On the stabilization of the Korteweg–de Vries equation

Vilmos Komornik Dedicated to D. L. Russell on the occasion of his 70th birthday. abstract: We consider the Korteweg–de Vries equation on a bounded interval with periodic boundary conditions. We prove that a natural mass conserving global feedback exponentially stabilizes the system in all Sobolev norms and we obtain explicit decay rates. The proofs are based on the family of conservation laws for the Korteweg–de Vries equation.

For k = 0 the equation (1.1) is a good model of shallow water: w(x, t) denotes the depth of water at a point x at time t; see [6], [9].The periodic boundary conditions correspond to a circular movement.In this model [w] denotes the total volume of water.
For k > 0 the action of the "feedback" −k(w − [w]) consists in balancing the level of water, conserving at the same time its total volume.Indeed, the latter property follows, at least formally, from (1.1): The following formal computation shows that w(t) converges exponentially to the constant M := [w 0 ] = [w] in L 2 (Ω) as t → ∞: Identifying L 2 (Ω) with its dual (L 2 (Ω)) ′ we obtain the algebraical and topological inclusions We recall from [8] that the problem (1.1) is well-posed in the following sense: has a unique solution ). (1.5) Furthermore, the mapping The purpose of this paper is to extend the estimate (1.3) on the asymptotic behaviour of the solutions of (1.5): Then for every fixed 0 < k ′ < k there exists a constant C = C(w 0 , k ′ ) such that the solution of (1.4) satisfies the estimate (1.6) In order to convince ourselves about the validity of these estimates let us consider for a moment the linearized problem Assuming that the solutions satisfy the regularity properties (1.6) (see [7]), the desired estimates follow by applying the multiplier method.Indeed, [w] is constant again because Denoting this constant by M , the function v := w − M has the same regularity properties as w and it solves the following problem: (1.7) By approximating the initial value by smoother functions it is sufficient to prove the estimates for periodic solutions belonging to H 2m p .Multiplying the differential equation in (1.7) by ∂ 2j v ∂x 2j for j = 0, . . ., m, integrating by parts in Ω and using the periodic boundary conditions we obtain that by the boundary conditions, we conclude that and hence for all t ≥ 0 and j = 0, . . ., m. Therefore v(t) H m p = e −kt v(0) H m p , t ≥ 0, and then, using the equation Taking the definition of v into account we obtain finally that The presence of the nonlinear term creates serious difficulties with respect to the linearized problems but as we will see, the final estimates are only slightly weaker than (1.8): we have a decay rate k − ε with arbitrarily small ε > 0 instead of k.
Theorem 1.2 has been planned to be a part of a joint work with D. L. Russell and B.-Y. Zhang (see [2], [3]) but, due to some mismanagement, it has never been published before.The author is indebted to D. L. Russell and B.-Y. Zhang for many helpful conversations on this subject.

Proof of Theorem 1.2 for m = 2
We shall often use the equality (1.2) and therefore we shall write [w] instead of [w 0 ].For brevity we shall write instead of Ω .
Applying a usual density argument it is sufficient to prove the estimates (1.6) for w 0 ∈ H 5 p .According to Theorem 1.1 thus we may assume that This regularity property is sufficient to justify all computations which follow.
It is convenient to introduce the notations then we deduce from (1.1), (1.2) and (2.1) that ) and the estimates (1.6) take the following form: is continuously differentiable, and (2.10) Proof: Since H 2 p is a Banach algebra, it follows from (2.4) that the function v 2 is continuously differentiable.Hence the function (2.9), being the composition of two function of class C 2 , is also continuously differentiable.
Using (2.6) and the periodicity of v (see (2.3)) we easily obtain the identity (2.10): is continuously differentiable and (2.12) It follows easily from (2.4) that the function (2.11) is continuously differentiable.Using (2.6) and the periodicity of v hence the identity (2.12) follows: Stabilization of the KdV equation 39 Lemma 2.3 The function is continuously differentiable and By (2.4) the function (2.13) is continuously differentiable.To show the identity (2.14) first we deduce from (2.6), using the periodicity of v, the following identity: It suffices to show that We have and Finally, we have In order to simplify the notation we shall write • p for the norm of L p (Ω), 1 ≤ p ≤ ∞.Since Ω is the unit interval, the Hölder inequality is particularly simple: We shall also use the Poincaré-Wirtinger inequality: The proof is simple: since v is continuous, there exists a ∈ Ω such that v(a) = 0.
Then for any y ∈ Ω we have Noe that Lemma 2.1 implies that v(t) 2 = v 0 2 e −kt for all t ∈ R + .
(2.17) Now let us show that for each fixed k ′ ∈ (0, k) there exists a positive constant C ′ such that v x (t) 2 = C ′ e −k ′ t for all t ≥ 0.
(2.18) Using (2.15)-(2.17)we have consequently, for any fixed ε > 0 (to be chosen later) there exists T ′ > 0 such that If ε is sufficiently small, then we also deduce from (2.19) that Thus, choosing a sufficiently small ε we deduce from (2.11), (2.20) and (2.21) that which implies (2.18) for all t > T ′ .The left-hand side of (2.18) being continuous, the estimate (2.18) remains valid for all t ≥ 0 with some bigger constant C ′ .Next we show similarly that for any fixed k ′ < k there exists a positive constant Using (2.15)-(2.18))we have and It follows that for any fixed ε > 0 (to be chosen later) there exists T ′′ > 0 such that Choosing ε > 0 sufficiently small we conclude from (2.23) that for all t > T ′′ .We deduce from (2.13), (2.24) and (2.25) that

Proof of Theorem 1.2 for m ≥ 3
The proof is constructive.It is based on an infinite sequence of polynomial conservation laws for the KdV equation obtained in [6].We begin by recalling four important properties concerning these laws, established in [4] and [6].
(i) There exists a sequence of polynomials P n = P n (v 0 , . . ., v n ) of n+1 variables, n = 0, 1, . .., such that setting also (ii) The highest order term of by the Leibniz rule and then replace each factor of the form Then the result may be written in the form where Y n is a suitable polynomial of n+3 variables, independent of the choice of the function w, n = −1, 0, 1, . . . .(We remark that our notation differs from that of [4] and [6]: with their notation we have P n = T n−2 and Y n = X n−2 ; we follow [5].) In order to simplify the notation in the sequel we shall denote the partial derivative ∂ j w ∂x j of the solution function w by w ,j ; in particular, w ,0 = w.Furthermore, all integrals will be taken on the interval (0, 1), i.e., • dx = 1 0 • dx.Finally, we write • p for the norm in the space L p (0, 1), 1 ≤ p ≤ ∞.
We need the following important lemma.
Lemma 3.1 Consider the solution w of (1.4) for some w 0 ∈ H m p .Then the following properties hold true: (a) We have in (0, ∞) the identity (b) For each n = 1, . . ., m there exist two polynomials of n variables such that the following identity is satisfied in (0, ∞): is the product of at least three, not necessarily different, factors, the exponent of v n−1 being always less than four.
(a) As we have seen in the introduction, [w(t)] does not depend on t: and the identity (3.1) is satisfied: the formal proofs given before are justified by the regularity (1.5) of the solution.
(b) Using properties (ii) and (iii) we see that P n has the form ) and c is a polynomial in v 0 ,. . ., v n−1 .Using the periodic boundary conditions it follows that where

This computation transforms w 2
,n into −2Kw (c) One can readily verify that each term of has the same rank as this proves the first statement.On the other hand, (3.7) implies that This proves the second statement.Now we turn to the proof of Theorem 1.2.First we observe that (3.1) immediately implies w V. Komornik We will prove by induction on j that w ,j (•, t) 2 ≤ ce −K ′ t , t ≥ 0 (3.9) for j = 1, . . ., m.
Let n ≤ m be a positive integer and assume that (3.9) is satisfied for j = 1, . . ., n − 1.We will prove that it is satisfied for j = n, too.Using the trivial inequalities and applying part (c) of Lemma 3.1 we obtain the estimates where C and N are positive integers, independent of t ≥ 0. Applying (3.8) if n = 1 and (3.9) for j = n − 1 if n ≥ 2, there follows the existence of a positive number T such that for all t > T the following inequalities hold: The validity of (3.9) for j = n now follows from (3.12) and (3.13) by taking We have thus proved (3.8) and (3.9) for j = 1, . . ., m. Equivalently, we have established that w(•, t) − [w] H m p ≤ Ce −K ′ t , t ≥ 0.

7 )
a term of one of these two polynomials (with some nonzero constant c), then Since k ≤ n − 1, we deduce from (3.7) the inequality
proving(2.22)for all t > T ′ .The left-hand side of (2.22) being continuous, the estimate (2.22) remains valid for every t ≥ 0 if we choose some larger constant C ′′ .Now we may easily complete the proof of the theorem.By (2.17), (2.18) and (2.22) for every fixed k ′ < k there exists a positive constant C 1 > 0 such that v(t) H 2 p ≤ C 1 e −k ′ t for all t ≥ 0.