GLOBAL WELL-POSEDNESS FOR CAUCHY PROBLEMS OF ZAKHAROV-KUZNETSOV EQUATIONS ON CYLINDRICAL SPACES

. We study the global well-posedness of the Zakharov-Kuznetsov equation on cylindrical spaces. Our goal is to establish the existence of global-in-time solutions below the energy class. To prove the results, we adapt the I-method to extend the local solutions globally in time. The main tool in our argument is multilinear estimates in the content of Bourgain’s spaces. Using modiﬁed energies induced by the I-method, we obtain polynomial bounds on the H s growth of global solutions.


Introduction
In this article, we study the Cauchy problem for the Zakharov-Kuznetsov equation on a cylinder, ∂ t u + ∂ x ∆u + u∂ x u = 0, (x, y) ∈ R × T, t ∈ R u(x, y, 0) = u 0 (x, y), (x, y) ∈ R × T, (1.1) where u = u(x, y, t) is a real-valued function, T = R/(2πZ), and ∆ = ∂ 2 x + ∂ 2 y is the Laplacian.This equation was introduced by Zakharov and Kuznetsov [26], as a model for the propagation of ionic-acoustic waves in magnetized plasma (see also [14] for the derivation of the equation).This equation is one of the two-dimensional extensions of the Korteweg-de Vries (KdV) equation The KdV equation is one of the famous nonlinear dispersive equation, which describes the behavior of shallow water waves.Another two dimensional generalizations of the KdV equation is the Kadomtsev-Petviashvili (KP) equations where the KP-I equation corresponds to minus sign, while the KP-II equation to plus sign.
We aim to prove that the initial value problem of the Zakharov-Kuznetsov equation in H s (R × T) is globally well-posed for some s < 1.This proof is based on bilinear estimates developed in [19] and I-method [5].
The main result of the paper is the following theorem.
Theorem 1.1.The initial value problem of (1.1) is globally well-posed in H s (R×T) for s > 29/31.In other words, for any u 0 ∈ H s (R × T), for all T > 0, there exists a unique solution u of (1.1) such that Moreover, for all 0 < T < T , there exists a neighborhood U of u 0 in H s (R × T) such that the data-to-solution map is smooth, where v(t) is a unique solution of (1.1) to the initial data v 0 .Here function space X s,b T is defined in Section 2. Remark 1.2.By time reversibility, we need to consider only the existence for positive time.
This article is organized as follows.In Section 2, we recall some harmonic analysis tools including Littlewood-Paley decompositions of functions.We also reviews some preliminary results for linear estimates in the Bourgain spaces X s,b [2,3,11].In Section 3, we follow the I-method scheme [5].We give several lemmas of bilinear estimates in conjunction with rescaling argument in [6] and give the local well-posedness results for rescaled data.In Section 4, we define modified energy functional E[Iu] in term of the H s -norm of the solution for s < 1. Section 5 is devoted to the proof of main theorem.One of the key steps in the construction of global solutions is to control the increment of the modified energy.We estimate the growth of the modified energy and provide a priori estimates for the solutions.

Notation and function spaces
In this section, we will introduce some function spaces that will be used throughout the paper.We denote the absolute value of (ξ, q) as |(ξ, q)| 2 = 3ξ 2 + q 2 .For positive real quantities a and b, the notation a b means that there is a constant c such that a ≤ cb.When a b and b a, we write a ∼ b.Moreover, b+ means that there exists δ > 0 such that b + δ.
Denote T λ = λT = R/(2πλZ) for λ > 0. Let f (x, y) be a function defined on R × T λ .The Fourier transform of f with respect to the variables (x, y) is denote by where (ξ, q) ∈ R × Z/λ.Let inverse Fourier inverse transform be denote by For simplicity, we abbreviate F λ xy and F −1,λ ξq by F λ and F −1,λ , respectively, when no confusion is likely.
Let u(x, y, t) be a function of R × T λ × R, and let u λ (ξ, q, τ ) be Fourier transform of u(x, y, t) in a similar manner as well 2πλ 0 e −i(tτ +ixξ+yq) u(x, y, t) dy dx dt.
We denote σ(ξ, q where (ξ, q, τ ) ∈ R×Z/λ×R.Here, we prepare notation of interval as I N = supp φ N for a dyadic number N .We define the Littlewood-Paley decomposition as where x = 1 + |x|.We also use L 2 λ = H 0 λ .We use the restriction operator R K with respect to the x variable as for dyadic number K, where F x is the Fourier transform with respect to the x variable.
We note that the following properties hold [6]: Next, we describe the Bourgain space via the Fourier transform.Following [11], we introduce a class of function spaces related to Bourgain space X s,b λ .Define the Bourgain space X s,b λ as follows.Definition 2.1.For s, b ∈ R and T > 0, When a norm of u introduced in (2.3) is bounded, we denote u ∈ X s,b λ .We also use the same notation L 2 λ = X 0,0 λ for functions on R × T λ × R as defined before for one on R × T λ , if there is no confusion.The space X s,b λ,T will be used for proof of the local well-posedness result, that is under restriction of time on [0, T ].
If λ = 1, we denote T , respectively.Remark 2.2.The space X s,b λ will be characterized by the formula , where e −t∂x∆ is the operator associated linear Zakharov-Kuznetsov equation on R × T λ , described by We summarize some inequalities which were stated in [8,11,19,21].
λ .The polynomial σ(ξ, q) which appeared in (2.3) plays an important role in characterizing of solution.Now, we define the resonance function which plays an important role in the control of the frequency instability range between nonlinear interactions.

Rescaled solutions
Throughout this paper, we assume that λ > 1. Recall T λ = λT = R/(2πλZ).By rescaling we consider the Cauchy problem for the Zakharov-Kuznetsov equation on R × T λ , If we construct the solution u λ (t) of (3.2) on the time interval [0, λ 3 T ], we have the solution u(t) of (1.1) on [0, T ].Let ζ be a combination of spatial variables, ζ = (ξ, q) ∈ R × Z/λ.Define the Fourier multiplier operator I, which was originally introduced in [5] to consider global well-posedness for KdV equation.For N ∈ 2 N , define the operator I by where m is a smooth, radially symmetric, non-increasing function satisfying On the low frequency part |ζ| ≤ N , the operator I is the identity operator, while on the high frequency, I is regarded as the integral operator.Remark that I maps H s λ functions to H 1 λ one.In this section, we prove the following local well-posedness results on  Lemma 3.7]).Denote a set Λ ⊂ R × Z/λ.Let the projection on the q be axis contained in a set I ⊂ Z/λ.Assume that there is a positive constant C such that for any fixed q 0 ∈ I ∩ Z/λ and |Λ ∩ {(ξ, q 0 ) | q 0 ∈ Z/λ}| ≤ C.Then, |Λ| ≤ λC(|I| + 1).
Using this lemma and the mean value theorem, we have the estimates for constructing bilinear estimates.Lemma 3.3 ([19,Lemma 3.8]).Let I and J be two intervals on the real line and f : J → R be a smooth function.Then Lemma 3.4.Let a = 0, b, c be real numbers and I an interval on the real line.Then Proof.Following [19, Lemma 3.9], we only consider the case of a > 0 and b = c = 0. Put I = [as 2 , at 2 ] for s, t ∈ R with s 2 < t 2 and s, t ≥ 0. From the shape of parabola curve and distribution of Z/λ points, we can say Using these lemmas, we obtain the following bilinear estimates.Lemma 3.5.For u, v ∈ L 2 λ , we have (3.4) where Proof.Estimate (3.6) follows from the interpolation argument with (3.3) and (3.5), so that we prove (3.3), (3.5) and (3.4) in this order.
Using the Plancherel's identity (2.1), convolution structure (2.2) and the Cauchy-Schwarz inequality, we obtain where By the definition of A ξ,q,τ and Lemma 3.
From the Cauchy-Schwarz inequality and the Plancherel's identity as before, it holds that where Using the triangle inequality, we have For the bound of B K ξ,q,τ , we calculate the second derivative of H as for all q 1 ∈ Z/λ.Finally, and we obtain (3.4) by Next we prove bilinear estimates in Lemma 3.6.For s > 9/10, we have Proof.We may assume the functions u λ and v λ are nonnegative, since by the definition of ) and v(ζ, τ ), respectively, and duality argument, one has to show that for w ∈ L 2 λ whose Fourier transform is nonnegative.Here Using dyadic decomposition, we rewrite J as the following where We decompose again J to five parts in frequencies: Estimate of J LL→L .In this case, we have We split this case two cases, when First, we consider the contribution of the case when In this case, by (3.5) and the Cauchy-Schwarz inequality, we have that the contribution of the case to J LL→L is bounded by Second, we consider the contribution of the case when Note that P N = P N P N for P N = P N/2 + P N + P 2N .In this case, the contribution of the case to J LL→L is bounded by Applying the Cauchy-Schwarz inequality again with (3.5), we have that the contribution of this case to J L0,L1,L2 N0,N1,N2 is bounded by (3.12) In a similar way to above, we have that the contribution of this case to J L0,L1,L2 N0,N1,N2 is bounded by Therefore, by (3.12) and (3.13) in conjunction with previous estimate, we have Estimate of J HL→H .The proof is same as for J LH→H , because of the symmetry.
Estimate of J HH→L .By symmetry, we assume N 0 N 1 ≤ N 2 .We separate this case into two cases that Applying the L 2 λ norm of functions (P N1 Q L1 u)(P N0 Q L0 w) and P N2 Q L2 v, we have that then the contribution of this case to J L0,L1,L2 N0,N1,N2 is bounded by where We apply (3.6) to this and have that the contribution of this case to J L0,L1,L2 N0,N1,N2 is bounded by Similarly, the contribution of this case to J L0,L1,L2 N0,N1,N2 is bounded by 1 Summing with respect to N 0 , N 1 , N 2 , L 0 , L 1 , L 2 , we obtain . Now, we separate this case into five subcases: where We apply the Cauchy-Schwarz inequality, and we have that the contribution of this case to J L0,L1,L2 N0,N1,N2 (k) is bounded by 2 For |ξ| ∼ 2 −k , we obtain by (2.4) Using (3.4), we obtain Summing with respect to dyadic numbers N 0 , N 1 , N 2 , L 0 , L 1 , L 2 and k ∈ N, we have the desired bound for the contribution of this case to J HH→H .
First we consider the case when |∂H/∂ξ 1 | ξ 2 .We shall use the dyadic decomposition |ξ| ∼ K.By the Cauchy-Schwarz inequality such as Case (i) above, we have that the contribution of this case to J L0,L1,L2 N0,N1,N2 is bounded by Recall the proof of Lemma 3.5, we have where From the triangle inequality, where Hence, we obtain Then, Combining the Cauchy-Schwarz inequality and bilinear estimate, we obtain that the contribution of this case to J L0,L1,L2 N0,N1,N2 is bounded by (3.17) Again summing with respect to dyadic numbers N 0 , N 1 , N 2 , L 0 , L 1 , L 2 , we have the desired bound for the contribution of this case to J HH→H .
Proof of Proposition 3.1.Define We show that the map Ψ defines a contraction in , In fact, from Lemmas 2.3, 2.4 and 2.5 and 3.6, it follows that for u λ ∈ Y δ,λ and choosing small δ > 0 which will depend on Iu λ 0 H s λ .Similarly, , for u λ , v λ ∈ Y δ,λ and small δ > 0. Then the contraction mapping theorem tells us that there is a unique solution u λ = Ψ(u λ ) ∈ Y δ,λ to the Cauchy problem (3.2).The persistence property u λ ∈ C([0, δ] : H s λ ) and the uniqueness in whole space follow in a similar way to [11, Proof of Theorem 1.5], by using a variant of (3.25), therefore we omit them.

Modified energy
The conserved quantities associated with (3.2) are From F λ u λ 0 (ξ, q) = Fu 0 (λξ, λq), it is easy to see that for λ 1.Here the choice of the large parameter N will be made latter, but λ > 1 is chosen by in which Moreover, we have Remark 4.1.By Gagliardo-Nirenberg inequality, the conservation law of L 2 λ -norm and (4.1), the solution u λ (t) satisfies where constants C 1 and C 2 are independent of λ.
Let us introduce the modified energy for proving global well-posedness.Using the Fundamental Theorem of Calculus, we obtain dt dt for δ > 0. We continue calculation of the integrand in the right-hand side.Then where we use the equation in (3.2).Taking the integral on [0, δ], we have We will estimate two terms on the right hand of (4.7) with Iu λ Lemma 4.2.For s > 9/10, we have Remark 4.3.Comparing to the estimate in Lemma 3.6, we have the small factor at the front of the right-hand, which corresponding to properties such as dispersion or the smoothing effects.
Proof.We recast the proof of Lemma 3.6 in the form of formula (4.8).Here we used the same symbol over as in the proof of Lemma 3.6.By Plancherel identity, it suffices to show that for u, v, w ∈ L 2 λ whose Fourier transform are nonnegative.In (4.9), we have Note the trivial bound Using dyadic decomposition in a similar way to the proof of Lemma 3.6, we rewrite J as the following where We repeat the same procedure as one of Lemma 3.6.Decompose again J to five parts in frequencies: We estimate each of them by case analysis.Estimate of J LH→H .We split the case into two cases, when N 1 N and when N 1 N .When N 1 N , we use the bound (4.10).On the other hand, when N 1 N , the mean value theorem gives By making use of the estimates in (3.12) and (3.13), we easily have the desired bounds for the contribution of this case to J LH→H .Indeed, when N 1 N , the computation in (3.12) with (4.10) leads the bound N 1/10− .By the computation in (3.13) with (4.12) we estimate this contribution by Then the desired estimate follows.The following lemma is not difficult to prove using Lemma 4.5 as in [5,6].

Proof of main theorem
In this section, we will give the proof of Theorem 1.1, by the local well-posedness results of Proposition 3.1 in conjunction with Lemmas 4.4 and 4.6.The proof follows the strategy described in [5].We prove the global well-posedness by the local well-posedness theory for solution Iu obtained in Proposition 3.1 and by the upper bound of increment of modified energy from (4.7).This completes the proof.
By performing the same calculation as [21, Proof of Proposition 3.1] and [11, Proof of Theorem 2.1], we use the Cauchy-Schwarz inequality to have that the contribution of this case to J HH→H is bounded by sup (ζ,τ )∈R×Z/λ×R

X 1
with the transition of energy in (4.7).