The Existence of Strong Solutions to the 3D Zakharov-Kuznestov Equation in a Bounded Domain

We consider the Zakharov-Kuznestov (ZK) equation posed in a limited domain (0,1)_{x}\times(-\pi /2, \pi /2)^d, d=1,2 supplemented with suitable boundary conditions. We prove that there exists a solution u \in \mathcal C ([0, T]; H^1(\dom)) to the initial and boundary value problem for the ZK equation in both dimensions 2 and 3 for every T>0. To the best of our knowledge, this is the first result of the global existence of strong solutions for the ZK equation in 3D. More importantly, the idea behind the application of anisotropic estimation to cancel the nonlinear term, we believe, is not only suited for this model but can also be applied to other nonlinear equations with similar structures. At the same time, the uniqueness of solutions is still open in 2D and 3D due to the partially hyperbolic feature of the model.


Introduction
The Zakharov-Kuznestov (ZK) equation where u = u(x, x ⊥ , t), x ⊥ = y or x ⊥ = (y, z), describes the propagation of nonlinear ionic-sonic waves in a plasma submitted to a magnetic field directed along the x-axis. Here c > 0 is the sound velocity. It has been derived formally in a long wave, weakly nonlinear regime from the Euler-Poisson system in [ZK74] and [LS82]. A rigorous derivation is provided in [LLS13]. For more general physical references, see [BPS81] and [BPS83]. When u depends only on x and t, (1.1) reduces to the classical Korteweg-de Vries (KdV) equation.
Recently the ZK equation has caught much attention, not only because it is closely related with the physical phenomena but also because it is the start to explore more general problems that are partly hyperbolic (such as the inviscid primitive equations).
Concerning the initial and boundary value problems of the Korteweg-de Vries equation posed on a bounded interval (0, L), we refer the interested readers to e.g. [BSZ03], [CG01a], [QT12] and [CG01b].
As for the existence of strong solutions, the global existence in space dimension 2 has been proven in a half strip {(x, y) : x > 0, y ∈ (0, L)} in [LT13]. The existence and exponential decay of regular solutions to the linearized ZK equation in a rectangle {(x, y) : x ∈ (0, L), y ∈ (0, B)} has been studied in [DL14]. The local existence of strong solutions in space dimensions 2 and 3 is established in [Wan]. In these previous works, the boundary conditions on x = 0, 1 are assumed to be u x=0 = u x=1 = u x x=1 = 0; however here we suppose different boundary conditions to serve our purposes.
To the best of our knowledge, the global existence and uniqueness of regular solutions in 3D is still an open problem. In this article, we prove that there exists a global solution u ∈ C([0, T ]; L 2 (M)) for the initial and boundary value problem of the ZK equation in both 2D and 3D, which we believe, will lead to the global well-posedness of strong of solutions in 3D eventually. It is interesting to observe that, for the 3D ZK equation, the nonlinear term has the same structure as the nonlinear term in the 3D Navier-Stkoes equations and that the basic a priori estimates (L ∞ (0, T ; L 2 (M)) and L 2 (0, T ; H 1 (M))) are the same, although the structure of the linear operator is totally different (e.g. not coercive as in (3.11) below).
For the proof we use the parabolic regularization as in [ST10], [STW12] and [Wan]. There are four main difficulties. Firstly, as in the case of 3D Navier-Stokes equation, the nonlinear term will pose a problem when we apply the Sobolev imbedding in 3D. Secondly, since the linear operator is not coercive, the L p estimations (see e.g. [CT07]) does not work. Thirdly, some assumption on the trace u xx x=1 x=0 is necessary for the estimate of ∇u ∈ L ∞ (0, T ; L 2 (M)). Finally, to pass to the limit on the boundary conditions, the methods in [ST10] and [STW12] are not applicable any more because of the change of the boundary conditions.
To overcome these difficulties, firstly we utilize the anisotropic resonance of the term u xxx and the nonlinear term uu x to cancel uu x , which leads to a bound of the H 1 norm over (0, T ) for u. This step of canceling the nonlinear term may also be applied to other nonlinear equations with similar structures. Next, we suppose periodic boundary conditions of u and u x j at x = 0, 1, j = 1, 2, so that the trace u xx x=1 x=0 now vanishes. Finally, we investigate a bound independent of ǫ for u ǫ xxx in L 3/2 (I x ; Y ), with Y a Banach space in x ⊥ and t, which facilitates the passage to the limit on the traces of u x j at x = 0, 1, j = 1, 2.
However the uniqueness of solutions is still open in both 2D and 3D, even with such a regularity and all the periodic boundary conditions satisfied. In particular, the methods in [ST10] and [STW12] can not be adapted to our case due to the lack of the boundary condition u x = 0 at x = 1.
The article is organized as follows. Firstly we introduce the basic settings of the equation in Section 2. Secondly we introduce the parabolic regularization as in [ST10] and [STW12] (Section 3.1). Then we derive the estimates independent of ǫ for u ǫ in L ∞ (0, T ; L 2 (M)) (Section 3.2.1), ∇u ǫ in L ∞ (0, T ; L 2 (M)) (Section 3.2.2) and for u ǫ xxx in L 3/2 (I x ; H −1 t (0, T ; H −4 (I x ⊥ ))) (Section 3.2.3). Eventually we can pass to the limit on the parabolic regularization and the traces and deduce the global existence of solutions u ∈ C([0, T ]; H 1 (M)) (Section 3.3). Finally, we discuss about the difficulties in the attempt of proving the uniqueness of solutions (Section 4).

ZK equation in a rectangle in dimensions and 3
We aim to study the ZK equation: in a rectangle or parallelepiped domain in R n with n = 2 or 3, denoted as M = (0, 1) x × (−π/2, π/2) d , with d = 1 or 2, ∆u = u xx + ∆ ⊥ u, ∆ ⊥ u = u yy or u yy + u zz depending on the dimension. In the sequel we will use the notations I x = (0, 1) x , I y = (−π/2, π/2) y , I z = (−π/2, π/2) z , and I x ⊥ = I y or I y × I z . We assume the boundary conditions of u, u x and u xx on x = 0, 1 to be periodic: For the boundary conditions in the y and z directions, we will choose either the Dirichlet boundary conditions u = 0 at y = ± π 2 and z = ± π 2 , (2.4) or the periodic boundary conditions (2.5) The initial condition reads: We study the initial and boundary value problem (2.1)-(2.3) and (2.6) supplemented with the boundary condition (2.4), that is, the Dirichlet case on the x ⊥ boundaries, and we will make some remarks on the extension to the periodic boundary condition case.
We denote by |·| and (·, ·) the norm and the inner product of L 2 (M), and by [·] 2 the following seminorm which will be useful in the sequel: 3 Existence of solutions u ∈ C([0, T ]; H 1 (M)) in dimensions 2 and 3 To prove this result, we use the parabolic regularization as in [STW12], but with different boundary conditions. For the sake of simplicity we only treat the more complicated case when d = 2.

Parabolic regularization
To begin with, we recall the parabolic regularization introduced in [ST10] and [STW12], that is, for ǫ > 0 "small", we consider the parabolic equation, supplemented with the boundary conditions (2.2)-(2.4) and the additional boundary conditions Note that from (2.3) and (3.2) we infer We also note that since u ǫ It is a classical result (see e.g. [Lio69], [LSU68] or also [STW12]) that there exists a unique solution to the parabolic problem which is sufficiently regular for all the subsequent calculations to be valid; in particular, we have (3.6)

Estimates independent of ǫ
We establish the estimates independent of ǫ for various norms of the solutions.

L 2 estimate independent of ǫ
We first show a bound independent of ǫ for u ǫ in L ∞ (0, T ; L 2 (M)).
Lemma 3.1. We assume that u 0 ∈ L 2 (M), (3.7) f ∈ L 2 (0, T ; L 2 (M)). (3.8) Then for every T > 0 the following estimates independent of ǫ hold: (3.10) Proof. As in [STW12], we multiply (3.1) with u, integrate over M and integrate by parts, dropping the superscript ǫ for the moment we find: Hence we find d dt |u ǫ (t)| 2 + 2ǫ[u ǫ ] 2 2 ≤ |f | 2 + |u ǫ | 2 . (3.12) Using the Gronwall lemma we classically infer where µ i indicates a constant depending only on the data u 0 , f , etc, whereas C ′ below is an absolute constant. These constants may be different at each occurrence. Let us admit for the moment the following: (3.14) By the previous lemma, we have ≤ (thanks to (3.13)) ≤ const := µ 2 , which implies (3.10). Thus Lemma 3.1 is proven once we have proven Lemma (3.2). Proof of Lemma 3.2. We first observe that using the generalized Poincaré inequality (see [Tem97]) we have Thanks to (2.2), we have 1 0 u ǫ x dx = u ǫ | x=1 x=0 = 0, and hence (3.15) implies Squaring both sides and integrating both sides on I x ⊥ , we find Similarly we can show that |u ǫ y | ≤ C ′ |u ǫ yy | and |u ǫ z | ≤ C ′ |u ǫ zz |, which implies

H 1 estimate independent of ǫ
Now we establish the key observation, a bound independent of ǫ for ∇u ǫ in L ∞ (0, T ; L 2 (M)). Proof. We multiply (3.1) with −∆u ǫ − 1 2 (u ǫ ) 2 , integrate over M and integrate by parts. Firstly we show the calculation details of the multiplication by ∆u ǫ , integration over M and integration by parts (dropping the super index of ǫ for the moment): Hence we find after changing the sign, (3.24) Next we show the calculation details of the multiplication by (u ǫ ) 2 , integrating over M and integrating by parts: = (thanks to (3.5) and (2.2)) = −2 M ∆u uu x dM, Hence we find (3.25) Adding (3.24) to (3.25) multiplied by −1/2, we observe that the terms M ∆u ǫ u ǫ u ǫ x dM get canceled, which yields Integrating both sides in time from 0 to t, we obtain for every t ∈ (0, T ), where We estimate each term on the right-hand-side of (3.26); we will use here the interpolation space H 1/2 (M) as defined in [LM72] where it is shown that H 1/2 (M) ⊂ L 3 (M) in dimension 3 with a continuous embedding. Dropping the superscript ǫ for the moment we then find: Collecting the above estimates, along with (3.26) we observe that the terms with third-order derivatives in the RHS of (3.27) and the following two inequalities can be canceled by a term on the LHS of (3.26). Thus (3.26) now yields ≤ (thanks to (3.13)) L 2 (0,T ;H 2 0 (M)) + (µ 1 + µ 2 1 )T + |f | 2 L 2 (0,T ;L ∞ (M)) . (3.28) In particular, setting σ ǫ (t) := 1 + C ′ ǫµ 1/4 (3.29)

3.2.3
Estimates independent of ǫ for u ǫ xxx and u ǫ u ǫ x For the sake of the passage to the limit on the boundary conditions and the compactness argument, we now derive bounds independent of ǫ for u ǫ xxx and u ǫ u ǫ x . In particular, to obtain the estimates for u ǫ xxx , we first deduce a bound independent of ǫ for ǫ u ǫ xxxx in L 2 (0, T ; L 2 (M)). and f xx assume the periodic boundary condition on x = 0, 1. Then we have the following bounds independent of ǫ, We estimate the first term on the right-hand side of (3.36) and find ǫ M u x u 2 xx dM ≤ ǫ |u x ||u xx | 2 L 4 (M) ≤ C ′ ǫ |u x ||u xx | 1/2 |∇u xx | 3/2 ≤ (by the intermediate derivative theorem |u xx | 2 ≤ |u x | |u xxx |) ≤ C ′ ǫ|u x | 5/4 |u xxx | 1/4 |∇u xx | 3/2 ≤ C ′ ǫ|u x | 5/4 |∇u xx | 7/4 ≤ (thanks to (3.30)) ≤ C ′ ǫµ Integrating both sides in t from 0 to T , we find ǫ|∇u ǫ xx | 7/4 dt + ǫ|f xxx | 2 L 2 (0,T ;L 2 (M)) + ǫµ 4 T.