The Zakharov-Kuznetsov equation in high dimensions: Small initial data of critical regularity

The Zakharov-Kuznetsov equation in spatial dimension $d\geq 5$ 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 $t \to \pm \infty$. The proof is based on i) novel endpoint non-isotropic Strichartz estimates which are derived from the $(d-1)$-dimensional Schr\"odinger 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 where d ≥ 2, u = u(t, x), (t, x) = (t, x 1 , . . . , x d ) ∈ R × R d , u is real-valued, and ∆ denotes the Laplacian with respect to x. In the following, H s (R d ) denotes the standard L 2 -based inhomogeneous Sobolev space and B s 2,1 (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 = 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, Faminskiȋ [3] established global well-posedness in the energy space H 1 (R 2 ). Later, Linares and Pastor [9] proved local well-posedness in H s (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 (R 3 ) for s > 9/8. Ribaud and Vento [12] proved local well-posedness for s > 1 and in B 1,1 2 (R 3 ). The global well-poseness in H s (R 3 ) for s > 1 was obtained by Molinet and Pilod in [11]. Recently, in dimensions d ≥ 3, local well-posedness in H s (R d ) in the full subcritical range s > s c was proved in [5], which implies global well-posedness in H 1 (R d ) if 3 ≤ d ≤ 5 and in L 2 (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 Here, e tS denotes the unitary group generated by the skew-adjoint linear operator S = −∂ x1 ∆, so that e tS u ± solves the linear homogeneous equation.
Our first main result covers small data in dimension d = 5. Note that in d = 6 this result includes the energy spaceḢ 1 (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 ≥ 4. Theorem 1.3. For d ≥ 4, the Cauchy problem (1.1) is globally well-posed for small data in H sc rad (R d ), and solutions scatter as t → ±∞. The same result holds for radial data inḢ sc rad (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 Subsection 1.1 we introduce notation. In Section 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 Section 3 we combine this with the bilinear transversal estimate and an interpolation argument, which leads to a proof of Theorem 1.2. Finally, in Section 4 we discuss an variation of these ideas under the additional radiality assumption and a proof of Theorem 1.3.
In addition, we define As usual, the Sobolev space H s (R d ) is defined as the completion of the Schwartz functions with respect to the norm Similarly, for s ≥ 0, the homogeneous Sobolev spaceḢ s (R d ) is defined as the completion of the Schwartz functions with respect to the norm and the homogeneous Besov spaceḂ s 2,1 (R d ) as the completion of the Schwartz functions with respect to the norm The radial subspaces H s rad (R d ) andḢ s rad (R d ) are defined by the requirement that Finally, the Duhamel operator is denoted by 2. Strichartz estimates and the proof of Theorem 1.1 which is (2.1). Similarly, by (2.4), Now, we can complete the proof of Theorem 1.1. Recall that d = 5 implies s c = 1/2.
, and the corresponding Banach spaces.
By the standard argument involving the contraction mapping principle, it suffices to prove the following: . For any N ∈ 2 Z , Theorem 2.1 gives

THE ZK EQUATION IN HIGH D: SMALL INITIAL DATA OF CRITICAL REGULARITY 5
Further, we obtain from the Kato-Ponce inequality and the Bernstein inequality. This can be summed up both in the homogeneous and in the inhomogeneous version.
This argument also implies the scattering claim, since it implies that the Duhamel integral converges to a free solutions as t → ±∞. We omit the details of this standard argument.
3. Transversal estimates and the proof of Theorem 1.2 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 ≤ p < ∞, we call a function a : Now, U p is defined as the space of all u : R → L 2 (R d ), such that there exists an atomic decomposition u = ∞ j=1 c j a j , where (c j ) ∈ ℓ 1 (N) and the a j 's are p−atoms. Then, u U p = inf ∞ j=1 |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 : where the supremum is taken over all increasing sequences (t j ). Now, we define the atomic space U p S = e · S U p with norm u U p S = e − · S u U p , and V p S = e · S V p with norm u V p S = e − · S u V p . There is the embedding V p S ⊂ U q S if p < q, see [8,Lemma 6.4]. Due to the atomic structure of U q S and the Strichartz estimate (2.1), we have for (d−1)−admissible pairs, and u U q S may be replaced by u V 2 S for non-endpoint pairs, i.e. when q > 2.
Again, the proof of Theorem 1.2 is a straight-forward application of the contraction mapping principle. The scattering claim follows from the well-known fact that functions in V 2 have limits at ±∞.

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 ′ .
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].
max , for all ξ ∈ supp f N1,λ1,M1 , η ∈ supp g N2,λ2,M2 . Then it holds that 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 rad andẎ s rad be the subspaces of Y s andẎ 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 Proof. For i = 1, 2, 3, we use u i := R Mi Q λi P Ni u. As in the proof of Proposition 3.4, it suffices to show λi,Mi Here and in the sequel, all functions are implicitely assumed to satisfy the radiality hypothesis. Let 1 Then we have By symmetry of (4.4) we may assume N 3 N 1 ∼ N 2 , λ 2 λ 1 , and then it is enough to consider the following three cases: (1) First, we assume M 1 ∼ N 1 , M 2 ∼ N 2 . By using (4.5) and (4.7) we obtain which completes (4.4).
As above, Theorem 1.3 follows by the standard argument.