LOCAL AND GLOBAL WELL-POSEDNESS IN THE ENERGY SPACE FOR THE DISSIPATIVE ZAKHAROV-KUZNETSOV EQUATION IN 3D

. In this paper, we consider the Zakharov-Kuznetsov equation in 3D, with a dissipative term of order 0 < α ≤ 2 in the x direction. We prove that the problem is locally well-posed in H s ( IR 3 ), for s > 1 − α 2 , and by an a priori energy estimate, we prove that the problem is globally well-posed in H 1 ( IR 3 ).

(3) Therefore L 2 and H 1 are two natural spaces to study the well-posedness for the Z-K equation.
In 2D, Faminskii [6] proved the global well-posedness of the Cauchy problem for the (Z-K) equation in H s (IR 2 ), for s ≥ 1, and the result was improved later by Linares and Pastor [17] who proved well-posedness in H s (IR 2 ), for s > 3 4 , where this result was improved also by Molinet and Pilot in [21] and Grunrock and Herr [10] to s > 1 2 . In the 3D case, the well-posedness problem was treated by many authors: Ribaud and vento [24], Linares and Saut [18] and Molinet and Pilot [21], where the last authors was proved that the problem is globally well-posed in H s (IR 3 ), for s > 1. Note that the global well-posedness in the energy space H 1 (IR 3 ) is still an open problem.
In this paper, we consider a dissipative version of the (Z-K) equation of the form where α ∈ (0, 2]. The term D α x represents the dissipation term, where D α x is the Lèvy operator defined through its Fourier transform by F(D α u)(ξ) = |ξ| α Fu(ξ).
In a recent work, Hirayama [11] have studied the (Z-K) burgers equation (i.e equation (4) with α = 2) in 2D, and he proved that the problem is locally and globally well-posed in H s,0 (IR 2 ) with s > −1/2 and up to our knowledge there is no other results for this equation.
The dissipative term will help to establish the global existence in the energy space H 1 (IR 3 ) and the quantities (2) and (3) are not longer conserved, more precisely we have: and We will establish the well-posedness of the problem (4) in H s (IR 3 ), s > 1 − and the paper is organized as follows : -In Section 2, we give some notations and definitions, and we derive estimates in Bourgain spaces on the linear operators W and L. The process is quite general and can be adapted to other dissipative dispersive semigroups.
-In Section 3, we prove a nonlinear estimate which enables us to obtain the Theorem 1, and section 4 is devoted to the proof of Theorem 2, by establishing an a priori energy estimate.
We state now our main results: Theorem 1. Let α ∈]0, 2], then ∀s > 1 − α 2 and u 0 ∈ H s (IR 3 ) there exist a positive T = T ( ϕ H s ) and a unique solution u to (4) in Moreover, the map ϕ → u is continuous from H s to Y T .

2.
Notations, definitions and linear estimate. For f ∈ S we denote byf or F(f ) the Fourier transform of f i.e.
where P (D x , D y , D z ) is the Fourier multiplier with symbol And by the Fourier transform, the linear part of the equation (4) can be written as : We denote by X s,b the Bourgain type space associated with the space H s,b for the equation (4), wich it endowed with the norm In [22], the authors performed the iteration process in the space X s,b equipped with the norm defined above, which take advantage of the mixed dispersive-dissipative part of the equation. We will rather work in its Besov version X s, 1 2 ,q (with q = 1) defined as the weak closure of the test functions that are uniformly bounded by the norm and to do this we define the decomposition of Littlewood-Paley: Any summations over capitalized variables such as N , L, M are presumed to be dyadic, i.e. these variables range over numbers of the form N = 2 j , j ∈ Z, M = 2 k , k ∈ Z and L = {0}∪{2 l }, l ∈ N (see [5], [11] and [21]). We set ϕ 0 (ξ) := η(ξ), ϕ N (ξ) = ϕ( ξ N ) and define the projections: Roughly speaking, the operators P N , R M and Q L localize respectively in the an- 1 2 ,1 T endowed with the Fourier restriction norm defined as follows: 1. We define the function space X s,b,1 as the completion of the Schwartz class S(IR 4 ) equipped with norm 2. For T ≥ 0, we consider the localized spaces X s,b,1 T endowed with the norm We will also use the space-time Lebesgue space L q,r endowed with the norm We denote by W (·) the semigroup associated with the free evolution of the equation (4) i.e.
and we extend W (·) to a linear operator defined on the whole real axis by setting By the Duhamel integral formulation, the equation (4) can be written as and to prove the local well posedness results, we shall apply a fixed point argument in X s, 1 2 ,1 T to the extension of (10), which is defined on whole the real axis by: where t ∈ IR and L is the operator defined as where ψ is a time cut-off function satisfying and ψ T (·) = ψ(·/T ) . Now, following Molinet-Ribaud [22], Baoxiang [27], Darwich [5] and exactly in the same way as the proof of the linear estimate etablished (2D case) in Section 2 in [11], we give the estimate of the linear term in the space X s, 1 2 ,1 , more precisely we have the following lemma: Lemma 1. Let s ∈ IR, then: a) For all ϕ ∈ H s , we have b) For all f ∈ S(IR 3 ), we have where 0 < γ < 1 2 .
3. Bilinear estimate and the local existence result. The aim of this section is to prove Theorem 1 and as it is standard for this type of problem, with the linear estimates in hand, we obtain the local existence result, once we estimate the nonlinear term ∂ x (u 2 ) in X s, −1 2 +γ,1 . More precisely we have the following proposition: Proposition 1. For all u, v ∈ X s,1/2,1 (R 4 ), s > 1 − α 2 with compact support in time included in the subset {(t, x, y, z) : t ∈ [−T, T ]}, there exists β > 0 such that the following bilinear estimate holds ||∂ x (uv)|| X s,−1/2+γ,1 CT β ||u|| X s,1/2,1 ||v|| X s,1/2,1 .
Remark 2. We will mainly use the following version of (15), which is a direct consequence of Proposition 1, together with the triangle inequality with µ(β) > 0.
To prove the bilinear estimate, we will need the following lemma, which can be obtained in the same way as for the (Z-K) equation see [21] and [18]: and for any couple (u, v) and any couple (N 1 , N 2 ) of dyadic numbers such that N 1 ≥ 4N 2 it holds: See [21] for the proof.
Remark 3. Following [8] it is easy to check that for any u ∈ L 2 (IR 4 ) supported in [−T, T ] and any δ > 0, there exists β = β(δ) such that: Now we are ready to prove our crucial nonlinear estimate (Proposition 1): Proof. We proceed by duality. Let w ∈ X 0,0,∞ , we will estimate the following term Now we need to separate two subcases: N −s , i = 1, 2, this gives that: where θ > 0 small enough such that − 1 2 + γ + θ < 0. Now by summing in L 1 , N 1 , M 1 , L 2 , N 2 , M 2 , L, M ≤ 1 and N and using Remark 3 we get: for ∈]0, 1] and that M ≤ N , inequality (20) becomes: we can sum on M > 1 and we get that: We assume without loss of generality that N 1 ≥ 4N 2 to get where we have used (18) in the last estimate. Now as in case 1, I is controled by : We need to separate in two subcases: Then we have that: If s ≥ 1 then N 2 1−s ≤ 1 and using that N N 1 to get Now as in case 1 and by separating the two case M ≤ 1 and M ≥ 1, we obtain that: If s < 1, then N 2 1−s ≤ N 1 1−s and this gives that: ,1 w N,M,L X 0,0,∞ and as above we get: ,1 w N,M,L X 0,0,∞ . Case 2.1 :N 1 < 1 Here then necessary M ≤ 1 and N ≤ 1, and using that M ≤ N , inequality (23) becomes: ,1 L −θ w N,M,L X 0,0,∞ where θ > 0 small enough such that − 1 2 +γ +θ < 0. Now by summing and by Remark 3, we obtain the estimate.
T (t )) u 2 (t ))dt , to obtain the well-posedness of (4), we prove that Ψ is contraction map on closed subset of X s,1/2,1 . Lemma 1 and Proposition 1 yield Now if we take T < T (4C 2 ϕ H s ) −1/β , we deduce that Ψ is strictly contractive in the ball of radius (2C ϕ H s ) in X s, 1 2 ,1 . This proves the existence of a unique solution u 1 to (11) in X s,1/2,1 T with T = T (||φ|| H s ). The above contraction argument gives the uniqueness of the solution to the truncated integral equation (11). The uniqueness of the solution to the integral equation (10) is obtained by the same argument as in section 4.2 of [22], then we omit it. Now with remark (2) in hand, we obtain that if s > s > 1 − α 2 and u 0 ∈ H s (IR 3 ), then there exists T = T (||u 0 || H s ) and u ∈ X s,1/2,1 T solution to (11). 4. Global existence result. We will prove in this section the global existence result in H 1 (IR 3 ) (Theorem 2). Now, for ϕ ∈ H 1 (IR 3 ), the local solution u of (4), can be extended on a maximal existence interval [0, We are going to see that the H 1 -norm of the solution can't blow up, which obviously ensures that T * = +∞. Let us first, prove that the energy control the norm ∇u 2 L 2 , more precisely we have the following lemma: Lemma 3. Let u(t) be the solution of (4), then : Proof. By interpolation we have: and by Young's inequality, we obtain that: this inequality with (3) and (5), give that: Proposition 2. Let u be the solution of (4), then ∀t ∈ [0, T ], we have : Proof. If we integrate in time the identity (5), we obtain ∀t ≥ 0: this give immediatly the first inequality of the proposition. Now let u 0 ∈ H 4 (IR 3 ), then there exists T = T (||u 0 || H 1 ) such that the solution exists in H 4 (IR 3 ) and since C ∞ c (IR 3 ) dense in H 4 (IR 3 ), then all calculations will be justified. Now by (6), integration by parts and Cauchy-Schwarz, we can write: noticing that the fractional Leibniz rule (see [15]) leads to : and by interpolation we obtain: and D and then for small enough and using Lemma 3, we obtain that: and by Gronwall inequality we obtain the desired estimate. Now we are ready to prove Theorem 2: Let u 0 in H 1 (IR 3 ) and T * be the maximal time of the existence of the solution emanating from u 0 .