Well-posedness of a class of non-homogeneous boundary value problems of the Korteweg-de Vries equation on a finite domain

DOI:10.1051/cocv/2012012 www.esaim-cocv.org

WELL-POSEDNESS OF A CLASS OF NON-HOMOGENEOUS BOUNDARY VALUE PROBLEMS OF THE KORTEWEG-DE VRIES EQUATION

ON A FINITE DOMAIN^∗,∗∗

Eugene Kramer

, Ivonne Rivas

and Bing-Yu Zhang

Abstract.In this paper, we study a class of Initial-Boundary Value Problems proposed by Colin and Ghidaglia for the Korteweg-de Vries equation posed on a bounded domain (0, L). We show that this class of Initial-Boundary Value Problems is locally well-posed in the classical Sobolev spaceH^s(0, L) fors >−³₄, which provides a positive answer to one of the open questions of Colin and Ghidaglia [Adv.

Differ. Equ.6(2001) 1463–1492].

Mathematics Subject Classification. 35Q53.

Received March 27, 2012. Revised November 18, 2011.

Published online January 10, 2013.

1. Introduction

In this paper we study a class of Initial-Boundary Value Problems (IBVP) for the Korteweg-de Vries (KdV) equation posed on a finite domain with nonhomogeneous boundary conditions:

ut+ux+uxxx+uux= 0, u(x,0) =φ(x), x∈(0, L), t∈R⁺,

u(0, t) =h₁(t), ux(L, t) =h₂(t), uxx(L, t) =h₃(t). (1.1) This IBVP can be considered as a model for propagation of surface water waves in the situation where a wave- maker is putting energy in a finite-length channel from the left (x = 0) while the right end (x = L) of the channel is free (corresponding the case ofh₂=h₃= 0) (see [16]). The problem was first proposed and studied by Colin and Ghidaglia in the late 1990’s [16–18] (cf.also [1–4,8,19,24,25] for other studies of boundary value problems of the KdV equation). In particular, they investigated the well-posedness of the IBVP in the classical Sobolev spaceH^s(0, L) and obtained the following results.

Keywords and phrases.The Kortweg-de Vries equation, well-posedness, non-homogeneous boundary value problem.

∗The authors would like to thank the referees for their comments and suggestions which have significantly improved the quality of the paper.

∗∗ This work was partially supported by a Grant from the Simons Foundation (#201615 to Bingyu Zhang).

1 Department of Mathematics, Physics, and Computer Science, Raymond Walters College, University of Cincinnati, Cincinnati, 45236 Ohio, USA.eugene.f.kramer@uc.edu

2 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, 45221 Ohio, USA.

rivasie@mail.uc.edu;zhangb@ucmail.uc.edu

3 IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brasil.

Article published by EDP Sciences c EDP Sciences, SMAI 2013

Theorem 1.1 ([18]).

(i) Given hj∈C¹([0,∞)), j= 1,2,3 andφ∈H¹(0, L)satisfying h₁(0) =φ(0), there exists a T >0 such that the IBVP(1.1)admits a solution (in the sense of distribution)

u∈L^∞(0, T;H¹(0, L))∩C([0, T];L²(0, L)).

(ii) The solutionuof the IBVP[(ii)] (1.1)exists globally inH¹(0, L)if the size of its initial valueφ∈H¹(0, L) and its boundary valueshj ∈C¹([0,∞)), j= 1,2,3 are all small.

In addition, they showed that the associate linear IBVP

ut+ux+uxxx= 0, u(x,0) =φ(x), x∈(0, L), t∈R⁺,

u(0, t) = 0, ux(L, t) = 0, uxx(L, t) = 0 (1.2)

possesses the Kato smoothing property: for anyφ∈L²(0, L), the linear IBVP (1.2)admits a unique solution u∈C(R⁺;L²(0, L))∩L²_loc(R⁺;H¹(0, L)).

Aided by this smoothing property Colin and Ghidaglia also showed that the homogeneous IBVP (1.1) is locally well-posed in the spaceL²(0, L).

Theorem 1.2 ([18]). Supposeh₁=h₂=h₃≡0, then for any φ∈L²(0, L), there exists aT >0 such that the IBVP (1.1)admits a unique weak solution

u∈C([0, T];L²(0, L))∩L²(0, T;H¹(0, L)).

In order to encourage further investigation, a series of open problems were proposed by Colin and Ghidaglia [18]. Included in the proposed problems are the following:

Problems:

(1) Is it possible to prove global existence of solutions of (1.1) for e.g.smooth solutions (as it is the case of the equation posed on a quarter plane or the whole line)?

It is remarked by Colin and Ghidaglia in [18]: “for these problems, uniqueness rely on a prioriestimate in H² that we are not able to extend here and therefore establish the existence of more regular solutions”.

(2) Is it possible to establish the existence of solutions of (1.1) with their initial value in the spaceH^s(0, L) for somes <0 as in the case of the whole line?

Colin and Ghidaglia expected the answer to be positive because of the Kato smoothing property of the associated linear IBVP (1.2).

In this paper, we will continue the work of Colin and Ghidaglia [16–18] by establishing the well-posedness of the IBVP (1.1) in the full strength of Hadamard includingexistence, uniqueness and continuous dependence and show that the IBVP (1.1) is (locally) well-posed in the space H^s(0, L) whens ≥0 and −³₄ < s < 0. In order to describe our results more precisely, let us first introduce some notations.

For givenT >0 ands∈R, let

H^s(0, T) :=H^s+13 (0, T)×H^s3(0, T)×H^s−13 (0, T), Ds,T :=H^s(0, L)×H^s(0, T) and

Zs,T =C([0, T];H^s(0, L))∩L²(0, T;H^s+1(0, L)).

For the well-posedness of the IBVP (1.1) that we intend to establish in this paper, some compatibility condi- tions relating the initial datumφ(x) and the boundary datahj(t), j= 1,2,3 are needed. A simple computation

shows if uis a C^∞-smooth solution of the IBVP (1.1), then its initial data u(x,0) = φ(x) and its boundary valuesh_j(t), j= 1,2,3 must satisfy the following compatibility conditions:

φk(0) =h⁽₁^k)(0), φ_k(L) =h⁽₂^k)(0), φ_k(L) =h⁽₃^k)(0) (1.3) fork= 0 ,1, . . . ,whereh⁽_j^k)(t) is thek-th order derivative ofhj and

φ₀(x) =φ(x) φk(x) =−

φ_k−1(x) +φ_k−1(x) +¹₂k−1

j=0 (k−1)!

j!(k−1−j)!(φj(x)φk−j−1(x)) (1.4) for k= 1,2, . . . When the well-posedness of (1.1) is considered in the space H^s(0, L) for s≥0, the following s-compatibility conditions arise naturally.

Definition 1.3 (s-compatibility).LetT >0 ands≥0 be given. A four-tuple (φ,h) = (φ, h₁, h₂, h₃)∈Ds,T

is said to bes-compatible with respect to the IBVP (1.1) if

φk(0) =h⁽₁^k)(0) (1.5)

whenk= 0,1, . . .[s/3] and ¹₂ < s−3[s/3]≤3/2,

φk(0) =h⁽₁^k)(0), φ_k(L) =h⁽₂^k)(0) (1.6) whenk= 0,1, . . .[s/3] and ³₂ < s−3[s/3]≤5/2 and

φk(0) =h⁽₁^k)(0), φ_k(L) =h⁽₂^k)(0) φ_k(L) =h⁽₃^k)(0) (1.7) whenk= 0,1, . . .[s/3] and

5/2< s−3[s/3]≤3, or whenk= 0,1, . . .[s/3]−1 and

0≤s−3[s/3]≤ 1 2·

The following theorem regarding the local well-posedness of the IBVP (1.1) in the space H^s(0, L) for any s≥0, is one of the main results of this paper.

Theorem 1.4. Let s≥0,T >0 andr >0 be given with s= 2j−1

2 , j= 1,2, . . . There exists aT^∗∈(0, T]such that for anys-compatible

(φ,h)∈Ds,T

with ⁴

(φ,h) D_0,T ≤r, (1.8)

the IBVP (1.1)admits a unique solution

u∈C([0, T^∗];H^s(0, L))∩L²(0, T^∗;H^s+1(0, L)).

Moreover, the corresponding solution map is Lipschitz continuous⁵.

4It is worth to point out here that it is not required that(φ,h)D_s,T ≤rwhens >0.

5The solution map, is in fact, real analytic (cf.[49–51]).

To consider the well-posedness of the IBVP (1.1) in the spaceH^s(0, L) withs <0, the following Bourgain-type spaces will be used (cf. [7,11,20,26–30,32]).

For any givens∈R, 0≤b≤1, 0≤α≤1 and functionw≡w(x, t) : R²→R, define Λs,b(w) =

_∞

−∞

_∞

−∞τ−(ξ³−ξ)^2bξ^2s|w(ξ, τˆ )|²dξdτ ¹₂

,

λα(w) = ^∞

−∞

|ξ|≤1τ^2α|w(ξ, τˆ )|²dξdτ ¹₂

(1.9) and

Gs(w) = ^∞

−∞ξ^2s _∞

−∞

|w(ξ, τˆ )|

1 +|τ−(ξ³−ξ)|dτ ₂

dξ ₁/2

where·:= (1 +| · |²)¹². LetXs,b be the space of all functionswsatisfying w _Xs,b :=Λ_s,b(w)<∞ whileYs,b be the space of all functionswsatisfying

w Y_s,b:=

Gs²(w) +Λ²_s,−b(w)₁/2<∞. In addition, letX_s,b^α be the space of all functionswsatisfying

w X_s,b^α :=

Λ²_s,b(w) +λ²_α(w)₁/2<∞ and letY_s,b^α be the space of allwsatisfying

w Y_s,b^α :=

λ²_α−1(w) +Gs²(w) +Λ²_s,−b(w)₁/2<∞.

The spacesXs,b,Ys,b,X_s,b^α andY_s,b^α are all Banach spaces.Xs,b andX_s,b^α are equivalent whenb≥α. The spaces Ys,b andXs,−b are also equivalent whenb < ¹₂. Furthermore, let

Xs,b^α ≡Cb(R;H^s(R))∩X_s,b^α with the norm

w _Xα s,b =

sup

t∈R w(·, t) ²H^s(R)+ w ²_Xα s,b

₁/2

.

The Bourgain-type spaces given above are defined for functions posed on the whole plane R×R. However the IBVP (1.1) is posed on the finite domain (0, L)×(0, T). It is thus natural to introduce a restricted version of the Bourgain spaceXs,b to the domain (0, L)×(0, T) as follows:

X_s,b^T =Xs,b

(0,L)×(0,T)

with the quotient norm u _XT

s,b ≡ inf

w∈X_s,b{ w X_s,b : w(x, t) =u(x, t) on (0, L)×(0, T)}

for anyu(x, t) defined on (0, L)×(0, T). The spaces Y_s,b^T, X_s,b^α,T,Y_s,b^α,T andX_s,b^α,T are defined similarly.

Another main result of this paper addresses the second problem listed previously from Colin and Ghidaglia [18]. The research presented establishes the local well-posedness of the IBVP (1.1) in the space H^s(0, L) for somes <0.

Theorem 1.5. Let s∈(−³₄,0),T >0 andr >0 be given. There existT^∗∈(0, T]andα > ¹₂ such that for any (φ,h)∈Ds,T

satisfying

(φ,h) D_s,T ≤r, the IBVP (1.1)admits a unique solutionu∈C([0, T^∗];H^s(0, L))with

u _Xα,T∗ s,1

2

<+∞.

Moreover, the corresponding solution map is Lipschitz continuous.

The following remarks are in order.

(i) Theorem 1.4 states that the IBVP (1.1) is well-posed in the spaceH^s(0, L) for anys≥0, not just fors= 0 or s= 1. In particular, it demonstrates the existence of classical solutions and shows that the smoother the initial value and boundary data, the smoother the corresponding solution.

(ii) To have solution u in the spaceC([0, T];H^s(0, L)), Theorem 1.4 only requires that the initial value φ∈ H^s(0, L) and the boundary data be

h₁∈H^s+13 (0, T), h₂∈H^s3(0, T), h₃∈H^s−13 (0, T). (1.10) In particular, ifs= 1, it is sufficient to require that

h₁∈H²³(0, T), h₂∈H¹³(0, T), h₃∈L²(0, T),

rather thanhj ∈C¹(0, T), j= 1,2,3 as in Theorem 1.1. Moreover, condition (1.10) is optimal in order to have the corresponding solutionu∈C([0, T];H^s(0, L)).

(iii) In Theorem 1.4, the life span (0, T^∗) of the solutions depends only (φ,h) D_0,T, not on (φ,h) D_s,T when s >0, in particular. Thus a solutionublows up in the spaceH^s(0, L) for anys >0 if and only if it blows up in the spaceL²(0, L).

(iv) Taking lead from the recent works of Bonaet al.[9], Molinet [35], and Molinet and Vento [37], we conjecture that the IBVP (1.1) is locally well-posed in the space H^s(0, L) for −1 < s≤ −³₄, but is ill-posed in the spaceH^s(0, L) for anys <−1.

It is also interesting and constructive to compare the study of the IBVP (1.1) with another class of IBVP of the KdV equation posed on the finite domain (0, L). The following problem has been well studied over the past few years [5,9,26,48]. ⎧

⎪⎨

⎪⎩

u_t+u_x+u_xxx+uu_x= 0, x∈(0, L), t∈R⁺, u(x,0) =φ(x),

u(0, t) =h₁(t), u(L, x) =h₂(t), ux(L, t) =h₃(t).

(1.11)

While the study of the IBVP (1.11) goes back as early as late 1970s [12,13], the nonhomogeneous IBVP (1.11) was first shown by Bonaet al.[5] to be locally well-posed in the spaceH^s(0, L) for anys≥0:

Lets≥0 ,r >0 andT >0be given. There exists T^∗∈(0, T]such that for any s-compatible⁶

φ∈H^s(0, L), h= (h₁, h₂, h₃)∈H^s+13 (0, T)×H^s+13 (0, T)×H^s3(0, T)

6See [5] for the exact definition ofs-compatibility for the IBVP (1.11).

satisfying⁷

φ _Hs(0,L)+ h

H^s+13 (0,T)×H^s+13 (0,T)×H^s3(0,T)≤r, the IBVP (1.11)admits a unique solution

u∈C([0, T^∗];H^s(0, L))∩L²(0, T^∗;H^s+1(0, L)).

Moreover, the corresponding solution map is Lipschitz continuous in the corresponding spaces.

Later Holmer [26] showed that the IBVP (1.11) is locally well-posed in the spaceH^s(0, L) for any−³₄ < s < ¹₂: Let s∈(−₄³,¹₂),r >0 andT >0 be given. There exists aT^∗∈(0, T] such that for any

φ∈H^s(0, L), h= (h₁, h₂, h₃)∈H^s+13 (0, T)×H^s+13 (0, T)×H^s3(0, T) satisfying

φ _Hs(0,L)+ h

H^s+13 (0,T)×H^s+13 (0,T)×H^s3(0,T)≤r, the IBVP (1.11)admits a unique mild solution⁸

u∈C([0, T^∗];H^s(0, L)).

Moreover, the corresponding solution map is Lipschitz continuous in the corresponding spaces.

7This condition can be weakened as

φ_L2(0,L)+h

H¹3(0,T)×H¹3(0,T)×L²(0,T)≤r.

8A functionu∈C([0, T^∗];H^s(0, L)) is said to be a mild solution of the IBVP (1.11) if there exist a sequence

un∈C¹([0, T^∗];L²(0, L))∩C[(0, T^∗];H³(0, L)), n= 1,2, . . . solving the equation in (1.11) and asn→ ∞,

un→u inC([0, T^∗];H^s(0, L)),

h1,n:=un(0,·)→h1, h2,n:=u(L,·)→h2 inH^s+13 (0, T^∗) and

h3,n:=∂xun(L,·)→h3 inH^s3(0, T^∗).

More recently, Bonaet al.[9] showed that the IBVP (1.11) is locally well-posedH^s(0, L) for any−1< s≤0.

Let r >0,−1< s≤0 andT >0 be given. There exists aT^∗∈(0, T]such that for any φ∈H^s(0, L), h= (h₁, h₂, h₃)∈H^s+13 (0, T)×H^s+13 (0, T)×H^s3(0, T) satisfying

φ _Hs(0,L)+ h

H^s+13 (0,T)×H^s+13 (0,T)×H^s3(0,T)≤r, the IBVP (1.11)admits a unique mild solution

u∈C([0, T^∗];H^s(0, L)).

Moreover, the corresponding solution map is Lipschitz continuous in the corresponding spaces.

Although there is only a slight difference between the boundary conditions of IBVP (1.1) and the IBVP (1.11), there is a big gap between their well-posedness results. For the IBVP (1.1), the well-posedness results presented in Theorems 1.4 and 1.5 are local in the sense that the time interval (0, T^∗) in which the solution exists depends onrand, in general, the larger ther, the smaller theT^∗. By contrast, the IBVP (1.11) is known to be globally well-posed in the spaceH^s(0, L) for any s≥0 in the sense that one always has (0, T^∗) = (0, T) no matter how large the value ofris (cf.[5,23]). This difference stems from the fact that the L²-energy of the solution of the homogeneous IBVP (1.11) (h= 0) is decreasing since

d dt

L 0

u²(x, t)dx=−u²_x(0, t) for t≥0,

while for the homogeneous IBVP (1.1) it is not clear, in general, whether the L²-energy of its solution is increasing or decreasing since

d dt

L 0

u²(x, t) =−u²(L, t)−2

3u³(L, t)−u²_x(0, t) for t≥0.

The approach used in the proofs of their results in [5,22,26] is very much different from what used in the proof of Theorem 1.1, but more or less along the line used in the proof of Theorem 1.2, in which the Kato smoothing property of the associated linear system plays an important role. In this paper, we will use a similar approach as that developed in [5,7] but with some modifications to prove Theorems 1.4 and 1.5. The key ingredients of the approach are listed below.

(1) An explicit solution formula will be derived for the following nonhomogeneous boundary value problem of the linear equation,

⎧⎪

⎨

⎪⎩

vt+vx+vxxx= 0, x∈(0, L), t≥0, v(x,0) = 0,

v(0, t) =h₁(t), vx(L, t) =h₂(t), vxx(L, t) =h₃(t),

(1.12) which not only enables us to establish the well-posedness of the IBVP (1.1) with the optimal regularity conditions imposed on the boundary data, but also plays an important role in obtaining the well-posedness of the IBVP (1.1) in the spaceH^s(0, L) with−³₄ < s <0.

(2) The Kato smoothing property of the associated linear problem

⎧⎪

⎨

⎪⎩

vt+vx+vxxx=f, x∈(0, L), t≥0, v(x,0) =φ(x),

v(0, t) =h₁(t), vx(L, t) =h₂(t), vxx(L, t) =h₃(t).

(1.13)

For givenT >0, there exists a constantC >0 such that the solutionv of(1.13)satisfies v Z_0,T ≤C

(φ,h) D_0,T + f _L1(0,T;L²(0,L))

for any (φ,h) ∈ D₀,T and f ∈ L¹(0, T;L²(0, L)). This property is an extension of the Kato smoothing property obtained by Colin and Ghidaglia [18] for the nonhomogeneous problem.

(3) Tartar’s nonlinear interpolation theorem [47].

LetB₀ and B₁ be two Banach spaces such thatB₁⊂B₀ with continuous inclusion map. Let f ∈B₀ and, fort≥0, define

K(f, t) = inf

g∈B₁{ f−g B₀+t g B₁}. For 0≤θ≤1, the (real) interpolation ofB₀ andB₁ is defined by

[B₀, B₁]θ=

f ∈B₀: f B_θ:=

_∞

0

K²(f, t)t^−2θ−1dt ¹₂

<+∞

.

ThenBθ is a Banach space with norm · B_θ.

Theorem 1.6 (Tartar [47]).Forj= 1,2, let B₀^j andB^j₁ be Banach spaces such thatB₁^j ⊂B^j₀ with contin- uous inclusion mappings. SupposeA is a mapping such that

(i) A: B¹₀→B²₀ and forf, g∈B₀¹,

Af −Ag _B2

0 ≤C₀( f _B1

0 + g _B1

0) f−g _B1 0

and

(ii) A:B¹₁→B₁² and for h∈B¹₁

Ah _B2

1 ≤C₁( h _B1 0) h _B1

1, whereCj:R⁺ →R⁺ are continuous non-decreasing functions,j= 0,1.

Then if0≤θ≤1,A maps B_θ¹ intoB_θ² and forf ∈B_θ¹, Af _B2

θ ≤C( f _B1 0) f _B1

θ

where for r >0, C(r) = 4C₀(4r)^1−θC₁(3r)^θ.

Note that ifB¹₀=D₀,T,B₁¹=D₃,T,B₁¹=Z₀,T andB²₁=Z₃,T. Then Ds,T = [B₀¹, B₁¹]θ, Zs,T = [B₀², B²₁]θ

withθ= ^s₃, 0≤s≤3. With the help of this theorem, we will only need to consider the case ofs= 3k, k= 0,1,2, . . .when proving Theorem 1.4.

(4) Following Bonaet al. [7], the IBVP (1.1) will be converted to an integral equation posed on the whole line Rwhich makes it possible to conduct Bourgain spaces analysis to obtain the well-posedness of the IBVP (1.1) inH^s(0, L) for−³₄ < s <0.

This paper is organized as follows. In Section 2, the various linear problems associated to the IBVP (1.1) are studied. Section 3 is devoted to the well-posedness of the nonlinear IBVP (1.1). Concluding remarks and open questions for further investigations are presented in Section 4.

2. Linear problems

2.1. The boundary integral operators

Consider the nonhomogeneous boundary-value problem

vt+vx+vxxx= 0, v(x,0) = 0, x∈(0, L), t≥0,

v(0, t) =h₁(t), vx(L, t) =h₂(t), vxx(L, t) =h₃(t). (2.1) We look for an explicit solution formula for the IBVP (2.1). Without loss of generality, we assume that L= 1 in this subsection. Applying the Laplace transform yo both sides of the equation in (2.1) with respect tot, the IBVP (2.1) is converted to

sˆv+ ˆvx+ ˆvxxx= 0,

ˆv(0, s) = ˆh₁(s), ˆvx(1, s) = ˆh₂(s), ˆvxx(1, s) = ˆh₃(s), (2.2) where

ˆv(x, s) = _+∞

0 e^−stv(x, t)dt and

ˆhj(s) = _∞

0

e^−sthj(t)dt, j= 1,2,3.

The solution of (2.2) can be written in the form v(x, s) =ˆ

3 j=1

cj(s)e^λj(s)x

whereλj(s), j= 1,2,3 are solutions of the characteristic equation s+λ+λ³= 0 andcj(s), j= 1,2,3, solve the linear system

⎛

⎝ 1 1 1 λ₁e^λ1 λ₂e^λ2 λ₃e^λ3 λ²₁e^λ1 λ²₂e^λ2 λ²₃e^λ3

⎞

⎠

A

⎛

⎝c₁ c₂ c₃

⎞

⎠=

⎛

⎝ ˆh₁ ˆh₂ ˆh₃

⎞

⎠ hˆ

. (2.3)

By Cramer’s rule,

cj= Δj(s)

Δ(s), j= 1,2,3,

whereΔ is the determinant ofAandΔj is the determinant of the matrixAwith the columnj replaced byh.

Taking the inverse Laplace transform ofvand following similar arguments as those in [5] yield the representation v(x, t) = ³

m=1

vm(x, t) with

v_m(x, t) =³

j=1

v_j,m(x, t)

and

v_j,m(x, t) =v⁺_j,m(x, t) +v_j,m⁻ (x, t), where

v⁺_j,m(x, t) = 1 2πi

₊i∞

0 e^stΔj,m(s)

Δ(s) ˆhm(s)e^λj(s)xds and

v_j,m⁻ (x, t) = 1 2πi

₀

−i∞e^stΔj,m(s)

Δ(s) ˆhm(s)e^λj(s)xds

for j, m = 1,2,3. Here Δj,m(s) is obtained from Δj(s) by letting ˆhm(s) = 1 and ˆhk(s) = 0 for k = m, k, m= 1,2,3. Making the substitutions=i(ρ³−ρ) with 1< ρ <∞in the characteristic equation

s+λ+λ³= 0 gives the following roots in terms ofρ

λ⁺₁(ρ) =iρ, λ⁺₂(ρ) =

3ρ²−4−iρ

2 λ⁺₃(ρ) = −

3ρ²−4−iρ

2 · (2.4)

Thusv_j,m⁺ (x, t) has the form

v_j,m⁺ (x, t) = 1 2π

_∞

1 e^i(ρ3−ρ)tΔ⁺_j,m(ρ)

Δ⁺(ρ) ˆh⁺_m(ρ)e^λ+j(ρ)x(3ρ²−1)dρ and

v⁻_j,m(x, t) =v⁺_j,m(x, t)

where ˆh⁺_m(ρ) = ˆhm(i(ρ³−ρ)), Δ⁺(ρ) and Δ⁺_j,m(ρ) are obtained from Δ(s) and Δj,m(s) by replacing s with i(ρ³−ρ) andλj(s) byλ⁺_j(ρ) =λj(i(ρ³−ρ)).

For givenm, j= 1,2,3, letWj,m be an operator onH₀^s(R⁺) defined as follows:

[Wj,mh](x, t)≡[Uj,mh](x, t) + [Uj,mh](x, t) (2.5) for anyh∈H₀^s(R⁺) with

[Uj,mh](x, t) ≡ 1 2π

_+∞

1 e^i(ρ3−ρ)te^{−λ+j(ρ)(1−x)}(3ρ²−1)[Q⁺_j,mh](ρ)dρ

= 1 2π

_+∞

1 e^i(ρ3−ρ)te^−λ+j(ρ)x(3ρ²−1)[Q⁺_j,mh](ρ)dρ, (x= 1−x), (2.6) forj= 1,2, m= 1,2,3 and

[U₃,mh](x, t)≡ 1 2π

_+∞

1 e^i(ρ3−ρ)te^λ+3(ρ)x(3ρ²−1)[Q⁺_3,mh](ρ)dρ (2.7) form= 1,2,3, where

[Q⁺_3,mh](ρ) := Δ⁺_3,m(ρ)

Δ⁺(ρ) ˆh⁺(ρ), [Q⁺_j,mh](ρ) =Δ⁺_j,m(ρ)

Δ⁺(ρ) e^λ+j(ρ)ˆh⁺(ρ)

for j = 1,2 and m = 1,2,3, ˆh⁺(ρ) = ˆh(i(ρ³−ρ)). The solution of the IBVP (2.1) then has the following representation.

Lemma 2.1. Given h₁, h₂ and h₃, definingh= (h₁, h₂, h₃). The solution v of the IBVP(2.1)can be written in the form

v(x, t) = [Wbdrh](x, t) :=

3 j,m=1

[Wj,mhm](x, t). (2.8)

2.2. Linear estimates

Consider the IBVP of the linear equation:

⎧⎪

⎨

⎪⎩

vt+vx+vxxx =f, x∈(0, L), t≥0, v(x,0) =φ(x),

v(0, t) = 0, vx(L, t) = 0, vxx(L, t) = 0.

(2.9)

By standard semigroup theory [39], for any φ ∈ L²(0, L) and f ∈ L¹(0, T;L²(0, L)), (2.9) admits a unique solutionv∈C([0, T];L²(0, L)), which can be written in the form

v(t) =W₀(t)φ+ t

0 W₀(t−τ)f(τ)dτ

whereW₀ is theC₀-semigroup in the spaceL²(0, L) generated by the linear operator Aψ=−ψ−ψ

with the domain

D(A) ={ψ∈H³(0, L) :ψ(0) =ψ(L) =ψ(L) = 0}.

Proposition 2.2. Let T > 0 be given. There exists a constant C such that for any φ ∈ L²(0, L) and f ∈ L¹(0, T;L²(0, L)), the corresponding solutionv of the IBVP(2.9)belongs to the space Z₀,T and

v Z_0,T ≤C

φ + f _L1(0,T;L²(0,L))

. (2.10)

Proof. First multiplying the both sides of the equation in (2.9) by 2v and integrating over (0, L) with respect toxgives

d dt

L

0 v²(x, t)dx+v²(L, t) +v²_x(0, t) = 2 L

0 xf(x, t)v(x, t)dx.

Thus

0<t<Tsup

v(·, t) L²(0,L)≤ φ L²(0,L)+C f L¹(0,T;L²(0,L)).

Then multiplying the both sides of the equation in (2.9) by 2xv and integrating over (0, L) with respect tox yields

d dt

L 0

xv²(x, t)dx+Lv²(L, t) + 3 L

0

v²_xdx= L

0

v²dx+ 2 L

0

f(x, t)v(x, t)dx.

Estimate (2.10) then follows easily.

Next we consider the nonhomogeneous boundary-value problem

⎧⎪

⎨

⎪⎩

vt+vx+vxxx = 0, x∈(0, L), t≥0, v(x,0) = 0,

v(0, t) =h₁(t), vx(L, t) =h₂(t), vxx(L, t) =h₃(t).

(2.11)

We have the following estimate for its solutions.

Proposition 2.3. For givenT >0, there exists a constantC such that for anyh∈H⁰(0, T), the corresponding solution v of the system(2.11)belongs to the space Z_0,T and

v Z_0,T ≤C h _H0(0,T). Proof. From Section 2.1 we have

λ⁺₁(ρ) =iρ, λ⁺₂(ρ) =

3ρ²−4−iρ

2 λ⁺₃(ρ) = −

3ρ²−4−iρ

2 ·

The asymptotic behaviors of the ratios ^Δ

+j,m(ρ)

Δ+(ρ) forρ→+∞are listed below.

Δ⁺_1,1(ρ) Δ⁺(ρ) ∼e⁻

√3

2 ρ Δ⁺_2,1(ρ)

Δ⁺(ρ) ∼e^{−√3ρ Δ}_Δ^+3,1₊₍⁽_ρ^ρ₎⁾∼1

Δ⁺_1,2(ρ)

Δ⁺(ρ) ∼ρ^{−1 Δ}_Δ^+2,2₊₍⁽_ρ^ρ₎⁾ ∼ρ⁻¹e⁻

√3

2 ρ Δ⁺_3,2(ρ) Δ⁺(ρ) ∼ρ⁻¹

Δ⁺_1,3(ρ)

Δ⁺(ρ) ∼ρ^{−2 Δ}_Δ^+2,3₊₍⁽_ρ^ρ₎⁾ ∼ρ⁻²e⁻

√3

2 ρ Δ⁺_3,3(ρ) Δ⁺(ρ) ∼ρ⁻² Form= 1,2,3 andj= 1,2, set

hˆ^∗+_3,m(ρ) := [Q⁺_3,mhm](ρ) = Δ⁺_3,m(ρ) Δ⁺(ρ) ˆh⁺_m(ρ) and

hˆ^∗+_j,m(ρ) := [Q⁺_j,mhm](ρ) = Δ⁺_j,m(ρ)

Δ⁺(ρ) e^λ+j(ρ)ˆh⁺_m(ρ).

Viewingh^∗_j,m as the inverse Fourier transform of ˆh^∗+_j,m, it is straightforward to see that for anys∈R,

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

h₁∈H₀^(s+1)/3(R⁺)⇒h^∗_j,1∈H^s+13 (R), j= 1,2,3, h₂∈H₀^s/3(R⁺) ⇒h^∗_j,2∈H^s+13 (R), j= 1,2,3, h₃∈H₀^(s−1)/3(R⁺)⇒h^∗_j,3∈H^s+13 (R), j= 1,2,3.

(2.12)

The proof is completed by the same arguments presented in [5] for their proofs of Propositions 2.7–2.9.

Now we consider the following IBVP of the linear KdV equation with variable coefficients.

⎧⎪

⎨

⎪⎩

vt+vx+vxxx+ (av)x=f, x∈(0, L), t >0, v(x,0) =φ(x),

v(0, t) =h₁(t), vx(L, t) =h₂(t), vxx(L, t) =h₃(t), t≥0

(2.13)

werea=a(x, t) is a given function.

Proposition 2.4. Let T > 0 be given and assume that a ∈ Z₀,T. Then for any (φ,h) ∈ D₀,T and f ∈ L¹(0, T;L²(0, L)), the IBVP(2.13)admits a unique solution v∈Z₀,T. Moreover, there exists a constantC >0 depending only on T and a Z_0,T such that

v Z_0,T ≤C

(φ,h) D_0,T + f L1(0,T;L2(0,L)) .

Proof. Letη >0 and 0< θ≤max{1, T}be constants to be determined. Set Sθ,η={w∈Z₀,θ, w Z_0,θ ≤η}. For given (φ,h)∈ D0,T define a map onSθ,ηby

v=Γ(w) withv being the unique solution of the IBVP

⎧⎪

⎨

⎪⎩

vt+vx+vxxx=−(aw)x+f, x∈(0, L), t∈(0, T), v(x,0) =φ(x),

v(0, t) =h₁(t), vx(L, t) =h₂(t), vxx(L, t) =h₃(t) forw∈Sθ,η. Using Propositions 2.2 and 2.3, we have

Γ(w) Z_0,θ ≤c₁ (φ,h) _D0,T +c₁ f L¹(0,T;L²(0,L))+c₁ (aw)x L¹(0,θ;L²(0,L))

≤c₁ (φ,h) D0,T +c₁ f _L1(0,T;L²(0,L))+c₁(θ¹³ +θ¹²) a Z_0,T w Z_0,θ. Here we have used the bilinear estimate (cf. [5,34])

T

0 (qp)x L²(0,L)dt≤C(T¹² +T¹³) q Z_0,T q Z_0,T

for anyp, q∈Z₀,T.

If we chooseηand θsuch that

η= 2c₁( (φ,h) _D0,T + f _L1(0,T;L²(0,L))) and

c₁(θ¹³ +θ¹²) a Z_0,T ≤1 2· Then we have

Γ(w) Z_0,θ ≤η for anyw∈Sθ,η. Moreover, for anyw₁, w₂∈Sθ,η, one has

Γ(w₁)−Γ(w₂) Z_0,θ ≤ 1

2 w₁−w₂ Z_0,θ.

Thus, Γ is a contraction onSθ,η whose fixed pointu∈Z₀,θ is the desired solution of the IBVP (2.13) which exists in the time interval (0, θ). Asθonly depends on a Z_0,T, by a standard argument the time interval (0, θ) in which the solution exists can be extended to (0, T) so that u∈Z₀,T. Proposition 2.4 will be sufficient for us to obtain the local well-posedness of the IBVP (1.1) in the space H^s(0, L) for s ≥ 0. However, in order to use Bourgain space analysis to obtain well-posedness in the space H^s(0, L) with s <0, we need to extend the problem posed on the finite domain (0, L)×(0, T) to an equivalent problem posed on the whole planeR×R.

Recall the solution of the following linear KdV equation,

vt+vx+vxxx = 0, x∈R, t∈R⁺,

v(x,0) =ψ(x) (2.14)