Modified Zakharov-Kuznetsov equation on rectangles

TL;DR

This paper investigates the initial-boundary value problem for the modified Zakharov-Kuznetsov equation on rectangles, addressing the challenges posed by the critical nonlinear power and analyzing the solutions' existence, uniqueness, and long-term behavior.

Contribution

It provides new results on existence, uniqueness, and asymptotic behavior for solutions of the modified Zakharov-Kuznetsov equation on bounded rectangles, handling the critical nonlinear power.

Findings

Proved existence of solutions under certain conditions

Established uniqueness of solutions

Analyzed asymptotic behavior of solutions

Abstract

Initial-boundary value problem for the modified Zakharov-Kuznetsov equation posed on a bounded rectangle is considered. The main difficulty is the critical power in nonlinear term. The results on existence, uniqueness and asymptotic behavior of solutions are presented.

Equations

$$u_t+u_x+u^2{u}_x+u_{xxx}+u_{xyy}=0.$$

$$\Omega =\{(x,y)\in {\mathbb{R}}^2:x\in (0,L),y\in (-B,B)\},Q_T=\Omega \times (0,T).$$

$$P$$

$$u(x,-B,t)=u(x,B,t)=0,x\in (0,L),t>0;$$

$$u(0,y,t)=u(L,y,t)=u_x(L,y,t)=0,y\in (-B,B),t>0;$$

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

$${\int }_0^L{\int }_{-B}^B{\omega }^2(x,y)dxdy\le \frac{4B^2}{{\pi }^2}{\int }_0^L{\int }_{-B}^B{\omega }_y^2(x,y)dxdy,$$

$${\int }_0^L{\int }_{-B}^B{\omega }^2(x,y)dxdy\le \frac{L^2}{{\pi }^2}{\int }_0^L{\int }_{-B}^B{\omega }_x^2(x,y)dxdy.$$

$$\Vert D^{j}u{\Vert }_{L^p(U)}\le C_U\Vert D^{m}u{\Vert }_{L^r(U)}^{\alpha }\Vert u{\Vert }_{L^q(U)}^{1-\alpha }$$

$$\frac{1}{p}=\frac{j}{n}+\alpha \left(\frac{1}{r}-\frac{m}{n}\right)+(1-\alpha )\frac{1}{q},$$

$$\frac{j}{m}\le \alpha \le 1.$$

$$\Vert u{\Vert }_{L^{2p}(\Omega )}\le C_{2p}\Vert \nabla u{\Vert }^{\frac{p-1}{p}}\Vert u{\Vert }^{\frac{1}{p}}$$

$$C_{2p}={\left(\frac{p!}{{\sqrt{2}}^{p-1}}\right)}^{\frac{1}{p}}.$$

$$\underset{(x,y)\in \Omega }{\sup }{u}^2(x,y)\le \Vert u{\Vert }_{H^1(\Omega )}^2+\Vert u_{xy}{\Vert }_{L^2(\Omega )}^2.$$

$$\left\{\begin{array}[]{c}u_{t}+Au=f,\\ u(0)=u_{0},\end{array}\right.$$

$$D(A)=\{u\in L^2(\Omega );\Delta u_x+u_x\in L^2(\Omega );u{|}_{\partial \Omega }=0\text{ and }{u}_x(L,y,t)=0,t\in (0,T)\},$$

$$u\in C([0,T];D(A)),u_t\in L^{\infty }(0,T;L^2(\Omega )),T>0.$$

$$u(t)=S(t)u_0+{\int }_0^{t}S(t-s)f(s)ds$$

$$u\in L^{\infty }\left(0,T;H_0^1(\Omega )\cap H^2(\Omega )\right),$$

$$\Vert u_t\Vert (t)\le \Vert A{u}_0\Vert +\Vert f\Vert (0)+\Vert f_t{\Vert }_{L_t^1{L}_{xy}^2},$$

$$\Vert Au\Vert (t)\le \Vert u_t\Vert (t)+\Vert f\Vert (t).$$

$$\displaystyle\|u\|_{L^{\infty}\left(0,T;H^{1}_{0}\cap H^{2}(\Omega)\right)}(t)\leq C\big{(}\|u\|_{L^{\infty}_{t}L^{2}_{xy}}+\|Au_{0}\|+\|f\|(0)+\|f_{t}\|_{L^{1}_{t}L^{2}_{xy}}+\|f\|_{L^{\infty}_{t}L^{2}_{xy}}\big{)}.$$

$$Y_{T}=\{f\in L^{1}\big{(}0,T;L^{2}(\Omega)\big{)}\;\text{such that}\;f_{t}\in L^{1}\big{(}0,T;L^{2}(\Omega)\big{)}\}$$

$$\Vert f{\Vert }_{Y_T}=\Vert f{\Vert }_{L_t^1{L}_{xy}^2}+\Vert f_t{\Vert }_{L_t^1{L}_{xy}^2}.$$

$$\underset{t\in [0,T]}{\sup }\Vert f{\Vert }_{L_{xy}^2}\le C_T\Vert f{\Vert }_{Y_T}$$

$$X_{T}^{0}=\left\{u\in L^{\infty}\big{(}0,T;H^{1}_{0}(\Omega)\big{)}\,;\,\nabla u_{y}\in L^{\infty}\big{(}0,T;L^{2}(\Omega)\big{)}\,\text{ and }\,u_{xx}\in L^{2}\big{(}0,T;L^{2}(\Omega)\big{)}\right\}$$

$$\Vert u{\Vert }_{X_T^0}=\Vert u{\Vert }_{L_t^{\infty }{H}_{0xy}^1}+\Vert \nabla u_y{\Vert }_{L_t^{\infty }{L}_{xy}^2}+\Vert u_{xx}{\Vert }_{L_t^2{L}_{xy}^2}.$$

$$\displaystyle X_{T}=\{u\in X_{T}^{0}:\ u_{t}\in L^{\infty}\big{(}0,T;L^{2}(\Omega)\big{)}\;\;\text{and}\;\;\nabla u_{t}\in L^{2}\big{(}0,T;L^{2}(\Omega)\big{)}\}.$$

$$\Vert u{\Vert }_{X_T}=\Vert u{\Vert }_{X_T^0}+\Vert u_t{\Vert }_{L_T^{\infty }{L}_{xy}^2}+\Vert \nabla u_t{\Vert }_{L_T^2{L}_{xy}^2}.$$

$$\Vert u{\Vert }_{L_T^{\infty }{H}_0^1\cap H_{xy}^2}+\Vert u_t{\Vert }_{L_T^{\infty }{L}_{xy}^2}\le C_1\Vert f{\Vert }_{Y_T}.$$

$$\Vert u{\Vert }_{X_T^0}+\Vert u_t{\Vert }_{L_T^{\infty }{L}_{xy}^2}\le C_2\Vert f{\Vert }_{Y_T}.$$

$$\frac{d}{dt}\left((1+x),u_t^2\right)(t)+\Vert \nabla u_t{\Vert }^2(t)+2\Vert u_{xt}{\Vert }^2+{\int }_{-B}^B{u}_{xt}^2(0,y,t)dy=\Vert u_t{\Vert }^2(t)+2{\int }_{\Omega }(1+x)f_t{u}_{t}d\Omega .$$

Full text

∗Partially supported by CAPES

1. Introduction

We are concerned with initial-boundary value problems (IBVPs) posed on bounded rectangles located at the right half-plane $\{(x,y)\in\mathbb{R}^{2}:\ x>0\}$ for the modified Zakharov-Kuznetsov (mZK) equation [13]

[TABLE]

This equation is a generalization [12] of the classical Zakharov-Kuznetsov (ZK) equation [20] which is a two-dimensional analog of the well-known modified Korteweg-de Vries (mKdV) equation [1].

Note that both ZK and mZK possess real plasma physics applications [20, 7].

As far as ZK is concerned, the results on both IVP and IBVPs can be found in [4, 5, 6, 10, 12, 14, 15, 17, 18, 19]. For IVP to mZK, see [13]; at the same time we do not know solid results concerning IBVP to mZK. The main difference between initial and initial-boundary value problems is that IVP on $\mathbb{R}^{2}$ provides (almost immediately) good estimates in $H^{1}({\mathbb{R}^{2}})$ by the conservation laws [13], while IBVP does not possesses this advantage.

Our work is motivated by [18] and provides a natural continuation of [2] where the original ZK equation was considered. There one can find out a more detailed background, descriptions of main features, and the deployed reference list.

In the present note we put forward an analysis of (1.1) posed on a bounded rectangle with homogeneous boundary conditions. Since the power is critical [12, 13], a challenge concerning the well-posedness of IBVPs appears. Section 4 provides the local results via fixed point arguments. In Section 5 we obtain global estimates which simultaneously provide the exponential decay rates of solution. These results have been proven for sufficiently small initial data, and under domain’s size restrictions. Restrictions upon the domain appear naturally due to the presence of a linear transport term $u_{x},$ see [2, 16] for details. For one-dimensional dispersive models the critical nonlinearity has been treated in [11].

2. Problem, notations and preliminaries

Let $L,B,T$ be finite positive numbers. Define $\Omega$ and $Q_{T}$ to be spatial and time-spatial domains

[TABLE]

In $Q_{T}$ we consider the following IBVP:

[TABLE]

where $u_{0}:\Omega\to\mathbb{R}$ is a given function.

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 $L^{2}(\Omega),$ and $\|\cdot\|_{H^{k}}$ stands for the norm in $L^{2}$-based Sobolev spaces. Abbreviations like $(L^{s}_{t};L^{l}_{xy})$ are also used for anisotropic spaces.

To prove the results we will apply

Lemma 2.1**.**

(V. A. Steklov)* Let $L,B>0$ and $\omega\in H_{0}^{1}(\Omega)$. Then*

[TABLE]

*and *

[TABLE]

See [2] for the proof.

The Nirenberg theorem (also often called as the Gagliardo-Nirenberg inequality) will be used in the following form:

Lemma 2.2**.**

(L. Nirenberg)* Let $U\subset\mathbb{R}^{n}$ be a bounded open set, $u\in L^{q}(U)$ and $D^{m}u\in L^{r}(U)$, $1\leq q,r\leq\infty.$ For $0\leq j\leq m,$ the following inequality holds:*

[TABLE]

where

[TABLE]

for all $\alpha$ from the interval

[TABLE]

The constant $C_{U}$ depends on $n,m,j,q,r,\alpha.$

For the proof see [3].

Corollary 2.1**.**

Let $u\in H_{0}^{1}(\Omega).$ Then for all $p\in{{\mathbb{N}}}$

[TABLE]

where

[TABLE]

The result can be proved by induction. We will also use the simple

Lemma 2.3**.**

Let $u\in H^{1}(\Omega)$ and $u_{xy}\in L^{2}(\Omega).$ Then

[TABLE]

See [2] for the proof.

3. Existence in sub-critical case

In this section we state the existence result in sub-critical case, i.e., for $\delta\in(0,1).$ Technically, we mainly follow [19]. A short motivation for this study is provided in subsection LABEL:motivation.

4. Local results

Consider the following Cauchy problem in abstract form:

[TABLE]

where $f\in L^{1}(0,T;L^{2}(\Omega))$ and $A:L^{2}(\Omega)\to L^{2}(\Omega)$ defined as $A\equiv\partial_{x}+\Delta\partial_{x}$ with the domain

[TABLE]

endowed with its natural Hilbert norm $\|u\|_{D(A)}(t)=\left(\|u\|^{2}_{L^{2}(\Omega)}(t)+\|\Delta u_{x}+u_{x}\|^{2}_{L^{2}(\Omega)}(t)\right)^{1/2}$ for all $t\in(0,T)$.

Proposition 4.1**.**

Assume $u_{0}\in D(A)$ and $f\in L^{1}_{loc}({\mathbb{R}}^{+};L^{2}(\Omega))$ with $f_{t}\in L^{1}_{loc}({\mathbb{R}}^{+};L^{2}(\Omega))$. Then problem (4.1) possesses the unique solution $u(t)$ such that

[TABLE]

Moreover, if $u_{0}\in L^{2}(\Omega)$ and $f\in L^{1}_{loc}({\mathbb{R}}^{+};L^{2}(\Omega)),$ then (4.1) possesses a unique (mild) solution $u\in C([0,T];L^{2}(\Omega))$ given by

[TABLE]

where $S(t)$ is a semigroup of contractions generated by $A.$

Corollary 4.1**.**

Under the hypothesys of Proposition 4.1, the solution $u$ in (4.2) satisfies

[TABLE]

For the proof, see [19].

Furthermore, one can get (see [8], for instance) the estimate for strong solution (4.2):

[TABLE]

and

[TABLE]

Since $D(A){\hookrightarrow}H^{1}_{0}(\Omega)\cap H^{2}(\Omega)$ compactly (see [19] for instance), we have the estimate

[TABLE]

where $C$ depends only on $\Omega$. Next, we define

[TABLE]

with the norm

[TABLE]

Remark 4.1**.**

If $f\in Y_{T},$ then $f\in C([0,T];L^{2}(\Omega))$ and the following inequality holds:

[TABLE]

where $C_{T}$ is proportional to $T$ and its positive powers [3].

Consider

[TABLE]

with the norm

[TABLE]

and define the Banach space

[TABLE]

with the norm

[TABLE]

Theorem 4.1**.**

Let $\delta=1$ and $u_{0}\in D(A)$. Then there exists $T>0$ such that IBVP (2.1)-(2.4) possesses a unique solution in $X_{T}$.

The proof of the Theorem consists in three lemmas below.

Lemma 4.1**.**

The function $Y_{T}\longrightarrow X_{T};\ \ f\mapsto\int_{0}^{t}S(t-s)f(s)ds$ is well defined and continuous.

For the proof, note that this function maps $f$ to the solution of homogeneous linear problem with zero initial datum. Estimates (4.5) and (4.7) then give

[TABLE]

Hence,

[TABLE]

Thus, it rests to estimate the term $\|\nabla u_{t}\|_{L^{2}_{T}L^{2}_{xy}}$ in (4.9).

Differentiate the equation in (4.1) with respect to $t,$ multiply it by $(1+x)u_{t}$ and integrate the outcome over $\Omega.$ The result reads

[TABLE]

Hölder’s inequality and (4.5) imply

[TABLE]

Using the equation from (4.1) and taking in mind that $u_{0}\equiv 0$, we get

[TABLE]

Inserting (4.14) into (4.13) provides

[TABLE]

where $K_{T}=\max\{1,C_{T}\}$. Therefore, estimates (4.10) and (4.15) read

[TABLE]

Lemma 4.2**.**

The function

[TABLE]

is well defined and continuous.

The proof follows the same steps as Lemma 4.1, taking into account that now $f\equiv 0$. The resulting estimate is

[TABLE]

where $M$ is given by

[TABLE]

and $C$ (which depends only on $\Omega$) is defined by continuous immersion $D(A)\hookrightarrow H^{1}_{0}(\Omega)\cap H^{2}(\Omega).$

Lemma 4.3**.**

Given $R>0$, consider the closed ball $B_{R}=\{u\in X_{T};\|u\|_{X_{T}}\leq R\}.$ If $R>0$ is sufficiently small, then the operator

[TABLE]

is the contraction.

Fix $R>0$ and $u,v\in B_{R}.$ We have

[TABLE]

so that (4.16) implies

[TABLE]

We study the right-hand norm in detail:

[TABLE]

First, we write

[TABLE]

For the integral $I_{1}$ one has

[TABLE]

Nirenberg’s inequality gives

[TABLE]

where $D$ is the Poincare’s constant from $\|w\|\leq D\|\nabla w\|.$ Since $u$ and $v$ lie in $B_{R},$ we conclude

[TABLE]

The integral $I_{2}$ can be treated in the similar way as $I_{1}$. It rests to estimate the integral $J$.

[TABLE]

For $J_{1}$ we have

[TABLE]

Niremberg’s inequality implies

[TABLE]

The integrals $J_{3}$ and $J_{4}$ are analogous to $J_{1}$. To get bound for $J_{5}$ we observe that

[TABLE]

The integral $J_{2}$ follows like $J_{5}$. Thus,

[TABLE]

Finally, choosing $T>0$ such that $KK^{*}T^{\frac{1}{3}}R^{2}<1,$ we conclude that $\Phi$ is a contraction map.

Lemma 4.3 is proved.

Let $u\in B_{R}$. If $R=2M\|u_{0}\|_{D(A)},$ then estimates (4.17) and (4.29) with $v\equiv 0$ assure

[TABLE]

Setting $T>0$ such that $KK^{*}T^{\frac{1}{3}}R^{3}<\frac{R}{2},$ one get

[TABLE]

Choose $T>0$ such that $KK^{*}T^{\frac{1}{3}}R^{2}<1$ and $KK^{*}T^{\frac{1}{2}}R^{3}<\frac{R}{2}.$ Then $\Phi$ is the contraction from the ball $B_{R}$ into itself. Therefore, the Banach fixed point theorem assures the existence of a unique element $u\in B_{R}$ such that $\Phi(u)=u.$

This completes the proof of Theorem 4.1.

5. Global estimates and decay

Theorem 5.1**.**

Let $B,L>0$ and $u_{0}(x,y)$ be such that

[TABLE]

Suppose $u_{0}\in D(A)$ satisfies

[TABLE]

and

[TABLE]

Then for all $T>0$ there exists a unique solution $u\in X_{T}$ to problem (2.1)-(2.4); more precisely,

[TABLE]

[TABLE]

Moreover, there exist constants $C>0$ and $\gamma>0$ such that

[TABLE]

and, in addition,

[TABLE]

[TABLE]

Let $u\in X_{T_{0}}$ be a local solution given by Theorem (4.1). We are going to obtain a priori estimates independent of $T_{0}$ in order to extend the solution to all $T>0.$

5.1. Estimate I

We start the proof of (5.2), multiplying (2.1) by $u$ and integrating over $Q_{t},$ which easily gives

[TABLE]

5.2. Estimate II

Multiplying (2.1) by $(1+x)u$ and integrating over $\Omega,$ we have

[TABLE]

For the integral $I_{1}=\frac{1}{2}\int_{\Omega}u^{4}=\frac{1}{2}\|u\|^{4}_{L^{4}(\Omega)}(t),$ Nirenberg’s inequality yields

[TABLE]

Take

[TABLE]

For all $\varepsilon>0$ we have

[TABLE]

Lemma 2.1 jointly with (5.2) and (5.5) provides

[TABLE]

Define

[TABLE]

The result for (5.2) reads

[TABLE]

If $0\leq\varepsilon-2\|u_{0}\|^{2},$ then

[TABLE]

and consequently

[TABLE]

5.3. Estimate III

Write (5.2) as

[TABLE]

Then

[TABLE]

Note for posterior use that $\nabla u$ is estimated by $u$ e $u_{t}$ provided $u_{0}$ be sufficiently small in $L^{2}(\Omega)$:

[TABLE]

where

[TABLE]

5.4. Estimate IV

Differentiate the equation with respect to $t$ and multiply the result by $(1+x)u_{t}$. Integrating over $\Omega$ then gives

[TABLE]

We have

[TABLE]

Hölder and Nirenberg’s inequalities provide

[TABLE]

Nirenberg’s inequality for $p=8$ then implies

[TABLE]

Taking $k=\frac{4}{3}$ requires $l=4$ and by Young’s inequality this reads

[TABLE]

where

[TABLE]

and by Corollary 2.1 we obtain that

[TABLE]

Thus,

[TABLE]

Using (5.12) we have

[TABLE]

Backing to (5.14), we get

[TABLE]

with

[TABLE]

The use of Steklov’s inequality gives

[TABLE]

Setting $z(t)=\left((1+x),u_{t}^{2}\right)(t),$ (5.4) reads

[TABLE]

where

[TABLE]

Next we compute $\frac{d}{dt}z(0)$ to show that $z(t)$ reaches a local (lateral) maximum at $t=0.$ In order $\frac{d}{dt}z(0)$ to be negative, it should be

[TABLE]

Without loss of generality one can assume $\left((1+x),u_{t}^{2}\right)(0)>0.$ Choose $u_{0}\in D(A)$ such that (5.1) holds, i.e., $\omega(0)<\frac{2\pi^{2}}{L^{2}}-1.$ Then

[TABLE]

which assures

[TABLE]

This means that $\left((1+x),u_{t}^{2}\right)(0)$ is a local (left-hand) straight maximum.

Observe that

[TABLE]

Therefore,

[TABLE]

Consequently,

[TABLE]

Integrating the inequality (5.4) gives

[TABLE]

Then $\frac{d}{dt}\left((1+x),u_{t}^{2}\right)(T_{0})<0.$

Using (5.33) in (5.4), we have

[TABLE]

Finally,

[TABLE]

where

[TABLE]

Integrating (5.4) over $[0,T],$ we obtain that

[TABLE]

Backing to(5.12) gives

[TABLE]

where

[TABLE]

with $\gamma_{0}$ defined in (5.9) and

[TABLE]

5.5. Estimate V

Multiply the equation by $(1+x)u_{yy}$ and integrate over $\Omega$. The result reads

[TABLE]

We have

[TABLE]

Hölder and Nirenberg’s inequalities imply

[TABLE]

In the same manner, the Hölder and Nirenberg inequalities with $p=8$ provide

[TABLE]

Setting $k=\frac{4}{3}$ and $l=4$ and applying generalized Young’s inequality we come to

[TABLE]

where

[TABLE]

Hence,

[TABLE]

In turn, (5.43) becomes

[TABLE]

where

[TABLE]

5.6. Estimate VI

Now we have to estimate traces $u_{x}(0,y,t),u_{xy}(0,y,t)$ e $u_{xx}(L,y,t)$ in order to obtain the estimate for $\nabla u_{x}\in L^{\infty}(0,T;L^{2}(\Omega)).$ From (5.11) we deduce that

[TABLE]

Now (5.50) becomes

[TABLE]

Multiply it by $x$ and integrate over $(0,L).$ The result reads

[TABLE]

Therefore

[TABLE]

Then

[TABLE]

For the latter right-hand norm we write

[TABLE]

with

[TABLE]

Finally,

[TABLE]

where

[TABLE]

5.7. Estimate VII

Differentiate the equation with respect to

$x$, multiply by $(1+x)u_{x}$ and integrate over $\Omega.$ The result is

[TABLE]

Inserting this into the equations gives

[TABLE]

Substituting (5.7) into (5.7) provides

[TABLE]

As a consequence, we have

[TABLE]

Thanks to (5.40), (5.52), (5.53), (5.6 ) and (5.59 ), one concludes

[TABLE]

where

[TABLE]

Integrate (5.7) in $t\in[0,T].$ The result reads

[TABLE]

Note that all the constants $K_{i}$ are proportional to $\|u_{0}\|_{D(A)}.$ Since all the estimates do not depend upon the $T_{0},$ the local solution $u\in X_{T_{0}}$ can be continued for all $T>0$ with the decay rate described above.

The uniqueness of solution $u\in X_{T}$ is proven by the usual way, using similar computations as in lemma 4.3.

The proof of Theorem 5.1 is completed.