Well-Posedness and Time Regularity for a System of Modiﬁed Korteweg-de Vries-Type Equations in Analytic Gevrey Spaces

: Studies of modiﬁed Korteweg-de Vries-type equations are of considerable mathematical interest due to the importance of their applications in various branches of mechanics and physics. In this article, using trilinear estimate in Bourgain spaces, we show the local well-posedness of the initial value problem associated with a coupled system consisting of modiﬁed Korteweg-de Vries equations for given data. Furthermore, we prove that the unique solution belongs to Gevrey space G σ × G σ in x and G 3 σ × G 3 σ in t . This article is a continuation of recent studies reﬂected. ﬁeld


Introduction and Main Results
A single initial value problem for the Korteweg-de Vries (KdV) equation is written as For x ∈ R , t ∈ [0, T] and u 0 (x) ∈ H s (R), it has been shown that (1) is locally well-posed for s > −3 4 in Sobolev spaces [1]. In [2], the authors extended results analyzed in Bourgain spaces [3] for s ≥ 0 to the case s > −3 4 for solution of (1) on time interval [0, δ], δ > 0. For s ≥ −1 2 , the local well-posedness of corresponding periodic R−valued initial value problem (1) was shown with x ∈ T. If we change the term u 2 in (1), the problem becomes a modified Korteweg-de Vries equation (mKdV).
For related problems in analytic Gevrey spaces, we review the results obtained in [4], where the Cauchy problem of the Ostrovsky equation is considered by Boukarou et al. with data in analytic Gevrey spaces on the line and the circle. Based on bilinear estimates in Bourgain spaces, the local well-posedness is proved and Gevrey regularity of the unique solution is provided. In [5], the question of well-posedness in Bourgain spaces for a class of Cauchy problem for fifth-order Kadomtsev-Petviashvili I equation has been treated in the system ∂ t u + α∂ 3 x u + ∂ 5 x u + ∂ −1 x ∂ 2 y u + u∂ x u = 0 u(x, y, 0) = ϕ(x, y), (2) where u = u(x, y, t), (x, y) ∈ R 2 or T 2 , (α, t) ∈ R 2 . The authors also obtained the regularity in t and x, y, where the solution is analytic in x, y and belongs to G 5 in t.
The dynamics of solutions in Korteweg-de Vries equation (KdV) and modified Korteweg-de Vries equation (mKdV) are well studied due to the complete integrability of these equations. A description, in the framework of modified Korteweg-de Vries equations, is given in many works [6][7][8] and for KdV the main results were published back in the 1970s, although many results have been obtained very recently [9]. We extend the previous results and propose a coupled system of modified Korteweg-de Vries equations on the line. Here we have a number of detailed articles and reviews, among which we note the work by [10], where the local and global well-posedness in H s (T) of the modified Korteweg-de Vries equation, for s ≥ 1/2, have been studied extensively and also the global well-posedness in L 2 (T) was established in [2]. To begin with, we consider the problem We are now in position to motivate our work. As a model example, we recall the following system [11] to study the nonlinear resonant interactions of long wavelength equatorial Rossby waves and barotropic Rossby waves with a significant mid-latitude projection, in the presence of suitable horizontally and vertically sheared zonal mean flows. For β = 1, the system (3) is reduced to a particular case of a large class of equations considered by [12]. In this case, the problems of well-posedness as well as the existence and the stability of the solitary waves for this type of system are widely studied by using the pioneer work in [13]. Oh [14] considered the problem where the local well posedness for data with regularity s ≥ 0 is showed by using the Fourier transform restriction norm method. For 0 < β < 1, the author in [9] proved that the initial value problem (3) is locally well posed for given data (u 0 , v 0 ) ∈ H s (R) × H s (R), s > − 1 2 . The modified KdV systems are important in the study of dispersive equations and are considered widely in the literature, as it is known, in mathematical modeling of wave processes in many problems of plasma physics, solid state physics, hydrodynamics, quantum field theory, biophysics, chemical kinetics, fiber optics, etc.
The novelty of our work lies primarily in the use of trilinear estimate in Bourgain spaces, to show the local well-posedness of initial value problem associated with coupled system consisting modified Korteweg-de Vries Equations (3) for given data. Furthermore, we prove that the unique solution belongs to Gevrey space G σ × G σ in x and G 3σ × G 3σ in t.
The first main result about the well-posedness of initial value problem related to coupled system of modified Korteweg-de Vries Equations (3) in analytic Gevrey spaces reads as follows.
Then for some real number b > 1 2 and a constant T = T( (u 0 , v 0 ) G σ,δ,s ×G σ,δ,s ), the Cauchy problem (3) admits a unique local in time solution The analytic spaces G θ,s have been introduced by Foias and Temam [15] by the norm and used by Grujic and Kalisch [16] to prove the well-posedness of non-periodic case for generalized Korteweg-de Vries equation.
The paper is organized as follows. Theorems 1 and 2 introduced in Section 1 are central. In Section 2, we define the function spaces, linear estimates and trilinear estimates. In Section 3, we prove Theorem 1, using the trilinear estimate and the linear estimate together with contraction mapping principle. The regularity in time variable is proved in the fourth section.

Linear Estimates
To present the proof of theorems, we start with trilinear estimate (8) defined in X σ,δ,s,b (R 2 ). Let us beginning by the embedded result.

Proof. Define the operator A by
It satisfies where X s,b is introduced in [9]. We observe that Aw belongs to C (R, H s ) and for some C > 0, we have Thus, it follows that w ∈ [0, T], G σ,δ,s and Owing to the Fourier transform with respect to the spatial variable x of the Cauchy problem (3), we obtain a differential equation and then solving it in t. We localize in t by using a cut-off function, . We consider, for the operators Λ, Γ, the following integral system which is equivalent to Cauchy problem (3) where S(t) = e −t∂ 3 x and S β (t) = e −tβ∂ 3 x , the unitary groups related to the linear problems are defined via Fourier transform as [ and for all v 0 , u 0 ∈ G σ,δ,s .
Proof. By definition, we have It follows that We use the fact that b > 1/2 to get and Proof. Define Let us consider the operator A given by (9), then we have Thus Using Lemma 2.1 in [9], we have The inequality (18) is similar.

Trilinear Estimates
The following Lemma states the desired trilinear estimate. and Proof. We observe, by considering the operator A in (9), that since Now, by using Proposition 2.3 of [9], there exists C > 0 such that

Existence of Solution
We are now ready to estimate all terms in (13) by using the trilinear estimates in the above Lemmas. We define the spaces and similar for N δ,s .

The Uniqueness
The uniqueness of solution for (Λ [u, v], Γ[u, v]) = (u, v) in B(0, R) comes from the argument used above (Fixed point). For the proof of uniqueness in the whole space B δ,s,b = X δ,s,b × X β δ,s,b can be seen in [24].

Continuous Dependence of the Initial Data
We will need to prove the next Lemma Taking (u, v), (u * , v * ) ∈ B(0, R) and T ≤ 1 4CR , we get This completes the prove of Theorem 1.

Gevrey-3σ Regularity in Time
We will now prove the temporal regularity of solution on the line .
The constant c will be chosen as in [25] so that the next inequality holds ∑ 0≤l≤k k l m l m k−l ≤ m k .
We remove 0 and k from the left hand side of (38). We use M p , to get Then, for any > 0, the sequence M p satisfies It is checked that for a given C > 1, there exists 0 > 0 such that By the definition of M 1 and M 2 in (37), we have for j = 1, that M 1 = a M 2 , where a = 9 4(2!) σ , for some C > 0. Now, let us define The next Lemma is the main idea for the proof of Theorem 2.
Lemma 9. Let (u, v) be the solution of (3) that satisfies (34) and (35), then there exists 0 > 0 such that for any 0 < ≤ 0 , we have for all x ∈ R, t ∈ [0, T] For this end, we need the next results.

Failure of Gevrey-D Regularity in Time
We replace t with −t, our system can be rewritten as The following Lemma will be used to estimate the higher-order derivatives of a solution with respect to t. Lemma 11. [19] Let (u, v) be solution of (60). Then we have and for every j ∈ {1, 2, ...}.

Definition 1.
Let {ω k } be a sequence of positive numbers. We denote by C(ω k ) the class of all functions g(x), infinitely differentiable on [−1, 1], for each of which there is a C > 0 such that there exists a function g(x) ∈ C(k kσ ) for which g (k) (0) = ϕ k .
This completes the proof of Theorem 2.

Conclusions
In mathematics, the Korteweg-de Vries equation is a mathematical model of waves on a shallow water surface, it was first introduced by Boussines (1877) and rediscovered by Kortewek and Gustav de Vries (1895). This is particularly visible as a prototype of an example of a precisely soluble model, that is to say a nonlinear partial differential equation whose solutions can be determined with precision. The mathematical theory underlying Korteweg-de Vries is the subject of active research. The main results in our paper are the following. In the first part of the manuscript, it is proved that (3) is locally well-posed in analytic Gevrey spaces (Theorem 1). In the second part of the manuscript, it is shown that the solutions of (3) obtained are Gevrey functions of order 3σ in time variable. Then, it is shown that the Gevrey regularity in time is sharp for the periodic case, that is, there exist initial data (u 0 , v 0 ) ∈ G σ,δ,s (R) × G σ,δ,s (R) such that the related solution (u, v) of (3) depending on real-valued initial data (u 0 , v 0 ) is not Gevrey in time of order d for any 1 ≤ d < 3σ (Theorem 2).