Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 11, pp. 1–41. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu SMOOTHING PROPERTIES FOR A COUPLED ZAKHAROV-KUZNETSOV SYSTEM JULIE L. LEVANDOSKY, OCTAVIO VERA Abstract. In this article we study the smoothness properties of solutions to a two-dimensional coupled Zakharov-Kuznetsov system. We show that the equations dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data (u0, v0) possesses certain regularity and sufficient decay as x → ∞, then the solution (u(t), v(t)) will be smoother than (u0, v0) for 0 < t ≤ T where T is the existence time of the solution. 1. Introduction The general form of the coupled Zakharov-Kuznetsov system [12] is ut + uxxx + uyyx − 6uux − vx = 0 (1.1) vt + δvxxx + λvyyx + ηvx − 6µvvx − ωux = 0. (1.2) This coupled system is a model describing two interacting weakly nonlinear waves in anisotropic media. Here, x and y are the propagation and transverse coordinates respectively, η is a group velocity shift between the coupled models, δ and λ are the relative longitudinal and transverse dispersion coefficients, and µ and ω are the relative nonlinear and coupled coefficients. In the absence of the transverse variation (i.e. - uy = vy = 0), this system reduces to the set of coupled KdV equations [7] which are known to describe the interaction of nonlinear long waves in certain fluid flows. In this article, we study (1.1)-(1.2) when the dispersion coefficients, δ and λ, and the coupling coefficient ω are positive. In that case, it suffices to consider the initial-value problem ut + uxxx + uyyx − 6uux − vx = 0 bvt + δvxxx + λvyyx + ηvx − 6µvvx − ux = 0 u(x, y, 0) = u0(x, y), v(x, y, 0) = v0(x, y) (1.3) where b > 0, δ > 0, and λ > 0. A number of results concerning gain of regularity for various nonlinear evolution equations have appeared. Cohen [4] considered the KdV equation, showing that “boxshaped” initial data φ ∈ L2(R2) with compact support lead to a solution u(t) which is smooth for t > 0. Kato [13] generalized this result, showing that if the initial data φ are in L2((1 + eσx)dx), the unique solution u(t) ∈ C∞(R2) for t > 0. 2020 Mathematics Subject Classification. 35Q53, 35Q35, 47J35. Key words and phrases. Coupled Zakharov-Kuznetsov system; gain in regularity; weighted Sobolev space. ©2023. This work is licensed under a CC BY 4.0 license. Submitted April 18, 2022. Published February 4, 2023. 1 2 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 Kruzhkov and Faminskii [15] replaced the exponential weight function with a poly- nomial weight function, quantifying the gain in regularity of the solution in terms of the decay at infinity of the initial data. Craig, Kappeler, and Strauss [6] expanded on the ideas from these earlier papers in their treatment of highly generalized KdV equations. Other results on gain of regularity for linear and nonlinear dispersive equations include the works of Hayashi, Nakamitsu, and Tsutsumi [9, 10], Hayashi and Ozawa [11], Constantin and Saut [5], Ponce [21], Ginibre and Velo [8], Kenig, Ponce, and Vega [14], and Vera [23]. Smoothing properties for coupled systems of nonlinear dispersive equations in one-spatial dimension were proven by Vera [22], Ceballos, Sepulveda, and Vera [3], and Alves and Vera Villagrán [1]. Here we treat a coupled system of nonlinear dispersive equations in two spatial dimensions. In studying propagation of singularities, it is natural to consider the bicharacter- istics associated with the differential operator. For the KdV equation, it is known that the bicharacteristics all point to the left for t > 0, and all singularities travel in that direction. Kato [13] makes use of this uniform dispersion, choosing a non- symmetric weight function decaying as x → −∞ and growing as x → ∞. In [6], Craig, Kappeler and Strauss also make use of a unidirectional propagation of sin- gularities in their results on infinite smoothing properties for generalized KdV-type equations for which fuxxx ≥ c > 0. For the two-dimensional case, Levandosky [16] proves smoothing properties for the KP-II equation. This result makes use of the fact that the bicharacteristics all point into one half-plane. Subsequently, Levandosky [17] considers generalized KdV-type equations in two dimensions, proving that if all bicharacteristics point into one half-plane, an infinite gain in regularity will occur, assuming sufficient de- cay at infinity of the initial data. Levandosky Sepulveda and Vera Villagran [19] proved a smoothing property for the KP-I equation. Since the bicharacteristics do not all point into the same half-plane, singularities may travel in all of R2. Conse- quently, the same proof techniques used above do not generalize to this equation. However, they are able to prove a finite gain in regularity. Levandosky In [18] proved smoothing properties for solutions to the fifth-order Kawahara equation in two spatial dimensions. In this paper, we extend the ideas discussed above to prove a gain in regularity result to a Zakharov-Kuznetsov system (1.3), a nonlinear dispersive system in two spatial dimensions. Specifically, we quantify the gain in regularity of the solution (u(t), v(t)) in relation to the decay of the initial data. In particular, we prove that if the initial data has sufficient regularity and decays sufficiently as x → ∞, then the solution (u(t), v(t)) ∈ C∞(R2)×C∞(R2) for 0 < t ≤ T where T is the existence time of the solution. We now state this more formally in a special case of our main theorem on the gain of regularity for the Zakharov-Kuznetsov system. Gain of regularity theorem. Consider a coupled system of the form (1.3) where b, δ, λ > 0. Let (u, v) be a solution of (1.3) in R2 × [0, T ] such that for all integers L ≥ 1, sup 0≤t≤T ∫ R2 (1 + x+)L ∑ |a|≤3 [(∂αu)2 + (∂αv)2] dx dy < +∞. Then our solution (u(t), v(t)) ∈ C∞(R2)× C∞(R2) for 0 < t ≤ T . EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 3 As will be shown, the assumption that sup 0 0, these terms have positive signs, thus, allowing us to prove a gain in regularity. We continue this procedure inductively. On each step, β, of the induction, we take α derivatives of (1.3)1 and (1.3)2 where |α| = α1 + α2 = β. We then multiply the differentiated equations by 2ξ(∂u) and 2ξ(∂v), respectively, where ∂α ≡ ∂α1 x ∂α2 y and ξ = ξβ is our weight function to be described below. Integrating over R2 and integrating by parts as described above, if our weight function ξ satisfies 0 < ∂jxξ ≤ C∂kxξ for all j ≥ k ≥ 0, we arrive at the following inequality. ∂t ∫ ξ((∂αu)2 + b(∂αv)2) + C ∫ ξx((∂αux)2 + (∂αvx)2) + C ∫ ξx((∂αuy)2 + (∂αvy)2) ≤ C ∫ ξt((∂ αu)2 + (∂αv)2) + C ∫ ξ((∂αu)2 + (∂αv)2) + C ∣∣ ∫ ξ(∂αu)∂α(uux) ∣∣ + C ∣∣ ∫ ξ(∂αv)∂α(vvx) ∣∣+ 2 ∫ ξ[(∂αu)(∂αv)]x. Choice of weight function. In what follows, we will be proving that if our initial data decays sufficiently as x → ∞, then the solution will experience a gain in regularity. Consequently, we will choose weight functions which behave like powers of x for x > 1. Since the bicharacteristics point into the left half-plane, it is natural to choose weight functions which decay as x→ −∞. We will choose weight functions which behave like eσx where σ ≥ 0 for x < −1. We define the classes of weight functions as follows. Definition 2.1. A function ξ = ξ(x, t) belongs to the weight class Wσ i k if it is a positive C∞ function on R × [0, T ], ξx > 0, and there are constants cj , 1 ≤ j ≤ 5 such that 0 < c1 ≤ t−ke−σ xξ(x, t) ≤ c2 ∀x < −1, 0 < t < T, 0 < c3 ≤ t−k x−iξ(x, t) ≤ c4 ∀x > 1, 0 < t < T, (t |ξt|+ |∂jxξ|)/ξ ≤ c5 ∀ (x, t) ∈ R× [0, T ], ∀ j ∈ Z+. (2.5) We now define weighted function spaces using the weight functions introduced above. Definition 2.2. Let N be a positive integer. Let Hβ(Wσ i k) be the space of functions with finite norm Hβ(Wσ i k) = { v : R2 → R : ‖v‖2Hβ(Wσ i k) = ∫ R2 ∑ |α|≤β (∂αv) 2 |ξ(x, y)| <∞ } (2.6) for any ξ ∈Wσ i k, β ≥ 0, and 0 ≤ t ≤ T . Remark 2.3. We note that although the norm above depends on ξ, all choices of ξ in this class lead to equivalent norms. The usual Sobolev space is HN (R2) without a weight. Definition 2.4. For each fixed ξ ∈Wσ i k, β ≥ 0, we define the space Lp([0, T ] : Hβ(Wσ i k)) = { v(x, y, t) : ‖v‖p Lp([0,T ]:Hβ(Wσ i k)) = ∫ T 0 ‖v(·, ·, t)‖p Hβ(Wσ i k) dt < +∞ } (2.7) EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 5 L∞([0, T ] : Hβ(Wσ i k)) = { v(x, y, t) : ‖v‖2L∞([0,T ]:Hβ(Wσ i k)) = sup t∈[0,T ] ‖v(·, ·, t)‖Hβ(Wσ i k)dt < +∞ } (2.8) Moreover, we define the spaces W̃σ i k = ∪j 1, an exponential eσx where σ > 0 for x < −1 and a power of t. As we proceed inductively, the powers of x for x > 1 decrease while the powers of t increase. In particular, for β = 1, ξβ ≈ txL−1 for x > 1. For β = 2, ξβ ≈ t2xL−2 for x > 1. We continue in this way, decreasing the power of x for x > 1 and increasing the power of t on each level of the induction. Proof of Lemma 3.1. Let β ≥ 1. Let α = (α1, α2) where |α| = β. Take α deriva- tives of (1.3)1, multiply the differentiated equation by 2ξβ(∂αu) where ξβ(x, t) = ∫ x −∞ χβ(z, t)dz (3.5) for χβ ∈ Wσ,L−β−1,β , and integrate over R2 × [0, t] for 0 ≤ t ≤ T . Letting ξ ≡ ξβ , we conclude that∫ ξ(·, t) (∂αu) 2 + 3 ∫ t 0 ∫ ξx (∂αux) 2 + ∫ t 0 ∫ ξx (∂αuy) 2 = ∫ ξ(·, 0)(∂αu0)2 + ∫ t 0 ∫ [ξt + ξxxx] (∂αu) 2 + 12 ∫ t 0 ∫ ξ(∂αu)∂α(uux) + 2 ∫ t 0 ∫ ξ(∂αu)(∂αvx). (3.6) Similarly, take α derivatives of (1.3)2, multiply the differentiated equation by 2ξβ(∂αv), and integrate over R2 × [0, t] for 0 ≤ t ≤ T . Doing so, we conclude that b ∫ ξ(·, t) (∂αv)2 + 3δ ∫ t 0 ∫ ξx (∂αvx) 2 + λ ∫ t 0 ∫ ξx (∂αvy) 2 = ∫ ξ(·, 0)(∂αv0)2 + ∫ t 0 ∫ [ξt + ξxxx + ηξx] (∂αv)2 + 12µ ∫ t 0 ∫ ξ(∂αv)∂α(vvx) + 2 ∫ t 0 ∫ ξ(∂αux)(∂αv). (3.7) Then, adding (3.6) and (3.7), we have∫ ξ(·, t)[(∂αu)2 + b(∂αv)2] + 3 ∫ t 0 ∫ ξx[(∂αux)2 + δ(∂αvx)2] + ∫ t 0 ∫ ξx[(∂αuy)2 + λ(∂αvy)2] = ∫ ξ(·, 0)[(∂αu0)2 + (∂αv0)2] + ∫ t 0 ∫ [ξt + ξxxx][(∂αu)2 + (∂αv)2] + ∫ t 0 ∫ ηξx(∂αv)2 + 12 ∫ t 0 ∫ ξ(∂αu)∂α(uux) + 12µ ∫ t 0 ∫ ξ(∂αv)∂α(vvx) + 2 ∫ t 0 ∫ ξ∂x[(∂αu)(∂αv)]. EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 7 Now using the fact that ∂jxξ ≤ Cξ and ξ(·, 0) = 0 for β ≥ 1, we obtain the identity∫ ξ(·, t)[(∂αu)2 + b(∂αv)2] + 3 ∫ t 0 ∫ ξx[(∂αux)2 + δ(∂αvx)2] + ∫ t 0 ∫ ξx[(∂αuy)2 + λ(∂αvy)2] ≤ C ∫ t 0 ∫ [ξt + ξ][(∂αu)2 + (∂αv)2] + 12 ∫ t 0 ∫ ξ(∂αu)∂α(uux) + 12µ ∫ t 0 ∫ ξ(∂αv)∂α(vvx) + 2 ∫ t 0 ∫ ξ∂x[(∂αu)(∂αv)]. The first term on the right-hand side above is bounded by terms of the form (3.3) and (3.4). Integrating by parts and using the Cauchy-Schwarz inequality, we see that the last term on the right-hand side satisfies∣∣ ∫ t 0 ∫ ξ∂x[(∂αu)(∂αv)] ∣∣ = ∣∣ ∫ t 0 ∫ ξx(∂αu)(∂αv) ∣∣ ≤ C ∫ t 0 ∫ ξx[(∂αu)2 + (∂αv)2]. Each of these terms is bounded by terms of the form (3.3) and (3.4). Therefore, we conclude that∫ ξ(·, t)[(∂αu)2 + b(∂αv)2] + 3 ∫ t 0 ∫ ξx[(∂αux)2 + δ(∂αvx)2] + ∫ t 0 ∫ ξx[(∂αuy)2 + λ(∂αvy)2] ≤ C + C ∣∣ ∫ t 0 ∫ ξ(∂αu)∂α(uux) ∣∣+ C ∣∣ ∫ t 0 ∫ ξ(∂αv)∂α(vvx) ∣∣ (3.8) where C depends only on (3.3) and (3.4). Therefore, it remains to look for bounds on the remainder terms C ∣∣ ∫ t 0 ∫ ξ(∂αu)∂α(uux) ∣∣+ C ∣∣ ∫ t 0 ∫ ξ(∂αv)∂α(vvx) ∣∣ (3.9) for each β ≥ 1. Case β = 1. Let ξ ≡ ξβ = ∫ x −∞ χβ(z, t)dz where χβ ∈ Wσ,L−2,1. Therefore, ξ ≈ txL−1 for x > 1 and ξ ≈ teσx for x < −1. Subcase α = (1, 0). The remainder terms satisfies∣∣ ∫ t 0 ∫ ξux(uux)x ∣∣ = ∣∣ ∫ t 0 ∫ ξ(u3x + uxuuxx) ∣∣ ≤ C ∣∣ ∫ t 0 ∫ ξu3x ∣∣+ C ∣∣ ∫ t 0 ∫ ξu(ux)2x ∣∣ ≤ C|ux|L∞ ∫ t 0 ∫ ξu2x + C ∫ t 0 ∫ ∣∣ξxu(ux)2 + ξu3x ∣∣ ≤ C|ux|L∞ ∫ t 0 ∫ ξu2x + C|u|L∞ ∫ t 0 ∫ ξxu 2 x ≤ C(|u|L∞ + |ux|L∞) ∫ t 0 ∫ ξu2x 8 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 ≤ C ∫ t 0 ∫ ξu2x where C depends only on ‖u‖H3 . Similarly for the remainder term involving v. Combining this bound with (3.8) and the fact that ξx = χβ we conclude that sup 0≤t≤T C ∫ ξ(·, t)[u2x + v2x] + C ∫ T 0 ∫ χ[u2xx + v2xx] + C ∫ T 0 ∫ χ[u2xy + v2xy] ≤ C where χ ≡ χβ and C depends only on ‖u‖H3 , ‖v‖H3 and terms of the form (3.3) and (3.4), as desired. Subcase α = (0, 1). In this case, the remainder terms in u satisfy∣∣ ∫ t 0 ∫ ξuy(uux)y ∣∣ = ∣∣ ∫ t 0 ∫ ξuy(uyux + uuxy) ∣∣ ≤ C|ux|L∞ ∫ t 0 ∫ ξu2y + C ∣∣ ∫ t 0 ∫ ξu(u2y)x ∣∣ ≤ C|ux|L∞ ∫ t 0 ∫ ξu2y + C ∣∣ ∫ t 0 ∫ ξxuu 2 y + ξuxu 2 y ∣∣ ≤ C|ux|L∞ ∫ t 0 ∫ ξu2y + C|u|L∞ ∫ t 0 ∫ ξxu 2 y ≤ C(|u|L∞ + |ux|L∞) ∫ t 0 ∫ ξu2y ≤ C ∫ t 0 ∫ ξu2y where C depends only on ‖u‖H3 . Similarly for the terms in v. Combining these estimates with (3.8), we conclude that sup 0≤t≤T C ∫ ξ(·, t)[u2y + v2y] + C ∫ T 0 ∫ χ[u2xy + v2xy] + C ∫ T 0 χ[u2yy + v2yy] ≤ C where C depends only on ‖u‖H3 , ‖v‖H3 and terms in (3.3) and (3.4). Case β = 2. In this case, let ξ ≡ ξβ = ∫ x −∞ χβ(z, t)dz where χβ ∈ Wσ,L−3,2. Therefore, ξ ≈ t2xL−2 for x > 1 and ξ ≈ t2eσx for x < −1. Subcase α = (2, 0). In this case, the remainder terms satisfy∣∣ ∫ t 0 ∫ ξuxx(uux)xx ∣∣ = ∣∣ ∫ t 0 ∫ ξuxx(3uxuxx + uuxxx) ∣∣ ≤ C(|u|L∞ + |ux|L∞) ∫ t 0 ∫ ξu2xx ≤ C ∫ t 0 ∫ ξu2xx where C depends only on ‖u‖H3 . Subcase α = (1, 1). In this case, the remainder terms satisfy∣∣ ∫ t 0 ∫ ξuxy(uux)xy ∣∣ EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 9 = ∣∣ ∫ t 0 ∫ ξuxy(2uxuxy + uyuxx + uuxxy) ∣∣ ≤ |ux|L∞ ∫ t 0 ∫ ξu2xy + C|uy|L∞ ∫ t 0 ∫ ξu2xx + |uy|L∞ ∫ t 0 ∫ ξu2xy ≤ C ∫ t 0 ∫ ξ(u2xx + u2xy) where C depends only on ‖u‖H3 . Subcase α = (0, 2). In this case, the remainder terms satisfy∣∣ ∫ t 0 ∫ ξuyy(uux)yy ∣∣ = ∣∣ ∫ t 0 ∫ ξuyy(uyyux + 2uyuxy + uuxyy) ∣∣ ≤ C|ux| ∫ t 0 ∫ ξu2yy + C|uy|L∞ ∫ t 0 ∫ ξ(u2xy + u2yy) ≤ C ∫ t 0 ∫ ξ(u2xy + u2yy) where C depends only on ‖u‖H3 . Similarly for v. Combining these estimates with (3.8), we conclude that∑ |α|=2 sup 0≤t≤T C ∫ ξ(·, t)[(∂αu)2 + (∂αv)2] + C ∑ |α|=3 ∫ T 0 ∫ χ[(∂αu)2 + (∂αv)2] ≤ C where C depends only on ‖u‖H3(R2), ‖v‖H3(R2) and terms in (3.3) and (3.4). Case β = 3. In this case, let ξ ≡ ξβ = ∫ x −∞ χβ(z, t)dz where χβ ∈ Wσ,L−4,3. Therefore, ξ ≈ t3xL−3 for x > 1 and ξ ≈ t3eσx for x < −1. We consider the subcase α = (3, 0). The other cases can be handled similarly. Subcase α = (3, 0). The remainder terms satisfy∣∣ ∫ t 0 ∫ ξuxxx(uux)xxx ∣∣ = ∣∣ ∫ t 0 ∫ ξuxxx(4uxuxxx + 3u2xx + uuxxxx) ∣∣ = ∣∣ ∫ t 0 ∫ 4ξuxu 2 xxx + 3ξu2xxuxxx + ξuuxxxuxxxx ∣∣ ≤ C|ux|L∞ ∫ t 0 ∫ ξu2xxx + C ∫ t 0 ∫ ξu2xxuxxx + C(|u|L∞ + |ux|L∞) ∫ t 0 ∫ ξu2xxx ≤ C(|u|L∞ + |ux|L∞) ∫ t 0 ∫ ξu2xxx + C ∫ t 0 ∫ ξu2xxuxxx ≤ C ∫ t 0 ∫ ξu2xxx + C ∫ t 0 ∫ ξu2xxuxxx where C depends only on ‖u‖H3 . To handle the last term on the right-hand side above, we use (2.12). In addition, we consider the cases x > 1 and x < −1 separately. Let A = {x > 1} × R and let B = {x < −1} × R. First, for x > 1,∫ t 0 ∫ A ξu2xxuxxx 10 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 = C ∫ t 0 ∫ A t3xL−3u2xxuxxx ≤ C (∫ t 0 ∫ A t3xL−3u4xx )1/2(∫ t 0 ∫ A t3xL−3u2xxx )1/2 ≤ C (∫ t 0 ∫ A t3[x(L−3)/4uxx]4 )1/2(∫ t 0 ∫ A t3xL−3u2xxx )1/2 ≤ C (∫ t 0 ∫ A t3(x(L−3)/4uxx)2 + t3([x(L−3)/4uxx]x)2 + t3([x(L−3)/4uxx]y)2 ) × (∫ t 0 ∫ A t3xL−3u2xxx )1/2 ≤ C (∫ t 0 ∫ A t3x(L−3)/2u2xx + t3x(L−3)/2u2xxx + t3x(L−3)/2u2xxy ) × (∫ t 0 ∫ A t3xL−3u2xxx )1/2 ≤ C (∫ t 0 ∫ A ξ(u2xx + u2xxx + u2xxy) )(∫ t 0 ∫ A ξu2xxx )1/2 . Second, for x < −1,∫ t 0 ∫ B ξu2xxuxxx = ∫ t 0 ∫ B t3eσxu2xxuxxx ≤ Ct3/2 (∫ t 0 ∫ B t3e2σxu4xx )1/2(∫ t 0 ∫ u2xxx )1/2 ≤ C‖u‖H3 (∫ t 0 ∫ B t3|eσx/2uxx|4 )1/2 ≤ C (∫ t 0 ∫ B t3[(eσx/2uxx)2 + ([eσx/2uxx]x)2 + ([eσx/2uxx]y)2] )2 ≤ C (∫ t 0 ∫ B t3eσx(u2xx + u2xxx + u2xxy) )2 . Combining these estimates, we have∣∣ ∫ t 0 ∫ ξuxxx(uux)xxx ∣∣ ≤ C where C depends only on ‖u‖H3 and terms of the form (3.3). The other terms on the level β = 3 can be handled similarly. Similarly for v. Combining these estimates with (3.8), we conclude that∑ |α|=3 sup 0≤t≤T C ∫ ξ(·, t)[(∂αu)2 + (∂αv)2] + C ∑ |α|=4 ∫ T 0 ∫ χ[(∂αu)2 + (∂αv)2] ≤ C where C depends only on ‖u‖H3(R2), ‖v‖H3(R2) and terms in (3.3) and (3.4). Case β ≥ 4. For β ≥ 4, let ξ ≡ ξβ = ∫ x −∞ χβ(z, t)dz where χβ ∈ Wσ,L−β−1,β . Therefore, ξ ≈ tβxL−β for x > 1 and ξ ≈ tβeσx for x < −1. We combine Lemma 3.2 given below, in which we estimate our remainder term (3.9) with our main EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 11 inequality (3.8) to conclude that∑ |α|=β sup 0≤t≤T C ∫ ξ[(∂αu)2 + (∂αv)2] + ∑ |α|=β+1 ∫ T 0 ∫ χ[(∂αu)2 + (∂αv)2] ≤ C where C depends only on ‖u‖H3 , ‖v‖H3 and terms in (3.2)-(3.4). � We now show the bounds on the remainder (3.9) for β ≥ 4. Lemma 3.2. For 4 ≤ β ≤ L, β = |α|, and (u, v) a solution of (1.3) sufficiently smooth and with sufficient decay at infinity, for 0 ≤ t ≤ T , we have∣∣ ∫ t 0 ∫ ξβ(∂αu)∂α(uux) ∣∣ ≤ C, ∣∣ ∫ t 0 ∫ ξβ(∂αv)∂α(vvx) ∣∣ ≤ C (3.10) where ξβ ∈ Wσ,L−β,β, χβ ∈ Wσ,L−β−1,β, σ > 0 arbitrary and C depends only on ‖u‖H3(R2), ‖v‖H3(R2), and on (3.2), (3.3), and (3.4). Before proving Lemma 3.2, we describe the form of each term in (3.9). Lemma 3.3. Every term in the integrand of∫ t 0 ∫ R2 ξ(∂αu)∂α(uux) dx dy dt∫ t 0 ∫ R2 ξ(∂αv)∂α(vvx) dx dy dt (3.11) is of the form Cξ(∂αu)(∂ru)(∂sux) Cξ(∂αv)(∂rv)(∂svx) (3.12) respectively, where r = (r1, r2), s = (s1, s2), ri + si = αi for i = 1, 2. The above lemma follows from the Leibniz formula applied to ∂α(uux) and ∂α(vvx). Proof of Lemma 3.2. By Lemma 3.3 we can write every term in the integrand of (3.11) in the form (3.12) where ξ ≡ ξβ ∈ Wσ,L−β,β . It remains to show that each of these terms is bounded by constants depending only on (3.2)-(3.4). In this part we draw our attention to the case when x > 1. For x > 1, our weight function ξ ' tk x` for k, ` ≥ 0. For the case x < −1, ξ ' tk eσ x for σ > 0 arbitrary. That case is even easier to handle. Let A = {(x, y) : x > 1}. Then for x > 1, we are looking to get bounds on (3.11) and (3.12) in terms of sup 0≤t≤T ∫ A tνxL−ν [(∂γu)2 + (∂γv)2], (3.13)∫ T 0 ∫ A tνxL−ν−1[(∂γux)2 + (∂γvx)2], (3.14)∫ T 0 ∫ A tνxL−ν−1[(∂γuy)2 + (∂γvy)2] (3.15) where β − 1 ≥ ν = |γ| ≥ 0. We need to break up each term of the form (3.12) into three parts, being sure to divide the weight function appropriately among the three terms. To do so, we combine ‖∂γu‖L∞(R2) ≤ (∫ [(∂γu)2 + (∂γuxx)2 + (∂γuyy)2] )1/2 (3.16) 12 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 with tkx`(∂γu) = tk γ1∑ j=0 (−1)j ( γ j ) ∂jx[(∂γ1−jx x`)(∂γ2y u)]. (3.17) In fact, ‖tkx`(∂γu)‖L∞(R2) ≤ tk γ1∑ j=0 ∥∥(γ j ) ∂jx[(∂γ1−jx x`)(∂γ2y u)] ∥∥ L∞(R2) . (3.18) Hence, combining (3.18) with (3.16) we conclude that sup 0≤t≤T ‖t(ν+1)/2 x(L−(ν+1))/2(∂γu)‖L∞(A) ≤ C (3.19) for ν = |γ| ≤ β − 3, and∫ T 0 ‖t(ν+1)/2 x(L−(ν+1)−1)/2 (∂γu)‖L∞(A) ≤ C (3.20) for ν = |γ| ≤ β − 2, where the constant C depends only (3.13), (3.14), (3.15). Similarly for v. Then using the above inequalities, we look at terms of the form (3.12). For x > 1, these terms can be expressed as follows∣∣ ∫ t 0 ∫ A ξ(∂αu)(∂ru)(∂sux) ∣∣ ≤ C ∫ T 0 ∣∣ ∫ A tβxL−β(∂αu)(∂ru)(∂sux) ∣∣. (3.21) For notation, let νr = r1 + r2 and νs = s1 + s2. Since β = α1 + α2, it follows that νr + νs = β. Case vs ≤ β − 4. In this case, we bound the remainder term as follows∫ T 0 ∣∣ ∫ A tβxL−β(∂αu)(∂ru)(∂sux) ∣∣ ≤ sup x>1 xN TM sup 0≤t≤T ‖t(νs+2)/2x(L−νs−1)/2(∂sux)‖L∞(A) × (∫ T 0 ∫ A t(νr−1)xL−(νr−1)−1(∂ru)2 )1/2( ∫ T 0 ∫ A tβ−1xL−(β−1)−1(∂αu)2 )1/2 . First, we show that M ≥ 0 and N ≤ 0, so that any extra powers of t or x can be thrown away. We see that β = M + νs + 2 2 + νr − 1 2 + β − 1 2 . Therefore, 2β = 2M + νs + 2 + νr − 1 + β − 1. Therefore, 2M = β − νs − νr = 0. Therefore, M = 0. Next L− β = N + L− µs − 1 2 + L− (νr − 1)− 1 2 + L− (β − 1)− 1 2 . Therefore, 2L− 2β = 2N + 3L− νs − νr − β − 1, which implies −L+ 1 = 2N . Therefore, N = −L+ 1 2 ≤ 0 EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 13 as long as L ≥ 1. The others three terms are bounded by (3.19), (3.14) and (3.15). Case β − 4 < νs ≤ β − 2. In this case, we have∫ T 0 ∫ tβxL−β(∂αu)(∂ru)(∂sux) ≤ CTMxN (∫ T 0 ∫ tβ−1xL−(β−1)−1(∂αu)2 )1/2(∫ T 0 ∫ tνrxL−νr−1(∂ru)4 )1/4 × (∫ T 0 ∫ tνs+1xL−(νs+1)−1(∂sux)4 )1/4 ≤ CTMxN (∫ T 0 ∫ tβ−1xL−(β−1)−1(∂αu)2 )1/2 × (∫ T 0 ∫ tνrxL−νr−1[(∂rux)2 + (∂ruy)2] )1/2 × (∫ T 0 ∫ tνs+1xL−(νs+1)−1[(∂suxx)2 + (∂suxy)2] )1/2 . Then β = M + β − 1 2 + νr 2 + νs + 1 2 . Therefore, M = 0. Also, L− β = N + L− (β − 1)− 1 2 + L− νr − 1 2 + L− (νs + 1)− 1 2 . Therefore, 2(L− β) = 2N + 3L− β − νr − νs − 3. Therefore, N ≤ 0 as long as 3 ≤ L. But L ≥ β ≥ 4. Therefore, L ≥ 3. Case νs = β − 1. In this case, νr = 1. Therefore r = (1, 0) or r = (0, 1). Consider first r = (1, 0). Then∫ T 0 ∫ A tβxL−β(∂ru)(∂sux)(∂αu) = ∫ T 0 tβxL−βux(∂(α1−1,α2)ux)(∂αu) = ∫ T 0 ∫ A tβxL−βux(∂αu)2 ≤ CT |ux|L∞ ∫ T 0 ∫ A tβ−1xL−β(∂αu)2 ≤ CT‖u‖H3 ∫ T 0 ∫ A tβ−1xL−(β−1)−1(∂αu)2. Then the term on the right-hand side is bounded as desired. Similarly, if r = (0, 1), we have ∫ T 0 ∫ A tβxL−β(∂ru)(∂sux)(∂αu) = ∫ T 0 ∫ A tβxL−βuy(∂(α1,α2−1)uy)(∂αu) ≤ CT |uy|L∞ ∫ T 0 ∫ A tβ−1xL−(β−1)−1[(∂(α1,α2−1)uy)2 + (∂αu)2]. 14 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 Case νs = β. In this case νr = 0. Then∫ T 0 ∣∣ ∫ A tβ xL−β(∂αu)(∂ru) (∂sux) ∣∣ = ∫ T 0 ∣∣ ∫ A tβxL−βu(∂αu)(∂αux) ∣∣ = C ∫ T 0 ∣∣ ∫ A tβ [xL−βu]x(∂αu)2 ∣∣ ≤ CT ( ‖u‖L∞(A) + ‖ux‖L∞(A) ) ( ∫ T 0 ∫ A tβ−1xL−β−1(∂αu)2 ) ≤ CT‖|u‖|H3 (∫ T 0 ∫ A tβ−1xL−(β−1)−1(∂αux)2 ) , where again the right-hand side is bounded by (3.14) or (3.15). Lemma 3.2 follows. � 4. A priori estimates In this section we prove two lemmas that will be used in a local-in-time existence theorem in Section 5. First we prove an a priori estimate for a linearized system related to (1.3). Second, we prove existence of a unique solution of that linearized system. For the lemma involving the a priori estimate, we introduce the following function space ZNT . We define ZNT = {u : u ∈ L∞([0, T ] : HN+3(R2)), ut ∈ L∞([0, T ] : HN (R2))}, (4.1) with the norm ‖u‖2ZNT = sup 0≤t≤T ∫ R2 [ u2+ ∑ |α|=N {(∂αuxxx)2+(∂αuyyy)2}+u2t + ∑ |α|=N (∂αut) 2 ] . (4.2) Lemma 4.1. Let u, v, w, z be functions in ZNt for all N and all t ≥ 0 such that u, v, w, z are solutions to ut + uxxx + uyyx − vx − 6wux = 0, bvt + δvxxx + λvyyx + ηvx − ux − 6µzvx = 0. (4.3) Then for all N ≥ 0, the following inequality holds ‖u‖2ZNt + b‖v‖2ZNt ≤ ‖u(·, ·, 0)‖2HN+3 + b‖v(·, ·, 0)‖2HN+3 + ‖ut(·, ·, 0)‖2HN + b‖vt(·, ·, 0)‖2HN + Ct‖w‖ZNt ‖u‖ 2 ZNt + Ct‖z‖ZNt ‖v‖ 2 ZNt (4.4) for all t ≥ 0. Proof. Fix N ≥ 0 and choose α such that |α| = N . Applying ∂α to (4.3)1 we have ∂αut + ∂αuxxx + ∂αuyyx − ∂αvx − 6∂α(wux) = 0. (4.5) EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 15 Multiplying (4.5) by 2∂αu and integrating over R2 we obtain ∂t ∫ (∂αu)2 = 12 ∫ (∂αu)∂α(wux) + 2 ∫ (∂αu)(∂αvx) ≤ C ∣∣ ∫ (∂αu)[(∂αw)ux + . . .+ w(∂αux)] ∣∣+ 2 ∫ (∂αu)(∂αvx) ≤ C ∣∣ ∫ (∂αw)ux(∂αu) ∣∣+ . . .+ ∣∣ ∫ R2 w(∂αux)(∂αu) ∣∣+ 2 ∫ (∂αu)(∂αvx) (4.6) The first term on the right-hand side of (4.6) is estimated using (2.11) together with the Cauchy-Schwarz inequality ∣∣ ∫ (∂αw)ux(∂αu) ∣∣ ≤ ‖ux‖L∞ (∫ (∂αw)2 )1/2(∫ (∂αu)2 )1/2 ≤ C (∫ [u2x + u2xxx + u2xyy] )1/2 ‖w‖H|α| ‖u‖H|α| ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t . For the second to last term on the right-hand side of (4.6), integrating by parts and using (2.11) we have ∣∣ ∫ w(∂αux)(∂αu) ∣∣ = C ∫ wx(∂αu)2 ≤ C‖wx‖L∞ ∫ (∂αu)2 ≤ C (∫ [w2 x + w2 xxx + w2 xy] )1/2 ‖u‖2 Z |α| t ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t . Therefore, ∂t ∫ ( ∂αu(·, ·, t̃) )2 ≤ C ‖w‖ Z |α| t ‖u‖2 Z |α| t + 2 ∫ (∂αu)(∂αvx). (4.7) In a similar way applying the same idea to (4.3)2, we obtain ∂tb ∫ ( ∂αv(·, ·, t̃ ) )2 ≤ C‖z‖ Z |α| t ‖v‖2 Z |α| t + 2 ∫ (∂αv)(∂αux). (4.8) Then adding (4.7) and (4.8) we obtain ∂t ∫ [(∂αu(·, ·, t̃))2 + b(∂αv(·, ·, t̃ ))2] ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t +C‖z‖ Z |α| t ‖v‖2 Z |α| t . (4.9) 16 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 Now, taking three x-derivatives of (4.5) and multiplying the result by 2∂αuxxx, we have ∂t ∫ (∂αuxxx) 2 ≤ ∣∣ ∫ (∂α (w ux)xxx) (∂αuxxx) ∣∣+ 2 ∫ (∂αuxxx)(∂αvxxxx) ≤ ∣∣ ∫ ∂α (wxxx ux + 3wxx uxx + 3wx uxxx + w uxxxx) (∂αuxxx) ∣∣ + 2 ∫ (∂αuxxx)(∂αvxxxx) ≤ ∣∣ ∫ ∂α (wxxxux) (∂αuxxx) ∣∣+ 3 ∣∣ ∫ ∂α (wxxuxx) (∂αuxxx) ∣∣ + 3 ∣∣ ∫ ∂α (wxuxxx) (∂αuxxx) ∣∣+ ∣∣ ∫ ∂α (wuxxxx) (∂αuxxx) ∣∣ + 2 ∫ R2 (∂αuxxx)(∂αvxxxx) = I1 + 3I2 + 3I3 + I4 + 2 ∫ (∂αuxxx)(∂αvxxxx). (4.10) Each term in the above expression is estimate separately. For the first term it follows that I1 = ∣∣ ∫ ∂α (wxxxux) (∂αuxxx) ∣∣ = ∣∣ ∫ [(∂αwxxx)ux (∂αuxxx) + · · ·+ wxxx (∂αux) (∂αuxxx)] ∣∣ ≤ C‖ux‖L∞ (∫ (∂αwxxx) 2 )1/2(∫ (∂αuxxx) 2 )1/2 + . . . + C‖∂αux‖L∞ (∫ w2 xxx )1/2(∫ (∂αuxxx) 2 )1/2 ≤ C (∫ [ u2x + u2xxx + u2xyy ] )1/2(∫ (∂αwxxx) 2 )1/2(∫ (∂αuxxx) 2 )1/2 + · · ·+ C (∫ [ (∂αux)2 + (∂αuxxx)2 + (∂αuxyy)2 ] )1/2 × (∫ w2 xxx )1/2(∫ R2 (∂αuxxx) 2 )1/2 ≤ C‖u‖Z0 t ‖w‖ Z |α| t ‖u‖ Z |α| t + · · ·+ C‖u‖ Z |α| t ‖w‖Z0 t ‖u‖ Z |α| t ≤ C‖u‖ Z |α| t ‖w‖ Z |α| t ‖u‖ Z |j| t + · · ·+ C‖u‖ Z |α| t ‖w‖ Z |α| t ‖u‖ Z |α| t ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t . Next, we consider I2. I2 = ∣∣ ∫ ∂α (wxxuxx) (∂αuxxx) ∣∣ = ∣∣ ∫ [(∂αwxx)uxx (∂αuxxx) + · · ·+ wxx (∂αuxx) (∂αuxxx)] ∣∣. (4.11) EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 17 Using the Cauchy-Schwarz inequality and (2.12), for the first term in (4.11), we have∣∣ ∫ (∂αwxx)uxx (∂αuxxx) ∣∣ ≤ (∫ ( ∂|α|wxx )4)1/4(∫ u4xx dx dy )1/4(∫ (∂αuxxx) 2 )1/2 ≤ (∫ [(∂αwxx) 2 + (∂αwxxx) 2 + (∂αwxxy) 2 ] )1/2(∫ [ u2xx + u2xxx + u2xxy ] )1/2 × (∫ (∂αuxxx) 2 )1/2 ≤ c‖w‖ Z |α| t ‖u‖2 Z |α| t . Integrating by parts the last term in (4.11), using the Cauchy-Schwarz inequality, and (2.12), we have∣∣ ∫ wxx (∂αuxx) (∂αuxxx) ∣∣ = 1 2 ∣∣ ∫ wxxx (∂αuxx) 2 ∣∣ ≤ 1 2 (∫ w2 xxx )1/2(∫ (∂αuxx) 4 )1/2 ≤ 1 2 ‖w‖ Z |0| t (∫ (∂αuxx) 2 + (∂αuxxx) 2 + (∂αuxxy) 2 )1/2 ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t . For the third term in (4.10) we have I3 = ∣∣ ∫ ∂α (wxuxxx) (∂αuxxx) ∣∣ = ∣∣ ∫ [(∂αwx)uxxx (∂αuxxx) + · · ·+ wx (∂αuxxx) (∂αuxxx)] ∣∣ ≤ C‖∂αwx‖L∞ (∫ u2xxx )1/2(∫ (∂αuxxx) 2 )1/2 + · · ·+ C‖wx‖L∞ ∫ (∂αuxxx) 2 ≤ C‖w‖ Z |α| t ‖u‖Z0 t ‖v‖ Z |α| t + · · ·+ C‖w‖Z0 t ‖u‖2 Z |α| t ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t . Now we estimate the last term in (4.10) as follows I4 = ∣∣ ∫ ∂α (wuxxxx) (∂αuxxx) ∣∣ = ∣∣ ∫ [(∂αw)uxxxx (∂αuxxx) + · · ·+ w (∂αuxxxx) ( ∂juxxx ) ] ∣∣. (4.12) If α = (0, 0), then integrating by parts and using (2.11),∣∣ ∫ ∂α (w uxxxx) (∂αuxxx) ∣∣ = ∣∣ ∫ wuxxxxuxxx ∣∣ = 1 2 ∣∣ ∫ R2 wx u 2 xxx ∣∣ ≤ C‖wx‖L∞ ∫ u2xxx 18 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 ≤ C (∫ [ w2 x + w2 xxx + w2 xyy ] )1/2 ∫ R2 u2xxx ≤ C‖u‖2Z0 t ‖w‖ Z |α| t . For |α| > 0, using (2.11) for the first term in (4.12), we have∣∣ ∫ (∂αw)uxxxx (∂αuxxx) ∣∣ ≤ C‖∂αw‖L∞ (∫ R2 u2xxxx )1/2(∫ (∂αuxxx) 2 )1/2 ≤ (∫ [ (∂αw) 2 + (∂αwxx) 2 + (∂αwyy) 2 ] )1/2(∫ u2xxxx )1/2(∫ (∂αuxxx) 2 )1/2 ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t . For the last term in (4.12), integrating by parts and using (2.11) we have∣∣ ∫ w (∂αuxxxx) (∂αuxxx) ∣∣ = 1 2 ∣∣ ∫ wx (∂αuxxx) 2 ∣∣ ≤ C‖wx‖L∞ ∫ (∂αuxxx) 2 ≤ C (∫ [ w2 x + w2 xxx + w2 xyy ] )1/2 ∫ R2 (∂αuxxx) 2 ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t . Consequently, ∂t ∫ (∂αuxxx) 2 ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t + 2 ∫ (∂αuxxx)(∂αvxxxx). (4.13) In a similar way, applying the same idea to (4.3)2, we obtain ∂tb ∫ (∂αvxxx) 2 ≤ C‖z‖ Z |α| t ‖v‖2 Z |α| t + 2 ∫ (∂αvxxx)(∂αuxxxx). (4.14) Then adding (4.13) and (4.14) we obtain ∂t ∫ [ (∂αuxxx) 2 + b (∂αvxxx) 2 ] ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t + C‖z‖ Z |α| t ‖v‖2 Z |α| t . (4.15) Similarly, applying ∂α∂3y to each equation in (4.3), and using similar analysis, we obtain ∂t ∫ [(∂αuyyy)2 + b(∂αvyyy)2] ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t + C‖z‖ Z |α| t ‖v‖2 Z |α| t . (4.16) On the other hand, applying one t-derivative to (4.5), multiplying by 2 (∂αut) and integrating over R2 we arrive at the inequality ∂t ∫ (∂αut) 2 ≤ C ∣∣ ∫ (∂α (wux)t) (∂αut) ∣∣+ 2 ∫ (∂αut)(∂ αvxt) ≤ C ∣∣ ∫ ∂α (wtux) (∂αut) ∣∣+ C ∣∣ ∫ ∂α (wuxt) (∂αut) ∣∣ = K1 +K2 + 2 ∫ (∂αut)(∂ αvxt). EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 19 For the first term on the right hand side, we use (2.11) and the Cauchy-Schwarz inequality K1 = ∣∣ ∫ ∂α (wtux) (∂αut) ∣∣ ≤ C ∣∣ ∫ (∂αwt)ux (∂αut) ∣∣+ · · ·+ C ∣∣ ∫ wt (∂αux) (∂αut) ∣∣ ≤ C‖ux‖L∞ (∫ (∂αwt) 2 )1/2(∫ (∂αut) 2 )1/2 + . . . + C‖∂αux‖L∞ (∫ R2 w2 t )1/2(∫ (∂αut) 2 )1/2 ≤ C‖u‖Z0 t ‖w‖ Z |α| t ‖u‖ Z |α| t + · · ·+ c‖u‖ Z |α| t ‖w‖Z0 t ‖u‖ Z |α| t ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t . We look at the second term on the right-hand side. If α = (0, 0) we have K2 = ∣∣ ∫ wuxt ut ∣∣ = 1 2 ∣∣ ∫ wxu 2 t ∣∣ ≤ C‖wx‖L∞ ∫ R2 u2t ≤ C‖w‖Z|α| t ‖u‖2 Z |α| t . If α 6= (0, 0), we have∣∣ ∫ ∂α (wuxt) (∂αut) ∣∣ = C ∣∣ ∫ (∂αw)uxt (∂αut) ∣∣+ · · ·+ C ∣∣ ∫ w (∂αuxt) (∂αut) ∣∣. (4.17) The first term in (4.17) is estimated as∣∣ ∫ (∂αw)uxt (∂αut) ∣∣ ≤ C‖∂αw‖L∞ (∫ u2xt )1/2(∫ (∂αut) 2 )1/2 ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t . Using integration by parts in the last term in (4.17) along with (2.11) and the Cauchy-Schwarz inequality, we have∣∣ ∫ w (∂αuxt) (∂αut) ∣∣ = 1 2 ∣∣ ∫ wx (∂αut) 2 ∣∣ ≤ C‖wx‖L∞ ∫ (∂αut) 2 ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t . Thus ∂t ∫ (∂αut) 2 ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t + 2 ∫ (∂αut)(∂ αvxt). (4.18) In a similar way, we obtain ∂tb ∫ (∂αvt) 2 ≤ C‖z‖ Z |α| t ‖v‖2 Z |α| t + 2 ∫ (∂αvt)(∂ αuxt). (4.19) Then adding (4.18) and (4.19) we obtain ∂t ∫ [(∂αut) 2 + b (∂αvt) 2 ] ≤ C‖w‖ Z |α| t ‖u‖2 Z |α| t + C‖z‖ Z |α| t ‖v‖2 Z |α| t . (4.20) 20 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 Then, for 0 ≤ t̃ ≤ t, it follows that ∂t ∫ [ ( ∂αu(·, ·, t̃) )2 + b ( ∂αv(·, ·, t̃) )2 + ( ∂αuxxx(·, ·, t̃) )2 + b ( ∂αvxxx(·, ·, t̃) )2 + ( ∂αuyyy(·, ·, t̃) )2 + b ( ∂αvyyy(·, ·, t̃) )2 + ( ∂αut(·, ·, t̃ ) )2 + b ( ∂αvt(·, ·, t̃) )2 ] ≤ c‖w‖ Z |α| t ‖u‖2 Z |α| t + C‖z‖ Z |α| t ‖v‖2 Z |α| t . Integrating with respect to t, and using the fact that this estimate is true for all α such that |α| = N , we obtain ‖u‖2ZNt + b‖v‖2ZNt ≤ ‖u(·, ·, 0)‖2HN+3 + b‖v(·, ·, 0)‖2HN+3 + ‖ut(·, ·, 0)‖2HN + b‖vt(·, ·, 0)‖2HN + Ct‖w‖ZNt ‖u‖ 2 ZNt + Ct‖z‖ZNt ‖v‖ 2 ZNt , as claimed. � Next we prove an existence result for a linearized version of (1.3). Consider the linear system u (n) t + u(n)xxx + u(n)yyx − v(n)x − 6u(n−1) u(n)x = 0, bv (n) t + δv(n)xxx + λv(n)yyx + ηv(n)x − u(n)x − 6µv(n−1)v(n)x = 0 (4.21) where the initial conditions are u(n)(x, y, 0) = u0(x, y) and v(n)(x, y, 0) = v0(x, y), and the first approximations are u(0)(x, y, t) = u0(x, y) and v(0)(x, y, t) = v0(x, y). Lemma 4.2. Given initial data u0, v0 ∈ ∩N≥0HN (R2), there exists a unique so- lution of system (4.21). The solution is defined in any time interval in which the coefficients are defined. Proof. The linear system (4.21) which is to be solved at each iteration has the form ut + uxxx + uyyx − vx − hux = 0 (4.22) bvt + δvxxx + λvyyx + ηvx − ux − h̃vx = 0 (4.23) where h, and h̃ are smooth bounded coefficients. Fix a time T > 0 and a constant M > 0. Define L = ∂tF +A∂xxx +B∂yyx + C∂x +D∂x (4.24) where F = ( 1 0 0 b ) , A = ( 1 0 0 δ ) , B = ( 1 0 0 λ ) , C = ( 0 −1 −1 η ) , D = ( −h 0 0 −h̃ ) , Ψ = ( u v ) . (4.25) Let L be defined on those functions (u, v) ∈ H3(R2) × H3(R2). For functions Φi = (νi, ζi) ∈ C ( [0, T ] : L2(R2) ) × C ( [0, T ] : L2(R2) ) which vanish at t = 0, we introduce the bilinear form B(Φ1,Φ2) = 〈Φ1,Φ2〉 = ∫ T 0 ∫ R2 e−MtΦ1 · Φ2 dx dy dt = ∫ T 0 ∫ R2 e−Mt (ν1ν2 + ζ1ζ2) dx dy dt. EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 21 Integrating by parts, we have∫ R2 LΨ ·Ψ dx dy ≥ 1 2 ∂t ∫ R2 |Ψ|2 dx dy − 1 2 ∫ R2 DxΨ ·Ψ dx dy ≥ 1 2 ∂t ∫ R2 |Ψ|2 dx dy − c 2 ∫ R2 |Ψ|2 dx dy (4.26) for some constant c large enough. Multiplying (4.26) by e−Mt and integrating in time from t = 0 to t = T , we obtain for Ψ ∈ C([0, T ] : H3(R2))×C([0, T ] : H3(R2)) with Ψ(x, y, 0) = (ν(x, y, 0), η(x, y, 0)) = (0, 0), 〈LΨ,Ψ〉 ≥ 1 2 e−M T ∫ |Ψ(x, y, T )|2 dx dy + 1 2 (M − c) ∫ T 0 ∫ e−Mt|Ψ|2 dx dy dt. (4.27) Therefore, 〈LΨ,Ψ〉 ≥ 〈Ψ,Ψ〉 provided M is large enough. Similarly, 〈L∗Φ,Φ〉 ≥ 〈Φ,Φ〉 for all Φ ∈ C([0, T ] : H3(R2)) × C([0, T ] : H3(R2)) with Φ(x, y, T ) ≡ (0, 0) where L∗ denotes the formal adjoint of L. Therefore, 〈L∗Φ,L∗Ψ〉 is an inner product on D = {Φ ∈ C([0, T ] : H3(R2)) × C([0, T ] : H3(R2)) : Φ(x, y, T ) ≡ (0, 0)}. We denote by Y the completion of D with respect to this inner product. By the Riesz representation theorem, there exists a unique solution V ∈ Y such that for any Φ ∈ D, 〈L∗V,L∗Φ〉 = (Ψ(0),Φ(x, y, 0)) (4.28) where we have used that (Ψ(0),Φ(x, y, 0)) is a bounded linear functional onD. Then Ψ = L∗V is a weak solution of LΨ = 0, with Ψ ∈ L2(R2×[0, T ])×L2(R2×[0, T ]). � Remark 4.3. To obtain higher regularity of the solution, we repeat the proof with higher derivatives included in the inner product. 5. Uniqueness and local existence In this section, we prove that for initial data (u0, v0) ∈ HN (R2) × HN (R2), for N ≥ 3 there exists a unique local solution (u, v) of (1.3) such that (u, v) ∈ L∞([0, T ] : HN (R2))×L∞([0, T ] : HN (R2)) where the time T of existence depends only on ‖u0‖H3 and ‖v0‖H3 . First we address the question of uniqueness. Theorem 5.1. Let 0 < T < ∞. Assume that (u0, v0) ∈ H3(R2) ×H3(R2). Then there is at most one solution (u, v) ∈ L∞([0, T ] : H3(R2))×L∞([0, T ] : H3(R2)) of (1.3) with initial data (u0, v0). Proof. Assume that (u, v) and (ũ, ṽ) are two solutions of (1.3) in L∞([0, T ] : H3(R2))×L∞([0, T ] : H3(R2)) with the same initial data (u0(x, y), v0(x, y)). From (1.3), ut, vt, ũt, ṽt ∈ L∞([0, T ] : L2(R2)), so the integrations below are justified. Therefore, the differences (u− ũ) and (v − ṽ) satisfy (u− ũ)t + (u− ũ)xxx + (u− ũ)yyx − (v − ṽ)x − 6(uux − ũũx) = 0. (5.1) Now, multiplying (5.1) by 2(u− ũ) and integrating for (x, y) ∈ R2, we have 2 ∫ (u− ũ)(u− ũ)t + 2 ∫ (u− ũ)(u− ũ)xxx + 2 ∫ (u− ũ)(u− ũ)yyx − 2 ∫ (u− ũ)(v − ṽ)x − 12 ∫ (u− ũ)(uux − ũũx) = 0. (5.2) 22 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 Then the 2nd and 3rd terms in (5.2) are shown to be identically zero. Hence ∂t ∫ (u− ũ)2 − 2 ∫ (u− ũ) (v − ṽ)x = 12 ∫ (u− ũ)(uux − ũũx). (5.3) On the other hand, 12 ∫ (u− ũ)(uux − ũũx) = 6 ∫ (u− ũ)(u2 − ũ2)x = − 6 ∫ (u− ũ)x(u2 − ũ2) = − 6 ∫ (u− ũ)x(u− ũ)(u+ ũ) = 3 ∫ (u− ũ)2(ux + ũx) ≤ 3 (‖ux‖L∞ + ‖ũx‖L∞) ∫ R2 (u− ũ)2 ≤ C ∫ (u− ũ)2. Combining this estimate with (5.3), we have ∂t ∫ R2 (u− ũ)2 ≤ C ∫ R2 (u− ũ)2 + 2 ∫ R2 (u− ũ)(v − ṽ)x. (5.4) Similarly we have b(v − ṽ)t + δ(v − ṽ)xxx + λ(v − ṽ)yyx + η(v − ṽ)x − ω(u− ũ)x − 6µ(vvx − ṽṽx) = 0. (5.5) Now, multiplying (5.1) by 2(v − ṽ) and integrating over R2 we have 2b ∫ (v − ṽ)(v − ṽ)t + 2δ ∫ (v − ṽ)(v − ṽ)xxx + 2λ ∫ (v − ṽ)(v − ṽ)xyy + 2η ∫ (v − ṽ)(v − ṽ)x − 2 ∫ (v − ṽ)(u− ũ)x − 12µ ∫ (v − ṽ)(v vx − ṽṽx) = 0. (5.6) Then the 2nd, 3rd, and 4th terms in (5.6) are shown to be identically zero. Hence ∂t ∫ b(v − ṽ)2 − 2 ∫ (v − ṽ)(u− ũ)x = 12µ ∫ (v − ṽ)(v vx − ṽṽx). (5.7) On the other hand, 12µ ∫ (v − ṽ)(vvx − ṽṽx) = 6µ ∫ (v − ṽ)(v2 − ṽ2)x = − 6µ ∫ (v − ṽ)x(v2 − ṽ2) = − 6µ ∫ (v − ṽ)x(v − ṽ)(v + ṽ) = 3µ ∫ (v − ṽ)2(vx + ṽx) ≤ 3µ (‖vx‖L∞ + ‖ṽx‖L∞) ∫ R2 (v − ṽ)2 ≤ C ∫ (v − ṽ)2. Combining this estimate with (5.3), we have ∂tb ∫ (v − ṽ)2 ≤ C ∫ (v − ṽ)2 + 2 ∫ (v − ṽ)(u− ũ)x. (5.8) EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 23 Adding (5.4) with (5.8) we obtain ∂t ∫ (u− ũ)2 + ∂tb ∫ (v − ṽ)2 ≤ C ∫ (u− ũ)2 + C ∫ (v − ṽ)2 + 2 ∫ (u− ũ)(v − ṽ)x + 2 ∫ (v − ṽ)(u− ũ)x. (5.9) The last two terms satisfy 2 ∫ (u− ũ) (v − ṽ)x + 2 ∫ (v − ṽ)(u− ũ)x = ∫ [(u− ũ)(v − ṽ)]x = 0. Therefore, ∂t ∫ [ (u− ũ)2 + b(v − ṽ)2 ] ≤ C ∫ [ (u− ũ)2 + b(v − ṽ)2 ] . (5.10) Using that u(x, y, 0) − ũ(x, y, 0) ≡ 0, v(x, y, 0) − ṽ(x, y, 0) ≡ 0 and Gronwall’s inequality it follows that ∫ (u− ũ)2 + ∫ b(v − ṽ)2 ≤ 0. We conclude that u ≡ ũ and v ≡ ṽ. This proves the uniqueness of the solution. � We now prove the existence of a local solution for (1.3). We show that for each (u0, v0) ∈ HN+3 ( R2 ) ×HN+3 ( R2 ) there exists a solution (u, v) in the space L∞ ( [0, T ] : HN+3(R2) ) × L∞ ( [0, T ];HN+3(R2) ) for a time T depending only on ‖u0‖H3(R2) and ‖v0‖H3(R2). Theorem 5.2. Let κ0, κ̃0 > 0 and N be an integer ≥ 0. Then there exists a time 0 < T < ∞, depending only on κ0 and κ̃0 such that for all u0, v0 ∈ HN+3(R2), with ‖u0‖H3(R2) ≤ κ0 and ‖v0‖H3(R2) ≤ κ̃0, there exists a solution of (1.3) with (u, v) ∈ L∞ ( [0, T ] : HN+3(R2) ) × L∞ ( [0, T ] : HN+3(R2) ) such that u(x, y, 0) = u0(x, y) and v(x, y, 0) = v0(x, y). The method of proof is as follows. We begin by approximating (1.3) by a sequence of linear equations. We then show that the sequence of solutions to our linear equations is bounded in L∞([0, T ];H3(R2)) × L∞([0, T ] : H3(R2)) for a time T depending only on ‖u0‖H3 , ‖v0‖H3 . Third, we prove that a subsequence of solutions to our approximate equations converges to a solution (u, v) ∈ L∞([0, T ];H3(R2))× L∞([0, T ];H3(R2)) of (1.3). Lastly, we show that if (u0, v0) ∈ HN+3(R2)×HN+3(R2) for N > 0, then our solution (u, v) is in L∞([0, T ];HN+3(R2)) × L∞([0, T ];HN+3(R2)), where the time T depends only on ‖u0‖H3 , ‖v0‖H3 . Proof of Theorem 5.2. It suffices to prove this result for u0, v0 ∈ ∩N≥0HN (R2). We can use the same approximation procedure as before to prove the result for general initial data. We begin by approximating (1.3) by the linear system (4.21) with initial data u(n)(x, y, 0) = u0(x, y), v(n)(x, y, 0) = v0(x, y), and where the first approximations are given by u(0)(x, y, t) = u0(x, y) and v(0)(x, y, t) = v0(x, y). By 24 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 Lemma 4.2, this system can be solved at each iteration. In particular, for each n there exists a unique solution (u(n), v(n)) and by Lemma 4.1 for N = 0 we have ‖u(n)‖2Z0 t + b‖v(n)‖2Z0 t ≤ ‖u(n)(·, ·, 0)‖2H3 + b‖v(n)(·, ·, 0)‖2H3 + ‖u(n)t (·, ·, 0)‖2L2 + b‖v(n)t (·, ·, 0)‖2L2 + Ct‖u(n−1)‖Z0 t ‖u(n)‖2Z0 t + Ct‖v(n−1)‖Z0 t ‖v(n)‖2Z0 t (5.11) for all t ≥ 0. By assumption, κ0 ≥ ‖u0‖H3(R2) and κ̃0 ≥ ‖v0‖H3(R2). On the other hand, ‖u(n)(·, ·, 0)‖2H3 + ‖u(n)t (·, ·, 0)‖2L2 = ‖u(n)(·, ·, 0)‖2H3 + ∫ [ u(n)xxx(·, ·, 0)− v(n)x (·, ·, 0) + u(n)xyy(·, ·, 0)− 6u(n−1) u(n)x (·, ·, 0) ]2 ≤ ‖u0‖2H3 + C ∫ [ u20xxx − v20x + (u0xyy)2 − (u0 u0x)2 ] ≤ K ( ‖u0‖2H3 + ‖v0‖2H3 ) ≤ K ( κ20 + κ̃20 ) . (5.12) In a similar way we have b‖v(n)(·, ·, 0)‖2H3 + b‖v(n)t (·, ·, 0)‖2L2 = b‖v(n)(·, ·, 0)‖2H3 + b ∫ R2 [ δ v(n)xxx(·, ·, 0) + ηv(n)x (·, ·, 0)− u(n)x (·, ·, 0) + λ∂−1x v(n)yy (·, ·, 0)− 6µ v(n−1) v(n)x (·, ·, 0) ]2 ≤ b‖v0‖2H3 + C ∫ R2 [δ v20xxx + η v20x − u20x + λ v0xyy)2 − (v0 v0x)2] ≤ K̃ ( ‖u0‖2H3 + ‖v0‖2H3 ) ≤ K̃ ( κ20 + κ̃20 ) , (5.13) where K and K̃ are independent of n. Without loss of generality, suppose K ≥ K̃. Let c20 = ( 2K(κ20 + κ̃20) + 1 ) . Let T (n) 0 = sup{t : ‖u(j)‖Z0 t ≤ c0 for 0 ≤ j ≤ n} T̃ (n) 0 = sup{t : ‖v(j)‖Z0 t ≤ c0 for 0 ≤ j ≤ n}. Let T ∗(n) 0 = min{T (n) 0 , T̃ (n) 0 }. Then, for t in the interval [0, T ∗(n) 0 ], from (5.11), (5.12), and (5.13), it follows that ‖u(n)‖2Z0 t + b‖v(n)‖2Z0 t ≤ ‖u(n)(·, ·, 0)‖2H3 + b‖v(n)(·, ·, 0)‖2H3 + ‖u(n)t (·, ·, 0)‖2L2 + b‖v(n)t (·, ·, 0)‖2L2 + Ct‖u(n−1)‖Z0 t ‖u(n)‖2Z0 t + Ct‖v(n−1)‖Z0 t ‖v(n)‖2Z0 t ≤ K ( κ20 + κ̃20 ) + K̃ ( κ20 + κ̃20 ) + Ctc30 + Ctc̃30 ≤ 2K(κ20 + κ̃20) + Ctc30. (5.14) Now choose T > 0 such that CTc30 = 1. We claim that T ∗(n) 0 ≥ T for all n and therefore, the sequence of approximate solutions {(u(n), v(n))} is bounded for the time T which is independent of n. If T ∗(n) 0 =∞ for all n, then clearly T ∗(n) 0 ≥ T for EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 25 all n. So, assume there exists n such that T ∗(n) 0 < ∞. Suppose T > T ∗(n) 0 . Then, by the continuity of ‖u(n)‖Z0 t , ‖v(n)‖Z0 t with respect to t, we have c20 = ‖u(j)‖2 Z0 T (n) 0 for some j ∈ [0, n] and c20 = ‖v(j̃)‖2 Z0 T̃ (n) 0 for some integer j̃ ∈ [0, n]. Without loss of generality, suppose T ∗(n) 0 = T (n) 0 . Therefore, by (5.14), c20 ≤ ‖u(j)‖2Z0 T (n) 0 + ‖v(j)‖2Z0 T (n) 0 ≤ 2K(κ20 + κ̃20) + CT (n) 0 c30 < 2K(κ20 + κ̃20) + CTc30 = c20. However, this implies c20 < c20 and we have a contradiction. Thus, we conclude that T ∗(n) 0 ≥ T for all n, and, therefore, sup 0≤t≤T ∫ ((u(n)(·, t))2 + (u(n)xxx)2 + (u(n)yyy)2 + (u (n) t )2 ≤ c20, sup 0≤t≤T ∫ ((v(n)(·, t))2 + (v(n)xxx)2 + (v(n)yyy)2 + (v (n) t )2 ≤ c20 for all n. Consequently, there exists a bounded sequence of solutions {(u(n), v(n))} ∈ Z0 T × Z0 T . Therefore, u(n)⇀u weak∗ in L∞ ( [0, T ] : H3(R2) ) u (n) t ⇀ut weak∗ in L∞ ( [0, T ] : L2(R2) ) (5.15) and v(n)⇀v weak∗ in L∞ ( [0, T ] : H3(R2) ) v (n) t ⇀vt weak∗ in L∞ ( [0, T ] : L2(R2) ) . (5.16) On the other hand, H3 loc(R2) c ↪→ H1 loc(R2) ↪→ L2(R2). Then by the Lions-Aubin compactness Theorem [20] there are subsequences u(nj) := u(n) and v(nj) := v(n) such that u(n) → u strongly in L∞ ( [0, T ] : H1 loc(R2) ) v(n) → v strongly in L∞ ( [0, T ] : H1 loc(R2) ) (5.17) Hence for subsequences u(nj) := u(n) and v(nj) := v(n), we have u(n) → u a.e. in L∞ ( [0, T ] : H1 loc(R2) ) v(n) → v a.e. in L∞ ( [0, T ] : H1 loc(R2) ) . (5.18) Moreover, from (5.15) we have u(n)xxx ⇀ uxxx, uxyy ⇀ uxyy weakly∗ in L∞ ( [0, T ] : L2(R2) ) v(n)xxx ⇀ vxxx, v(n)xyy ⇀ vxyy weakly∗ in L∞ ( [0, T ] : L2(R2) ) . (5.19) Now we show that the nonlinear term converges to its correct limit. From (5.17), u(n−1) → u strongly in L∞ ( [0, T ] : H1 loc(R2) ) ↪→ L∞ ( [0, T ] : L2 loc(R2) ) v(n−1) → v strongly in L∞ ( [0, T ] : H1 loc(R2) ) ↪→ L∞ ( [0, T ] : L2 loc(R2) ) . (5.20) 26 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 Moreover, u(n−1)x ⇀ux weak∗ in L∞ ( [0, T ] : L2(R2) ) v(n−1)x ⇀vx weak∗ in L∞ ( [0, T ] : L2(R2) ) . (5.21) Therefore, u(n−1)u(n)x ⇀uux weakly∗ in L∞ ( [0, T ] : L1 loc(R2) ) v(n−1)v(n)x ⇀vvx weakly∗ in L∞ ( [0, T ] : L1 loc(R2) ) . (5.22) Therefore, (u, v) is a solution to (1.3). Now, we prove that if (u0, v0) ∈ HN+3(R2)×HN+3(R2) for some integer N > 0, then the solution (u, v) satisfies (u, v) ∈ L∞ ( [0, T ] : HN+3(R2) ) × L∞ ( [0, T ] : HN+3(R2) ) for the time T chosen above. We already know that there is a solution (u, v) ∈ L∞ ( [0, T ] : H3(R2) ) × L∞([0, T ] : H3(R2)). Therefore, we only need to show that the approximating sequence (u(n), v(n)) is bounded in ZNT × ZNT and thus, by the convergence arguments above, our solution (u, v) is in L∞ ( [0, T ] : HN+3(R2) ) × L∞([0, T ] : HN+3(R2)). We use the same argument as before. By Lemma 4.2, we know our linearized equation can be solved in any interval of time in which the coefficients are defined. Therefore, for each iterate, ‖u(n)‖ZNt and ‖v(n)‖ZNt are continuous in t ∈ [0, T ]. Using Lemma 4.1 it follows that ‖u(n)‖2ZNt + b‖v(n)‖2ZNt ≤ ‖u (n)(·, ·, 0)‖2HN+3(R2) + b‖v(n)(·, ·, 0)‖2HN+3(R2) + ‖u(n)t (·, ·, 0)‖2HN (R2) + b‖v(n)t (·, ·, 0)‖2HN (R2) + ct‖u(n−1)‖ZNt ‖u (n)‖2ZNt + ct‖v(n−1)‖ZNt ‖v (n)‖2ZNt . As before, we have ‖u(n)(·, ·, 0)‖2HN+3(R2) + ‖u(n)t (·, ·, 0)‖2HN (R2) ≤ Kκ 2 N , (5.23) b‖v(n)(·, ·, 0)‖2HN+3(R2) + b‖v(n)t (·, ·, 0)‖2HN (R2) ≤ K̃κ̃ 2 N (5.24) where κN and κ̃N are independent of n. Without loss of generality, assume K ≥ K̃ and define c2N = ( 2K(κ2N + κ̃2N ) + 1 ) . Let T (n) N = sup{t : ‖u(j)‖ZNt ≤ cN for 0 ≤ j ≤ n} T̃ (n) N = sup{t : ‖v(j)‖ZNt ≤ cN for 0 ≤ j ≤ n}. Let T ∗(n) N = min{T (n) N , T̃ (n) N }. Then, for t in the interval [0, T ∗(n) N ], it follows that ‖u(n)‖2ZNt + b‖v(n)‖2ZNt ≤ 2KN (κ2N + κ̃2N ) + Ctc3N . Now choosing TN such that CTNc 3 N = 1, by the same arguments as in the case N = 0, we conclude that T (n) N ≥ TN , and, therefore, ‖u(n)‖2ZNTN ≤ c2N , ‖v(n)‖2ZNTN ≤ c2N . Now, let T ∗N ≡ sup{t : u, v ∈ ZNt }. EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 27 We claim that T ∗N ≥ T , and, therefore, a time of existence can be chosen depending only on ‖u0‖H3 , ‖v0‖H3 . By Lemma 4.2 the linear equation (4.21) can be solved in any interval of time in which the coefficients are defined, and, thus T ∗N ≥ T . � Corollary 5.3. Let u0, v0 ∈ HN+3(R2) for some N ≥ 0 and let u (n) 0 be a sequence converging to u0 in HN+3(R2), v (n) 0 be a sequence converging to v0. Let (u, v) and (u(n), v(n)) be the corresponding unique solutions, given by Theorems 5.1 and 5.2 in L∞([0, T ] : HN+3(R2)) × L∞([0, T ] : HN+3(R2)) for a time T depending only on supn ‖u (n) 0 ‖H3(R2) and supn ‖v (n) 0 ‖H3(R2). Then u(n)⇀u weak* in L∞ ( [0, T ] : HN+3(R2) ) v(n)⇀v weak* in L∞ ( [0, T ] : HN+3(R2) ) . (5.25) Proof. By assumption u(n), v(n) ∈ L∞([0, T ] : HN+3(R2)), then there exist weak* convergent subsequences, still denoted {u(n)} and {v(n)} such that u(n)⇀ũ weak* in L∞ ( [0, T ] : HN+3(R2) ) v(n)⇀ṽ weak* in L∞ ( [0, T ] : HN+3(R2) ) . Moreover, by equation (1.3), u(n), v(n) ∈ L∞ ( [0, T ] : HN+3(R2) ) implies u (n) t , v (n) t ∈ L∞ ( [0, T ] : L2(R2) ) . By the Lions-Aubin Compactness theorem [20] we have u(n) → ũ strongly in L∞([0, T ] : H 1/2 loc (R2)) v(n) → ṽ strongly in L∞([0, T ] : H 1/2 loc (R2)). Now we just to show that each term in (1.3) converges to its correct limit, and thus u (n) t → ũt and v (n) t → ṽt for ũ, ṽ ∈ L∞ ( [0, T ] : HN+3(R2) ) . The only thing we need to show is that the nonlinear term converges to its correct limit, namely that u(n) u (n) x → ũ ũx. We know that u (n) x ∗ ⇀ ũx weakly in L∞([0, T ] : H1(R2)) and u(n) → ũ strongly in L∞([0, T ] : H 1/2 loc (R2)). Therefore, their product converges in L2([0, T ] : L1 loc(R2)). Clearly the linear terms also converge in L2([0, T ] : L1 loc(R2)) and therefore, we conclude that u (n) t → ũt in L2([0, T ] : L1 loc(R2)). In a similar way we conclude that v (n) t → ṽt in L2([0, T ] : L1 loc(R2)). We also know that (ũ, ṽ) ∈ L∞([0, T ] : HN+3(R2)) × L∞([0, T ] : HN+3(R2)). By the uniqueness theorem, Theorem 5.1, (ũ, ṽ) = (u, v). � 6. Weighted estimates and main estimates of error terms At the end of this section, we state and prove our main theorem, Theorem 6.4. First, however, as a starting point for the a priori gain of regularity results that will be discussed in Theorem 6.4, we need to develop some estimates for solutions of the coupled system (1.3) in weighted Sobolev spaces. The existence of these weighted estimates is often called a persistence property of the initial data (u0, v0). Indeed, we prove that if our initial data (u0, v0) ∈ H3(R2) × H3(R2) also lies in some weighted space HK(W0 i 0)×HK(W0 i 0), for integers K ≥ 0 and i ≥ 1, then our solution (u, v) also lies in L∞ ( [0, T ] : HK(W0 i 0) ) × L∞ ( [0, T ] : HK(W0 i 0) ) . 28 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 Theorem 6.1. Assume (u, v) is the solution to (1.3) in L∞([0, T ] : H3(R2)) × L∞([0, T ] : H3(R2)) with initial data (u0, v0) ∈ H3(R2)×H3(R2) such that (u0, v0) also lie in the weighted space HK(W0 i 0) × HK(W0 i 0) for some integers K ≥ 0, i ≥ i. Then u ∈ L∞([0, T ] : H3(R2) ∩HK(W0 i 0)), v ∈ L∞([0, T ] : H3(R2) ∩HK(W0 i 0)) (6.1) and ∫ T 0 ∫ χ(∂γux)2 + ∫ T 0 ∫ χ(∂γuy)2 ≤ C,∫ T 0 ∫ χ(∂γvx)2 + ∫ T 0 ∫ χ(∂γvy)2 ≤ C (6.2) for |γ| ≤ K, where χ is a weight function in Wσ,i−1,0 for σ > 0 arbitrary, and C depends only on T and the norms of u0, v0 ∈ H3(R2) ∩HK(W0 i 0). Proof. We will prove this result by induction on β, for 0 ≤ β ≤ K. As before, we need to derive a priori estimates for smooth solutions (u, v) which depend only on the noms of u, v ∈ L∞([0, T ];H3(R2)) and u0, v0 ∈ HK(W0 i 0). Then, we can apply convergence arguments to show that the result holds true for general solutions. In order to do so, we need to approximate general solutions u, v ∈ H3(R2) by smooth solutions and approximate general weight functions ξ ∈ W0 i 0 by smooth, bounded weight functions. We have discussed approximating solutions in the previous section, so we will concentrate on the approximation of the weight function here. We begin by taking a sequence of bounded weight functions χν , which decay as |x| → ∞ and which approximate χ ∈ Wσ,i−1,0 from below, uniformly on any half-line (−∞, c). Let ξν = 1 + ∫ x −∞ χν(z, t)dz. (6.3) Hence, the functions ξν are bounded weight functions which approximate a weight function ξ ∈W0 i 0 from below, uniformly on compact sets. Now we will follow the same methodology as in the development of the Lemma 3.1. Indeed, for the βth induction step, we take α derivatives of (1.3)1, where |α| = β, multiply the result by 2ξν (∂αu), and integrate over R2. Performing straightforward calculations and using (ξν)t, (ξν)x ≤ Cξν we obtain the following estimate ∂t ∫ ξν (∂αu)2 + 3 ∫ (ξν)x (∂αux) 2 + ∫ (ξν)x (∂αuy) 2 ≤ C ∫ ξν (∂αu) 2 + 2 ∫ ξν (∂αu) ∂α (uux) + 2 ∫ ξν(∂αu)(∂αvx). (6.4) Similarly, we take α derivatives of (1.3)2, where |α| = β, multiply the result by 2ξν (∂αv), and integrate over R2. Performing straightforward calculations as in (6.4) we obtain the estimate ∂tb ∫ ξν(∂αv)2 + 3δ ∫ (ξν)x (∂αvx) 2 + λ ∫ R2 (ξν)x (∂αvy) 2 ≤ C ∫ ξν(∂αv)2 + 2 ∫ ξν (∂αv) ∂α (vvx) + 2 ∫ ξν(∂αux)(∂αv). (6.5) EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 29 Adding (6.4) and (6.5) and using the Cauchy-Schwarz inequality we have ∂t ∫ ξν [ (∂αu)2 + b(∂αv)2 ] + 3 ∫ (ξν)x [ (∂αux) 2 + δ (∂αvx) 2 ] + ∫ (ξν)x [ (∂αuy) 2 + λ (∂αvy) 2 ] ≤ ∫ ξν (∂αu) 2 + ∫ ξν (∂αv) 2 + ∫ 2 (ξν) (∂αu) ∂α (uux) + ∫ 2 (ξν) (∂αv) ∂α (vvx) + 2 ∫ ξν∂x[(∂αu)(∂αv)] ≤ ∫ ξν (∂αu) 2 + ∫ ξν (∂αv) 2 + ∫ 2ξν (∂αu) ∂α (uux) + ∫ 2ξν (∂αv) ∂α (vvx) + C ∣∣ ∫ (ξν)x (∂αu)(∂αv) ∣∣ ≤ C ∫ ξν [ (∂αu) 2 + (∂αv) 2 ] + ∫ 2ξν (∂αu) ∂α (uux) + ∫ 2ξν (∂αv) ∂α (vvx) . (6.6) Case β = 0. We need to estimate the terms ∣∣ ∫ ξν (∂αu) ∂α (uux) ∣∣, ∣∣ ∫ ξν (∂αv) ∂α (vvx) ∣∣. (6.7) For (6.7)1, we have ∣∣ ∫ ξν (∂αu) ∂α (uux) ∣∣ = ∣∣ ∫ ξν u 2ux ∣∣ ≤ ‖ux‖L∞(R2) ∫ ξν u 2 ≤ C‖u‖H3 ∫ ξνu 2 ≤ C ∫ ξνu 2 where C depends only on the norm of u ∈ L∞([0, T ] : H3(R2)) (which depends only on the norms of u0, v0 ∈ H3(R2)). Similarly, ∣∣ ∫ ξνv 2vx ∣∣ ≤ C ∫ ξνv 2 Combining these estimates with (6.6), we conclude that ∂t ∫ ξν(u2 + bv2) + 3 ∫ (ξν)x(u2x + δv2x) + ∫ (ξν)x(u2y + λv2y) ≤ C ∫ (ξν)x(u2 + v2) + ∫ 2ξνu 2ux + ∫ 2ξνv 2vx ≤ C ∫ ξν(u2 + bv2) (6.8) 30 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 where C depends only on ‖u0‖H3 and ‖v0‖H3 . Integrating (6.8) on t ∈ [0, T ] we obtain∫ ξν(u2 + bv2) + 3 ∫ T 0 ∫ (ξν)x(u2x + δv2x) + ∫ T 0 ∫ (ξν)x(u2y + λv2y) ≤ ∫ ξν(·, ·, 0)(u20 + v20) + C ∫ t 0 ∫ ξν(u2 + v2) ≤ C + C (∫ t 0 ∫ ξν(u2 + bv2) ) . (6.9) Therefore, using Gronwall’s inequality, sup 0≤t≤T C ∫ ξν(u2 + v2) + 3 ∫ T 0 ∫ (ξν)x(u2x + δv2x) + ∫ T 0 ∫ (ξν)x(u2y + λv2y) ≤ C where C does not depend on ν, but only on T and the norm of u0, v0 ∈ H3(R2) ∩ H0(W0 i 0). Passing to the limit, sup 0≤t≤T C ∫ ξ(u2 + v2) + 3 ∫ T 0 ∫ χ(u2x + δv2x) + ∫ T 0 ∫ χ(u2y + λv2y) ≤ C. (6.10) Case β = 1. Consider α = (1, 0). In fact, for the case α = (1, 0) we have∣∣ ∫ ξνux(uux)x ∣∣ = ∣∣ ∫ ξνux(u2x + uuxx) ∣∣ = ∣∣ ∫ ξν(u3x + uux uxx) ∣∣ ≤ ∣∣ ∫ R2 ξνu 3 x ∣∣+ ∣∣ ∫ ξνuuxuxx ∣∣ = ∣∣ ∫ ξνu 3 x ∣∣+ ∣∣1 2 ∫ ξνu(u2x)x ∣∣ ≤ C ∣∣ ∫ ξνu 3 x ∣∣+ C ∣∣ ∫ (ξν)xuu 2 x ∣∣ ≤ C(|u|L∞ + |ux|L∞) ∫ ξνu 2 x ≤ C ∫ ξνu 2 x (6.11) where C depends only on the norms of u0, v0 ∈ H3(R2). Performing similar calcula- tions as in the case above, along with (6.6), and Gronwall’s inequality we conclude that sup 0≤t≤T C ∫ ξν(u2x + v2x) + 3 ∫ T 0 ∫ (ξν)x(u2xx + δv2xx) + ∫ T 0 ∫ (ξν)x(u2xy + λv2xy) ≤ C (6.12) where C does not depend on ν, but only on T and the norm of u0, v0 ∈ H3(R2) ∩ H1(W0 i 0). Passing to the limit, sup 0≤t≤T C ∫ ξ (u2x + v2x) + 3 ∫ T 0 ∫ χ(u2xx + δv2xx) + ∫ T 0 ∫ χ(u2xy + λv2xy) ≤ C where C depends only on the norms of u0, v0 ∈ H3(R2) ∩H1(W0 i 0). Next, we consider the case α = (0, 1). For the case α = (0, 1), we have∣∣ ∫ ξνuy(uux)y ∣∣ = ∣∣ ∫ ξνuy(uyux + uuxy) ∣∣ EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 31 ≤ ∣∣ ∫ ξνuxu 2 y ∣∣+ ∣∣ ∫ ξνuuyuxy ∣∣ ≤ C(|u|L∞ + |ux|L∞) ∫ ξνu 2 y ≤ C ∫ ξνu 2 y. Therefore, using the same idea as above, we conclude that sup 0≤t≤T C ∫ ξ(u2y + v2y) + 3 ∫ T 0 ∫ χ(u2xy + δv2xy) + ∫ T 0 ∫ χ(u2yy + λv2yy) ≤ C. Case β = 2. We have α = (2, 0), α = (1, 1), and α = (0, 2). First, for the case α = (2, 0) we have∣∣ ∫ ξνuxx(uux)xx ∣∣ = ∣∣ ∫ ξνuxx(3uxuxx + uuxxx) ∣∣ = ∣∣ ∫ ξν(3uxu 2 xx + uuxxuxxx) ∣∣ = ∣∣3∫ R2 ξνuxu 2 xx ∣∣+ ∣∣1 2 ∫ ξνu(u2xx)x ∣∣ ≤ C(|u|L∞ + |ux|L∞) ∫ ξνu 2 xx ≤ C ∫ ξνu 2 xx. (6.13) Performing similar calculations as in the cases given above together with (6.6) and using Gronwall’s inequality we have sup 0≤t≤T C ∫ R2 ξν(u2xx + v2xx) + 3 ∫ T 0 ∫ R2 (ξν)x(u2xxx + δv2xxx) (6.14) + ∫ T 0 ∫ R2 (ξν)x(u2xxy + λv2xxy) ≤ C (6.15) where C depends on T and the norm of u0, v0 ∈ H3(R2)∩H2(W0 i 0), and does not depend on ν. Passing to the limit, sup 0≤t≤T C ∫ ξ(u2xx + v2xx) + 3 ∫ T 0 ∫ χ(u2xxx + δv2xxx) + ∫ T 0 ∫ χ(u2xxy + λv2xxy) ≤ C. For the case α = (1, 1) we have∣∣ ∫ ξνuxy(uux)xy ∣∣ = ∣∣ ∫ ξνuxy(2uxuxy + uyuxx + uuxxy) ∣∣ = ∣∣ ∫ ξν(2uxu 2 xy + uyuxyuxx + uuxyuxxy) ∣∣ ≤ ∣∣2 ∫ ξνuxu 2 xy ∣∣+ ∣∣ ∫ ξνuyuxyuxx ∣∣+ ∣∣ ∫ ξνuuxyuxxy ∣∣ ≤ C|ux|L∞ ∫ ξνu 2 xy + C|uy|L∞ ∫ ξνu 2 xx dx dy + C|uy|L∞ ∫ ξνu 2 xy + (|u|L∞ + |ux|L∞) ∫ ξνu 2 xy ≤ C + C ∫ ξνu 2 xy 32 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 where C depends only on u0, v0 ∈ H3(R2) ∩ H2(W0 i 0). Consequently, using the same ideas as above, sup 0≤t≤T C ∫ ξ(u2xy + v2xy) + 3 ∫ T 0 ∫ χ(u2xxy + δv2xxy) + ∫ T 0 ∫ χ(u2xyy + λv2xyy) ≤ C where C depends only on u0, v0 ∈ H3(R2) ∩H2(W0 i 0). For the case α = (0, 2), our remainder term satisfies∣∣ ∫ ξνuyy(uux)yy ∣∣ = ∣∣ ∫ ξνuyy(uyyux + 2uyuxy + uuxyy) ∣∣ ≤ C|ux|L∞ ∫ ξνu 2 yy + C|uy|L∞ ∫ ξνu 2 xy + C|uy|L∞ ∫ ξνu 2 yy + C(|u|L∞ + |ux|L∞) ∫ ξνu 2 yy. Therefore, sup 0≤t≤T C ∫ ξ(u2yy + v2yy) + 3 ∫ T 0 ∫ χ(u2xyy + δv2xyy) + ∫ T 0 ∫ χ(u2yyy + λv2yyy) ≤ C where C depends only on u0, v0 ∈ H3(R2) ∩H2(W0 i 0). Case β = 3. For β = 3, we consider the case α = (3, 0). The other cases can be handled similarly. For α = (3, 0), our remainder terms satisfy∣∣ ∫ ξν uxxx(uux)xxx ∣∣ = ∣∣ ∫ ξνuxxx(3u2xx + 4uxuxxx + uuxxxx) ∣∣ ≤ ∣∣3 ∫ ξνu 2 xxuxxx ∣∣+ ∣∣4 ∫ ξνuxu 2 xxx ∣∣+ ∣∣ ∫ R2 ξνuuxxxuxxxx ∣∣ ≡ I1 + I2 + I3. First, we consider I1. We consider the case x > 1 and x < −1 separately. For x > 1, we use the fact that ξ 1/2 ν ≤ Cξν for i ≥ 1. In addition, we will use (2.13).∣∣3 ∫ A ξνu 2 xxuxxx ∣∣ = C ∣∣ ∫ A ξν(u3xx)x ∣∣ = C ∣∣ ∫ A (ξν)xu 3 xx ∣∣ ≤ C (∫ A ξνu 4 xx )1/2(∫ A ξνu 2 xx )1/2 ≤ C (∫ A ([ξ1/4ν ux]x)4] )1/2(∫ A ξνu 2 xx )1/2 ≤ C (∫ A ([ξ1/4ν ux]x)2 + ([ξ1/4ν ux]xx)2 + ([ξ1/4ν ux]xy)2 )(∫ A ξνu 2 xx )1/2 ≤ C (∫ A ξ1/2ν (u2x + u2xxx + u2xxy )(∫ A ξνu 2 xx )1/2 ≤ C (∫ A ξν(u2x + u2xxx + u2xxy )(∫ A ξνu 2 xx )1/2 . Further, we note that∫ T 0 (∫ A ξν(u2x + u2xxx + u2xxy) )(∫ A ξνu 2 xx )1/2 dt EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 33 ≤ C sup 0≤t≤T (∫ A ξνu 2 xx )1/2 ∫ T 0 ∫ A ξν(u2x + u2xxx + u2xxy) ≤ C ∫ T 0 ∫ A ξν(u2x + u2xxx + u2xxy) where C depends only on the norms of u0, v0 ∈ H3(R2)∩H2(W0 i 0) by the previous step of the induction. On the other hand, for x < −1, we use the fact that ξν ' C to show ∣∣ ∫ B ξνu 2 xxuxxx ∣∣ ≤ C∣∣ ∫ B u2xxuxxx ∣∣ ≤ C (∫ B u4xx )1/2(∫ B u2xxx )1/2 ≤ C (∫ B [u2xx + u2xxx + u2xxy] )(∫ B u2xxx )1/2 ≤ C‖u‖H3 ∫ B u2xxx ≤ C ∫ B ξδu 2 xxx. (6.16) For the term I2, we have∣∣ ∫ ξνuxu 2 xxx ∣∣ ≤ C‖ux‖L∞ ∫ ξνux u 2 xxx ≤ C‖u‖H3(R2) ∫ ξνu 2 xxx. (6.17) Lastly for I3, we have∣∣ ∫ ξν uuxxx uxxxx ∣∣ ≤ ∣∣ ∫ [ξν u]x u 2 xxx ∣∣ ≤ C (|u|L∞ + |ux|L∞) ∫ ξνu 2 xxx ≤ C‖u‖H3(R2) ∫ ξνu 2 xxx. (6.18) Combining these estimates with (6.4) and using similar estimates for v, we conclude that sup 0≤t≤T C ∫ ξν(·, ·, t)(u2xxx + v2xxx) + 3 ∫ T 0 ∫ (ξν)x(u2xxxx + u2xxxy) ≤ C ∫ ξν(·, ·, t)u20xxx + C ∫ ξν(·, ·, t)v20xxx + C ∫ T 0 ∫ ξν(u2xxx + v2xxx + u2xxy + v2xxy) (6.19) for 0 ≤ t ≤ T . Using similar estimates for other derivatives on the level β = 3, we conclude that∑ |α|=3 sup 0≤t≤T C ∫ ξν(·, ·, t)[(∂αu)2 + (∂αv)2] + C ∑ |α|=4 ∫ T 0 ∫ (ξν)x[(∂αu)2 + (∂αv)2] ≤ ∑ |α|=3 C ∫ ξν(·, ·, t)[(∂αu0)2 + (∂αv0)2] + C ∑ |α|=3 ∫ T 0 ∫ ξν [(∂αu)2 + (∂αv)2]. 34 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 Therefore, by Gronwall’s inequality,∑ |α|=3 sup 0≤t≤T ∫ Cξν(·, ·, t)[(∂αu)2 + (∂αv)2] + C ∑ |α|=4 ∫ T 0 ∫ (ξν)x[(∂αu)2 + (∂αv)2] ≤ C where C does not depend on ν, but only on T and the norms of u0, v0 ∈ H3(R2) ∩ H3(W0 i 0). Passing to the limit, we conclude that∑ |α|=3 sup 0≤t≤T C ∫ ξ(·, ·, t)[(∂αu)2 + (∂αv)2] + C ∑ |α|=4 ∫ T 0 ∫ χ[(∂αu)2 + (∂αv)2] ≤ C. Case β ≥ 4. For K ≥ β ≥ 4, 0 ≤ t ≤ T , we use Lemma 6.2 below, where we prove that ∑ |α|=β ∣∣ ∫ t 0 ∫ ξν(∂αu)∂α(uux) ∣∣+ ∣∣ ∫ t 0 ∫ ξν(∂αv)∂α(vvx) ∣∣ ≤ C + C ∑ |α|=β (∫ t 0 ∫ ξν(∂αu)2 ) (6.20) where C depends only on terms bounded in previous steps of the induction. Con- sequently, sup 0≤t≤T ∑ |α|=β ∫ ξν [(∂αu)2 + (∂αv)2] + C ∑ |α|=β+1 ∫ T 0 ∫ (ξν)x[(∂αu)2 + (∂αv)2] ≤ C where C does not depend on ν, but only on T and the norms of u0, v0 ∈ H3(R2) ∩ Hβ(W0 i 0). Passing to the limit, we obtain the desired estimates. � Lemma 6.2. For ξν as defined in (6.3), α = (α1, α2) such that |α| = β, 4 ≤ β ≤ K, the following holds:∑ |α|=β ∣∣ ∫ t 0 ∫ ξν(∂αu)∂α(uux) ∣∣+ ∣∣ ∫ t 0 ∫ ξν(∂αv)∂α(vvx) ∣∣ ≤ C + C ∑ |α|=β (∫ t 0 ∫ ξν(∂αu)2 ) + C ∑ |α|=β (∫ t 0 ∫ ξν(∂αv)2 ) (6.21) for 0 ≤ t ≤ T , where C depends only on sup 0≤t≤T ∫ ξν(∂γu)2, sup 0≤t≤T ∫ ξν(∂γv)2, (6.22)∫ T 0 ∫ (ξν)x(∂γux)2, ∫ T 0 ∫ (ξν)x(∂γvx)2, (6.23)∫ T 0 ∫ (ξν)x(∂γuy)2, ∫ T 0 ∫ (ξν)x(∂γvy)2 (6.24) for γ = (γ1, γ2) where |γ| ≤ β − 1. EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 35 The proof uses the same ideas as in the proof of Lemma 3.2. The primary difference is in our weight function ξν . First, our weight function here, ξν is ap- proximately constant for x < −1, whereas in Lemma 3.2, the weight function decayed exponentially for x < −1. Consequently, in our inductive proof of Lemma 6.2 below, we are not able to use the estimates we obtained on∫ T 0 ∫ B ∫ (ξν)x{(∂γux)2 + (∂γuy)2} ≤ C ∫ T 0 ∫ B eσx{(∂γux)2 + (∂γuy)2} from the previous step of the induction. In addition, for x > 1, the weight function ξν ≈ xi at all levels of the induction. Proof. We estimate only the terms∣∣ ∫ T 0 ∫ ξν (∂αu) ∂α (uux) ∣∣. (6.25) The terms ∣∣ ∫ T 0 ∫ ξν (∂αv) ∂α (vvx) ∣∣ are bounded in the same way. Each term in (6.25) is of the form∣∣ ∫ T 0 ∫ ξν (∂αu) (∂ru) (∂sux) ∣∣ where ri + si = αi for i = 1, 2. We use the notation qr = r1 + r2, qs = s1 + s2, With this notation, it follows that β = qr + qs. Remark 6.3. In what follows, we combine the fact that ξν(∂γu) = γ1∑ j=0 (−1)j ( γ j ) ∂jx((∂γ1−jx ξν)(∂γ2y u)). with (2.11) to conclude that sup 0≤t≤T ‖ξν(∂γu)2‖L∞(R2) ≤ C, for q ≤ β − 3, (6.26)∫ T 0 ‖ξν(∂γu)2‖L∞(R2) ≤ C, forq ≤ β − 2, (6.27) where γ1 + γ2 = q and C depends only on (6.22)-(6.24). We will use estimates (6.26) and (6.27) below in bounding each term in the integrand. Case qs ≤ β − 4. For this case we have∣∣ ∫ t 0 ∫ ξν (∂αu) (∂ru) (∂sux) ∣∣ = ∣∣ ∫ t 0 ∫ ξ1/2ν (∂αu) (∂ru) ξ1/2ν (∂sux) ∣∣ ≤ sup 0≤t≤T ‖ξ1/2ν (∂sux) ‖L∞ (∫ t 0 ∫ R2 (∂ru) 2 )1/2(∫ t 0 ∫ ξν (∂αu) 2 )1/2 . 36 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 The first term is bounded by (6.26). If qr ≤ β−1, then the second term is bounded by (6.22). If qr = β, then the third term is bounded by∑ α1+α2=β (∫ t 0 ∫ ξν ( ∂αu )2)1/2 . In either case, we obtain∣∣ ∫ t 0 ∫ ξν (∂αu) (∂ru) (∂sux) ∣∣ ≤ C + C ∑ |α|=β (∫ t 0 ∫ ξν (∂αu) 2 ) for qs ≤ β − 4. Case qs = β − 3. If qs = β − 3, then qr = 3. For this case, we consider x > 1 and x < −1 separately. For x > 1, ξν ≤ Cxi, while for x < −1, ξν ' C(1 + eσ x). Again, let A = {x > 1} × R and B = {x < −1} × R. First, we consider x > 1. For x > 1, if β ≥ 5, we use the estimate∣∣ ∫ t 0 ∫ A ξν (∂αu) (∂ru) (∂sux) ∣∣ ≤ (∫ t 0 ‖(ξν)x (∂ru) 2 ‖L∞(A) dt )1/2( sup 0≤t≤T ∫ A ξν (∂sux)2dx dy )1/2 × (∫ t 0 ∫ A ξν (∂αu)2 )1/2 . The first term is bounded by (6.27) because qr ≤ β−2. The second term is bounded by (6.22) because qs + 1 = β − 2. If β = 4 and qs = β − 3, we have qs = 1, qr = 3, in which case,∣∣ ∫ t 0 ∫ A ξν (∂αu) (∂ru) (∂sux) ∣∣ ≤ (∫ t 0 ‖(ξν)x (∂sux) 2 ‖L∞(A)dt )1/2( sup 0≤t≤T ∫ A ξν(∂ru)2 )1/2(∫ t 0 ∫ A ξν(∂αu)2 )1/2 . The first time is bounded by (6.27) because qs + 1 = 2 ≤ β − 2. The second term is bounded by (6.22) because qr ≤ β − 1. We now consider x < −1. In that case ξν ≈ 1 + eσx ≤ C. Since qs = β − 3, we know that qr = 3 ≤ β − 1. Therefore,∣∣ ∫ t 0 ∫ B ξν(∂αu)(∂tu)(∂sux) ∣∣ ≤ C ∫ t 0 |∂sux|L∞(B) (∫ B (∂ru)2 )1/2(∫ B (∂αu)2 )1/2 ≤ sup 0≤t≤T (∫ B (∂ru)2 )1/2(∫ t 0 |∂sux|2L∞(B) )1/2(∫ t 0 ∫ B (∂αu)2 )1/2 ≤ C sup 0≤t≤T (∫ B ξν(∂ru)2 )1/2(∫ t 0 ∫ B ξν(∂αu)2 )1/2 × (∫ t 0 ∫ B ξν{(∂sux)2 + (∂suxxx)2 + (∂suxyy)2} )1/2 . EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 37 The first term is bounded by (6.22) since qr = 3 ≤ β − 1. The other two terms are bounded by C ∫ t 0 ∫ B ξν{(∂sux)2 + (∂suxxx)2 + (∂suxyy)2}+ C ∫ t 0 ∫ ξν(∂αu)2 ≤ C ∑ |α|=β ∫ t 0 ∫ B ξν(∂αu)2, as desired. Case qs = β − 2. If qs = β − 2, then qr = 2 = 4 − 2 ≤ β − 2. We consider x > 1 and x < −1 separately. First, for x > 1, as in the case qs = β − 3, we have∣∣ ∫ t 0 ∫ A ξν(∂αu)(∂ru)(∂sux) ∣∣ ≤ ∫ t 0 |∂ru|L∞(A) (∫ A ξν(∂sux)2 )1/2(∫ A ξν(∂αu)2 )1/2 ≤ sup 0≤t≤T (∫ A ξν(∂sux)2 )1/2(∫ t 0 |(ξν)x(∂ru)2|L∞(A) )1/2(∫ t 0 ∫ ξν(∂αu)2 )1/2 . The first term is bounded by (6.22). The second term is bounded by (6.27). There- fore, ∣∣ ∫ t 0 ∫ ξν(∂αu)(∂ru)(∂sux) ∣∣ ≤ C + C ∫ t 0 ∫ ξν(∂αu)2 where C depends only on (6.22), (6.23), and (6.24). Next, we consider x < −1. In this case, ξν ≈ 1 + eσx ≤ C. In the case when qs = β − 2 and β ≥ 5, we can bound it as follows:∣∣ ∫ t 0 ∫ B ξν(∂αu)(∂ru)(∂sux) ∣∣ ≤ ∫ t 0 |∂ru|L∞(B) (∫ B (∂sux)2 )1/2(∫ B (∂αu)2 )1/2 ≤ C sup 0≤t≤T |∂ru|L∞(B) sup 0≤t≤T (∫ B (∂sux)2 )1/2 ∫ T 0 (∫ B (∂αu)2 )1/2 . Since qs + 1 = β − 1, the first two terms on the right-hand side are bounded by (6.22). It remains to consider x < −1 when qs = β − 2 and β = 4. In that case, qs = 2 and qr = 2. Then∣∣ ∫ t 0 ∫ B ξν(∂αu)(∂ru)(∂sux) ∣∣ ≤ C ( sup 0≤t≤T ∫ B (∂sux)2 )1/2(∫ t 0 |∂ru|L∞(B) )1/2(∫ t 0 ∫ B (∂αu)2 )1/2 . Since qs = 2, it follows that qs+1 = 3 = β−1. Therefore, the first term is bounded by (6.22). Since qr = 2, the second term satisfies∫ t 0 |∂ru|2L∞(B) ≤ C ∫ t 0 ∫ B (∂ru)2 + (∂ruxx)2 + (∂ruyy)2 38 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 ≤ C ∫ t 0 ∫ B ξν { (∂ru)2 + (∂ruxx)2 + (∂ruyy)2 } ≤ C ∑ α1+α2=β ∫ t 0 ∫ B ξν(∂αu)2. Therefore, we conclude that for qs = β − 2,∣∣ ∫ t 0 ∫ B ξν(∂αu)(∂ru)(∂sux) ∣∣ ≤ C ∑ |α|=β ∫ t 0 ∫ B ξν(∂αu)2 where C depends only on (6.22). Case qs = β − 1. If qs = β − 1, then qr = 1. Therefore,∣∣ ∫ t 0 ∫ B ξν(∂αu)(∂ru)(∂sux) ∣∣ ≤ C sup 0≤t≤T |∂ru|L∞(B) (∫ t 0 ∫ B ξν(∂sux)2 )1/2(∫ t 0 ∫ B ξν(∂αu)2 )1/2 ≤ C sup 0≤t≤T (∫ B (∂ru)2 + (∂ruxx)2 + (∂ruyy)2 )(∫ t 0 ∫ B ξν(∂sux)2 )1/2 × (∫ t 0 ∫ B ξν(∂αu)2 )1/2 ≤ C ∑ |α|=β ∫ t 0 ∫ B ξν(∂αu)2. Case qs = β. If qs = β, then qr = 0 and s = α. Therefore,∣∣ ∫ t 0 ∫ B ξν(∂αu)(∂ru)(∂sux) ∣∣ = C ∣∣ ∫ t 0 ∫ B ξνu[(∂αu)2]x ∣∣ ≤ C(‖u‖L∞ + ‖ux‖L∞) ∫ t 0 ∫ B ξν(∂αu)2 ≤ C ∫ t 0 ∫ B ξν(∂αu)2. � We now state and prove our main theorem, that if our initial data (u0, v0) has minimal regularity and sufficient decay as x → ∞, then the solution (u, v) is smoother than (u0, v0). For simplicity, we introduce the following space which will be used in the proof. Let ZL = H3(R2) ∩H0(W0L 0) with the accompanying norm ‖f‖2ZL = ∫ (1 + ξ)f2 + ∑ |α|=3 (∂αf)2 where ξ ∈W0L 0. Theorem 6.4. Let T > 0 and (u, v) be a solution of (1.3) in the region R2× [0, T ] such that (u, v) ∈ L∞([0, T ] : ZL)× L∞([0, T ] : ZL) (6.28) EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 39 for some L ≥ 1. Then sup 0≤t≤T ∫ ξβ [(∂αu)2 + (∂αv)2] <∞,∫ T 0 ∫ χβ [(∂αux)2 + (∂αuy)2] <∞,∫ T 0 ∫ χβ [(∂αvx)2 + (∂αvy)2] <∞, for 0 ≤ β ≤ L where β = α1 + α2, ξβ ∈ Wσ,L−β,β, χβ ∈ Wσ,L−β−1,β, σ > 0 arbitrary, with the exception that for β = L, Wσ,−1,L is replaced by W̃σ,−1,L. Remark 6.5. If assumption (6.28) holds for all L ≥ 1, then the solution is infinitely differentiable in the x and y variables. In that case, from (1.3), the solution is C∞ in all of its variables. Proof of Theorem 6.4. By assumption, u, v ∈ L∞([0, T ];ZL). Recall this means u, v ∈ H3(R2) and ∫ ξ(u2 + v2) < ∞ for ξ ∈ W0L 0. The equations imply ut, vt ∈ L∞([0, T ];L2(R2)). Therefore, u, v are weakly continuous functions of t with values in ZL, and, in particular, u(·, ·, t), v(·, ·, t) are in ZL for every t. Let {u(n)0 }, {v (n) 0 } be sequences of functions in C∞0 (R2) which converge to u(·, ·, t0), v(·, ·, t0) strongly in ZL, for 0 ≤ t0 < T . Let (u(n)(x, y, t), v(n)(x, y, t)) be the unique solution of (1.3) with initial data (u (n) 0 (x, y), v (n) 0 (x, y)) at time t = t0. By Theorem 5.2, the solution is guaranteed to exist in a time interval [t0, t0 + δ] where δ does not depend on n. By Theorem 6.1, u(n), v(n) ∈ L∞([t0, t0 + δ];ZL) and∫ t0+δ t0 ∫ χ[(u(n)x )2 + (u(n)y )2] dx dy dt + ∫ t0+δ t0 ∫ χ[(v(n)x )2 + (v(n)y )2] dx dy dt ≤ C, (6.29) where χ ∈ Wσ,L−1,0 and C depends only on the norms of u (n) 0 , v (n) 0 ∈ ZL. Also by Theorem 6.1, we have (non-uniform) bounds on sup t∈[t0,t0+δ] sup (x,y)∈R2 (1 + |x+|k) ∣∣∣∂αu(n)(x, y, t)∣∣∣ < +∞, (6.30) sup t∈[t0,t0+δ] sup (x,y)∈R2 (1 + |x+|k) ∣∣∣∂αv(n)(x, y, t)∣∣∣ < +∞ (6.31) for each n, k, and α. Therefore, the main estimates in Lemma 3.1 are justified for each u(n) and v(n) in the interval [t0, t0 + δ]. The multiplier χ may be chosen arbitrarily in its weight class and ξ is defined by (3.5). We start our induction with β = 1, in which case α = (1, 0) or α = (0, 1). Take χ ∈ Wσ,L−2,1 and let ξ = ∫ x −∞ χ(z, t)dz. As shown in Lemma 3.1, we have the following bounds on the higher derivatives of u(n), v(n), sup [t0,t0+δ] ∫ ξ[(u(n)x )2 + (v(n)x )2] + ∫ t0+δ t0 ∫ χ[(u(n)xx )2 + (u(n)xy )2] + ∫ t0+δ t0 ∫ χ[(v(n)xx )2 + (v(n)xy )2] ≤ C, (6.32) 40 J. L. LEVANDOSKY, O. VERA EJDE-2023/11 sup [t0,t0+δ] ∫ ξ[(u(n)y )2 + (v(n)y )2] + ∫ t0+δ t0 ∫ χ[(u(n)xy )2 + (u(n)yy )2] + ∫ t0+δ t0 ∫ χ[(v(n)xy )2 + (v(n)yy )2] ≤ C, (6.33) where C depends only on the norms of u(n), v(n) ∈ L∞([0, T ];ZL) and the terms in (6.29). We conclude, therefore, that the constants C in (6.32) and (6.33) depend only on ‖u(n)0 ‖ZL and ‖v(n)0 ‖ZL . We continue this process inductively. For the βth step, let χ ∈Wσ,L−β−1,β , and define ξ = ∫ x −∞ χ(z, t)dz. The non-uniform bounds on u(n) and v(n) in (6.30) and (6.31) allow us to use Lemma 3.1 and our inductive hypothesis to conclude that sup [t0,t0+δ] ∫ ξ[(∂αu(n))2 + (∂αv(n))2] + ∫ t0+δ t0 ∫ χ[(∂αu(n)x )2 + (∂αu(n)y )2] + ∫ t0+δ t0 ∫ χ[(∂αv(n)x )2 + (∂αv(n)y )2] ≤ C, where again C does not depend on n, but only on the norms of u (n) 0 , v (n) 0 ∈ ZL. By Corollary 5.3, u(n) ⇀ u weak* in L∞([t0, t0 + δ];H3(R2)) and v(n) ⇀ v weak* in L∞([t0, t0 + δ];H3(R2)). Therefore, we can pass to the limit and conclude that sup [t0,t0+δ] ∫ ξ[(∂αu)2 + (∂αv)2] + ∫ t0+δ t0 ∫ χ[(∂αux)2 + (∂αuy)2] + ∫ t0+δ t0 ∫ χ[(∂αvx)2 + (∂αvy)2] ≤ C. We continued the process inductively up to β = L, with the exception that on the last level, β = L, we replace Wσ,−1,L with W̃σ,−1,L. Since δ is fixed, this result is valid over the whole interval [0, T ]. � 7. Concluding remarks In this article, we proved that if the initial data decays faster than any polynomial as x→∞, the solution of a coupled Zakharov-Kuznetsov system lies in C∞(R2)× C∞(R2). We quantified the gain in regularity of the solution depending on the amount of decay of the initial data as x→∞. In particular, we showed that if the initial data (u0, v0) lies in H3(R2) ×H3(R2) and lies in a weighted L2 × L2 space with a weight function that behaves like xL as x→∞, then the solution (u, v) lies in a weighted Sobolev space HL ×HL for 0 < t ≤ T where T is the existence time of the solution. Acknowledgments. This research was partially supported by project Fondecyt 1191137 and UTA MAYOR 2022-2023, 4764-22. EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 41 References [1] M. S. Alves, O. Vera Villagrán; Smoothing properties for a coupled system of nonlinear evolution dispersive equations, Indag. Math., 20 no. 2 (2009), 285-327. [2] O. V. Besov, V. P. II’in, S. M. 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USSR Sb., 48 (1984), 93-138. [16] J. Levandosky; Gain of regularity for the KP-II equation, Indiana Univ. Math. J., 49 (2000), 353-403. [17] J. Levandosky; Smoothing properties of nonlinear dispersive equations in two spatial dimen- sions, J. Differential Equations, 175 (2001), 275-352. [18] J. Levandosky; Smoothing properties for a two-dimensional Kawahara equation, J. Differen- tial Equations, 316 (2022), 158-196. [19] J. Levandosky, M. Sepulveda, O. Vera Villagran; Gain of regularity for the KP-I equation, J. Differential Equations, 245, 3 (2008), 762-808. [20] J. L. Lions; Quelque méthodes de résolution des problemes aux limites non linéaires, Gauthiers-Villars. [21] G. Ponce; Regularity of solutions to nonlinear dispersive equations, J. Differential Equations, 78 (1989), 122-135. [22] O. Vera; Gain of regularity for a generalized coupled system of nonlinear evolution dispersive equations type, PhD thesis, UFRJ, Rio de Janeiro, Brazil, 2001. [23] O. Vera; Gain of regularity for a Korteweg-de Vries-Kawahara type equation, Electron. J. Differential Equations, 2004, 71 (2004), 1-24. [24] Y. Zhu, R. Grimshaw; Oblique interactions between internal solitary waves., Stud. Appl. math. Vol. 92, 3 (1994), 249-270. Julie L. Levandosky Department of Mathematics, Framingham State University, Framingham, MA 01701 USA Email address: jlevandosky@framingham.edu Octavio Vera Departamento de Matemáticas, Universidad de Tarapaca, Casilla 7-D, Arica, Chile Email address: opverav@academicos.uta.cl 1. Introduction Gain of regularity theorem 2. Preliminaries Choice of weight function 3. Main inequality 4. A priori estimates 5. Uniqueness and local existence 6. Weighted estimates and main estimates of error terms 7. Concluding remarks Acknowledgments References