi \in {\text {supp}}_{\xi } u_{N_1,\lambda _1}\), \(\eta \in {\text {supp}}_{\xi } u_{N_2,\lambda _2}\) or \(|\nabla \varphi (\eta )-\nabla \varphi (\zeta )| > rsim \lambda _{\max }N_{\max }\) for all \(\eta \in {\text {supp}}_{\xi } u_{N_2,\lambda _2}\) and \(\zeta \in {\text {supp}}_{\xi } u_{N_3,\lambda _3}\). For the former case, it follows from the Hölder’s inequality, the Strichartz estimate (3.2), and the bilinear estimate (3.3) that

$$\begin{aligned}&\sum _{\lambda _1, \lambda _2, \lambda _{3} \in 2^{{\mathbb Z}}}\lambda _3 \Bigl | \iint u_{N_1,\lambda _1} u_{N_2,\lambda _2} u_{N_3,\lambda _3} dx dt \Bigr |\\&\quad \le \sum _{\lambda _i \le N_i (i=1,2,3)}\lambda _3 \Vert P_{N_3} Q_{\lambda _3} \left( u_{N_1,\lambda _1} u_{N_2,\lambda _2}\right) \Vert _{L_t^{q'} L_{x'}^{r'} L_{x_1}^2} \Vert u_{N_3,\lambda _3}\Vert _{L_t^{q} L_{x'}^{r} L_{x_1}^2}\\&\quad \lesssim \sum _{\lambda _i \le N_i (i=1,2,3)} \lambda _{\max }^{\frac{d-1}{d-3}\varepsilon } \lambda _{\min }^{\frac{1}{d-3}-\frac{d-1}{d-3} \varepsilon } N_{\max }^{-\frac{1}{2} + \frac{1}{d-3}} N_{\min }^{\frac{d-3}{2} - \frac{2}{d-3}} \Vert u_{N_1,\lambda _1} \Vert _{V^2_S} \Vert u_{N_2,\lambda _2}\Vert _{V_S^{2}} \Vert u_{N_3,\lambda _3}\Vert _{V^2_S}\\&\quad \le N_{\min }^{s_c + \frac{1}{2} -\frac{1}{d-3}- \frac{d-1}{d-3} \varepsilon }N_{\max }^{- \frac{1}{2} + \frac{1}{d-3}+\frac{d-1}{d-3} \varepsilon } \Vert u_{N_1} \Vert _{V^2_S} \Vert u_{N_2}\Vert _{V_S^{2}} \Vert u_{N_3}\Vert _{V^2_S}. \end{aligned}$$

Here, the pair (qr) should satisfy the hypothesis of Proposition 3.2, and we have used \(\lambda _{\max } \le N_{\max }\) and \(\lambda _{\min }\le N_{\min }\). In the similar way, the latter case is treated as follows:

$$\begin{aligned}&\sum _{\lambda _1, \lambda _2, \lambda _{3} \in 2^{{\mathbb Z}}}\lambda _3 \Bigl | \iint u_{N_1,\lambda _1} u_{N_2,\lambda _2} u_{N_3,\lambda _3} dx dt \Bigr |\\&\quad \le \sum _{\lambda _i \le N_i (i=1,2,3)} \lambda _3 \Vert P_{N_1} Q_{\lambda _1} \left( u_{N_2,\lambda _2} u_{N_3,\lambda _3}\right) \Vert _{L_t^{q'} L_{x'}^{r'} L_{x_1}^2} \Vert u_{N_1,\lambda _1}\Vert _{L_t^{q} L_{x'}^{r} L_{x_1}^2}\\&\quad \le N_{\min }^{s_c + \frac{1}{2} -\frac{1}{d-3}- \frac{d-1}{d-3} \varepsilon }N_{\max }^{- \frac{1}{2} + \frac{1}{d-3}+\frac{d-1}{d-3} \varepsilon } \Vert u_{N_1} \Vert _{V^2_S} \Vert u_{N_2}\Vert _{V_S^{2}} \Vert u_{N_3}\Vert _{V^2_S}. \end{aligned}$$

Finally, we explain why (3.12) implies Proposition 3.4. By duality, see, e.g., [1, Lemma 7.3], we obtain

$$\begin{aligned} \Vert P_{N_3} \mathcal {I}\left( \partial _{x_1}\left( P_{N_1} u_1 P_{N_2} u_2\right) \right) \Vert _{V^2_S} \lesssim N_{\min }^{s_c}\left( \frac{N_{\min }}{N_{\max }}\right) ^{ \varepsilon } \Vert P_{N_1}u_{1} \Vert _{V^2_S}\Vert P_{N_2}u_{2} \Vert _{V^2_S}. \end{aligned}$$

This can be easily summed up. \(\square \)

Again, the proof of Theorem 1.2 is a straightforward application of the contraction mapping principle. The scattering claim follows from the well-known fact that functions in \(V^2\) have limits at \(\pm \infty \).

Radial Strichartz estimates and the proof of Theorem 1.3

We first prove a variant of the Strichartz estimates in 2.1 for functions which, for fixed \(x_1\), are radial in \(x'\).

Theorem 4.1

Let \(d \ge 3\) and \(2 \le q\),\(r \le \infty \) satisfy

$$\begin{aligned} \frac{2}{q} \le (2 d -3) \left( \frac{1}{2} - \frac{1}{r} \right) , \quad (d,q,r)\not =(3,2,\infty ), \quad (q,r)\not =\left( 2,\frac{2(2d-3)}{2d-5}\right) , \end{aligned}$$

and let \(\sigma =-\frac{d-1}{2}+\frac{d-1}{r} +\frac{2}{q}\). Then, for all functions \(f \in L^2_{\text {rad}}({\mathbb {R}}^d)\), we have

$$\begin{aligned} \Vert D_{x_1}^{\frac{1}{q}} |\nabla _{x'}|^\sigma e^{t S} f \Vert _{L_t^{q} L_{x'}^{r} {{L}_{x_1}^2}} \lesssim \Vert f \Vert _{L_x^2}. \end{aligned}$$
(4.1)

The proof follows the exact same lines as the proof of Theorem 2.1, but with the Strichartz estimates for the \((d-1)\)-dimensional Schrödinger equation from [6] replaced by the radial version obtained in [2, Theorem 1.1].

Lemma 4.2

Let \(d\ge 2\) and \(f_{N_1,\lambda _1,M_1} = R_{M_1} Q_{\lambda _1} P_{N_1} f\), \(g_{N_2,\lambda _2,M_2} = R_{M_2} Q_{\lambda _2} P_{N_2} g\). (i) Suppose that there exists \(\ell \in \{2,\ldots ,d\}\) such that

$$\begin{aligned} |\partial _{\ell } \varphi (\xi )-\partial _{\ell } \varphi (\eta )| > rsim N_{\max }^2, \end{aligned}$$

for all \(\xi \in {\text {supp}}\widehat{f}_{N_1,\lambda _1,M_1}\), \(\eta \in {\text {supp}}\widehat{g}_{N_2,\lambda _2,M_2}\). Then, it holds that

$$\begin{aligned} \begin{aligned}&\Vert P_{N_3}\left( e^{tS}f_{\lambda _1,N_1,M_1} \, e^{tS}g_{\lambda _2,N_2,M_2}\right) \Vert _{L_t^2 L_x^2} \\&\quad \lesssim \left( \frac{\min \left\{ \lambda _1,\lambda _2 \right\} \min \left\{ M_{1},M_2\right\} ^{d-2}}{ N_{\max }^2}\right) ^{\frac{1}{2}} \Vert f_{M_1,\lambda _1,N_1}\Vert _{L^2} \Vert g_{M_2,\lambda _2,N_2} \Vert _{L^2}. \end{aligned} \end{aligned}$$
(4.2)

(ii) Suppose that

$$\begin{aligned} |\partial _{1} \varphi (\xi )-\partial _{1} \varphi (\eta )| > rsim N_{\max }^2, \end{aligned}$$

for all \(\xi \in {\text {supp}}\widehat{f}_{N_1,\lambda _1,M_1}\), \(\eta \in {\text {supp}}\widehat{g}_{N_2,\lambda _2,M_2}\). Then, it holds that

$$\begin{aligned} \begin{aligned}&\Vert P_{N_3}(e^{tS}f_{\lambda _1,N_1,M_1} \, e^{tS}g_{\lambda _2,N_2,M_2}) \Vert _{L_t^2 L_x^2} \\&\quad \lesssim \left( \frac{ \min \{M_{1},M_2\}^{d-1}}{ N_{\max }^2}\right) ^{\frac{1}{2}} \Vert f_{M_1,\lambda _1,N_1}\Vert _{L^2} \Vert g_{M_2,\lambda _2,N_2} \Vert _{L^2}. \end{aligned} \end{aligned}$$
(4.3)

As above, the proof of this lemma follows from [1, Lemma 2.6]. As above, it immediately extends to \(U^2_S\)-functions.

Let \(Y^{s}_{\text {rad}}\) and \(\dot{Y}^{s}_{\text {rad}}\) be the subspaces of \(Y^{s}\) and \(\dot{Y}^{s}\) of functions which, for fixed \(x_1\), are radial in \(x'\), with the same norms. Then, the key for the proof of Theorem 1.3 is the following

Proposition 4.3

Let \(d \ge 4\). Then, we have

$$\begin{aligned} \Vert \mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert _{Y^{s_c}_{\text {rad}}} \lesssim \Vert u_1 \Vert _{Y^{s_c}_{\text {rad}}} \Vert u_2 \Vert _{Y^{s_c}_{\text {rad}}}, \quad \Vert \mathcal {I} \left( \partial _{x_1}\left( u_1u_2\right) \right) \Vert _{\dot{Y}^{s_c}_{\text {rad}}} \lesssim \Vert u_1 \Vert _{\dot{Y}^{s_c}_{\text {rad}}} \Vert u_2 \Vert _{\dot{Y}^{s_c}_{\text {rad}}}. \end{aligned}$$

Proof

For \(i=1,2,3\), we use \(u_{i} := R_{M_i} Q_{\lambda _i} P_{N_i} u \). As in the proof of Proposition 3.4, it suffices to show

$$\begin{aligned} \sum _{\lambda _i, M_i} \lambda _{\max } \Bigl | \iint u_1 u_{2} u_{3} dx dt \Bigr | \lesssim N_{\min }^{s_c + \varepsilon } N_{\max }^{- \varepsilon } \prod _{i=1}^3 \Vert u_{N_i} \Vert _{V^2_S}. \end{aligned}$$
(4.4)

Here and in the sequel, all functions are implicitly assumed to satisfy the radiality hypothesis. Let

$$\begin{aligned} \left( \frac{1}{q_1}, \frac{1}{r_1} \right) =&\left( \frac{1}{2} - \varepsilon , \frac{2d-5}{2(2d-3)} \right) ,\\ \left( \frac{1}{q_2}, \frac{1}{r_2} \right) =&\left( \frac{(d-1)(2 d-3)}{2 \left( d-1+ 2 (2d-3) \varepsilon \right) } \varepsilon , \frac{d-1}{2 \left( d-1+ 2 (2d-3) \varepsilon \right) } \right) ,\\ \left( \frac{1}{q_3}, \frac{1}{r_3} \right) =&\left( 2 \varepsilon , \frac{2}{2d-3}\right) . \end{aligned}$$

Then, we have

$$\begin{aligned}&\Vert R_{M} Q_{\lambda } P_{N} u \Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \lesssim \lambda ^{- \frac{1}{q_1}} M^{-\frac{d-2}{2 d-3} + 2 \varepsilon } \Vert R_{M} Q_{\lambda } P_{N} u\Vert _{U^{q_1}_S},\end{aligned}$$
(4.5)
$$\begin{aligned}&\Vert R_{M} Q_{\lambda } P_{N} u\Vert _{L_t^{q_2} L_{x'}^{r_2} L_{x_1}^2} \lesssim \lambda ^{- \frac{1}{q_2}} \Vert R_{M} Q_{\lambda } P_{N} u\Vert _{U^{q_2}_S}, \end{aligned}$$
(4.6)
$$\begin{aligned}&\Vert R_{M} Q_{\lambda } P_{N} u\Vert _{L_t^{q_3} L_{x'}^{r_3} L_{x_1}^2} \lesssim \lambda ^{- \frac{1}{q_3}} M^{\frac{(d-1)(2 d-7)}{2 (2 d-3)} - 4 \varepsilon }\Vert R_{M} Q_{\lambda } P_{N} u\Vert _{U^{q_3}_S}. \end{aligned}$$
(4.7)

By symmetry of (4.4), we may assume \(N_3 \lesssim N_1 \sim N_2\), \(\lambda _2 \lesssim \lambda _1\), and then, it is enough to consider the following three cases:

  1. (1)

    \(M_1 \sim N_1\), \(M_2 \sim N_2\). (2) \(M_1 \sim N_1\), \(M_2 \ll N_1\), (3) \(M_1 \ll N_1\), \(M_2 \ll N_1\).

    1. (1)

      First, we assume \(M_1 \sim N_1\), \(M_2 \sim N_2\). By using (4.5) and (4.7), we obtain

      $$\begin{aligned}&\Bigl | \iint u_1 u_{2} u_{3} dx dt \Bigr |\\&\quad \lesssim \lambda _{\min }^{\frac{1}{2}} \Vert u_1 \Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \Vert u_{2} \Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \Vert u_{3} \Vert _{L_t^{q_3} L_{x'}^{r_3} L_{x_1}^2} \\&\quad \lesssim \lambda _{\min }^{\frac{1}{2}} \lambda _1^{-\frac{1}{q_1}} \lambda _2^{-\frac{1}{q_1}} \lambda _3^{-2 \varepsilon } M_3^{\frac{(d-1)(2 d-7)}{2 (2 d-3)} - 4 \varepsilon } N_1^{-\frac{2(d-2)}{2d-3}+4 \varepsilon } \prod _{i=1,2,3} \Vert u_i \Vert _{V^2_S}\\&\quad \lesssim \lambda _{\min }^{\varepsilon } \lambda _{\max }^{-1}M_3^{s_c+\varepsilon }N_1^{-2 \varepsilon } \prod _{i=1,2,3} \Vert u_i \Vert _{V^2_S}, \end{aligned}$$

      which completes (4.4).

  2. (2)

    In the case \(M_1 \sim N_1\), \(M_2 \ll N_1\), it is observed that \(\lambda _2 \sim M_3 \sim N_1\). Then, without loss of generality, we may assume \(\lambda _3 \lesssim \lambda _1 \sim N_1\). In the case \(\lambda _3 \ll \lambda _1\), for all \(\xi \in {\text {supp}}_{\xi } \widehat{u}_{1}\), \(\eta \in {\text {supp}}_{\xi } \widehat{u}_{2}\) such that \(\xi +\eta \in {\text {supp}}_{\xi } \widehat{u}_3\), we observe

    $$\begin{aligned}&|\tau _1- \varphi (\xi )|+|\tau _2 - \varphi (\eta )|+ |\tau _1+\tau _2 - \varphi (\xi +\eta )|\\&\quad > rsim {} |\varphi (\xi +\eta ) - \varphi (\xi ) - \varphi (\eta )| \\&\quad > rsim {} \bigl | \xi _1|\xi '|^2 \bigr | - |\xi _1+\eta _1| \, \bigl | |\xi + \eta |^2 + \xi _1^2 - \xi _1 \eta _1 + \eta _1^2 \bigr | - \bigl |\eta _1 | \eta '|^2 \bigr | > rsim N_1^3. \end{aligned}$$

    Thus, we can assume that at least one of \(u_1\), \(u_2\), \(u_3\) satisfies \({\text {supp}}\widehat{u}_i \subset \{ (\tau ,\xi )\, |\, |\tau -\varphi (\xi )| > rsim N_1^3\}\). We easily see that this condition verifies the claim by utilizing Theorem 2.1 and (3.2). For example, if \({\text {supp}}\widehat{u}_1 \subset \{ (\tau ,\xi )\, |\, |\tau -\varphi (\xi )| > rsim N_1^3\}\), using Bernstein’s inequality and Theorem 4.1 we obtain

    $$\begin{aligned} \Bigl | \iint u_1 u_{2} u_{3} dx dt \Bigr |&\lesssim \Vert u_1 \Vert _{L_t^2 L_x^2} \Vert u_2 \Vert _{L_t^4 L_{x'}^{2(d-1)} L_{x_1}^2} \Vert u_3\Vert _{L_t^4 L_{x'}^{\frac{2(d-1)}{d-2}} L_{x_1}^{\infty }}\\&\lesssim N_1^{-\frac{3}{2}} \Vert u_1 \Vert _{V_S^2} M_2^{\frac{d-3}{2}} \Vert u_2 \Vert _{L_t^4 L_{x'}^{\frac{2(d-1)}{d-2}} L_{x_1}^2} \lambda _3^{\frac{1}{2}} \Vert u_3 \Vert _{L_t^4 L_{x'}^{\frac{2(d-1)}{d-2}} L_{x_1}^2}\\&\lesssim \lambda _3^{\frac{1}{4}} M_2^{s_c + \frac{1}{2}} N_1^{- \frac{7}{4}} \prod _{i=1,2,3}\Vert u_i \Vert _{V_S^2}. \end{aligned}$$

    Next we consider the case \(\lambda _1 \sim \lambda _2 \sim \lambda _3 \sim N_1\). Since \(M_2 \ll M_1 \sim \lambda _1\), we may assume that there exists \(\ell \in \{2, \ldots , d\}\) such that \(|\partial _{\ell } \varphi (\xi )-\partial _{\ell } \varphi (\eta )| > rsim N_1^2\) for \(\xi \in {\text {supp}}_{\xi } \widehat{u}_1\), \(\eta \in {\text {supp}}_{\xi } \widehat{u}_2\). Then, from (4.2) we get

    $$\begin{aligned} \Vert u_1 u_2 \Vert _{L_t^2 L_x^2} \lesssim M_2^{\frac{d-2}{2}} N_1^{- \frac{1}{2}} \Vert u_1\Vert _{U^2_S} \Vert u_2\Vert _{U_S^2}. \end{aligned}$$
    (4.8)

    On the other hand, for \((\frac{1}{\alpha }, \frac{1}{\beta }) = (\frac{1}{q_1}+\frac{1}{q_2}, \frac{1}{r_1} + \frac{1}{r_2})\), we have

    $$\begin{aligned} \Vert u_1 u_2 \Vert _{L_t^{\alpha } L_{x'}^{\beta }L_{x_1}^2}&\lesssim \lambda _1^{\frac{1}{2}} \Vert u_1\Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \Vert u_2\Vert _{L_t^{q_2} L_{x'}^{r_2} L_{x_1}^2} \end{aligned}$$
    (4.9)
    $$\begin{aligned}&\lesssim \lambda _1^{\frac{1}{2} -\frac{1}{q_1}-\frac{1}{q_2}} N_1^{-\frac{d-2}{2d-3}+2 \varepsilon } \Vert u_1\Vert _{U_S^{q_1}} \Vert u_2\Vert _{U^{q_2}_S}\end{aligned}$$
    (4.10)
    $$\begin{aligned}&\sim N_1^{-\frac{1}{q_2}-\frac{d-2}{2d-3}+3 \varepsilon } \Vert u_1\Vert _{U_S^{q_1}} \Vert u_2\Vert _{U^{q_2}_S}. \end{aligned}$$
    (4.11)

    We notice that \( \alpha , \beta \ge 1\) and \(q_1,q_2>2\) if \( \varepsilon >0\) is chosen sufficiently small. Let \(\theta = \frac{2(d-1)+ 4 (2 d-3)\varepsilon }{(d-1)(2 d -5) - 4 (2 d-3)\varepsilon }\). Then, since

    $$\begin{aligned} \frac{\theta }{\alpha } + \frac{1- \theta }{2} = \frac{1}{q_1'}, \quad \frac{\theta }{\beta } + \frac{1- \theta }{2} = \frac{1}{r_1'}, \end{aligned}$$

    by interpolating the above two estimates (with a similar argument as in the proof of Proposition 3.4), we have

    $$\begin{aligned} \Vert u_1 u_2 \Vert _{L_t^{q_1'} L_{x'}^{r_1'}L_{x_1}^2} \lesssim M_2^{\frac{d-2}{2}(1-\theta )} N_1^{-\frac{1}{2} - \frac{\theta }{q_2} + \frac{\theta }{2 (2 d-3)}+ 3 \varepsilon \theta }\Vert u_1\Vert _{V^2_S} \Vert u_2\Vert _{V_S^2}. \end{aligned}$$
    (4.12)

    This and (4.5) yield

    $$\begin{aligned} \Bigl | \iint u_1 u_{2} u_{3} dx dt \Bigr |&\lesssim \Vert u_1 u_2 \Vert _{L_t^{q_1'} L_{x'}^{r_1'}L_{x_1}^2} \Vert u_{3} \Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \\&\lesssim M_2^{\frac{d-2}{2}(1-\theta )} N_1^{-\frac{3 d-5}{2 d-3}- \frac{\theta }{q_2} + \frac{\theta }{2 (2 d-3)}+ 3 \varepsilon (1+\theta )}\prod _{i=1,2,3}\Vert u_i\Vert _{V^2_S}\\&\lesssim M_2^{s_c+\varepsilon }N_1^{-1 - \varepsilon } \prod _{i=1,2,3} \Vert u_i \Vert _{V^2_S}. \end{aligned}$$
  3. (3)

    We deal with the last case \(M_1 \ll N_1\), \(M_2 \ll N_1\). By symmetry, we assume \(M_2 \le M_1\). Assume first that \(M_1 > rsim (\lambda _3 N_1)^{\frac{1}{2}}\) which implies \(\lambda _3 \ll \lambda _1 \sim \lambda _2 \sim N_1\). Thus, we observe that \(|\partial _1 \varphi (\xi )-\partial _1 \varphi (\eta )| > rsim N_1^2\) for \(\xi \in {\text {supp}}_{\xi } \widehat{u}_2\), \(\eta \in {\text {supp}}_{\xi } \widehat{u}_3\). (4.3) implies

    $$\begin{aligned} \Vert u_2 u_3 \Vert _{L_t^2 L_x^2} \lesssim M_{\min }^{\frac{d-1}{2}} N_1^{-1} \Vert u_2\Vert _{U^2_S} \Vert u_3\Vert _{U_S^2}. \end{aligned}$$

    While, similarly to the above observation, we get

    $$\begin{aligned} \Vert u_2 u_3 \Vert _{L_t^{\alpha } L_{x'}^{\beta }L_{x_1}^2} \lesssim \lambda _2^{-\frac{1}{q_1}} \lambda _3^{\frac{1}{2}-\frac{1}{q_2}} M_{\min }^{-\frac{d-2}{2d-3}+2 \varepsilon } \Vert u_2\Vert _{U_S^{q_1}} \Vert u_3\Vert _{U^{q_2}_S}. \end{aligned}$$

    Interpolating the above two, we get

    $$\begin{aligned} \Vert u_2 u_3 \Vert _{L_t^{q_1'} L_{x'}^{r_1'}L_{x_1}^2} \lesssim \lambda _2^{-\frac{\theta }{q_1}} \lambda _3^{\frac{\theta }{2}-\frac{\theta }{q_2}} M_{\min }^{\frac{d-1}{2}(1-\theta )-\frac{d-2}{2 d-3} \theta + 2 \varepsilon \theta } N_1^{-1+\theta } \Vert u_1\Vert _{V^2_S} \Vert u_2\Vert _{V_S^2}. \end{aligned}$$

    Consequently, it follows from \(M_1 > rsim \max \{M_{\min },(\lambda _3 N_1)^{\frac{1}{2}}\}\) that

    $$\begin{aligned}&\Bigl | \iint u_1 u_{2} u_{3} dx dt \Bigr |\\&\quad \lesssim \Vert u_1 \Vert _{L_t^{q_1} L_{x'}^{r_1} L_{x_1}^2} \Vert u_2 u_3 \Vert _{L_t^{q_1'} L_{x'}^{r_1'}L_{x_1}^2}\\&\quad \lesssim \lambda _1^{-\frac{1+\theta }{q_1}} \lambda _3^{\frac{\theta }{2}-\frac{\theta }{q_2}} M_1^{-\frac{d-2}{2d-3}+2 \varepsilon } M_{\min }^{\frac{d-1}{2}(1-\theta )-\frac{d-2}{2 d-3} \theta + 2 \varepsilon \theta } N_1^{-1+\theta } \prod _{i=1,2,3}\Vert u_i\Vert _{V^2_S}\\&\quad \lesssim \lambda _1^{-1}\lambda _3^{\varepsilon } M_{\min }^{s_c+\varepsilon }N_1^{-2 \varepsilon } \prod _{i=1,2,3} \Vert u_i \Vert _{V^2_S}. \end{aligned}$$

    In the case \(M_1 \ll (\lambda _3 N_1)^{\frac{1}{2}}\), we easily observe that at least one of \(u_1\), \(u_2\), \(u_3\) satisfies \({\text {supp}}\widehat{u}_i \subset \{ (\tau ,\xi ) \, |\, |\tau -\varphi (\xi )| > rsim \lambda _3 N_1^2\}\). Indeed, \(M_2 \lesssim M_1 \ll (\lambda _3 N_1)^{\frac{1}{2}}\) yields

    $$\begin{aligned}&|\tau _1- \varphi (\xi )|+|\tau _2 - \varphi (\eta )|+ |\tau _1+\tau _2 - \varphi (\xi +\eta )|\\&\quad > rsim {} |\varphi (\xi +\eta ) - \varphi (\xi ) - \varphi (\eta )| \\&\quad \ge {} x 3| \xi _1 \eta _1\left( \xi _1+\eta _1\right) | - 10\left( |\xi _1|+|\eta _1|\right) \left( |\xi '|^2 + |\eta '|^2\right) > rsim \lambda _3 N_1^2, \end{aligned}$$

    for all \(\xi \in {\text {supp}}_{\xi } \widehat{u}_1\), \(\eta \in {\text {supp}}_{\xi } \widehat{u}_2\) which satisfy \(\xi +\eta \in {\text {supp}}_{\xi } \widehat{u}_3\). In the case \({\text {supp}}\widehat{u}_1 \subset \{ (\tau ,\xi )\, |\, |\tau -\varphi (\xi )| > rsim \lambda _3 N_1^2\}\) and \(M_2 \lesssim M_3\), since \(M_2 \lesssim M_1 \ll (\lambda _3 N_1)^{\frac{1}{2}}\), it follows from the Strichartz estimates (3.2) and Bernstein’s inequality that

    $$\begin{aligned} \Bigl | \iint u_1 u_{2} u_{3} dx dt \Bigr |&\lesssim \Vert u_1 \Vert _{L_t^2 L_x^2} \Vert u_2 \Vert _{L_t^3 L_{x'}^{3(d-1)} L_{x_1}^2} \Vert u_3\Vert _{L_t^{6} L_{x'}^{\frac{6(d-1)}{3 d-5}} L_{x_1}^{\infty }}\\&\lesssim \lambda _3^{-\frac{1}{2}} N_1^{-1} \Vert u_1 \Vert _{V_S^2} M_{2}^{\frac{d-3}{2}} \Vert u_2 \Vert _{L_t^3 L_{x'}^{\frac{6 (d-1)}{3 d -7}} L_{x_1}^2} \lambda _3^{\frac{1}{2}} \Vert u_3\Vert _{L_t^{6} L_{x'}^{\frac{6(d-1)}{3 d-5}} L_{x_1}^{2}}\\&\lesssim \lambda _3^{-\frac{1}{6}} M_1^{\frac{1}{3} + 2 \varepsilon }M_2^{s_c + \frac{1}{6}-2 \varepsilon } N_1^{-\frac{4}{3}} \prod _{i=1,2,3} \Vert u_i \Vert _{V_S^2}\\&\lesssim \lambda _3^{\varepsilon } M_2^{s_c + \frac{1}{6}-2 \varepsilon } N_1^{-\frac{7}{6}+ \varepsilon } \prod _{i=1,2,3} \Vert u_i \Vert _{V_S^2}. \end{aligned}$$

    The other cases are treated similarly. \(\square \)

As above, Theorem 1.3 follows by the standard argument.

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Herr, S., Kinoshita, S. The Zakharov–Kuznetsov equation in high dimensions: small initial data of critical regularity. J. Evol. Equ. (2021). https://doi.org/10.1007/s00028-021-00671-9

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Keywords

  • Global well-posedness
  • scattering
  • Zakharov–Kuznetsov equation

Mathematics Subject Classification

  • 35Q53
  • 35A01