The 2D Zakharov-Kuznetsov-Burgers equation on a strip

An initial-boundary value problem for the 2D Zakharov-Kuznetsov-Burgers equation posed on a channel-type strip was considered. The existence and uniqueness results for regular and weak solutions in weighted spaces as well as exponential decay of small solutions without restrictions on the width of a strip were proven both for regular solutions in an elevated norm and for weak solutions in the $L^2$-norm.


Introduction
We are concerned with an initial-boundary value problem (IBVP) for the two-dimensional Zakharov-Kuznetsov-Burgers (ZKB) equation posed on a strip modeling an infinite channel {(x, y) ∈ R 2 : x ∈ R, y ∈ (0, B), B > 0}. This equation is a two-dimensional analog of the well-known Korteweg-de Vries-Burgers (KdV) equation which includes dissipation and dispersion and has been studied by various researchers due to its applications in Mechanics and Physics [1,2,3]. One can find extensive bibliography and sharp results on decay rates of solutions to the Cauchy problem (IVP) for (1.2) in [1]. Exponential decay of solutions to the initial problem for (1.2) with additional damping has been established in [3]. Equations (1.1) and (1.2) are typical examples of so-called dispersive equations which attract considerable attention of both pure and applied mathematicians in the past decades. Quite recently, the interest on dispersive equations became to be extended to multi-dimensional models such as Kadomtsev-Petviashvili (KP) and Zakharov-Kuznetsov (ZK) equations [23]. As far as the ZK equation and its generalizations are concerned, the results on IVPs can be found in [5,10,16,17,18,19,22] and IBVPs were studied in [4,6,9,14,15,22]. In [14,15] was shown that IBVP for the ZK equation posed on a half-strip unbounded in x direction with the Dirichlet conditions on the boundaries possesses regular solutions which decay exponentially as t → ∞ provided initial data are sufficiently small and the width of a half-strip is not too large. This means that the ZK equation may create an internal dissipative mechanism for some types of IBVPs. The goal of our note is to prove that the ZKB equation on a strip also may create a dissipative effect without adding any artificial damping. We must mention that IBVP for the ZK equation on a strip (x ∈ (0, 1), y ∈ R) has been studied in [4,21] and IBVPs on a strip (y ∈ (0, L), x ∈ R) for the ZK equation were considered in [8] and for the ZK equation with some internal damping in [7]. In the domain (y ∈ (0, B), x ∈ R, t > 0), the term u x in (1.1) can be scaled out by a simple change of variables. Nevertheless, it can not be safely ignored for problems posed both on finite and semi-infinite intervals as well as on infinite in y direction bands without changes in the original domain [4,20].
The main results of our paper are the existence and uniqueness of regular and weak global-in-time solutions for (1.1) posed on a strip with the Dirichlet boundary conditions and the exponential decay rate of these solutions as well as continuous dependence on initial data. The paper has the following structure. Section 1 is Introduction. Section 2 contains formulation of the problem. In Section 3, we prove global existence and uniqueness theorems for regular solutions in some weighted spaces and continuous dependence on initial data. In Section 4, we prove exponential decay of small regular solutions in an elevated norm corresponding to the H 1 (S)-norm. In Section 5, we prove the existence, uniqueness and continuous dependence on initial data for weak solutions as well as the exponential decay rate of the L 2 (S)-norm for small solutions without limitations on the width of the strip.
Hereafter subscripts u x , u xy , etc. denote the partial derivatives, as well as ∂ x or ∂ 2 xy when it is convenient. Operators ∇ and ∆ are the gradient and Laplacian acting over S. By (·, ·) and · we denote the inner product and the norm in L 2 (S), and · H k stands for norms in the L 2 -based Sobolev spaces. We will use also the spaces H s ∩ L 2 b , where L 2 b = L 2 (e 2bx dx), see [11]. Consider the following IBVP:

Existence of regular solutions
Approximate solutions. We will construct solutions to (2.1)-(2.3) by the Faedo-Galerkin method: let w j (y) be orthonormal in L 2 (S) eigenfunctions of the following Dirichlet problem: Define approximate solutions of (2.1)-(2.3) as follows: where g j (x, t) are solutions to the following Cauchy problem for the system of N generalized Korteweg-de Vries equations: It is known that for g j (x, 0) ∈ H s , s ≥ 3, the Cauchy problem (3.4)-(3.5) has a unique regular solution [1,11,12]. To prove the existence of global solutions for (2.1)-(2.3), we need uniform in N global in t estimates of approximate solutions u N (x, y, t). Estimate I. Multiply the j-th equation of (3.4) by g j , sum up over j = 1, ..., N and integrate the result with respect to x over R to obtain d dt It follows from here that for N sufficiently large and ∀t > 0 In our calculations we will drop the index N where it is not ambiguous. Estimate II. For some positive b, multiply the j-th equation of (3.4) by e 2bx g j , sum up over j = 1, ..., N and integrate the result with respect to x over R. Dropping the index N, we get In our calculations, we will frequently use the following multiplicative inequalities [13]: ii) For all u ∈ H 1 (D)

10)
where the constant C D depends on a way of continuation of u ∈ H 1 (D) asũ(R 2 ) such thatũ(D) = u(D).
Extending u N (x, y, t) for a fixed t into exterior of S by 0 and exploiting the Gagliardo-Nirenberg inequality (3.9), we find Substituting this into (3.8), we come to the inequality By the Gronwall lemma, (e 2bx , u 2 )(t) ≤ C(b, T, u o )(e 2bx , u 2 0 ). Returning to (3.11) gives It follows from this estimate and (3.6) that uniformly in N and for any r > 0 and t ∈ (0, T ) where C does not depend on N. Estimates (3.12), (3.13) make it possible to prove the existence of a weak solution to (2.1)-(2.3) passing to the limit in (3.4) as N → ∞. For details of passing to the limit in the nonlinear term see [11].
Estimate III. Multiplying the j-th equation of (3.4) by −(e 2bx g jx ) x , and dropping the index N, we come to the equality 14) Making use of Proposition 3.1, we estimate Similarly, Substituting I 1 , I 2 into (3.14) and taking 2δ = b, we obtain for ∀t ∈ (0, T ) : Estimate IV. Multiplying the j-th equation of (3.4) by −2(e 2bx λg j ), and dropping the index N, we come to the equality Making use of Proposition 3.1, we estimate Taking . Making use of (3.7) and the Gronwall lemma, we get ∀t ∈ (0, T ) : This and (3.15) imply that for all finite r > 0 and all t ∈ (0, T ) Estimate V. Multiplying the j-th equation of (3.4) by (e 2bx g jxx ) xx , and dropping the index N, we come to the equality Using (3.9), we find Taking 2δ = b and substituting I into (3.18), we obtain Taking into account (3.7), we find Estimate VI. Differentiate (3.4) by t and multiply the result by e 2bx gj t to obtain Making use of (3.9), we estimate . This implies ∀t ∈ 0, T ): where Multiplying the j-th equation of (3.4) by −e 2bx g jx , we come, dropping the index N, to the equality Making use of (3.9), we estimate . Taking 4δ = 1, using (3.15)-(3.21) and substituting I into (3.22), we get Estimate VIII. We will need the following lemma : where δ, δ 1 are arbitrary positive numbers.
Returning to the function u(x, y, t), we prove Lemma 3.2 Multiplying the j-th equation of (3.4) by e 2bx g jxxx , we come, dropping the index N, to the equality Using Lemma 3.2 and (3.7), we estimate (3.26) Taking ǫ and δ sufficiently small, positive and substituting I into (3.25), we find Consequently, it follows from the equality In other words, and these inclusions are uniform in N. Estimate IX. Differentiating the j-th equation of (3.4) with respect to x and multiplying the result by e 2bx ∂ 4 x g j , we come, dropping the index N, to the equality Applying the Young inequality, taking ǫ 1 , δ sufficiently small positive, substituting I 1 , I 2 into (3.31) and integrating the result, we come to the following inequality: Estimate X. Multiplying the j-th equation of (3.4) by −e 2bx λ 2 g jx , we come, dropping the index N, to the equality (e 2bx , u 2 xxyy + u 2 xyyy )(t) = −(e 2bx , u ty , u 2 xyyy )(t) + (b + 2b 2 )(e 2bx , u 2 xyy )(t) − (e 2bx u y u x , u xyyy )(t) + (e 2bx uu xy , u xyyy )(t). (3.34) We estimate Choosing ǫ, ǫ 1 , δ sufficiently small, positive, after integration, we transform (3.34) into the form Acting similarly, we get from the scalar product Then there exists a regular solution to (2.1)-(2.3) u(x, y, t) : which for a.e. t ∈ (0, T ) satisfies the identity where φ(x, y) is an arbitrary function from L 2 (S).

Proof. Rewrite (3.4) in the form
where Φ N (y) is an arbitrary function from the set of linear combinations N i=1 α i w i (y) and Ψ(x) is an arbitrary function from H 1 (R). Taking into account estimates (3.7), (3.37) and fixing Φ N , we can easily pass to the limit as N → ∞ in linear terms of (3.39). To pass to the limit in the nonlinear term, we must use (3.17) and repeat arguments of [11]. Since linear combinations [ N i=1 α i w i (y)]Ψ(x) are dense in L 2 (S), we come to (3.38). This proves the existence of regular solutions to (2.1)-(2.3). Remark 1. Estimates (3.7),(3.37) are valid also for the limit function u(x, y, t) and (3.7) obtains its sharp form: Uniqueness of a regular solution.

Decay of regular solutions
In this section we will prove exponential decay of regular solutions in an elevated weighted norm corresponding to the H 1 (S) norm. We start with Theorem 4.1 which is crucial for the main result.
After a corresponding process of scaling we prove Proposition 4.2.
The last inequality implies (4.1). The proof of Theorem 4.1 is complete.
Observe that differently from [14,15], we do not have any restrictions on the width of a strip B.
The main result of this section is the following assertion.
Proof. We start with the following lemma.
Summing I 1 + I 2 + I 3 , we obtain (4.9). In turn, multiplying it by e χt and integrating the result over (0, t), we come to (4.7). The proof of Lemma 4.4 is complete.
In order to estimate the right-hand side of (4.10), we will need the following By Proposition 3.1, we estimate ](e 2bx , u 2 )(t).
Using this estimate, we substitute I 1 into (4.12) and come to the following inequality: , u 2 0 ). Since b > 0, the proof of Proposition 4.5 is complete.

Weak solutions
Here we will prove the existence, uniqueness and continuous dependence on initial data as well as exponential decay results for weak solutions of (2.1)-(2.3) when the initial function u 0 ∈ L 2 (S).
Then for all finite positive T and B there exists at least one function u(x, y, t) ∈ L ∞ (0, T ; L 2 (S)), u x ∈ L 2 (0, T ; L 2 (S)) such that e bx u ∈ L ∞ (0, T ; L 2 (S)) ∩ L 2 (0, T ; H 1 (S)) and the following integral identity takes a place: where v ∈ C ∞ (S T ) is an arbitrary function.
Proof. In order to justify our calculations, we must operate with sufficiently smooth solutions u m (x, y, t). With this purpose, we consider first initial functions u 0m (x, y), which satisfy conditions of Theorem 3.3, and obtain estimates (3.7), (3.17) for functions u m (x, y, t). This allows us to pass to the limit as m → ∞ in the following identity: and come to (5.2).
Uniqueness of a weak solution.
Theorem 5.2. A weak solution of Theorem 5.1 is uniquely defined.
Hence, Since e bx z(x, y, t) is a weak limit of regular solutions {e bx z m (x, y, t)}, then (e 2bx , z 2 )(t) ≤ (e 2bx , z 2 m )(t) = 0. This implies u 1 ≡ u 2 a.e. in S T . The proof of Theorem 5.2 is complete.
We have in this Theorem a more strict condition u 0 ≤ 3π 16B instead of u 0 ≤ 3π 8B in the case of decay for regular solution because for weak solutions we do not have the sharp estimate (3.40), but only (3.7).