Asymptotic stability of a Korteweg-de Vries equation with a two-dimensional center manifold

Local asymptotic stability analysis is conducted for an initial-boundary-value problem of a Korteweg-de Vries equation posed on a finite interval $\left[0, 2\pi \sqrt{7/3}\right]$. The equation comes with a Dirichlet boundary condition at the left end-point and both of the Dirichlet and Neumann homogeneous boundary conditions at the right end-point. It is known that the associated linearized equation around the origin is not asymptotically stable. In this paper, the nonlinear Korteweg-de Vries equation is proved to be locally asymptotically stable around the origin through the center manifold method. In particular, the existence of a two-dimensional local center manifold is presented, which is locally exponentially attractive. By analyzing the Korteweg-de Vries equation restricted on the local center manifold, a polynomial decay rate of the solution is obtained.


Introduction.
The Korteweg-de Vries (KdV) equation y t + y x + yy x + y xxx = 0 (1.1) was first derived by Boussinesq in [2, Equation (283 bis)] and by Korteweg and de Vries in [13], for describing the propagation of small amplitude long water waves in a uniform channel. This equation is now commonly used to model unidirectional propagation of small amplitude long waves in nonlinear dispersive systems. An excellent reference to help understand both physical motivation and deduction of the KdV equation is the book by Whitham [21].
By means of multiplier technique and the Hilbert Uniqueness Method (HUM) [14], he proved that (1.3) is exactly controllable if and only if the length of the spatial domain is not critical, i.e., L / ∈ N , where N denotes following set of critical lengths N := 2π j 2 + l 2 + jl 3 ; j, l ∈ N * . (1.4) Then, by employing the Banach fixed point theorem, he derived that the nonlinear KdV control system (1.2) is locally exactly controllable around 0 provided that L / ∈ N . In the cases with critical lengths L ∈ N , Rosier demonstrated in [19] that there exists a finite dimensional subspace M of L 2 (0, L) which is unreachable for the linear system (1.3) when starting from the origin. In [7], Coron and Crépeau treated a critical case of L = 2kπ (i.e., taking j = l = k in N ), where k is a positive integer such that (see, [6,Theorem 8
Here, the uncontrollable subspace M for the linear system (1.3) is one-dimensional. However, through a third order power series expansion of the solution, they showed that the nonlinear term yy x always allows to "go" in small-time into the two directions missed by the linearized control system (1.3), and then, using a fixed point theorem, they deduced the small-time local exact controllability around the origin of the nonlinear control system (1.2). In [4], Cerpa studied the critical case of L ∈ N ′ , where N ′ := 2π j 2 + l 2 + jl 3 ; j, l ∈ N * satisfying j > l and j 2 + jl + l 2 = m 2 + mn + n 2 , ∀m, n ∈ N * \{j} . (1. 6) In this case, the uncontrollable subspace M for the linear system (1.3) is of dimension 2, and the author used a second order expansion of the solution to the nonlinear control system (1.2) to prove the local exact controllability in large time around the origin of the nonlinear control system (1.2) (the local controllability in small time for this length L is still an open problem). Furthermore, Cerpa and Crépeau considered in [6] the cases when the dimension of M for the linear system (1.3) is higher than 2. They implemented a second order expansion of the solution to (1.2) for the critical lengths L = 2kπ for any k ∈ N * , and implemented an expansion to the third order if L = 2kπ for some k ∈ N * . They showed that the nonlinear term yy x always allows to "go" into all the directions missed by the linearized control system (1.3) and then proved the local exact controllability in large time around the origin of the nonlinear control system (1.2).
Consider the case when there is no control, i.e., u = 0, in (1.2), which gives the following initial-boundary-value KdV problem posed on a finite interval [0, L]: where the boundary conditions are homogeneous. For the Lyapunov function Thus, 0 ∈ L 2 (0, L) is stable (see (P 1 ) below for the definition of stable) for the KdV equation (1.7). Moreover, it has been proved in [17] that, if L / ∈ N , then 0 is exponentially stable for the corresponding linearized equation around the origin which gives the local asymptotic stability around the origin for the nonlinear equation (1.7). However, when L ∈ N , Rosier pointed out in [19] that the equation (1.10) is not asymptotically stable. Inspired by the fact that the nonlinear term yy x introduces the local exact controllability around the origin into the KdV control system (1.2) with L ∈ N , we would like to discuss whether the nonlinear term yy x could introduce local asymptotic stability around the origin for (1.7). This paper is devoted to investigating the local asymptotic stability of 0 ∈ L 2 (0, L) for (1.7) with the critical length (1.11) L = 2π 7 3 , corresponding to j = 1 and l = 2 in (1.4). Let us recall that this local asymptotic stability means that the following two properties are satisfied. (P 1 ) Stability: for every ε > 0, there exists η = η(ε) > 0 such that, if y 0 L 2 (0,L) < η, then As mentioned above, the stability property (P 1 ) is implied by (1.9). Our main concern is thus the local attractivity property (P 2 ). We prove the following theorem, where the precise definition of a solution to (1.7) is given in Definition 2.1 and the precise definition of the finite dimensional vector space M ⊂ L 2 (0, L) when L = 2π 7/3 is given in (2.16). Theorem 1.1. Consider the KdV equation (1.7) with L = 2π 7/3. There exist δ ∈ (0, +∞), K > 0, ω > 0 and a map g : M → M ⊥ , where M ⊥ ⊂ L 2 (0, L) is the orthogonal of M for the L 2 -scalar product, satisfying (1.15) such that, with the following three properties hold for every solution y to (1.7) with y 0 L 2 (0,L) < δ, where d(χ, G) denotes the distance between χ ∈ L 2 (0, L) and G: If y 0 ∈ G, then y(t, ·) ∈ G, ∀t ≥ 0.
3. If y 0 is in G, then there exists C > 0 such that In particular, 0 ∈ L 2 (0, L) is locally asymptotically stable in the sense of the L 2 (0, L)norm for (1.7). Remark 1.1. It can be derived from [8, Theorem 1 and Comments] that, for every L > 0, there are non-zero stationary solutions with the period of L to the following ordinary differential equation (ODE): That is, besides the origin, there also exist other steady states of the nonlinear KdV equation (1.7). Therefore, 0 ∈ L 2 (0, L) is not globally asymptotically stable for (1.7): Property (P 2 ) does not hold for arbitrary ε 0 > 0.
Our proof of Theorem 1.1 relies on the center manifold approach. This center manifold is G in Theorem 1.1. Center manifold theory plays an important role in studying dynamic properties of nonlinear systems near "critical situations". The center manifold theorem was first proved for finite dimensional systems by Pliss [18] and Kelley [11], and the readers could refer to [12,16] for more details of this theory. Analogous results are also established for infinite dimensional systems, such as partial differential equations (PDEs) [3,1] and functional differential equations [9]. The center manifold method usually leads to a dimension reduction of the original problems. Then, in order to derive stability properties (asymptotic stable, or, unstable) of the full nonlinear equations, one only needs to analyze the reduced equation (restricted on the center manifold). When dealing with the infinite dimensional problems, this method can be extremely efficient if the center manifold is finite dimensional. Following the results on existence, smoothness and attractivity of a center manifold for evolution equations in [20], Chu, Coron and Shang studied in [5] the local asymptotic stability property of (1.7) with the critical length L = 2kπ for any positive integer k such that (1.5) holds. They proved the existence of a one-dimensional local center manifold. By analyzing the resulting one-dimensional reduced equation, they obtained the local asymptotic stability of 0 for (1.7). For L = 2π 7/3, we get, following [5], the existence of a two-dimensional local center manifold. It is predictable that the two-dimensional local center manifold introduces more complexity than the one-dimensional local center manifold case.
The organization of this paper is as follows. In Section 2, some basic properties of the linearized KdV equation (1.10) and the KdV equation (1.7) are given. Then, in Section 3, we recall a theorem on the existence of a local center manifold for the KdV equation (1.7) and analyze the dynamics on the local center manifold. Theorem 1.1 follows from this analysis. In Section 4, we present the conclusion and some possible future works. Finally, we end this article with an appendix that contains computations which are important for the study of the dynamics on the center manifold.
then the linearized equation (1.10) can be written as an evolution equation in L 2 (0, L): The following lemma can be immediately obtained. Proof. By calculation, we get Hence we get the existence of A −1 and that, by the Sobolev embedding theorem, this operator is compact on L 2 (0, L). Therefore, σ(A), the spectrum of A, consists of isolated eigenvalues only.
The following proposition is proved.
, that is, for any given initial data y 0 ∈ L 2 (0, L), S(t)y 0 is the mild solution of the linearized equation (1.10), and If Re (λ) < 0, ∀λ ∈ σ (A), then it follows directly from the ABLP (Arendt-Batty-Lyubich-Phong) Theorem [15] that the semigroup S(t) is asymptotically stable on L 2 (0, L). Since we only have Re (λ) ≤ 0, ∀λ ∈ σ (A), the main concern needs to be put on the eigenvalues on the imaginary axis and their corresponding eigenfunctions. Following the proofs for [5, Lemma 2.6] and [19,Lemma 3.5], the following lemma is proved. Lemma 2.2. There exists a unique pair of conjugate eigenvalues of A on the imaginary axis, that is, Moreover, the corresponding eigenfunctions of A with respect to λ = ±iq are respectively, where C is an arbitrary constant, and ϕ 1 , ϕ 2 are two nonzero real-valued functions: Remark 2.1. The equations satisfied by ϕ 1 and ϕ 2 are (2.14) and, with the definition of Θ given in (2.11), where ϕ 1 , ϕ 2 are defined in (2.9), (2.10) and (2.11). Then the following decomposition holds: . There exists C > 0 such that, for every solutions y 1 and y 2 , corresponding to every initial conditions (y 10 , y 20 ) ∈ (L 2 (0, L)) 2 respectively, to the equation (1.7) on [0, T ], one has the following inequalities: Let us also mention that for every solution y to (1.7) on [0, T ] or on [0, +∞), This can be easily seen by multiplying the first equation of (1.7) with y, integrating on [0, L] and performing integration by parts. One then gets, if y is smooth enough, which gives (2.26). The general case follows from a smoothing argument. As a consequence of Proposition 2.2, Proposition 2.3 and (2.26), one sees that (1.7) has one and only one solution defined on [0, +∞) if y 0 L 2 (0,L) < ε(1).

Existence of a center manifold and dynamics on this manifold.
Let us start this section by recalling why, as it is classical, the property "0 ∈ L 2 (0, L) is locally asymptotically stable in the sense of the L 2 (0, L)-norm for (1.7)" stated at the end of Theorem 1.1 is a consequence of the other statements in this theorem. For convenience, let us recall the argument. Let y 0 ∈ L 2 (0, L) be such that y 0 L 2 (0,L) < δ and let y be the solution to (1.7). It suffices to check that By (1.17), (2.26) and the fact that M is of finite dimension, there exists an increasing sequence of positive real numbers (t n ) n∈N and z 0 ∈ L 2 (0, L) such that t n → +∞ as n → +∞, (3.2) y(t n , ·) → z 0 in L 2 (0, L) as n → +∞, (3.3) z 0 ∈ G and z 0 L 2 (0,L) < δ. Let η > 0. By (3.5), there exists τ > 0 such that (3.6) z(τ, ·) L 2 (0,L) ≤ η 2 .
By (3.6) and (3.7), there exists n 0 ∈ N such that which, together with (2.26), implies that which concludes the proof of (3.1).
The remaining parts of this section are organized as follows. We first recall in Section 3.1 a theorem (Theorem 3.1) on the existence of a local center manifold for (1.7). Then in Section 3.2 we analyze the dynamics of (1.7) on this center manifold and deduce Theorem 1.1 from this analysis.

Existence of a local center manifold.
In [5, Theorem 3.1], following [20], the existence of a center manifold for (1.7) was proved for the first critical length, i.e., L = 2π. The same proof applies for our L (i.e., the L defined by (1.11)) and allows us to get the following theorem. where d(χ, G) denotes the distance between χ ∈ L 2 (0, L) and G: If y 0 ∈ G, then y(t, ·) ∈ G, ∀t ≥ 0.
Using Definition 2.1 with φ(t, x) := ψ(x), (3.24) and integration by parts, we get Letting τ → 0 + in (3.25), and using (3.20), (3.21) and (3.23), we get We expand g in a neighborhood of 0 ∈ M . Using (1.14) and (1.15), there exist As usual, by (3.28), we mean that, for every ς 1 > 0, there exists ς 2 > 0 such that Similar definitions are used in (3.29), (3.30) and later on. We now expand the left hand side of (3.26) in terms of m 0 1 , m 0 2 , (m 0 1 ) 2 , m 0 1 m 0 2 and (m 0 2 ) 2 as |m 0 1 | + |m 0 2 | → 0. For the functions ϕ 1 and ϕ 2 defined by (2.9), (2.10) and (2.11), the following equalities can be derived from (2.12), (2.13) and using integrations by parts:  a ′ + a ′′′ + ϕ 1 ϕ ′ 1 − c 1 ϕ 2 + qb = 0, a(0) = a(L) = 0, a ′ (L) = 0, Let us now study the local asymptotic stability property of 0 ∈ R 2 for (3.21). We propose two methods for that. The first one is a more direct one, which relies on normal forms for dynamical systems on R 2 . The second one, which relies on a Lyapunov approach related to the physics of (1.7), is less direct. However, there is a reasonable hope that this second method can be applied to other critical lengths L ∈ N \ 2πN for which the dimension of M is larger than 2.
Method 1: normal form. Let and it follows from (3.21) and (3.43) that, as |z| → 0, where P j (z, z) are polynomials in z, z of degree j.
To be more precise, we have
Method 2: Lyapunov function. Let us start with a formal motivation. Recall that, by (1.9) and with E defined in (1.8), we have, along the trajectories of (1.7), It is therefore natural to consider the following candidate for a Lyapunov function where µ > 0 is small enough. Indeed, one then gets and one may hope to absorb −µKK with −(1/2)K 2 − µ K 2 and getV < 0 on G \ {0}, at least in a neighborhood of 0. We follow this strategy together with the approximation of g previously found. For m = (m 1 , m 2 ) ∈ Ω, let (see (3.28))  Let us emphasize that, even if "along the trajectories of (3.21)" might be misleading, E is just a function of m ∈ Ω. It is the same forV ,K,K which appear below. Using (1.15) and (3.20), we have, along the trajectories of (3.21), Using (3.20), we get the existence of C > 0 such that, along the trajectories of (3.21), We can now define our Lyapunov functionṼ . Let µ ∈ (0, 1/4]. LetṼ : Ω → R be defined by From (3.84), we have the existence of η 0 > 0 such that, for every m ∈ R 2 satisfying |m| < η 0 and along the trajectories of (3.21), Let us assume for the moment that, for every m = (m 1 , m 2 ) ∈ R 2 , Then, by homogeneity, there exists η 1 > 0 such that From (3.85) and (3.88), we get the existence of η 3 > 0 such that, for every µ ∈ (0, η 3 ), Moreover, straightforward estimates show that there exists η 4 > 0 such that, for every µ ∈ (0, η 4 ), which, together with (3.89), proves the existence of C > 0 such that, at least if m 0 ∈ R 2 is small enough, the solution to (3.21) satisfies It only remains to prove (3.86). From the Appendix, one gets that c ′ (0) ≈ 0.0118 = 0, then (3.86) holds if m 1 = 0. Let us now deal with the case m 1 = 0. Dividing both the polynomials on the two equations on the left hand side of (3.86) by m 2 1 , then the two resulting polynomials have a common root if and only if their resultant is zero. This resultant is the determinant of the Sylvester matrix S: Hence, the two resulting polynomials do not have a common root. Thus, (3.86) is proved. Remark 3.1. It follows from our proof of Theorem 1.1 that the decay rate stated in (1.20) is optimal in the following sense: there exists ε > 0 such that, for every y 0 ∈ G such that y 0 L 2 (0,L) ≤ ε, .

Conclusion and future works.
In this article, we have proved that for the critical case of L = 2π 7/3, 0 ∈ L 2 (0, L) is locally asymptotically stable for the KdV equation (1.7). First, we recalled that the equation has a two-dimensional local center manifold. Next, through a second order power series approximation at 0 ∈ M of the function g defining the local center manifold, we derived the local asymptotic stability of 0 ∈ L 2 (0, L) on the local center manifold and obtained a polynomial decay rate for the solution to the KdV equation (1.7) on the center manifold.
Since the KdV equation (1.7) also has other (periodic) steady states than the origin (see Remark 1.1), it remains an open and interesting problem to consider the (local) stability property of these steady states for the KdV equation (1.7). Furthermore, it remains to consider all the other critical cases with a two-dimensional (local) center manifold as well as all the last remaining critical cases, i.e., when the equation has a (local) center manifold with a dimension larger than 2.  Here, 13) and each A +l is the matrix formed by replacing the l-th column of A + with a column vector −b + , where Similarly, by employing the method of undetermined coefficients, the (unique) solution to the nonhomogeneous ODE system (A.6) is