Smoothing properties for a coupled Zakharov-Kuznetsov system

. 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 ( u 0 ,v 0 ) possesses certain regularity and suﬃcient decay as x → ∞ , then the solution ( u ( t ) ,v ( t )) will be smoother than ( u 0 ,v 0 ) for 0 < t ≤ T where T is the existence time of the solution.


Introduction
The general form of the coupled Zakharov-Kuznetsov system [12] is (1.1) 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.-u y = v y = 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 u(x, y, 0) = u 0 (x, y), v(x, y, 0) = v 0 (x, y) 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 ∈ L 2 (R 2 ) 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 L 2 ((1 + e σx )dx), the unique solution u(t) ∈ C ∞ (R 2 ) for t > 0.
As will be shown, the assumption that sup 0<t≤T R 2 2 ] dx dy < +∞ for all integers L ≥ 1 may be reduced to assuming this property holds for some integer L ≥ 1.The smoothing phenomenon will still occur, but the amount of smoothing will depend on the size of L, thereby showing the relationship between the decay at infinity of the initial data and the gain in regularity of the solution.
The plan of the paper is the following.In Section 2, we introduce the weighted Sobolev spaces which will be used to describe the gain in regularity.In Section 3, we state and prove the main inequality used to show the gain in regularity.In Sections 4 and 5, we prove an existence result showing that if the initial data (u 0 , v 0 ) is sufficiently smooth, there exists a unique solution (u(t), v(t)) of (1.3) with the same amount of regularity for a time interval [0, T ] depending only on a Sobolev norm involving the initial data.In Section 6, we prove that if the initial data also possesses sufficient decay at infinity, the solution (u(t), v(t)) possesses similar decay at infinity.Finally, in Theorem 6.4, we state and prove the main result.Using induction we show that the decay at infinity of the initial data leads to a gain in regularity for the solution (u(t), v(t)).

Preliminaries
The idea for the proof of the gain in regularity is the following.For the first step of the induction, we multiply (1.3) 1 by 2ξu where ξ is our weight function, to be specified later, and integrate over R 2 .Upon doing so, we obtain 2 ξ uu t + 2 ξ uu xxx + 2 ξuu yyx − 12 ξ u(u u x ) − 2 ξ uv x = 0. (2.1) where = R 2 dx dy.Then, integrating by parts, we see that If our weight function ξ satisfies 0 < ∂ j x ξ ≤ C∂ k x ξ for all j ≥ k ≥ 0, then we arrive at the inequality Similarly, multiplying (1.3) 2 by 2ξv, integrating by parts, and using the same assumption on the weight function ξ, we see that (2.4) Combining these two inequalities, we have Notice the second and third terms on the left-hand side.Assuming ξ x > 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| = α 1 + α 2 = β.We then multiply the differentiated equations by 2ξ(∂u) and 2ξ(∂v), respectively, where ∂ α ≡ ∂ α1 x ∂ α2 y and ξ = ξ β is our weight function to be described below.Integrating over R 2 and integrating by parts as described above, if our weight function ξ satisfies 0 < ∂ j x ξ ≤ C∂ k x ξ for all j ≥ k ≥ 0, we arrive at the following inequality.
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 e σx 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 W σ i k if it is a positive C ∞ function on R × [0, T ], ξ x > 0, and there are constants c j , 1 ≤ j ≤ 5 such that 0 < c 1 ≤ t −k e −σ x ξ(x, t) ≤ c 2 ∀ x < −1, 0 < t < T, (2.5) We now define weighted function spaces using the weight functions introduced above.
Definition 2.2.Let N be a positive integer.Let H β (W σ i k ) be the space of functions with finite norm for any ξ ∈ W σ 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 H N (R 2 ) without a weight.
Definition 2.4.For each fixed ξ ∈ W σ i k , β ≥ 0, we define the space Moreover, we define the spaces We shall use those spaces only in the case when i = −1.
In this article, we make use of the following Sobolev embedding estimates (2.12) In general, we have the anisotropic imbedding.For 2 ≤ n < 6, (2.13)

Main inequality
In this section we state and prove the main lemma that will be used in our main theorem on the gain of regularity.Specifically, we prove that if there exists a solution (u, v) of (1.3) sufficiently smooth and with sufficient decay at infinity, the weighted Sobolev norms for (u, v) are bounded above by other weighted Sobolev norms involving less derivatives of u and v. Lemma 3.1., For (u, v) a solution of (1.3) sufficiently smooth and with sufficient decay at infinity, ) The idea of the proof is the following.We would like to bound terms on the lefthand side of (3.1) in terms of integrals of the same form, but with a lower number of derivatives.In particular, we hope to bound the left-hand side of (3.1) in terms of (3.2) - (3.4).On each level of the induction, the weight function ξ behaves like a power of x for x > 1, an exponential e σx 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, ξ β ≈ tx L−1 for x > 1.For β = 2, ξ β ≈ t 2 x L−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.
where |α| = β.Take α derivatives of (1.3) 1 , multiply the differentiated equation by 2ξ β (∂ α u) where Similarly, take α derivatives of (1.3) 2 , multiply the differentiated equation by 2ξ β (∂ α v), and integrate over R 2 × [0, t] for 0 ≤ t ≤ T .Doing so, we conclude that Then, adding (3.6) and (3.7), we have Now using the fact that ∂ j x ξ ≤ Cξ and ξ(•, 0) = 0 for β ≥ 1, we obtain the identity 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 Each of these terms is bounded by terms of the form (3.3) and (3.4).Therefore, we conclude that where C depends only on (3.3) and (3.4).Therefore, it remains to look for bounds on the remainder terms for each β ≥ 1. Case Subcase α = (1, 0).The remainder terms satisfies where C depends only on u H 3 .Similarly for the remainder term involving v.
Combining this bound with (3.8) and the fact that ξ x = χ β we conclude that where χ ≡ χ β and C depends only on u H 3 , v H 3 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 where C depends only on u H 3 .Similarly for the terms in v. Combining these estimates with (3.8), we conclude that where C depends only on u H 3 , v H 3 and terms in (3.3) and (3.4).
Subcase α = (2, 0).In this case, the remainder terms satisfy where C depends only on u H 3 .
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 Combining these estimates, we have where C depends only on u H 3 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 where C depends only on u H 3 (R 2 ) , v H 3 (R 2 ) and terms in (3.3) and (3.4).
Before proving Lemma 3.2, we describe the form of each term in (3.9).
The above lemma follows from the Leibniz formula applied to ∂ α (uu x ) and ∂ α (vv x ).
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 ≡ ξ β ∈ W σ,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 t k x for k, ≥ 0. For the case x < −1, ξ t k e σ 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 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 In fact, Hence, combining (3.18) with (3.16) we conclude that for ν = |γ| ≤ β − 3, and 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 For notation, let ν r = r 1 + r 2 and In this case, we bound the remainder term as follows 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 as long as L ≥ 1.The others three terms are bounded by (3.19), (3.14) and (3.15).
In this case, we have Therefore, M = 0. Also, Then the term on the right-hand side is bounded as desired.Similarly, if r = (0, 1), we have Case ν s = β.In this case ν r = 0. Then where again the right-hand side is bounded by (3.14) or (3.15).Lemma 3.2 follows.

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 Z N T .We define with the norm Lemma 4.1.Let u, v, w, z be functions in Z N t for all N and all t ≥ 0 such that u, v, w, z are solutions to Then for all N ≥ 0, the following inequality holds for all t ≥ 0.
Proof.Fix N ≥ 0 and choose α such that |α| = N .Applying ∂ α to (4.3) 1 we have Multiplying (4.5) by 2∂ α u and integrating over R 2 we obtain 6) The first term on the right-hand side of (4.6) is estimated using (2.11) together with the Cauchy-Schwarz inequality For the second to last term on the right-hand side of (4.6), integrating by parts and using (2.11) we have Therefore, In a similar way applying the same idea to (4.3) 2 , we obtain Then adding (4.7) and (4.8) we obtain . (4.9) Now, taking three x-derivatives of (4.5) and multiplying the result by 2∂ α u xxx , we have Each term in the above expression is estimate separately.For the first term it follows that .
Next, we consider I 2 .
Using the Cauchy-Schwarz inequality and (2.12), for the first term in (4.11), we have Integrating by parts the last term in (4.11), using the Cauchy-Schwarz inequality, and (2.12), we have For the third term in (4.10) we have . Now we estimate the last term in (4.10) as follows If α = (0, 0), then integrating by parts and using (2.11), For |α| > 0, using (2.11) for the first term in (4.12), we have .
For the last term in (4.12), integrating by parts and using (2.11) we have Consequently, In a similar way, applying the same idea to (4.3) 2 , we obtain Then adding (4.13) and (4.14) we obtain Similarly, applying ∂ α ∂ 3 y to each equation in (4.3), and using similar analysis, we obtain On the other hand, applying one t-derivative to (4.5), multiplying by 2 (∂ α u t ) and integrating over R 2 we arrive at the inequality For the first term on the right hand side, we use (2.11) and the Cauchy-Schwarz inequality .
We look at the second term on the right-hand side.If α = (0, 0) we have .
If α = (0, 0), we have The first term in (4.17) is estimated as Using integration by parts in the last term in (4.17) along with (2.11) and the Cauchy-Schwarz inequality, we have In a similar way, we obtain Then adding (4.18) and (4.19) we obtain Then, for 0 ≤ t ≤ t, it follows that .
Integrating with respect to t, and using the fact that this estimate is true for all α such that |α| = N , we obtain as claimed.
, there exists a unique solution 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 where h, and h are smooth bounded coefficients.Fix a time T > 0 and a constant where ) which vanish at t = 0, we introduce the bilinear form Integrating by parts, we have for some constant c large enough.Multiplying (4.26) by e −M t and integrating in time from t = 0 to t = T , we obtain for Ψ ∈ C([0, T ] : )) 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 ] : )) : 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 on D.
Remark 4.3.To obtain higher regularity of the solution, we repeat the proof with higher derivatives included in the inner product.

Uniqueness and local existence
In this section, we prove that for initial data (u where the time T of existence depends only on u 0 H 3 and v 0 H 3 .First we address the question of uniqueness. )), so the integrations below are justified.Therefore, the differences (u − u) and (v − v) satisfy (5.1) Now, multiplying (5.1) by 2(u − u) and integrating for (x, y) ∈ R 2 , we have (5.2) the 2nd and 3rd terms in (5.2) are shown to be identically zero.Hence On the other hand, Combining this estimate with (5.3), we have (5.4) Similarly we have Now, multiplying (5.1) by 2(v − v) and integrating over R 2 we have (5.6) Then the 2nd, 3rd, and 4th terms in (5.6) are shown to be identically zero.Hence On the other hand, Combining this estimate with (5.3), we have (5.8) Adding (5.4) with (5.8) we obtain (5.9) The last two terms satisfy Therefore, (5.10) Using that u(x, y, 0) − u(x, y, 0) ≡ 0, v(x, y, 0) − v(x, y, 0) ≡ 0 and Gronwall's inequality it follows that We conclude that u ≡ u and v ≡ v.This proves the uniqueness of the solution.
We now prove the existence of a local solution for (1.3).We show that for each (u 0 , v 0 ) ∈ H N +3 R 2 × H N +3 R 2 there exists a solution (u, v) in the space L ∞ [0, T ] : ) for a time T depending only on u 0 H 3 (R 2 ) and v 0 H 3 (R 2 ) .Theorem 5.2.Let κ 0 , κ 0 > 0 and N be an integer ≥ 0. Then there exists a time 0 < T < ∞, depending only on κ 0 and κ 0 such that for all u 0 , v 0 ∈ H N +3 (R 2 ), with u 0 H 3 (R 2 ) ≤ κ 0 and v 0 H 3 (R 2 ) ≤ κ 0 , there exists a solution of (1.3) with 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 )) for a time T depending only on u 0 H 3 , v 0 H 3 .Third, we prove that a subsequence of solutions to our approximate equations converges to a solution , where the time T depends only on u 0 H 3 , v 0 H 3 .
Proof of Theorem 5.2.It suffices to prove this result for u 0 , v 0 ∈ ∩ N ≥0 H N (R 2 ).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) = u 0 (x, y), v (n) (x, y, 0) = v 0 (x, y), and where the first approximations are given by u (0) (x, y, t) = u 0 (x, y) and v (0) (x, y, t) = v 0 (x, y).By 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 for all t ≥ 0. By assumption, . On the other hand, (5.12) In a similar way we have where K and K are independent of n.Without loss of generality, suppose ], from (5.11), (5.12), and (5.13), it follows that (5.14) Now choose T > 0 such that CT c 3 0 = 1.We claim that T all n.So, assume there exists n such that T * (n) 0 < ∞.Suppose T > T * (n) 0 . Then, by the continuity of u (n) However, this implies c 2 0 < c 2 0 and we have a contradiction.Thus, we conclude that T * (n) 0 ≥ T for all n, and, therefore, for all n.Consequently, there exists a bounded sequence of solutions {(u (5.16) On the other hand, Then by the Lions-Aubin compactness Theorem [20] there are subsequences u (nj ) := u (n) and v (nj ) (5.17) Hence for subsequences u (nj ) := u (n) and v (nj ) := v (n) , we have (5.18)Moreover, from (5.15) we have (5.19) Now we show that the nonlinear term converges to its correct limit.From (5.17), (5.20) We claim that T * N ≥ T , and, therefore, a time of existence can be chosen depending only on u 0 H 3 , v 0 H 3 .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 u 0 , v 0 ∈ H N +3 (R 2 ) for some N ≥ 0 and let u be a sequence converging to v 0 .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 ] : (5.25) )), then there exist weak* convergent subsequences, still denoted {u (n) } and {v (n) } such that By the Lions-Aubin Compactness theorem [20] we have ).Now we just to show that each term in (1.3) converges to its correct limit, and thus u ) .The only thing we need to show is that the nonlinear term converges to its correct limit, namely that u Clearly the linear terms also converge in L 2 ([0, T ] : L 1 loc (R 2 )) and therefore, we conclude that u ).In a similar way we conclude that v . By the uniqueness theorem, Theorem 5.1, ( u, v) = (u, v).

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 (u 0 , v 0 ).Indeed, we prove that if our initial data (u 0 , v 0 ) ∈ H 3 (R 2 ) × H 3 (R 2 ) also lies in some weighted space H K (W 0 i 0 ) × H K (W 0 i 0 ), for integers K ≥ 0 and i ≥ 1, then our solution (u, v) also lies in L ∞ [0, T ] : and for |γ| ≤ K, where χ is a weight function in W σ,i−1,0 for σ > 0 arbitrary, and C depends only on T and the norms of 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 ]; H 3 (R 2 )) and u 0 , v 0 ∈ H K (W 0 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 ∈ H 3 (R 2 ) by smooth solutions and approximate general weight functions ξ ∈ W 0 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 χ ν , which decay as |x| → ∞ and which approximate χ ∈ W σ,i−1,0 from below, uniformly on any half-line (−∞, c).Let Hence, the functions ξ ν are bounded weight functions which approximate a weight function ξ ∈ W 0 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 β th induction step, we take α derivatives of (1.3) 1 , where |α| = β, multiply the result by 2ξ ν (∂ α u), and integrate over R 2 .Performing straightforward calculations and using (ξ ν ) t , (ξ ν ) x ≤ Cξ ν we obtain the following estimate Similarly, we take α derivatives of (1.3) 2 , where |α| = β, multiply the result by 2ξ ν (∂ α v), and integrate over R 2 .Performing straightforward calculations as in (6.4) we obtain the estimate Adding (6.4) and (6.5) and using the Cauchy-Schwarz inequality we have Case β = 0. We need to estimate the terms For (6.7) 1 , we have where C depends only on the norm of u ∈ L ∞ ([0, T ] : Combining these estimates with (6.6), we conclude that where C depends only on u 0 H 3 and v 0 H 3 .Integrating (6.8) on t ∈ [0, T ] we obtain (6.9) Therefore, using Gronwall's inequality, where C does not depend on ν, but only on T and the norm of u 0 , v 0 ∈ H 3 (R 2 ) ∩ H 0 (W 0 i 0 ).Passing to the limit, Case β = 1.Consider α = (1, 0).In fact, for the case α = (1, 0) we have where C depends only on the norms of u 0 , v 0 ∈ H 3 (R 2 ).Performing similar calculations as in the case above, along with (6.6), and Gronwall's inequality we conclude that where C does not depend on ν, but only on T and the norm of u 0 , v 0 ∈ H 3 (R 2 ) ∩ H 1 (W 0 i 0 ).Passing to the limit, where C depends only on the norms of u 0 , v 0 ∈ H 3 (R 2 ) ∩ H 1 (W 0 i 0 ).Next, we consider the case α = (0, 1).For the case α = (0, 1), we have Therefore, using the same idea as above, we conclude that Case β = 2.We have α = (2, 0), α = (1, 1), and α = (0, 2).First, for the case α = (2, 0) we have Performing similar calculations as in the cases given above together with (6.6) and using Gronwall's inequality we have where C depends on T and the norm of u 0 , v 0 ∈ H 3 (R 2 ) ∩ H 2 (W 0 i 0 ), and does not depend on ν.Passing to the limit, For the case α = (1, 1) we have where C depends only on u 0 , v 0 ∈ H 3 (R 2 ) ∩ H 2 (W 0 i 0 ).Consequently, using the same ideas as above, where C depends only on u 0 , v 0 ∈ H 3 (R 2 ) ∩ H 2 (W 0 i 0 ).For the case α = (0, 2), our remainder term satisfies Therefore, where C depends only on u 0 , v 0 ∈ H 3 (R 2 ) ∩ H 2 (W 0 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 First, we consider I 1 .We consider the case x > 1 and x < −1 separately.For x > 1, we use the fact that ξ 1/2 ν ≤ Cξ ν for i ≥ 1.In addition, we will use (2.13). 3 .
Further, we note that where C depends only on the norms of u 0 , v 0 ∈ H 3 (R 2 )∩H 2 (W 0 i 0 ) by the previous step of the induction.On the other hand, for x < −1, we use the fact that ξ ν C to show (6.16) For the term I 2 , we have Lastly for I 3 , we have Combining these estimates with (6.4) and using similar estimates for v, we conclude that for 0 ≤ t ≤ T .Using similar estimates for other derivatives on the level β = 3, we conclude that Therefore, by Gronwall's inequality, where C does not depend on ν, but only on T and the norms of u 0 , v 0 ∈ H 3 (R 2 ) ∩ H 3 (W 0 i 0 ).Passing to the limit, we conclude that Case β ≥ 4. For K ≥ β ≥ 4, 0 ≤ t ≤ T , we use Lemma 6.2 below, where we prove that where C depends only on terms bounded in previous steps of the induction.Consequently, where C does not depend on ν, but only on T and the norms of u 0 , v 0 ∈ H 3 (R 2 ) ∩ H β (W 0 i 0 ).Passing to the limit, we obtain the desired estimates.Lemma 6.2.For ξ ν as defined in (6.3), α = (α 1 , α 2 ) such that |α| = β, 4 ≤ β ≤ K, the following holds: for 0 ≤ t ≤ T , where C depends only on The proof uses the same ideas as in the proof of Lemma 3.2.The primary difference is in our weight function ξ ν .First, our weight function here, ξ ν is approximately 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 from the previous step of the induction.In addition, for x > 1, the weight function ξ ν ≈ x i at all levels of the induction.
Proof.We estimate only the terms

.25)
The terms are bounded in the same way.Each term in (6.25) is of the form where r i + s i = α i for i = 1, 2. We use the notation q r = r 1 + r 2 , q s = s 1 + s 2 , With this notation, it follows that β = q r + q s .Remark 6. 3.In what follows, we combine the fact that where γ 1 + γ 2 = 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 q s ≤ β − 4. For this case we have The first term is bounded by (6.26).If q r ≤ β − 1, then the second term is bounded by (6.22).If q r = β, then the third term is bounded by In either case, we obtain Case q s = β − 3.If q s = β − 3, then q r = 3.For this case, we consider x > 1 and x < −1 separately.For x > 1, ξ ν ≤ Cx i , while for x < −1, First, we consider x > 1.For x > 1, if β ≥ 5, we use the estimate The first term is bounded by (6.27) because q r ≤ β −2.The second term is bounded by (6.22) because q s + 1 = β − 2. If β = 4 and q s = β − 3, we have q s = 1, q r = 3, in which case, The first time is bounded by (6.27) because q s + 1 = 2 ≤ β − 2. The second term is bounded by (6.22) because q r ≤ β − 1.
Case q s = β − 1.If q s = β − 1, then q r = 1.Therefore, Case q s = β.If q s = β, then q r = 0 and s = α.Therefore, We now state and prove our main theorem, that if our initial data (u 0 , v 0 ) has minimal regularity and sufficient decay as x → ∞, then the solution (u, v) is smoother than (u 0 , v 0 ).For simplicity, we introduce the following space which will be used in the proof.Let with the accompanying norm where ξ ∈ W 0 L 0 .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.