To one problem of Saut-Temam for the 3D Zakharov-Kuznetsov equation

TL;DR

This paper proves the existence, uniqueness, and exponential decay of solutions for a specific initial-boundary value problem related to the 3D Zakharov-Kuznetsov equation on unbounded domains, for small initial data.

Contribution

It establishes the global regularity and decay properties of solutions to the 3D Zakharov-Kuznetsov equation with new analytical techniques.

Findings

Existence and uniqueness of global regular solutions.

Exponential decay of the $H^2$-norm for small initial data.

Results applicable to unbounded domains.

Abstract

An initial-boundary value problem for the 3D Zakharov-Kuznetsov equation posed on an unbounded domain is considered. Existence and uniqueness of a global regular solution as well as exponential decay of the $H^2$-norm for small initial data are proven.

Equations

$$u_t+(1+u)u_x+u_{xxx}+u_{xyy}+u_{xzz}=0$$

$$u_t+u{u}_x+u_{xxx}=0.$$

$$u_t+u_x+u{u}_x+u_{xxx}=0.$$

$$Lu=u_t+u_x+u{u}_x+\Delta u_x=0;$$

$$u(0,y,z,t)=u(L,y,z,t)=u_x(L,y,z,t)=0,$$

$$u(x,y,z,0)=u_0(x,y,z),(x,y,z)\in \Omega .$$

$$u_0(0,x,y,z)=u_0(L,y,z,t)=u_{0x}(L,y,z,t)=0,$$

$$\Vert u_0{\Vert }^2+\Vert u_{0yy}{\Vert }^2+\Vert u_{0yz}{\Vert }^2+\Vert u_{0zz}{\Vert }^2+J_0<\infty ,$$

$$\Vert u_0{\Vert }^4\le \frac{{\pi }^2}{8K_1{L}^2},J_0^2\le \frac{{\pi }^2}{200K_2{L}^2},$$

$$u\in L^{\infty }(0,+\infty ;H^2(\Omega ))\cap L^2(0,+\infty ;H^3(\Omega )),$$

$$u_t\in L^{\infty }(0,+\infty ;L^2(\Omega ))\cap L^2(0,\infty ;H^1(\Omega )),$$

$$\Delta u_x\in L^{\infty }(0,+\infty ;L^2(\Omega ))\cap L^2(0,\infty ;H^1(\Omega ))$$

$$\Vert u{\Vert }_{H^2(\Omega )}^2(t)+\Vert u_t{\Vert }^2(t)\le C(L,J_0)e^{-\frac{\chi }{2}t},\forall t>0,$$

$$\Vert u{\Vert }_{L^q(\Omega )}\le 4^{\theta }\Vert \nabla u{\Vert }^{\theta }\Vert u{\Vert }^{1-\theta },$$

$$\Vert u{\Vert }_{L^q(\Omega )}\le 4^{\theta }{C}_{\Omega }\Vert u{\Vert }_{H^1(\Omega )}^{\theta }\Vert u{\Vert }^{1-\theta },$$

$$\Vert v_x{\Vert }^2(t)\ge \frac{{\pi }^2}{L^2}\Vert v{\Vert }^2(t).$$

$$f^{\prime }(t)+(\alpha -k{f}^n(t))f(t)\le 0.$$

$$\alpha -k{f}^n(0)>0.$$

$$f(t)<f(0)$$

$${\int }_0^{\tau }(\alpha -k{f}^n(t))f^n(t)dt<0$$

$$L_{ϵ}{u}_m^{ϵ}=u_{mt}^{ϵ}+u_{mx}^{ϵ}+u_m^{ϵ}{u}_{mx}^{ϵ}+\Delta u_{mx}^{ϵ}+ϵ{\partial }^4{u}_m^{ϵ}=0,$$

$$u_m^{ϵ}(0,y,z,t)=u_m^{ϵ}(L,y,z,t)=u_{mx}^{ϵ}(L,y,z,t)$$

$$=ϵu_{mxx}^{ϵ}(0,y,z,t)=0,$$

$$u_m^{ϵ}(x,y,z,0)=u_{0m}(x,y,z),$$

$$u_{0m}(0,y,z)=ϵu_{0mxx}(0,y,z)=u_{0m}(L,y,z)=u_{0mx}(L,y,z)=0,$$

$${\partial }^4{u}_m^{ϵ}=\frac{{\partial }^4}{\partial x^4}{u}_m^{ϵ}+\frac{{\partial }^4}{\partial y^4}{u}_m^{ϵ}+\frac{{\partial }^4}{\partial z^4}{u}_m^{ϵ},$$

$$J_{0m}^{ϵ}=((1+x),|u_{0m^{ϵ}}+u_{0mx}^{ϵ}+u_{0m}^{ϵ}{u}_{0mx}^{ϵ}+\Delta u_{0mx}^{ϵ}+ϵ{\partial }^4{u}_{0m}^{ϵ}{|}^2)$$

$$J_{0m}=((1+x),|u_{0m}+u_{0mx}+u_{0m}{u}_{0mx}+\Delta u_{0mx}{|}^2).$$

$$J_{0m}\to J_0,,\Vert u_{0myy}\Vert \to \Vert u_{0yy}\Vert \Vert u_{0myz}\Vert \to \Vert u_{0yz}\Vert ,\Vert u_{0mzz}\Vert \to \Vert u_{0zz}\Vert ,$$

$$\{\begin{array}{l}{u}_m^{ϵ}\text{ is bounded in }{L}^{\infty }(0,T;L^2(\Omega ))\cap L^2(0,T;H^1(\Omega )),\\ u_{mx}^{ϵ}\text{ is bounded in }{L}^2(0,T;L^2(\mathcal{S})).\end{array}$$

$$\Vert u_m^{ϵ}{\Vert }^2(t)+{\int }_0^t{\int }_{\mathcal{S}}(u_{mx}^{ϵ}{)}^2(0,y,z,\tau )dydzd\tau +2ϵ{\int }_0^t\Vert u_m^{ϵ}{\Vert }_{[2]}^2(t)ds\le \Vert u_{0m}{\Vert }^2$$

$$\Vert u_m^{ϵ}{\Vert }^2(t)+{\int }_0^t{\int }_{\mathcal{S}}(u_{mx}^{ϵ}{)}^2(0,y,z,\tau )d\tau ds\le 2\Vert u_0{\Vert }^2.$$

$$2(L_{ϵ}{u}_m^{ϵ},(1+x)u_m^{ϵ})(t)=0$$

$$\frac{d}{dt}((1+x),u^2)(t)+(1-ϵ){\int }_{\mathcal{S}}{u}_x^2(0,y,z,t)dydz-\Vert u{\Vert }^2(t)$$

Full text

† Corresponding author

1. Introduction

We are concerned with the existence, uniqueness and exponential decay of the $H^{2}$-norm for global regular solutions to an initial-boundary value problem (IBVP) for the 3D Zakharov-Kuznetsov (ZK) equation

[TABLE]

which describes the propagation of nonlinear ionic-sonic waves in a plasma submitted to a magnetic field directed along the $x$ axis. This equation is a three-dimensional analog of the well-known Korteweg-de Vries (KdV) equation

[TABLE]

Equations (1.1), (1.2) are typical examples of so-called dispersive equations which attract considerable attention of both pure and applied mathematicians in the past decades. The KdV equation is probably most studied in this context. The theory of the initial-value problem (IVP henceforth) for (1.2) is considerably advanced today [1, 14, 15, 34, 37].

Recently, due to physics and numerics needs, publications on initial-boundary value problems to (1.2) both in bounded and unbounded domains for dispersive equations have appeared [2, 20, 25, 40]. In particular, it has been discovered that the KdV equation posed on a bounded interval possesses an implicit internal dissipation. This allowed to prove the exponential decay rate of small solutions for (1.2) posed on unbounded intervals without adding any artificial damping term [20]. Similar results were proved for a wide class of dispersive equations of any odd order with one space variable [12].

However, (1.2) is a satisfactory approximation for real waves phenomena while the equation is posed on the whole line ($x\in\mathbb{R}$); if cutting-off domains are taken into account, (1.2) is no longer expected to mirror an accurate rendition of reality. The correct equation in this case (see, for instance, [2]) should be written as

[TABLE]

Indeed, if $x\in{\mathbb{R}},\ t>0$, the linear traveling term $u_{x}$ in (1.3) can be easily scaled out by a simple change of variables, but it can not be safely ignored for problems posed both on finite and semi-infinite intervals without changes in the original domain.

Once bounded domains are considered as a spatial region of waves propagation, their sizes appear to be restricted by certain critical conditions. We recall, however, that if the transport term $u_{x}$ is neglected, then (1.3) becomes (1.2), and it is possible to prove the exponential decay rate of small solutions for (1.2) posed on any bounded interval. More results on control and stabilizability for the KdV equation can be found in [32, 33].

Later, the interest on dispersive equations became to be extended for multi-dimensional models such as Kadomtsev-Petviashvili (KP) and ZK equations. As far as the ZK equation is concerned, results both on IVP and IBVP can be found in [10, 11, 13, 27, 28, 29, 30, 35]. The biggest part of these publications is devoted to study of well-posedness of the Cauchy problem and initial-boundary value problems for the 2D ZK equation [10, 11, 13, 27, 28]. In the case of the 3D ZK equation, there are results on local well- posedness for the Cauchy problem [29, 30]; the existence of local strong solutions to an initial- boundary value problem posed on a bounded domain, [40], as well as the existence of global weak solutions [35].

Our work has been inspired by [35] where (1.1) posed on an unbounded domain was considered. A thorough analysis of these papers has revealed that an implicit dissipativity of the terms $u_{xyy}+u_{xzz}$ may help to establish a global well-posedness of initial-boundary value problems in classes of regular solutions. Yearlier this dissipativity has been used in order to prove exponential decay for the 2D ZK equation [19, 26].

The main goal of our work is to prove the existence and uniqueness of global-in-time regular solutions of (1.1) posed on unbounded domains and the exponential decay rate of these solutions for sufficiently small initial data. To cope with this problem, we exploited the strategy completely different from the standard schemes: first to prove the existence result and after that to study uniqueness and decay properties of solutions. In our case, we prove simultaneously existence of global regular solutions and their exponential decay.

The paper is outlined as follows. Section I is Introduction. Section 2 contains formulation of the problem and auxiliaries. In Section 3, we prove the existence of global regular solutions and, simultaneously, exponential decay of the $H^{2}$-norm. In section 4 uniqueness of a regular solutions and continuous dependence on initial data are proven.

2. Problem, Preliminares and Main Result

Let $L>0$ be a finite number. Define $\Omega=\{(x,y,z)\in{\mathbb{R}}^{3}:x\in(0,L),y\in{\mathbb{R}};z\in{\mathbb{R}}\},\;\;\mathcal{S}={\mathbb{R}}^{2}$.

Consider in $\Omega\times(0,t)$ the following initial- boundary value problem for the Zakharov-Kuznetsov equation:

[TABLE]

Hereafter subscripts $u_{x},\ u_{xy},$ etc. denote the partial derivatives, as well as $\partial_{x}$ or $\partial_{xy}^{2}$ when it is convenient. Operators $\nabla$ and $\Delta$ are the gradient and Laplacian acting over $\Omega.$ By $(\cdot,\cdot)$ and $\|\cdot\|$ we denote the inner product and the norm in $L^{2}(\Omega),$ $\|\cdot\|_{H^{k}}$ stands for the norm in $L^{2}$-based Sobolev spaces, and $\|u\|_{[2]}^{2}=\|u_{xx}\|^{2}+\|u_{yy}\|^{2}+\|u_{zz}\|^{2}$.

Theorem 2.1**.**

Let $L\leq\frac{\pi}{2}$ and $u_{0}(x,y,z)$ satisfying the following conditions:

[TABLE]

[TABLE]

[TABLE]

where $K_{1}=2^{16}3^{3}(1+L)(\frac{2^{3}}{25}C_{1}+1)$, $C_{1}=2+\frac{2^{13}}{3}\|u_{0}\|^{4}$, $K_{2}=2^{19}3^{3}(1+L)^{6}$ and $J_{0}=((1+x),|u_{0}+u_{0x}+u_{0}u_{0x}+\Delta u_{0x}|^{2})$.

Then there exists a unique global regular solutions to (2.1 - 2.3):

[TABLE]

[TABLE]

[TABLE]

such that

[TABLE]

where $\chi=\frac{\pi^{2}}{2L^{2}(1+L)}$.

We will need the following results:

Lemma 2.1**.**

Let $u\in H^{1}(\Omega)$ and $\gamma$ be the boundary of $\Omega.$

If $u|_{\gamma}=0,$ then

[TABLE]

where \theta=3\big{(}\frac{1}{2}-\frac{1}{q}\big{)}.

If $u|_{\gamma}\neq 0,$ then

[TABLE]

where $C_{\Omega}$ does not depend on a size of $\Omega$.

Proof.

See [16, 17].

Lemma 2.2**.**

Let $v\in H^{1}_{0}(0,L).$ Then

[TABLE]

Proof.

The proof is based on the Steklov inequality: let $v(t)\in H^{1}_{0}(0,\pi)$, then $\int_{0}^{\pi}v_{t}^{2}(t)\,dt\geq\int_{0}^{\pi}v^{2}(t)\,dt.$ Inequality (2.11) follows by a simple scaling. ∎

Lemma 2.3**.**

Let $f(t)$ be a continuous positive function such that

[TABLE]

Then

[TABLE]

for all $t>0$.

Proof.

Obviously, $f^{\prime}(0)+(\alpha-kf^{n}(0))f^{n}(0)\leq 0$. Since $f$ is continuous, there exists $T>0$ such that $f(t)<f(0)$ for every $t\in[0,T]$. Suppose that there is $\tau>0$ and $f(0)=f(\tau)$. Integrating (2.12), we find

[TABLE]

that contradicts (2.13). Therefore, $f(t)<f(0)$ for all $t>0.$

(See also [38]).

The proof of Lemma 2.2 is complete. $\Box$

3. Existence of regular solutions

Regularized problem. To solve (2.1)-(2.3), we exploit the parabolic regularization of this problem as follows:

For $\epsilon>0$ (small), consider in $\Omega\times(0,t)$ the following parabolic problem:

[TABLE]

where

[TABLE]

$u_{0}$ is an independent of $\epsilon$ approximation of $u_{0}$ such that for all $m\in{\mathbb{N}}$.

Define

[TABLE]

It is known [17, 37, 36] that there exists a unique regular solution of (3.1)-(3.4), provided $u_{0m}$ is sufficiently smooth.

Our goal is to obtain estimates for the $u_{m}^{\epsilon}$ independent of $m$ and $\epsilon$ with $u_{0m}$ sufficiently smooth, fixed; then to pass the limit as $\epsilon$ tends to [math] getting a solution to (2.1)-(2.3) with initial data $u_{0m}$. After that we pass to the limit as $m$ tends to $\infty$ and $u_{0m}$ tends to $u_{0}$, obtaining a solution to the original problem.

We will assume that $u_{0m}$ converges to $u_{0}$ in the following sense:

[TABLE]

as $m\to\infty$.

We assume that $m\geq m^{*}$, where $m^{*}$ is af natural number such that $J_{0m}\leq 2J_{0}$. In turn, $\epsilon>0$ is sufficiently small such that $J_{0m}^{\epsilon}\leq 4J_{0}$.

Lemma 3.1**.**

Under the conditions of Theorem 2.1, for $m$ sufficiently large and $\epsilon$ sufficiently small, the following independent of $\epsilon$ and $m$ estimates hold:

[TABLE]

Proof.

Estimate I. Multiply (3.1) by $u_{m}^{\epsilon}$ and integrate over $\Omega\times(0,t)$ to obtain

[TABLE]

and for $m$ sufficiently large

[TABLE]

Estimate II. Dropping the indices $m,\epsilon$, we transform the scalar product

[TABLE]

into the following equality:

[TABLE]

where $\mathcal{P}_{1}=((1+x),u_{xx}^{2}+u_{yy}^{2}+u_{zz}^{2})(t).$

Making use of (2.9), we find

[TABLE]

By Lemma 2.2,

[TABLE]

Substituting (3.8), (3.11) and (3.12) into (3.10), we obtain for a fixed, sufficiently large $m$ that

[TABLE]

Under conditions of Theorem 2.1, we have

[TABLE]

Hence

[TABLE]

and

[TABLE]

where $\chi=\frac{\pi^{2}}{L^{2}(1+L)}$. Returning to (3.10), using (3.11) and (3.13), we obtain

[TABLE]

Lemma 3.2**.**

Under the conditions of Theorem (2.1), for $m$ sufficiently large, the following independent of $\epsilon$ and $m$ estimates hold:

[TABLE]

Proof.

Estimate III. Dropping the indices $\epsilon$, $m$, transform the inner product

[TABLE]

into the equality

[TABLE]

By Holder and Young inequalities,

[TABLE]

Making use of (2.9), (3.14) and the last inequality, we reduce (3.16) to the form

[TABLE]

where $C_{1}=2+\frac{2^{13}}{3}\|u_{0}\|^{4}$ is independent of $t>0$.

Returning to (3.16), we get

[TABLE]

Estimate IV. Consider the inner product

[TABLE]

We calculate

[TABLE]

Exploiting Lemma 2.1 and the Young inequality, we obtain

[TABLE]

Substituting (3.20) and (3.22) into (3.19), we get

[TABLE]

Since by (3.18),

[TABLE]

and by Lemma 2.2,

[TABLE]

then (3.23) reads

[TABLE]

According to Theorem 2.1 notations,

[TABLE]

For $\epsilon$ small and $m$ sufficiently large fixed $2\epsilon^{2}((1+x),|\partial^{4}u_{0m}^{\epsilon}|^{2})\leq J_{0}$ and

[TABLE]

By Lemma 2.3, $((1+x),|u_{mt}^{\epsilon}|^{2})(t)\leq J_{0m}^{\epsilon}$ for all $t>0$ and making use of (3.24), (3.25), we obtain

[TABLE]

By (2.4) and (3.25), we get $\frac{\pi^{2}}{2L^{2}}-1-K_{1}\|u_{0}\|^{4}-25K_{2}J_{0}^{2}\geq 0$.

Hence

[TABLE]

and

[TABLE]

Returning to (3.23), we obtain

[TABLE]

and from (3.14), (3.17),

[TABLE]

for $\epsilon>0$ sufficiently small and $m$ sufficiently large fixed.

Lemma 3.3**.**

Under the assumptions of Theorem 2.1, we find

[TABLE]

Proof.

Estimate V. Dropping the indices $\epsilon$, $m$, transform the scalar product

[TABLE]

into the following equality:

[TABLE]

where \mathcal{P}_{2}(t)=\big{(}(1+x),|u_{xxy}^{2}+u_{xxz}^{2}+u_{yyy}+u_{yyz}^{2}+u_{zzz}^{2}+u_{zzy}^{2}|\big{)}(t).

We estimate

[TABLE]

Similarly,

[TABLE]

Substituting $I_{1},I_{2}$ into (3.29), we find

[TABLE]

Making use of (3.28),

[TABLE]

To prove that $u$ is bounded in $L^{\infty}(0,T;H^{2}(\Omega))$, it is sufficiently to estimate $\|u_{mxx}^{\epsilon}\|(t)$.

Estimate VI. From the inner product

[TABLE]

dropping the indices $\epsilon$ and $m$, we find

[TABLE]

Substituting $J_{1}$-$J_{8}$ into (3.31), we get

[TABLE]

where \mathcal{P}_{3}=\big{(}(1+x),\big{[}u_{yyyy}^{2}+2u_{yyzz}^{2}+u_{yyyz}^{2}+u_{zzzz}^{2}+u_{zzzy}^{2}+u_{yyxx}^{2}+u_{zzxx}^{2}\big{]}\big{)}(t).

We estimate the nonlinear term in the following manner:

[TABLE]

Making use of (3.33), we find

[TABLE]

where $\delta_{1},\delta_{2}$ are arbitrary positive constants and $C$ is a constant independent of $\epsilon,m$ and $t$. Substituting $I_{1}$-$I_{3}$ into (3.33), we get

[TABLE]

Similarly,

[TABLE]

and

[TABLE]

Using (3.17), (3.30), we estimate

[TABLE]

where $\delta_{3},\delta_{4}$ are arbitrary positive constants and $C$ is a constant independent of $\epsilon,m$ and $t$. Substituting $J_{1}$-$J_{4}$ into (3.37) and making use of (3.32), (3.33), we reduce (3.32) to the form

[TABLE]

Integrating over $(0,t)$ , we obtain

[TABLE]

for $\epsilon$ sufficiently small and $m$ fixed and sufficiently large, with the constant $C(L,J_{0})$ independent of $t>0.$

Estimate VII. Dropping the indices $\epsilon,m$ and the variables $y,z,t$, rewrite (3.1)-(3.4) in the form

[TABLE]

By (3.38), $g\in L^{2}(0,T;L^{2}(\Omega))$. Multiplyng (3.39) by $x$ and integrating over $x\in(0,1)$, we get

[TABLE]

Integrating (3.39) over $(x,1)$ gives

[TABLE]

Subtracting (3.42) from (3.41), we obtain

[TABLE]

Define

[TABLE]

Then (3.39) reads

[TABLE]

Multiplying (3.43) by $u_{xx}$ and integrating over $(0,1)$, we find

[TABLE]

Hence, (3.43) becomes

[TABLE]

Therefore

[TABLE]

and

[TABLE]

Passage to the limit as $\epsilon\to 0$. Using the estimates obtained in Lemmas 3.1, 3.2 3.3 and compactness arguments, we can pass to the limit as $\epsilon\to 0$ in (3.1-3.4) and get a solution $u_{m}$ for (2.1)-(2.3) with initial data $u_{0m}$ for $m$ large and fixed:

[TABLE]

such that

[TABLE]

Rewriting (3.45) as

[TABLE]

and making use of (3.26), (3.27), (3.38), we find that

[TABLE]

Passage to the limit as $m\to\infty$. Since the constants of estimates in Lemmas 3.1, 3.2 3.3 do not depend on $\epsilon,m,t$, we can pass to the limit in (3.45)-(3.47) as $m\to\infty$ and obtain a solution $u(x,y,z,t)$ for (2.1)-(2.3) such that

[TABLE]

[TABLE]

Regularity of u. Making use of (3.32) and (3.52), write (2.1)-(2.3) in the form

[TABLE]

Denoting $v=u_{x}-\phi(y,z,t)(L-x)$, we get

[TABLE]

where $F\in L^{\infty}(0,T,H^{-1}(\Omega))$.

From the inner product

[TABLE]

we calculate

[TABLE]

[TABLE]

and come to the inequality

[TABLE]

Since $\|u\|_{L^{4}}^{4}(t)\leq 4^{3}\|u\|(t)\|\nabla u\|^{3}(t)$, making use of (3.52), we get

[TABLE]

Returning to (3.53)-(3.55), we can see that $F\in L^{2}(0,\infty;L^{2}(\Omega))$. This implies that $u\in L^{2}(0,\infty;H^{2}(\Omega))$. Hence $u\in L^{\infty}(0,\infty;H^{2}(\Omega))\cap L^{2}(0,\infty;H^{3}(\Omega))$.

Regularity of $\Delta u_{x}$. Writing $\Delta u_{x}=-u_{t}-u_{x}-uu_{x}$, and recalling that

[TABLE]

we estimate

[TABLE]

where C is a constant independent of $t$. By (3.48), (3.49) and (3.57) read

[TABLE]

Hence

[TABLE]

This proves the existence part of Theorem 2.1.

4. Uniqueness of a regular solution and continuous dependence on initial data

Theorem 4.1**.**

A global regular solution to (2.1)-(2.3) is uniquelly defined.

Proof.

Let $u_{1},u_{2}$ be two distinct solutions to (2.1)-(2.3) and $w=u_{1}-u_{2}$. Then

[TABLE]

and (2.1)-(2.3) can be rewritten in the form

[TABLE]

Transform the inner product

[TABLE]

into the following equality

[TABLE]

Using Lemma 2.1, we find

[TABLE]

and

[TABLE]

For $\delta>0$ sufficiently small, we find

[TABLE]

By the Grownwall Lemma,

[TABLE]

Remark 4.1**.**

If $w(x,y,z,0)=w_{0}(x,y,z)\neq 0,$ then

[TABLE]

This means continuous dependence of solutions to (2.1)-(2.3) on initial data.

Remark 4.2**.**

The geometrical restriction $L\leq\frac{\pi}{2}$ in Theorem 2.1 is caused by the presence of the term $u_{x}$ in (2.1) and is connected with spectral properties of the linear spatial operator $u_{x}+\Delta u_{x}$ and existing of critical size domains (see [19] in 2D case). On the other hand, there are some boundary conditions under which there are not critical size domains [8]. We need also small initial data in order to suppress destabilizing effects of the nonlinear convective term $uu_{x}$. We must note that in [21] such restrictions for $c_{s}u_{x}$ and initial data did not appear while establishing the existence of weak solutions for the 3D ZK equation, but there was an open problem, still unresolved, on uniqiueness of this weak solution.

Conclusions. We have established the existence and uniqueness of global regular solutions to (2.1)-(2.3) as well as exponential decay of the $H^{2}$-norm exploiting an approach of proving simultaneously existence and exponential decay. Therefore geometrical restrictions and “smallness” conditions for initial data have appeared. Of course, theses restrictons are not necessary while proving only existence and uniqueness of global regular solutions for the 2D ZK equation. Nevertheless. similar restrictions appear while proving exponential decay of the existing global regular solutions [19].