General Boundary Value Problems of the Korteweg-de Vries Equation on a Bounded Domain

In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite interval \begin{equation} u_t+u_x+u_{xxx}+uu_x=0,\qquad u(x,0)=\phi(x), \qquad 00 \qquad (1) \end{equation} subject to the nonhomogeneous boundary conditions, \begin{equation} B_1u=h_1(t), \qquad B_2 u= h_2 (t), \qquad B_3 u= h_3 (t) \qquad t>0 \qquad (2) \end{equation} where \[ B_i u =\sum _{j=0}^2 \left(a_{ij} \partial ^j_x u(0,t) + b_{ij} \partial ^j_x u(L,t)\right), \qquad i=1,2,3,\] and $a_{ij}, \ b_{ij}$ $ (j,i=0, 1,2,3)$ are real constants. Under some general assumptions imposed on the coefficients $a_{ij}, \ b_{ij}$, $ j,i=0, 1,2,3$, the IBVPs (1)-(2) is shown to be locally well-posed in the space $H^s (0,L)$ for any $s\geq 0$ with $\phi \in H^s (0,L)$ and boundary values $h_j, j=1,2,3$ belonging to some appropriate spaces with optimal regularity.


Introduction
In this paper we consider the initial-boundary value problems (IBVP) of the Korteweg-de Vries (KdV) equation posed on a finite domain (0, L) u t + u x + u xxx + uu x = 0, u(x, 0) = φ(x), 0 < x < L, t > 0 (1.1) with general non-homogeneous boundary conditions posed on the two ends of the domain (0, L), where a ij ∂ j x u(0, t) + b ij ∂ j x u(L, t) , i = 1, 2, 3, and a ij , b ij , j, i = 0, 1, 2, are real constants. We are mainly concerned with the following question: Under what assumptions on the coefficients a kj , b kj in (1.2) is the IBVP (1.1)-(1.2) wellposed in the classical Sobolev space H s (0, L)?
Theorem C [5]: Let s ≥ 0 , r > 0 and T > 0 be given. There exists T * ∈ (0, T ] such that for any s−compatible 1 the IBVP (1.7) admits a unique solution Moreover, the corresponding solution map is analytic in the corresponding spaces.
1 See [5] for exact definition of s−compatibility.
As for the IBVP (1.8), its study began with the work of Colin and Ghidalia in late 1990's [14,15,16]. They obtained in [16] the following results.
In addition, they showed that the associate linear IBVP possesses a strong smoothing property: For any φ ∈ L 2 (0, L), the linear IBVP (1.9) admits a unique solution Aided by this smoothing property, Colin et al. showed that the homogeneous IBVP (1.9) is locally well-posed in the space L 2 (0, L).
Recently, Kramer et al.,in [32], and Jia et al., in [22], have shown that the IBVP (1.8) is locally well-posedness in the classical Sobolev space H s (0, L) for s > −1, which provide a positive answer to one of the open question in [16].
Theorem F [22,32]: Let s > −1, T > 0 and r > 0 be given with There exists a 0 < T * ≤ T such that for any φ ∈ H s (0, L), Moreover, the corresponding solution map is analytically continuous.
In addition, Rivas et al.,in [33], shown that the solutions of the IBVP (1.8) exist globally as long as their initial value and the associate boundary data are small. Moreover, those solutions decay exponentially if their boundary data decay exponentially.
There exist positive constants δ and T such that for any s−compatible 2 φ ∈ H s (0, L) and for any t ≥ 0, and sup If, in addition to these conditions, there exist γ 1 > 0, C 1 > 0 and g ∈ B s (t,t+T ) such that and h B s (t,t+T ) ≤ g(t)e −γ1t , for t ≥ 0, then there exists γ with 0 < γ ≤ γ 1 and C 2 > 0 such that the corresponding solution u of the IBVP (1.8) satisfies t+T ) )e −γt , for t ≥ 0.
Moreover, the solution u depends continuously on its initial data φ and the boundary values h j , j = 1, 2, 3, in the respective spaces.
In this paper we continue to study the general IBVP (1. and (1.12) We have the following well-posedness results for the IBVP (1.1)-(1.2).
Moreover, the corresponding solution map is analytically continuous.
The following remarks are now in order.
(ii) The assumptions imposed on the boundary conditions in the Theorems 1.1-1.4 can be reformulated as follows: Here, and As a comparison, note that the assumptions of Theorem A are satisfied if and only if one of the following boundary conditions are imposed on the equation in (1.3).
Then, it follows of our results that the conditions (1.13), (1.14) and (1.15) for Theorem A can be removed completely.
(iii) In Theorem 1.1, we replace the s−compatibility of (φ, h) (cf. Theorem C) by assuming To prove our theorems, we rewrite the boundary operators B k , k = 1, 2, 3, 4 as and To prove our main result, we will first study the linear IBVP 16) to establish all the linear estimates needed later for dealing with the nonlinear IBVP (1.1)-(1.2).
Here δ k = 0 for k = 1, 2, 3 and δ 4 = 1. Then we will consider the nonlinear map Γ defined by the following IBVP We will show that Γ is a contraction in an appropriated space whose fixed point will be the desired solution of the nonlinear IBVP (1.1)-(1.2). The key to show that Γ is a contraction in an appropriate space is the sharp Kato smoothing property of the solution of the IBVP (1.16) as described below, for example, for s = 0: For given φ ∈ L 2 (0, L) and f ∈ L 1 (0, T ; L 2 (0, L)) and h ∈ H 0 In order to demonstrate the sharp Kato smoothing properties for solutions of the IBVP (1.16), we need to study the following IBVP The plan of the present paper is as follows.
-In Section 2 we will study the linear IBVP (1.16) The explicit representation formulas for the boundary integral operators W (k) bdr , for k = 1, 2, 3, 4, will be first presented. The various linear estimates for solutions of the IBVP (1.16) will be derived including the sharp Kato smoothing properties.
-The Section 3 is devoted to well-posedness of the nonlinear problem (1.1)-(1.2) will be established.
-Finally, in the Section 4, some conclusion remarks will be presented together with some open problems for further investigations.

Linear problems
This section is devoted to study the linear IBVP (1.16) which will be divided into two subsections. In subsection 2.1, we will present an explicit representation for the boundary integral operators W

Boundary integral operators and their applications
In this subsection, we first derive explicit representation formulas for the following four classes of nonhomogeneous boundary-value problems Without loss of generality, we assume that L = 1 in this subsection.
The following lemma is helpful in deriving various linear estimates for solutions of the IBVP (1.16) in the next subsection.
and view h * j,m,k as the inverse Fourier transform ofĥ * j,m,k . Then for any s ∈ R, (2.25) Proof: Recall that for k = 1, 2, 3, we have for ρ ≥ 0, and for k = 4, as ρ → +∞. Thus, the following asymptotic estimates of where W 0,k (t) is the C 0 -semigroup associated with the IBVP (2.26) with f ≡ 0. Recall the solution of the Cauchy problem of the linear KdV equation, Hereψ denotes the Fourier transform of φ. In terms of the C 0 -group W R,k (t) and the boundary integral operator W Lemma 2.7. For given φ ∈ L 2 (0, L) and f ∈ L 1 loc (R + ; L 2 (0, L)), let Then the solution of the IBVP (2.26) can be written as bdr h k .

Linear estimates
In this subsection we consider the following IBVP of the linear equations: and present various linear estimates for its solutions. For given s ≥ 0 and T > 0, let us consider: for k = 1, 2, 3, 4, j = 1, 3 and m = 1, 2, 3.
Proof. We only consider the case that h = (h 1 , 0, 0) and k = 4; the proofs for the others cases are similar. Note that, the solution z 4 can be written as .
Next we consider the following initial boundary-value problem: for k = 1, 2, 3, 4. Recall that for any s ∈ R, ψ ∈ H s (R) and g ∈ L 1 loc (R + ; H s (R)), the Cauchy problem of the following linear KdV equation posed on R, admits a unique solution v ∈ C(R + ; H s (R)) and possess the well-known sharp Kato smoothing properties.
Lemma 2.9. Let T > 0, L > 0 and s ∈ R be given. For any ψ ∈ H s (R), g ∈ L 1 (0, T ; H s (R)), the solution w of the system (2.31) admits a unique solution w ∈ Z s,T with Moreover, there exists a constant C > 0, depending only s and T , such that for l = 0, 1, 2.
The following two propositions follow from Proposition 2.8 and Lemma 2.9.

Nonlinear problems
In this section, we will consider the IBVP of the nonlinear KdV equation on (0, L) with the general boundary conditions where the boundary operators B k , k = 1, 2, 3, 4, are introduced in the introduction. For given s ≥ 0 and T > 0, let The next lemma is helpful in establishing the well-posedness of (3.1) whose proof can be found in [5,31].
There exists a C > 0 and µ > 0 sucht for any T > 0 and u, v ∈ Y 0,T , and Consider the following linear IBVPs for k = 1, 2, 3, 4. The following lemma follows from the discussion in the Section 2, therefore, the proof will be omitted.
Next, we consider the following linearized IBVP associated to (3.1) for k = 1, 2, 3, 4 and a(x, t) is a given function.
Moreover, there exists a constant C > 0 depending only on T and a Y 0,T such that Proof. Let r > 0 and 0 < θ ≤ T be a constant to be determined. Set which is a bounded closed convex subset of Y 0,θ . For given (φ, h) ∈ X k 0,T , a ∈ Y 0,T and f ∈ L 1 (0, T ; L 2 (0, L)), define a map Γ on S θ,r by v = Γ(u) for any u ∈ S θ,r where v is the unique solution of By Lemma 3.2 (see also Propositions 2.11 and 2.12), for any u, w ∈ S θ,r , Thus Γ is a contraction mapping from S r,θ to S r,θ if one chooses r and θ by Its fixed point v = Γ(u) is desired solution of (3.5) in the time interval [0, θ]. Note that θ only depends on a Y 0,T thus by standard extension argument, the solution v can be extended to the time interval [0, T ]. Thus, the proof is completed. Now, we turn to consider the well-posedness problem of the nonlinear IBVP (3.1).
Proof. We only prove the theorem in the case of 0 ≤ s ≤ 3. When s > 3 it follows from a standard procedure developed in [3]. First we consider the case of s = 0. As in the proof of Proposition 3.3, let r > 0 and 0 < θ ≤ T be a constant to be determined. Set where v is the unique solution of t ≥ 0.
(3.6) By Proposition 3.3, for any u, w ∈ S θ,r , Choosing r and θ with Γ is a contraction whose critical point is the desired solution.
Next we consider the case of s = 3. Let v = u t we have v solves where φ * (x) = −φ (x) − φ (x) and a(x, t) = 1 2 u(x, t). Applying Proposition 3.3 implies that v = u t ∈ Y 0,T * . Then it follows from the equation that u xxx ∈ Y 0,T * and u ∈ Y 3,T * . The case of 0 < s < 3 follows using Tartart's nonlinear interpolation theory [34] and the proof is archived.

Concluding remarks
In this paper we have studied the nonhomogenous boundary value problem of the KdV equation on the finite interval (0,L) with general boundary conditions, and have shown that the IBVP (4.1) is locally well-posed in the space H s (0, L) for any s ≥ 0 with s = 2j−1 2 , j = 1, 2, 3..., and (φ, h) ∈ X s k,T . Two important tools have played indispensable roles in approach; one is the explicit representation of the boundary integral operators W (k) bdr associated to the IBVP (4.1) and the other one is the sharp Kato smoothing property. We have obtained our results by first investigating the associated linear IBVP The local well-posedness of the nonlinear IBVP (4.1) follows via contraction mapping principe.
While the results reported in this paper has significantly improved the earlier works on general boundary value problems of the KdV equation on a finite interval, there are still many questions to be addressed regarding the IBVP (4.1). Here we list a few of them which are most interesting to us. u t + u x + uu x + u xxx = f, 0 < x < L, t > 0 u(x, 0) = φ(x) u(0, t) = h 1 (t), u(L, t) = h 2 (t), u x (L, t) = h 3 (t).
(2) Is the IBVP well-posed in the space H s (0, L) for some s ≤ −1?
We have shown that the IBVP (4.1) is locally well-posed in the space H s (0, L) for any s ≥ 0. Our results can also be extended to the case of −1 < s ≤ 0 using the same approach developed in [8]. For the pure initial value problems (IVP) of the KdV equation posed on the whole line R or on torus T, u t + uu x + u xxx = 0, u(x, 0) = φ(x), x, t ∈ R (4.4) and u t + uu x + u xxx = 0, u(x, 0) = φ(x), x ∈ T, t ∈ R, (4.5) it is well-known that the IVP (4.4) is well-posed in the space H s (R) for any s ≥ − 3 4 and is (conditionally) ill-posed in the space H s (R) for any s < − 3 4 in the sense the corresponding solution map cannot be uniformly continuous. As for the IVP (4.5), it is well-posed in the space H s (T) for any s ≥ −1. The solution map corresponding to the IVP (4.5) is real analytic when s > − 1 2 , but only continuous (not even locally uniformly continuous) when −1 ≤ s < − On the other hand, by contrast, the IVP of the KdV-Burgers equation is known to be well-posed in the space H s (R) for any s ≥ −1, but is known to be ill-posed for any s < −1. We conjecture that the IBVP (4.1) is ill-posed in the space H s (0, L) for any s < −1.  A common feature for these two boundary value problems is that the L 2 −norm of their solutions are conserved: The IBVP (4.6) is equivalent to the IVP (4.5) which was shown by Kato [23,24] to be well-posed in the space H s (T) when s > 3 2 as early as in the late 1970s. Its well-posedness in the space H s (T) when s ≤ 3 2 , however, was established 24 years later in the celebrated work of Bourgain [9,10] in 1993. As for the IBVP (4.7), its associated linear problem      u t + u xxx = 0, x ∈ (0, L), u(x, 0) = φ(x), u(0, t) = 0, u(L, t) = 0, u x (0, t) = u x (L, t) (4.8) has been shown by Cerpa (see, for instance, [13]) to be well-posed in the space H s (0, L) forward and backward in time. However, whether the nonlinear IBVP (4.7) is well-posed in the space H s (0, L) for some s is still unknown.