Global Well-Posedness for Coupled System of mKdV Equations in Analytic Spaces

For an effective approach to solving problems arising in modern science and technology, one cannot do without researching nonlinear problems of mathematical physics. The rapid development of new technology and the emergence of its high speed allow researchers to build and consider increasingly complex multidimensional models describing various phenomena, which are modeled, as a rule, using nonlinear partial differential equations (systems). However, now it has become clear that without the development of analytical methods, it is impossible to get a complete idea of the essence of the phenomenon. Analytical methods provide not only a reliable tool for debugging and comparing various numerical methods but also sometimes anticipate some scientific discoveries, make it possible to study the properties of models, to detect the presence of certain effects as a result of the existence or nonexistence of objects (solutions) with the required properties. Therefore, at present, fundamental research is being intensively carried out aimed at proving theorems of existence, uniqueness, and regularity of solutions of nonlinear partial differential equations. In the present paper, a coupled system of modified Korteweg-de Vries equations is considered as follows:


Introduction and Main Results
For an effective approach to solving problems arising in modern science and technology, one cannot do without researching nonlinear problems of mathematical physics. The rapid development of new technology and the emergence of its high speed allow researchers to build and consider increasingly complex multidimensional models describing various phenomena, which are modeled, as a rule, using nonlinear partial differential equations (systems). However, now it has become clear that without the development of analytical methods, it is impossible to get a complete idea of the essence of the phenomenon. Analytical methods provide not only a reliable tool for debugging and comparing various numerical methods but also sometimes anticipate some scientific discoveries, make it possible to study the properties of models, to detect the presence of certain effects as a result of the existence or nonexistence of objects (solutions) with the required properties. Therefore, at present, fundamental research is being intensively carried out aimed at proving theorems of existence, uniqueness, and regularity of solutions of nonlinear partial differential equations.
In the present paper, a coupled system of modified Korteweg-de Vries equations is considered as follows: The dynamics of solutions in the Korteweg-de Vries equations (KdV) and the modified Korteweg-de Vries equations (mKdV) are well studied due to the complete integrability of these equations (see [1][2][3][4][5][6]). For KdV equations, the studies date back to the 1970s, although some results have been obtained very recently (please see [7]). We extend the results in [7] and consider a coupled system of mKdV-type equations on the line in Equation (1).
For mKdV equations, many problems have been studied. It is proved that the mKdV equation is locally [8] and globally [9] well-posed in H s ðT Þ for s ≥ 1/2. Global wellposedness in L 2 ðT Þ is shown in [10].
For 0 < β < 1, the author in [7] proved that the IVP (Equation (1)) is locally well-posed for the given data ðu 0 , v 0 Þ ∈ H s ðℝÞ × H s ðℝÞ, s > −1/2. Oh in [11] used the Fourier transform restriction norm method and proved that the next IVP is locally well-posed for data with regularity s ≥ 0.
For β = 1, the system (Equation (1)) reduces to a special case of a broad class of nonlinear evolution equations considered by Ablowitz et al. [12] in the inverse scattering context. In this case, the well-posedness issues along with existence and stability of solitary waves for this system are widely studied in the literature, using the technique developed by Kenig et al. in [13,14].
Well-posedness for the nonperiodic gKdV equation in spaces of analytic functions has been proved by Grujic and Kalisch [15].
A class of suitable analytic functions for our analysis is the analytic Gevrey class G δ,s ðℝÞ = G δ,s introduced by Foias and Temam [16], defined as follows: for s ∈ ℝ and δ > 0 with h·i = ð1 + j·jÞ. For δ = 0, the space G δ,s coincides with the standard Sobolev space H s . For all 0 < δ ′ < δ and s, s ′ ∈ ℝ, we have which is the embedding property of the Gevrey spaces. New minimal conditions are used to show the local wellposedness of solution by using linear and trilinear estimates, together with contraction mapping principle. By imposing a more appropriate conditions with the help of the approximate conservation law, we obtain an unusual global existence result in Gevery spaces. Proposition 1 (Paley-Wiener Theorem) [17]. Let δ > 0, s ∈ ℝ. Then, f ∈ G δ,s if and only if it is the restriction to the real line of a function F which is holomorphic in the strip fx + iy : x, y ∈ ℝ, jyj < δg and satisfies Remark 2. In the view of the Paley-Wiener Theorem, it is natural to take initial data in G δ,s , to obtain the best behavior of solution and may be extended to be globally in time. It means that given ðu 0 , v 0 Þ ∈ G δ,s × G δ,s for some initial radius δ > 0, we then estimate the behavior of the radius of analyticity δðTÞ over time.
The first main result on local well-posedness of Equation (1) in analytic spaces reads as follows.
An effective method for studying lower bounds on the radius of analyticity, including this type of problem, was introduced in [18] for 1D Dirac-Klein-Gordon equations. It was applied in [19] to the modified Kawahara equation and in [20] to the nonperiodic KdV equation (for more details, please see [20][21][22][23]).
The second result for the problem (Equation (1)) is given in the next theorem. Theorem 4. Let s > −1/2, 0 < β < 1, and δ 0 > 0. Assume that ðu 0 , v 0 Þ ∈ G δ,s × G δ,s , then the solution in Theorem 3 can be extended to be global in time and for any T ′ > 0, we have the following: with where σ 0 > 0 can be taken arbitrarily small and C 1 > 0 is a constant depending on w 0 , δ 0 , s, and σ 0 .
The third result is Gevrey's temporal regularity of the unique solution obtained in the Theorem 3. A nonperiodic function f ðxÞ is the Gevrey class of order r, i.e., f ðxÞ ∈ G r , if there exists a constant C > 0 such that if r = 1f ðxÞ is analytic.
T, G δðTÞ,s Þ given by Theorem 4 belongs to the Gevrey class The proof of Theorem 5 is similar to that in [1]. The paper is organized as follows. In Section 2, we define the function spaces and linear and trilinear estimates. In Section 3, we prove Theorem 3, using the linear and trilinear estimates, together with contraction mapping principle. In Section 4, we prove the existence of fundamental approximate conservation law. In the last section, Theorem 4 will be proved using the approximate conservation law.

Preliminary Tools and Analytic
Function Spaces 2.1. Function Spaces. We define the analytic Bourgain spaces related to the modified Korteweg-de Vries type equations. The completion of the Schwartz class Sðℝ 2 Þ is given by We often use without mention, the definition For any interval I, we define the localized spaces X β δ,s,b ðℝ × IÞ = X β,I δ,s,b with norm: 2.2. Linear Estimates. We have the trilinear estimate (Equations (15) and (16)) defined in the analytic Bourgain spaces. Since the spaces X β δ,s,b is continuously embedded in Cð½0, T, G δ,s Þ, provided b > 1/2. Lemma 6. Let b > 1/2, s ∈ ℝ, and δ > 0. Then, for all T > 0, we have the following: Proof. First, we note that the operator A defined by where X β s,b is introduced in [7]. We observe that Aw belongs to Cðℝ, H s Þ and for some C > 0, we have the following: Thus, it follows that w ∈ Cð½0, T, G δ,s Þ and Taking the Fourier transform with respect to x of the Cauchy problems (Equation (1)), after an ordinary calculation, we localize in t by using a cut-off function, satisfying ψ ∈ C ∞ 0 , with ψ = 1 in ½−1, 1, suppψ ⊂ ½−2, 2, and ψ T ðtÞ = ψðt/TÞ. We consider the operator Λ, Γ given by the following: where SðtÞ = e −t∂ 3 x and S β ðtÞ = e −tβ∂ 3 x are the unitary groups associated with the linear problems.
The nonlinear terms defined by F 1 = ðuv 2 Þ and F 2 = ðu 2 vÞ will be treated in the next lemmas.

Journal of Function Spaces
Lemma 7. Let s, b ∈ ℝ and δ > 0. For some constant C > 0, we have the following: for all u 0 , v 0 ∈ G δ,s .
Proof. By definition, we have the following: It follows that Since b > 1/2, we have the following: and δ > 0, then for some constant C > 0, we have the following: Proof. Define We have, by Equation (19), the following: Thus, Owing to Lemma 6 in [7], we get the following: This completes the proof.
Proof. The proof of Lemma 9 for δ = 0 can be found in Lemma 13 of [14], for δ > 0 as one merely has to replace w by Aw, where the operator is defined in Equation (19).

Trilinear Estimates.
We have the trilinear estimate in the following lemmas.

Journal of Function Spaces
Thus, Λ × Γ maps Bð0, RÞ into Bð0, RÞ, which is a contraction, since  (1) with ðvð·, 0Þ, uð·, 0ÞÞ = ðv * ð·, 0Þ, u * ð·, 0ÞÞ in G δ,s × G δ,s . Setting ϑ = u − u * and ω = v − v * , we see that ϑ, ω solves the Cauchy problem: Thus, by Equation (50), we have the following: since we have the following: Thanks to Equation (53), we have the following: Integrating by parts of the last integral, we obtain the following: from which we deduce the inequality as follows: Since u, u * ∈ Cð½0, T, G δ,s Þ, we have that u and u * are continuous in t on the compact set ½0, T and are G δ,s in x. Thus, we can conclude that Therefore, from Equations (56) and (57), we obtain the differential inequality: Solving it gives the following: Since ∥ϑð0Þ∥ 2 L 2 = 0, from Equation (59), we obtain that ϑðtÞ = 0, 0 ≤ t ≤ T, or u = u * . Now by Equation (51), we have the following: Solving it gives the following: Since ∥ωð0Þ∥ 2 L 2 = 0, from Equation (61), we obtain that 3.3. Continuous Dependence of the Initial Data. To prove continuous dependence of the initial data, we will prove the following.

Approximate Conservation Law
We have the following: which is conserved for a solution ðu, vÞ of Equation (1). We are going to show an approximate conservation law for a solution to Equation (1) Moreover, we have the following: We need the following estimate.

Journal of Function Spaces
Now by taking s = −κ ∈ ð−1/2, 0, we obtain the following: Then, Now let L 2 = ðAuAuAvÞ − Aðu 2 vÞ. Then, By Equations (76) and (77), we have the following: Proof (Theorem 15). Let Uðt, xÞ = Auðt, xÞ, Vðt, xÞ = Avðt, xÞ which are real-valued since the multiplier A is even and u, v are real-valued. Applying A to Equation (1), we obtain the following: where G 1 = ∂ x ½ðAuAvAvÞ − Aðuv 2 Þ and G 2 = ∂ x ½ðAuAuAvÞ − Aðu 2 vÞ. By multiplying both sides of Equation (79) by U and Equation (80) by V and integrating with respect to space variable, we get the following: Then, Noting that ∂ j x Uðx, tÞ ⟶ 0 as |x | ⟶∞ (see [20]), we use integration by parts to obtain the following: Integrating the last equality with respect to t ∈ ½0, T 1 , we obtain the following: Thus, By using Holder's inequality, Lemma 10, Lemma 9, and the fact that