On the periodic Zakharov-Kuznetsov equation

We consider the Cauchy problem associated with the Zakharov-Kuznetsov equation, posed on $\mathbb{T}^2$. We prove the local well-posedness for given data in $H^s(\mathbb{T}^2)$ whenever $s>5/3$. More importantly, we prove that this equation is of quasi-linear type for initial data in any Sobolev space on the torus, in sharp contrast with its semi-linear character in the $\mathbb{R}^2$ and $\mathbb{R}\times \mathbb{T}$ settings.

The model presented in (1.1) is a bi-dimensional generalization of the Korteweg-de Vries (KdV) equation, and was introduced in [29] to describe the propagation of nonlinear ion-acoustic waves in magnetized plasma (for its rigorous derivation we refer to [17]). This model is widely known as the Zakharov-Kuznetsov (ZK) equation and is extensively studied in the literature, see for example [4,5,18,19,21,22,23] and the references therein.
If (1.1) is posed on R 2 , it is known that it is locally well-posed in H s (R 2 ), s > 1/2 (see [22,5]). Furthermore, if (1.1) is posed on R × T then it is well-posed in H s (R × T), s ≥ 1 (see [22]). In all these results the dispersion in the x direction plays a crucial role, the solutions are constructed by the Picard iteration and the flow map is smooth on H s .
Our interest here is in studying the local well-posedness issue to the IVP (1.1) for given data in the periodic Sobolev space H s (T 2 ), s ∈ R. We note that for s > 2 one can ignore the dispersive effects and solve (1.1) as a quasi-linear hyperbolic equation leading to the local well-posedness of (1.1) in H s (T 2 ), s > 2. Our first goal is to prove a well-posedness result in spaces of lower regularity.
Theorem 1.1. Let s > 5/3. Then for every w 0 ∈ H s (T 2 ) there exist a time T = T ( w 0 H s ) and a unique solution w ∈ C([0, T ] : H s (T 2 )) to the IVP (1.1) such that w, ∂ x w, ∂ y w ∈ L 1 T L ∞ xy . Moreover, the map that takes the initial data to the solution w 0 → w ∈ C([0, T ] : H s (T 2 )) is continuous.
This local well-posedness result for the IVP associated with the ZK equation (1.1) posed in a purely periodic domain is the first one exploiting in a non trivial way the dispersive effect. The method of proof of Theorem 1.1 is nowadays standard (see e.g. [3,6,9,10,14] ). It uses in a crucial way short time Strichartz estimates for the linear part of the equation, see Lemma 2.1.
Our short time Strichartz estimate is derived by purely local in space considerations in the spirit of [3]. It may be that global in space considerations based on number theory arguments may improve the result of Lemma 2.1. To the best of our knowledge, the works [12,28] are the first papers where short time Strichartz estimates on a compact spatial domain were considered. We also expect that further improvements can be obtained by using the ideas introduced in [7].
Our main purpose in this work is to show that the flow map constructed in Theorem 1.1 is not locally uniformly continuous, which highlights the quasilinear behaviour of the ZK equation (1.1) when considered with periodic boundary conditions. For previous related contributions, we refer to [15,16,25]. Theorem 1.2. Let s > 5/3. There exist two positive constants c and C and two sequences (u m )

2)
and satisfying initially Let us finally mention that since we work in the periodic setting, in the proof of Theorem 1.2 we cannot use the localization in space arguments as in [15,16]. This makes that in order to construct the approximation of the true solution, we exploit a "curvature property" of the resonance function associated with the ZK equation (see also (4.6) and Remark 4.1 below).

PRELIMINARY ESTIMATES
In this section we derive some preliminary estimates that are useful to prove the local wellposedness result of this work. We begin with the preparation to obtain the Strichartz estimate in localized form.
For N ∈ 2 N ∪ {0}, we define a projection operator P N as a Fourier multiplier operator by where m = (m 1 , m 2 ) ∈ Z 2 , and χ I is the indicator function of the interval I. In terms of the projection operator, we can write an equivalence for the Sobolev norms on the torus in the following where we use the notation a ∨ b = max(a, b).

Consider now the linear problem
and let W (t) defined by be the unitary group that describes the solution to (2.2).
In what follows we prove the localized version of the Strichartz estimate satisfied by the unitary group W (t).
Lemma 2.1. Let W (t) be as defined in (2.3) and P N be the operator defined in (2.1). Then for

4)
and Proof. We start by proving (2.4). Since it is straightforward for N = 0, we assume N ≥ 1 in the following. For x = (x, y), using the definition of the group W (t) and the projection operator P N , we have that In order to decouple the frequencies in the time oscillation, we will use the symmetrization argument of [5]. First, to have good localizations in frequency after symmetrizing, we observe Since P N is a bounded operator with norm one, it suffices to prove that (2.4) holds with P N replaced with where φ 4N is a smooth cut-off defined as follows : Indeed, with this definition and the remark above, we have P N = P N P N .
We can now define the operator where χ I stands for the indicator function of the interval I.
To prove that T is bounded from L 2 (T 2 ) to L 2 (I : L ∞ (T 2 )) with norm less than N − 1 3 , the classical "T T * " argument reduces the problem to show that is bounded with norm less than N − 2 3 . This last operator can be written as an integral operator, whose kernel is given by Note that the time localizations imply that |t − t | N −2 . Therefore the whole matter reduces to proving the following pointwise estimate on the kernel for any x ∈ T 2 and N −3 |t| N −2 .
Indeed, with (2.6) at hand, the bound on TT * follows from Young's inequality and the fact that To obtain (2.6), we use Poisson summation formula [27] ∑ m∈Z 2 Using (2.7), the sum in the LHS of (2.6) becomes Here we symmetrize the linear evolution : by using the linear transformation in [5, Section 2.1] we can write the integrals within the sum above as Thus (2.6) can be expressed as First, note that from the time and frequency localizations, we have |ξ 2 t| 1, thus for |X| large enough the phase in the above integral has no stationary point. As a result, we can bound |F N (X)| for |X| > C for some (large) constant C > 0 by successive integrations by parts. Indeed, by writing F N as an abstract oscillatory integralˆR ψ(ξ )e iΦ(ξ ) dξ , using the definition of φ 4N and the localizations in ξ and t we get that for any k ∈ N we have ψ (k) In view of the last bound, to prove (2.8) it remains to treat the contribution of the sum for |n 1 + √ 3n 2 | < M and |n 1 − √ 3n 2 | < M. In this case we have in particular |n| ≤ 2M so that the sum is finite (uniformly in N), thus it is enough to get the bound We first write F N as The first term is From [12] we have that the last integral is bounded (see also Lemma 7.2 in [20] or Lemma 3.6 in [13]), so it remains to estimate the second term. We observe that the phase Φ(ξ ) = Xξ + ξ 3 2 t satisfies |Φ (ξ )| = 3|tξ | N|t| on the support of (1 − φ 4N 2 ). Thus we can conclude from van der Corput lemma (see e.g. [26,Chapter 8]) that where in the last step we have used the lower bound on |t|. This proves (2.9).
The proof of the estimate (2.5) follows from the short time Strichartz estimate using a Littlewood-Paley procedure. In fact, splitting the interval [0, 1] on (1 ∨ N) 2 intervals of size (1 ∨ N) −2 we obtain, using the short time Strichartz estimate (and that P 2 N = P N ) Summing the last estimate over N ≥ 0 and using the equivalence of norms, we conclude that This completes the proof of Lemma 2.1.
Next, we move to prove a version of the Kato-Ponce Commutator estimate in T 2 , which is an extension of the one proved for the T-case in [6].
First, let us define For s ∈ R we also define the Fourier multiplier Lemma 2.2. Let s ≥ 1 and f , g ∈ H ∞ (T 2 ). Then In this section we use the estimates established in the previous section to establish some more estimates and prove the main result regarding local well-posedness. We begin by proving an a priori estimate that plays a fundamental role in our argument. Then for any T ∈ [0, 1] and s ≥ 1, we have Proof. We apply the operator J s to (1.1), multiply the resulting equation by J s w and then integrate by parts, to obtain In the light of estimate (2.10) in Lemma 2.2, we have Using Gronwall's inequality, (3.5) yields which gives the required estimate (3.1).
Following the approach in [9] we obtain the following estimate that will be useful in our argument.
Lemma 3.2. Let F ∈ L 1 ([0, T ] : L 2 (T 2 )). Then any solution of the equation Proof. Let P N be the projection operator defined in (2.1), and fix such a number N ∈ 2 N ∪ {0}.
Proof. Observe that w, ∂ x w and ∂ y w satisfy Lemma 3.2 with F given, respectively, by 1 2 w 2 , 1 2 ∂ x (w 2 ) and 1 2 ∂ y (w 2 ). Hence, for s > 2/3, using (3.7) for w, ∂ x w and ∂ y w with respective F, we obtain (3.14) It follows that, for s > 5/3, On the other hand, From (3.17) we can deduce the required result analogously as in [9], so we omit the details.
Proof of Theorem 1.1. The main tools in the proof are the results from Lemmas 3.1 and 3.3.
Let s > 5/3 and consider an initial datum w 0 ∈ H s (T 2 ). As already mentioned, for s > 2, we can prove the local well-posedness in H s of (1.1) by solving it as a quasi-linear hyperbolic PDE.
So, we may consider w 0 ∈ H s , 5/3 < s ≤ 2. Density of H ∞ in H s allows one to find w ε 0 ∈ H ∞ such that w ε 0 − w 0 H s → 0. Moreover, one has w ε 0 H s ≤ c w 0 H s . For 0 < ε < 1, let w ε be the solution to the IVP (1.1) corresponding to the initial data w ε 0 ∈ H ∞ on [0, τ], τ > 0. One can use Lemma 3.3 to extend w ε on a time interval [0, T ], T = T ( w 0 H s ) > 0 and to show that there exists a constant c T such that Following the idea in [15,16,25], we will construct a family of approximate solutions whose  In particular, note that for any choice of (m, n) ∈ Z 2 , we have R(m, 0, n, 2n) = 0. Consequently, we will exploit this resonant interaction to construct the solutions in Theorem 1.2.
Proof of Theorem 1. Note that all three modes above are solutions to the linear part of (1.1) modulated by a time oscillating factor, and the third one corresponds to the main part of the nonlinear interaction of the first two (see below). The last term is given by It allows to cancel the remaining interactions (due to the requirement of working with non localized real-valued solutions) in order for these approximate solutions to satisfy the equation up to a sufficiently small error. Indeed, we have the estimate To prove (4.5), let us write u i , i = 1, 2, 3 for the first three terms in the definition of u θ ,m (4.3), then the term inside the norm on the left-hand side of (4.5) reads A key ingredient here is that Next, straightforward computations give and Since ϕ(m, −1) = ϕ(m, 1) (as the symbol is even in n), we see that the first and third terms in the nonlinear interaction (4.8) are counterbalanced by the linear evolution (4.7). To cancel the two remaining terms, we finally compute by using the definition of ϕ and R. This proves (4.5).
With this estimate at hand, we can control the difference between u θ ,m and the genuine solution where G is the term in (4.5). Proceeding then as in Lemma 3.1 at the L 2 level, we get From the definition of the approximate solution and with the use of (4.6), we finally get  (a) It is not difficult to see that u θ ,m is still a sufficiently good approximate solution even for lower values of s : one can check that the last bound holds with ε > 0 for any s > 1/2.
However, since there exists no flow map defined on H s (T 2 ) when 1/2 < s ≤ 5/3, we restricted our construction to those s > 5/3 treated by Theorem 1.1.
(b) The construction above is independent of the choice of the periods in x and y, thus one can repeat the argument for any torus T 2 λ = (R/2πλ 1 Z) × (R/2πλ 2 Z) and recover the quasi-linear behaviour likewise. In particular, there is another interesting resonant interaction : R(m, m 1 , √ 3m, − √ 3m 1 ) = 0. To perform the same construction as above, this requires to work on a torus T 2 λ whose periods λ = (λ 1 , λ 2 ) satisfy the condition √ 3 λ 2 λ 1 ∈ Q.
(c) The frequency m 1 = 0 in the aforementioned resonant interaction corresponds to the xmean valueˆT u(t, x, y) dx, which is preserved by the flow : T u(t, x, y) dx =ˆT u 0 (x, y) dx, ∀(t, y) ∈ R × T.
We do not know the existence of a gauge transformation allowing to get rid of the contribution of this frequency. This is in contrast with the periodic KP type equations (see e.g. [2,24]), whereˆT u 0 (x, y) dx must be independent of y to define the anti-derivative, and where the Galilean transform reduces the problem to initial data with this constant being equal to zero.
(d) At last, let us notice that the above construction uses in a crucial way the "curvature"property (4.6), meaning that for a given resonant interaction R(m, m 1 , n, n 1 ) = 0 we need R(m, m + m 1 , n, n + n 1 ) to be sufficiently large.