Non-homogeneous boundary value problems for the Korteweg-de Vries and the Korteweg-de Vries-Burgers equations in a quarter plane

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www.elsevier.com/locate/anihpc

Non-homogeneous boundary value problems for the Korteweg–de Vries and the Korteweg–de Vries–Burgers

equations in a quarter plane

Jerry L. Bona

^a,^∗

, S.M. Sun

^b

, Bing-Yu Zhang

^c

aDepartment of Mathematics, Statistics & Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA bDepartment of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-4097, USA

cDepartment of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USA Received 28 April 2006; received in revised form 2 May 2007; accepted 22 July 2007

Available online 24 January 2008

Abstract

Attention is given to the initial-boundary-value problems (IBVPs) ut+ux+uux+uxxx=0, forx, t0,

u(x,0)=φ(x), u(0, t)=h(t)

(0.1) for the Korteweg–de Vries (KdV) equation and

u_t+u_x+uu_x−u_xx+u_xxx=0, forx, t0, u(x,0)=φ(x), u(0, t)=h(t)

(0.2) for the Korteweg–de Vries–Burgers (KdV-B) equation. These types of problems arise in modeling waves generated by a wavemaker in a channel and waves incoming from deep water into near-shore zones (see [B. Boczar-Karakiewicz, J.L. Bona, Wave dominated shelves: a model of sand ridge formation by progressive infragravity waves, in: R.J. Knight, J.R. McLean (Eds.), Shelf Sands and Sandstones, in: Canadian Society of Petroleum Geologists Memoir, vol. 11, 1986, pp. 163–179] and [J.L. Bona, W.G. Pritchard, L.R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Roy. Soc. London Ser. A 302 (1981) 457–510] for example). Our concern here is with the mathematical theory appertaining to these problems. Improving upon the existing results for (0.2), we show this problem to be (locally) well-posed inH^s(⁺)when the auxiliary data(φ, h)is drawn fromH^s(⁺)× H

s+1 3

loc (⁺), provided only thats >−1 ands=3m+ ¹₂ (m=0,1,2, . . .). A similar result is established for (0.1) inH_ν^s(⁺) provided(φ, h)lies in the spaceH_ν^s(⁺)×H

s+1 3

loc (⁺). Here,H_ν^s(⁺)is the weighted Sobolev space H_ν^s

⁺

= f∈H^s

⁺

;e^νxf∈H^s ⁺

with the obvious norm (cf. Kato [T. Kato, On the Cauchy problem for the (generalized) Korteweg–de Vries equations, in: Ad- vances in Mathematics Supplementary Studies, in: Studies Appl. Math., vol. 8, 1983, pp. 93–128]). Both local and global in time results are derived. An added outcome of our analysis is a very strong smoothing property associated with the problems

* Corresponding author.

E-mail addresses:[email protected] (J.L. Bona), [email protected] (S.M. Sun), [email protected] (B.-Y. Zhang).

doi:10.1016/j.anihpc.2007.07.006

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(0.1) and (0.2) which may be expressed as follows. Supposeh∈H_loc^∞and that for someν >0 ands >−1 withs=3m+¹₂ (m=0,1,2, . . .),φlies inH_ν^s(⁺)(respectivelyH^s(⁺)). Then the corresponding solutionuof the IBVP (0.1) (respectively the IBVP (0.2)) belongs to the spaceC(0,∞;H_ν^∞(⁺))(respectivelyC(0,∞;H^∞(⁺))). In particular, for anys >−1 with s=3m+¹₂(m=0,1,2, . . .), ifφ∈H^s(⁺)has compact support andh∈H_loc^∞(⁺), then the IBVP (0.1) has a unique solution lying in the spaceC(0,∞;H^∞(⁺)).

Keywords:Korteweg–de Vries equation; Korteweg–de Vries–Burgers equation; Initial-boundary-value problems; Nonlinear dispersive wave equations

1. Introduction

In this paper, we continue the study of the initial-boundary-value problem (IBVP) for the Korteweg–de Vries (KdV) equation and the Korteweg–de Vries–Burgers (KdV-B) equation posed in a quarter plane, namely

u_t+u_x+uu_x−μu_xx+u_xxx=0, forx, t0, u(x,0)=φ(x), u(0, t )=h(t ),

(1.1) whereμ=0 for KdV andμ >0 for KdV-B. These IBVP’s arise in mathematical descriptions of water waves generated by a wave maker in a channel and in other situations where a wavetrain is generated at or impinges upon one end of a stretch of a medium of propagation that suffers both non-linear and dispersive effects in response to disturbances (see [2,3,5,35]). Our main concern is the well-posednessof (1.1) in the sense to be specified now, in the classical Sobolev spacesH^s(⁺)for small values ofs.

Definition 1.1 (Well-posedness). For a given real numbers, the IBVP (1.1) is said to be well-posed in the space H^s(⁺)×H

s+1 3

loc (⁺) if for any r >0, there exists a T =T (r) >0 depending only on s and r such that for s-compatible(φ, h)∈H^s(⁺)×H^s⁺³¹(0, T )satisfying

(φ, h)

H^s(⁺)×H^s+1³ (0,T )r, the IBVP (1.1) admits a unique solution

u∈C

[0, T];H^s ⁺

.

Moreover, the solution depends continuously in this latter space on variations of the auxiliary data in their respective function spaces.

A pair(φ, h)∈H^s(⁺)×H^s⁺³¹(0, T )is said to bes-compatiblefor (1.1) if, in cases > ¹₂, φ_k(0)=h_k(0)

fork=0,1, . . . ,[s/3]−1 whens−3[s/3]¹₂and fork=0,1, . . . ,[s/3]whens−3[s/3]>¹₂, whereh_k(t )≡h^(k)(t ) is thekth order derivative ofh,

φ₀(x)=φ(x), and φ_k(x)= −

φ_k₋₁(x)+φ_k₋₁(x)−μφ_k₋₁(x)+_k₋₁

j=0 φ_j(x)φ_k₋_j₋₁(x)

(1.2) fork=1,2, . . . .Here, for non-negative numbersr,[r]is the largest integer less than or equal tor.

The well-posedness described by Definition 1.1 is usually calledlocalwell-posedness since the life-spanT of the solution depends on the sizerof the auxiliary data. If, instead,T is independent ofr, then (1.1) is said to beglobally well-posed in the spaceH^s(⁺)×H

s+13

loc (⁺). Whens3, the term ‘solution’ in Definition 1.1 is understood to refer to a strong solution, which is to sayusatisfies the equation in (1.1) in the spaceL²(⁺)for allt∈ [0, T]. When s <3, the solutionuin Definition 1.1 is understood to be a mild solution as defined below in Definition 1.2. One of the advantages of using the concept of mild solution instead of solution in the sense of distributions is that one does not need to be concerned with whether or not the non-linear term uu_x in the equation makes sense in the relevant function class, a point that is especially telling whensis negative.

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Definition 1.2 (mild solution).Lets∈ andT >0 be given. For a givens-compatible pair (φ, h)∈H^s(⁺)× H

s+1 3

loc (⁺), a function u∈C([0, T];H^s(⁺)) is said to be a mild solution of the IBVP (1.1) on the time interval [0, T]if there exists a sequence{u_n}^∞_n₌₁in the space

C

[0, T];H³ ⁺

∩C¹

[0, T];L² ⁺ with

φ_n(x)=u_n(x,0), h_n(t )=u_n(0, t ), n=1,2, . . . , such that

(i) unsolves the equation in (1.1) inL²(⁺)for 0< t < T; (ii) limn→∞u_n−uC([0,T];H^s(⁺))=0;

(iii) limn→∞φ_n−φH^s(⁺)=0 and limn→∞h_n−h

H^s⁺³¹(0,T )=0.

Remark.Ifs3 anduis a strong solution of (1.1), then the constant sequenceu_n=ufor allnsuffices to determine uas a mild solution.

Boundary-value problems for non-linear, dispersive wave equations began with the work of Bona and Bryant [3]

for the BBM-equation. The study of the IBVP (1.1) withμ=0 was initiated by Ton in [54], where existence and uniqueness were established assuming that the initial dataφis smooth and the boundary datah≡0. The first well- posedness result in the sense of Definition 1.1 for the IBVP (1.1) was presented by Bona and Winther [13,14]; they showed that the IBVP (1.1) is (globally) well-posed in the spaceH^3k⁺¹(⁺)with(φ, h)∈H^3k⁺¹(⁺)×H_loc^k⁺¹(⁺) for k=1,2, . . .. There have been many works on (1.1) since then. The reader is referred to [4,8–12,20,26–29,33, 32,31,30,34] and the references therein for an overall literature review. In particular Bona, Sun and Zhang in [8]

extended the theory of Kenig, Ponce and Vega in [41,42] on the initial value problem (IVP) for the KdV equation posed on the whole line to the IBVP (1.1), showing that it is well-posed in the spaceH^s(⁺)×H

s+13

loc (⁺)for s > ³₄. In [20], Colliander and Kenig demonstrated how the powerful theory developed by Kenig, Ponce and Vega, Bourgain and others for the pure initial value problems for non-linear dispersive wave equations can be adapted to deal with initial boundary value problems for the same equations. They showed in [20] that for a givens-compatible pair (φ, h)∈H^s(⁺)×H

s+1 3

loc (⁺)with 0s1, s=¹₂, the IBVP (1.1) admits a solutionu∈C([0, T];H^s(⁺))which depends continuously on(φ, h). This result was strengthened later by Holmer [34] to include the case−³₄< s <0.In a recent paper [12], Bona, Sun and Zhang showed that the IBVP (1.1) possesses a strong global smoothing property that comes about because of the dissipative mechanism introduced through imposition of the boundary condition at x=0. With the aid of this boundary smoothing property and the use of restricted Bourgain spaces, they resolved the uniqueness issue left open in [20] and showed that the IBVP (1.1) is unconditionally well-posed in the space H^s(⁺)×H

s+1 3

loc (⁺)for anys >−³₄. The following question then arises naturally.

Question 1.3.Is the IBVP (1.1) well-posed in the spaceH^s for any values ofs <−³₄?

The issue of how large can be the spaces of auxiliary data and still maintain well-posedness also arises for the pure initial value problem (IVP) for the KdV-equation posed on the whole line, viz.

u_t+uu_x+u_xxx=0, x, t∈ u(x,0)=φ(x),

(1.3) or posed with periodic boundary conditions, which is to say, posed on the unit circleSin the plane,

u_t+uu_x+u_xxx=0, x∈S, t∈ u(x,0)=φ(x), x∈S.

(1.4)

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After considerable effort (see [7,6,15,17–19,21,22,37,39–44,46,50,51,53] and the references contained therein), it is understood that the IVP (1.3) is well-posed in the spaceH^s()fors >−³₄ and the IVP (1.4) is well-posed in the spaceH^s(S)fors−¹₂. The analogue of Question 1.3 in the context of (1.3)–(1.4) is the following.

Question 1.4. Is the IVP (1.3) well-posed inH^s()for somes <−³₄ or is the IVP (1.4) well-posed inH^s(S)for somes <−¹₂?

The answer is instructive. For the IVP (1.3), whens <−³₄, the IVP (1.3) has been shown to be ill-posed inH^s() in the sense that the solution map, if it were to exist, cannot be locally uniformly continuous (see [16,55,19,45]). The same can be said for the IVP (1.4); it is ill-posed inH^s(S)whens <−¹₂ in the sense that the solution map cannot be locally uniformly continuous. Whens= −³₄, a weaker form of local well-posedness has been established for (1.3) in [21]. Thus, the indications were that the answer to Question 1.4 was almost certainly negative. It came as quite a surprise when Kappeler and Topalov [36] demonstrated recently that the IVP (1.4) is (globally) well-posed in the spaceH^s(S)fors−1. In addition, there is another recent and very interesting paper [47] of Molinet and Ribaud on the pure initial-value problem

ut+uux+uxxx−uxx=0, x∈ , t >0

u(x,0)=ψ (x), x∈

(1.5) for the KdV–Burgers equation. They showed that (1.5) is well-posed in the spaceH^s()fors >−1 and is ill-posed whens <−1 in the sense that the corresponding solution map is notC². This is also a surprising result since the pure initial-value problem for the Burgers equation, namely

u_t+uu_x−u_xx=0, x∈ , t >0 u(x,0)=ψ (x), x∈ ,

(1.6) is known to be well-posed in the spaceH^s()fors−¹₂and is ill-posed inH^s()for anys <−¹₂ [1,25]. Molinet and Ribaud achieved their result by taking full advantage of the combination of the dispersion introduced through the term u_xxx and the dissipation inherent in the Burgers’ term−u_xx, though their analysis is informed by the work of Bourgain, and Kenig, Ponce and Vega. The corresponding solution map is real analytic whens >−1. In contrast, the approach of Kappeler and Topalov is completely different from those of Bourgain and Kenig, Ponce and Vega, being based on the classical inverse scattering transform. The corresponding solution map associated with the IVP (1.4) is continuous, but not locally uniformly continuous whens <−¹₂.

An implication of the work of Kappeler–Topalov and Molinet–Ribaud is that it is reasonable to conjecture the IBVP (1.1) is well-posed in the spaceH^s(⁺)for at least some range ofs <−³₄. Indeed, their works not only indicate that an affirmative answer to Question 1.3 is possible, but also suggest two possible approaches:

(a) seeking to use the dissipative mechanism inherent in imposing the boundary condition for (1.1) atx=0 in a way reminiscent of what Molinet and Ribaud did with the Burgers term in (1.6);

(b) using the inverse scattering methodology as did Kappeler and Topalov.

Recent work in the direction of an inverse scattering transform for (1.1) by Fokas, Its and Pelloni [33,32,31,30]

shows promise for its use in rigorous studies. In the present essay, we have elected approach (a), in part because results obtained by such considerations are likely to have more scope as they rely upon a less rigid structure. A theory based on the inverse scattering transform is being investigated separately.

There are two important auxiliary points arising in the present analysis that are worth singling out, as they have independent interest. One is an accurate appraisal of the damping implied by imposition of the boundary condition at x=0. The other is a relation between the IBVP’s for the KdV-equation and the KdV–Burgers equation. These points are explained now.

It is a simple matter to see that the imposition of the boundary conditionu(0, t )=h(t )in (1.1) results in dissipation.

Indeed, suppose for example thath(t )≡0 for allt0 and thatuis a suitably smooth solution of (1.1) which, along

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with its first few partial derivatives, decays rapidly to zero asx→ ∞. Multiplying the KdV-equation byu, integrating over the half-line⁺, and integrating by parts, leads to the formula

d dt

∞ 0

u(x, t )²dx= −1

2u²_x(0, t ) for allt0.

Thus the L²-norm of the solution is seen to be decreasing, and strictly so wheneveru_x(0, t )=0. This dissipative mechanism is somewhat more subtle than that induced by a Burgers-type term in the equation itself, or other forms of point dissipation. These have been studied and quantified in the works of Bona, Sun and Zhang [12] and Russell and Zhang [48]. In particular, it was shown in [12] that the IBVP for the linear problem

u_t+u_x+u_xxx=0, x, t >0, u(x,0)=0, u(0, t )=h(t ), x, t >0

(1.7) associated to (1.1) has the following smoothing property; its solutionu(x, t )is the restriction to⁺×⁺of a function w(x, t )defined on × which is such that

_∞

−∞

∞

−∞

1+ |ξ|2s

1+τ−ξ³^2bw(ξ, τ )ˆ ²dξ dτ 1/2

Ch

H

3b+s−1/2

3 (⁺) (1.8)

wherebis any value in[0,¹₂−₃^s)if−³₂s <1,b is any value in[0,⁵₆−^s₃]if−¹₂< s <1 andC is a constant depending only onsandb. As a direct consequence of this estimate, we derive the corollary that

h∈H

s+1 3

loc

⁺

implies that the solutionuof (1.7)belongs to the spaceL²

0, T;H^s⁺³² ⁺

.

This is a much stronger smoothing result than the well-known Kato smoothing property which only implies that u∈L²(0, T;H_loc^s⁺¹(⁺))(see [12]). This global boundary smoothing property is the key to our proof in [12] that the IBVP (1.1) is unconditionally well-posed inH^s(⁺)for anys >−³₄.

We also mention that it is not only the imposition of a boundary condition atx=0 that can result in smoothing of solutions of (1.1). Indeed, for both the pure initial-value problem (1.3) and the initial-boundary-value problem for the KdV-equation ((1.1) withμ=0), more rapid decay of the initial value asx→ +∞leads to an increase in the smoothness of the solution for t >0. This well-known fact (see, e.g. [23,24,26,27,38,46,49]) has, for example, been exploited by Faminskii [26,27] and Kruzhkov and Faminskii [46] in their study of well-posedness issues for the KdV-equation.

Now, attention is directed to a connection between the KdV equation and the KdV–Burgers equation. Letα, β∈ be given and consider the transformation

u(x, t )=e^αx⁺^βtv(x, t ).

A direct calculation shows thatuis a solution of the KdV equation

u_t+u_x+uu_x+u_xxx=0 (1.9)

if and only ifvis a solution of the equation v_t+

α+α³+β v+

3α²+1

v_x+v_xxx+3αvxx+e^βt⁺^αx

αv²+vv_x

=0. (1.10)

If we choose

α <0, β= −α−α³,

thenv(x, t )is a solution of the IBVP for the KdV–Burgers type equation

vt+(3α³+1)vx+e⁻^(α⁺^α³^)t⁺^αx(vvx+αv²)+vxxx+3αvxx=0, x, t >0,

v(x,0)=e⁻^αxφ(x), x0,

v(0, t )=e^(α⁺^α³^)th(t ), t0

⎫⎪

⎬

⎪⎭ (1.11)

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if and only if u(x, t )is a solution of (1.1). Thus, one is led to consider the IBVP for the variable-coefficient KdV–

Burgers equation posed on the half line⁺,viz.

v_t+ρv_x+γ vv_x+a₁(x)b₁(t )vv_x+a₂(x)b₂(t )v²+v_xxx−v_xx=0, x, t >0,

v(x,0)=ψ (x), x0,

v(0, t )=g(t ), t0

⎫⎪

⎬

⎪⎭ (1.12)

whereρ >0 andγ∈R are constants,a_j ∈H^∞[0,∞)andb_j ∈C^∞[0,∞)for j =1,2. Note that well-posedness of (1.12) in the spaceH^s(⁺), thes-compatibility of the initial valueψ and the boundary valueg, and the concept of mild solution of the IBVP (1.12) can be defined just as described in Definitions 1.1 and 1.2 for the IBVP (1.1).

Using the approach developed in our earlier paper [12,56], we can extend Molinet and Ribaud’s work [47] on the pure initial-value problem (1.5) to the IBVP (1.12) as follows.

Theorem 1.5(local well-posedness). Lets >−1 be given withs=3m+¹₂, m=0,1,2, . . .. For anyr >0, there exists a T =T (r) >0 such that if(ψ, g)∈H^s(⁺)×H^s⁺³¹(0, T )iss-compatible with respect to the equation in (1.12)and satisfies

(ψ, g)

H^s(⁺)×H^s⁺³¹(0,T )< r,

then the IBVP(1.12)admits a unique solutionu∈C([0, T];H^s(⁺)). Moreover, the corresponding solution map is (real)analytic from the spaceH^s(⁺)×H^s⁺³¹(0, T )to the spaceC([0, T];H^s(⁺)).

Theorem 1.5 is a local result since the asserted life span(0, T )of the solution depends on the sizer of its initial and boundary data. As in [8], the following global well-posedness results obtain for the IBVP (1.12).

Theorem 1.6(global well-posedness). Assume that the system(1.12)satisfies b₂(t )a₂(x)−1

3b₁(t )a₁(x)≡0 x, t∈(0,+∞). (1.13)

(i) Lets0andT >0be given withs=3m+¹₂,m=0,1,2, . . .. For anys-compatible pair (ψ, g)∈H^s

⁺

×H^s⁺³¹⁺^η(s)(0, T ) where

η(s)=

>0, if0s <3, 0, ifs3,

the IBVP(1.12)admits a unique solutionu∈C([0, T];H^s(⁺)). Moreover, the corresponding solution map is (real)analytic from the spaceH^s(⁺)×H^s+³¹⁺^η(s)(0, T )to the spaceC([0, T];H^s(⁺)).

(ii) Lets∈(−1,0)andT >0be given. For any(ψ, g)∈H^s(⁺)×H¹³⁺(0, T ), the IBVP(1.12)admits a unique solutionu∈C([0, T];H^s(⁺))∩C((0, T];L²(⁺)). Moreover, ifg∈H^∞(0, T ), thenu∈C((0, T];H^∞(⁺)).

Introduce some natural weighted Sobolev-spaces following Kato [39]. For given ν >0 ands∈ , letH_ν^s(⁺) denote the Hilbert space

H_ν^s ⁺

≡

f ∈H^s ⁺

; e^νxf∈H^s ⁺ with the norm

fH_ν^s⁺≡

f²_Hs(⁺)+e^νxf²

H^s(⁺)

¹

2.

The following local well-posedness result for the IBVP (1.1) then follows as a corollary to Theorem 1.5.

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Theorem 1.7(local well-posedness). Letν >0ands >−1be given withs=3m+¹₂,m=0,1,2, . . .. For anyr >0, there exists aT =T (r) >0such that if ans-compatible pair

(φ, h)∈H_ν^s ⁺

×H^s⁺³¹(0, T ) satisfying

(φ, h)

H_ν^s(⁺)×H^s+1³ (0,T )< r

is posed as auxiliary data, then the IBVP(1.1)admits a unique solutionu∈C([0, T];H_ν^s(⁺)). Moreover, the corre- sponding solution map is(real)analytic fromH_ν^s(⁺)×H^s⁺³¹(0, T )toC([0, T];H_ν^s(⁺)).

In addition, the following global well-posedness result holds for the IBVP (1.1).

Theorem 1.8(global well-posedness).

(i) Lets0andT >0be given withs=3m+¹₂,m=0,1,2, . . .. For any(φ, h)∈H_ν^s(⁺)×H^s⁺³¹⁺^η(s)(0, T ), the IBVP(1.1)admits a unique solution

u∈C

[0, T];H_ν^s ⁺

.

Moreover, the corresponding solution map is(real)analytic from the spaceH_ν^s(⁺)×H^s+1³ ⁺^η(s)(0, T )to the spaceC([0, T];H_ν^s(⁺)).

(ii) Lets∈(−1,0),ν >0andT >0be given. For >0and any(φ, h)∈H_ν^s(⁺)×H¹³⁺(0, T ), the IBVP(1.1) admits a unique solution

u∈C

[0, T];H_ν^s ⁺

∩C

(0, T];L²_ν ⁺

. Moreover, ifh∈H^∞(0, T ), thenu∈C((0, T];H_ν^∞(⁺)).

Note that Theorem 1.8 does not follow directly from Theorem 1.6 since system (1.11) does not satisfy assump- tion (1.13). However, (1.11) can be rewritten in the form

vt+(3α³+1)vx+¹₂αuv+¹₂(uv)x+vxxx+3αvxx=0, x, t >0,

v(x,0)=e⁻^αxφ(x), x0,

v(0, t )=e^(α⁺^α³^)th(t ), t0

⎫⎪

⎬

⎪⎭ (1.14)

whereu=u(x, t ) is the solution of (1.1), which is known (cf. [12,29]) to exist globally in the space H^s(⁺)if (φ, h)∈H^s(⁺)×H

s+1 3 +η()

loc (⁺)iss-compatible with respect to the system (1.1). Since the equation in (1.14) is a linear equation with a variable coefficientu=u(x, t )that exists globally in the spaceH^s(⁺), the solutionvof (1.14) exists globally in the spaceH^s(⁺), from which Theorem 1.8 follows.

The paper is organized as follows. In Section 2, explicit representation formulas for solutions of initial-boundary- value problems for the linear KdV–Burgers equation are presented. These are developed along the same lines as those in our earlier paper [8]. Various estimates will be established for the linear problems associated to (1.1) and (1.12) and these play a central role in the analysis of the non-linear problems. In Section 3, the well-posedness results for the IBVP (1.1) and (1.12) as described in Theorems 1.5–1.8 are established. The last section is an Appendix A containing explanations of some technical lemmas that are central to the analysis in Sections 2 and 3.

2. Linear problems

This section is divided into two subsections. In the first, consideration is given to linear problems associated to the KdV–Burgers equation. As mentioned above, explicit representation formulas for solutions of initial-boundary-value problems for this equation will be derived. Then, the boundary integral operator that arises in the solution formulas will be extended from the quarter plane⁺× ⁺to the whole plane × using the approach developed in [12]. The extended boundary integral operator will play a crucial role in our analysis. The second subsection containsa priori estimates of norms of solutions of the linear problems and of norms of the boundary integral operators.

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2.1. Solution formulas for linear problems

Consider first the non-homogeneous problem u_t+ρu_x−u_xx+u_xxx=0, forx, t0, u(x,0)=0, u(0, t )=h(t ),

(2.1) whereρ0 is a given constant.

Proposition 2.1.The solution of (2.1)may be written in the form u(x, t )= W_bdr(t )h

(x)= U_b(t )h

(x)+ U_b(t )h

(x) (2.2)

where, forx, t0, Ub(t )h

(x)= 1 2π

∞

√ρ

e^{it (μ}³⁻^ρμ)e^(α(μ)⁺^iβ(μ))x

3μ²−ρ^∞

0

e⁻^iξ(μ³⁻^ρμ)h(ξ ) dξ dμ.

Here, bothα(μ)andβ(μ)are real-valued functions withα(μ) <0for allμ∈ ⁺and α(μ)∼ −

3μ²−4ρ

2 , β(μ)∼ −μ

2 (2.3)

asμ→ +∞.

Proof. Letu˜andh˜ denote the Laplace transform ofuandhwith respect tot, respectively. By applying the Laplace transform to both sides of the equation in (2.1), one obtains

λu(x, λ)˜ +ρu˜_x(x, λ)+ ˜u_xxx(x, λ)− ˜u_xx(x, λ)=0, u(0, λ)˜ = ˜h(λ).

As bothu(x, λ)˜ andu˜_x(x, λ)tend to zero asx→ +∞, it is concluded that for anyλwith Reλ >0,

˜

u(x, λ)= ˜h(λ)e^r¹^(λ)x

wherer₁(λ)is the unique solution of λ+r³+ρr−r²=0

satisfying Rer₁(λ) <0. Thus, for any fixedγ >0, one has the representation u(x, t )= 1

2π i

i∞+ γ

−i∞+γ

e^λth(λ)e˜ ^r¹^(λ)xdλ.

Using the fact that the right-hand side of this relation is continuous inγup toγ=0, there obtains the simpler looking formula

u(x, t )= 1 2π i

i∞

0

e^λth(λ)e˜ ^r¹^(λ)xdλ+ 1 2π i

0

−i∞

e^λth(λ)e˜ ^r¹^(λ)xdλ.

For eachλon the positive imaginary axis, writeλin the formλ=i(μ³−ρμ)for the unique value ofμin the interval

√ρμ <+∞. In terms ofμ, the quantityr1(λ)has the form r₁(λ)=α(μ)+iβ(μ)

withα(μ) <0 and α(μ)∼ −

3μ²−4ρ

2 , β(μ)∼ −μ

2

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asμ→ +∞. Recalling that h(λ)˜ =

∞ 0

e⁻^λth(t ) dt,

a change of variables and straightforward calculation then reveals that forx, t0, 1

2π i

i∞

0

e^λth(λ)e˜ ^r¹^(λ)xdλ= Ub(t )h (x)

and, similarly, that 1

2π i 0

−i∞

e^λth(λ)e˜ ^r¹^(λ)xdλ= Ub(t )h (x),

thus completing the proof. 2

Next, consider the same linear problem posed with zero boundary condition, but non-trivial initial data,viz.

ut+ρux−uxx+uxxx=0, forx, t0, u(x,0)=φ(x), u(0, t )=0.

(2.4) By semigroup theory, its solution may be obtained in the form

u(t )=W_c(t )φ, (2.5)

where the spatial variable is suppressed andW_c(t )is theC₀-semigroup in the spaceL²(⁺)generated by the operator Af = −f−ρf+f

with the domain D(A)=

f ∈H³ ⁺

|f (0)=0 .

Moreover, by Duhamel’s formula, one may use the semigroupW_c(t )to formally write the solution of the forced linear problem with zero auxiliary data,

u_t+ρu_x−u_xx+u_xxx=f, forx, t0, u(x,0)=0, u(0, t )=0,

(2.6) in the form

u(t )= t 0

W_c(t−τ )f (·, τ ) dτ. (2.7)

Recall the explicit solution formula for the pure initial-value problem for the linear KdV–Burgers equation posed on the whole line,viz.

u_t+ρu_x−u_xx+u_xxx=0, x, t∈ ,

u(x,0)=φ(x), x∈ ,

(2.8) namely

u(x, t )=W(t )φ(x)= 1 2π

∞

−∞

e^i(ξ³⁻^{ρξ )t}⁻^ξ²^te^ixξ ∞

−∞

e⁻^iyξφ(y) dy dξ, (2.9)

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obtained by taking the Fourier transform in the spatial variablex. As the formula forW(t )is explicit and simple, it is advantageous to give a representation ofW_c(t )in terms ofW(t )with a correction that involvesWbdr(t ). Because of the Burgers-type term, notice that ifφ∈H^s()for anys∈ , thenu(x, t )is anH^∞()-function ofxfor anyt >0, and so isC^∞in the domain{(x, t ): x∈ , t >0}.Hence, the trace ofuatx=0, say, is a well-defined function.

Let a functionφbe defined on the half line⁺and letφ^∗be an extension ofφto the whole line. The mapping φ→φ^∗can be organized so that it defines a bounded linear operatorBfromH^s(⁺)toH^s(). Henceforth,φ^∗=Bφ will refer to the result of such an extension operator applied toφ∈H^s(⁺). Assume thatv=v(x, t )is the solution of

v_t+ρv_x−v_xx+v_xxx=0, v(x,0)=φ^∗(x),

for x ∈ , t 0. If g(t )=v(0, t ), then vg =vg(x, t )=Wbdr(t )g is the corresponding solution of the non- homogeneous boundary-value problem (2.1) with boundary conditionh(t )=g(t )fort0. It is clear that forx >0, the functionv(x, t )−v_g(x, t )solves the IBVP (2.4), and this leads directly to a representation of the semigroupW_c(t ) in terms ofWbdr(t )andW(t ).

Proposition 2.2.For a givensandφ∈H^s(⁺)withφ(0)=0, ifs > ¹₂, ifφ^∗is its extension toas described above, thenW_c(t )φmay be written in the form

W_c(t )φ=W(t )φ^∗−Wbdr(t )g (2.10)

for anyx, t >0, wheregis the trace ofW(t )φ^∗atx=0.

In a similar manner, one may derive an alternative representation for solutions of the inhomogeneous IBVP (2.6).

Proposition 2.3.Iff^∗(·, t )=Bf (·, t )is an extension off from⁺× ⁺to × ⁺, say, then the solutionuof(2.6) may be written in the form

u(·, t )= t 0

W(t−τ )f^∗(·, τ ) dτ−Wbdr(t )v for anyx, t0wherev≡v(t )is the trace oft

0W(t−τ )f^∗((·, τ ) dτ atx=0.

The solution formulas in Propositions 2.2 and 2.3 hold only forx >0 andt >0. It will be convenient to extend them in such a way that they hold for allx, t∈ . This will provide a context in which to establish the well-posedness of the non-linear problem in the framework of Bourgain spaces. Note that the termW(t )can be redefined to be

W(t )φ= 1 2π

∞

−∞

e^i(ξ³⁻^{ρξ )t}⁻^ξ²^|^t^|e^ixξ ∞

−∞

e⁻^iyξφ(y) dy dξ

for allx, t∈ without disturbing the validity of (2.10) in⁺× ⁺. Thus it is only necessary to extend the second term in both formulas from⁺× ⁺to × to effect an extension ofW_c.

Presented now are two different types of extensions of the boundary integral operatorWbdr(t ). To begin, rewrite Wbdr(t )in the form

Wbdr(t )h

(x)= 1 2πRe

∞

√ρ

e^{it (μ}³⁻^ρμ)e^(α(μ)⁺^iβ(μ))x

3μ²−ρ^∞

0

e⁻^i(μ³⁻^ρμ)ξh(ξ ) dξ dμ

= 1 2πRe

4√ρ

√ρ

eîμ³^t⁻îρμte^(α(μ)⁺îβ(μ))φ³^(x)

3μ²−ρ φ₁(μ)

∞ 0

+ 1 2πRe

∞

2√ρ

√3

eîμ³^t⁻îρμte^(α(μ)⁺îβ(μ))x

3μ²−ρ φ₂(μ)

∞ 0

:= 1 2π

I₁(x, t )+I₂(x, t )

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whereφ₁(μ)andφ₂(μ)are non-negative cut-off functions satisfying φ1(μ)+φ2(μ)=1 for allμ∈ ⁺

with suppφ₁⊂(−1,4√

ρ ), suppφ₂⊂(3√ρ,∞)andφ₃(x)is a smooth function onsuch that φ₃(x)=

x forx0, 0 forx−1.

The integralI₁(x, t )is naturally defined for all values ofx andt and, viewed as a function defined on × , is in factC^∞-smooth there, with all its derivatives decreasing rapidly asx→ ±∞. Thus, no complicated extension ofI1

is required as the obvious one suffices. It is otherwise forI2. To discussI2(x, t ), it is convenient to letμ(λ)denote the positive solution of the cubic equation

μ³−ρμ=λ forλ0 andμ√

ρ, whileμ(λ)= −μ(−λ)forλ <0. By a change of variables, the integralI2can be rewritten in the form

I₂(x, t )=Re ∞

2√ρ 3√

3

∞ 0

eîλte^(α^μ^(λ)⁺îβ^μ^(λ))xe⁻îλsφ₂ μ(λ)

h(s) ds dλ

= ∞

2√ρ 3√

3

∞ 0

e^α^μ^(λ)xcos

λ(t−s)+β_μ(λ)x φ₂

μ(λ)

h(s) ds dλ

:=E(x, t )

forx0 withαμ(λ):=α(μ(λ))andβμ(λ):=β(μ(λ)). Denote the extension ofE(x, t )tox <0 byg(x, t )so that I2(x, t )=

E(x, t ), x0, g(x, t ), x <0,

whereg(x, t )is to be defined. As in [12], the Fourier transform ofI₂may be decomposed as follows:

Fx,t[I2] =Ft

_∞

0

E(x, t )cos(xξ )+g(−x, t )cos(xξ ) dx

+ i π

∞

−∞

1 ξ−ηFt

_∞

0

cos(ηx)E(x, t ) dx− ∞ 0

cos(ηx)g(−x, t ) dx

dη.

Note that different choices of g(x, t ) give different extensions ofE(x, t )tox <0. The following two choices of g(x, t )will be used in this article.

(i) Forx >0, chooseg(−x, t )such that Ft

∞ 0

g(−x, t )cos(xξ ) dx

(τ )= −Ft

∞ 0

E(x, t )cos(xξ )

(τ )Θ(ξ, τ )

+Ft

_∞

0

E(x, t )cos(xξ ) dx

(τ )

1−Θ(ξ, τ )

ν(ξ )ω(τ ) (2.11)

whereΘ(ξ, τ )=χ (|ξ| −δ|τ|^1/3)withδ >0 fixed, 0χ (ξ )1 everywhere, χ (ξ )=

1, ξ <0, 0, ξ >0,

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whilst ν(ξ )=

1 if|ξ|1, 0 if|ξ|<1,

andω(τ )is a smooth and bounded function to be specified momentarily. It is easy to see that such agis a combination of even and odd extensions,viz.

Fx,t[I2] := ˆI21(ξ, τ )+ ˆI22(ξ, τ ) where

Iˆ₂₁(ξ, τ )=Ft

∞ 0

(τ )

1−Θ(ξ, τ )

1+ν(ξ )ω(τ ) and

Iˆ₂₂(ξ, τ )= i π

∞

−∞

1 ξ−ηFt

∞ 0

E(x, t )cos(xη) dx

(τ )

2Θ(η, τ )+

1−Θ(η, τ )

1+ν(ξ )ω(τ ) dη

= i π

∞ 0

1

ξ−η+ 1 ξ+η

Ft

_∞

0

(τ )

×

2Θ(η, τ )+

1−Θ(η, τ )

1+ν(ξ )ω(τ ) dη.

Because of the algebraic identity 1

ξ−η+ 1 ξ+η=2

ξ

1+ η² ξ²−η²

,

we may writeIˆ₂₂(ξ, τ )as Iˆ₂₂(ξ, τ )= 2i

π ξ ∞ 0

Ft

∞ 0

(τ )

2Θ(η, τ )+

1−Θ(η, τ )

1+ν(ξ )ω(τ ) dη

+ 2i π ξ

∞ 0

(η/ξ )² 1−(η/ξ )²Ft

∞ 0

(τ )

×

2Θ(η, τ )+

1−Θ(η, τ )

1+ν(ξ )ω(τ ) dη :=Q₁(ξ, τ )+Q₂(ξ, τ ).

Here, the Fourier transformFt[_∞

0 E(x, t )cos(xη) dx]may be expressed in the form Ft

_∞

0

(τ )= 4 m=1

Km1(η, τ )φ2

μ(τ )h(τ )ˆ + 4 m=1

Km2(η,−τ )φ2

μ(−τ )h(ˆ −τ ) (2.12)

where h(τ )ˆ =

∞ 0

e⁻^{iτ s}h(s) ds,

K₁₁(η, λ)= −α_μ(λ)

2(α²_μ(λ)+(η+β_μ(λ))²), K₂₁(η, λ)= −α_μ(λ)

2(α²_μ(λ)+(η−βμ(λ)²),

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K31(η, λ)= α²_μ(λ)β_μ(λ)i

(α_μ²(λ)+(η+β_μ(λ))²)(α_μ²(λ)+(η−β_μ(λ))²), K₄₁(η, λ)= (β_μ²(λ)−η²)βμ(λ)i

(α_μ²(λ)+(η+βμ(λ))²)(α_μ²(λ)+(η−βμ(λ))²) and K₁₂(η, λ)=K₁₁(η, λ), K₂₂(η, λ)=K₂₁(η, λ),

K32(η, λ)= −K31(η, λ), K42(η, λ)= −K41(η, λ).

A straightforward calculation reveals that

∞ 0

4 m=1

K_m1(η, τ )

1−Θ(η, τ ) dη

C₂>0,

for a fixed constantC₂independent ofτ, and so one can choose aC^∞smooth functionω(τ )(see [12]) such that for allτ,

Q1(ξ, τ )≡0, for all|ξ|1.

Hence, for|ξ|>1,

Iˆ₂₂(ξ, τ )=Q₂(ξ, τ ).

Thus, when|ξ|>1 andτ0, Iˆ₂₂(ξ, τ )=2iC2

ξ ∞

0

η² ξ²−η²

₄

m=1

K_m1(η, τ )φ₂

μ(τ )h(τ )ˆ

2Θ(η, τ )+

1−Θ(η, τ )

1+ω(τ ) dη,

whereas

Iˆ22(ξ, τ )=2iC2

ξ ∞

0

η² ξ²−η²

₄

m=1

Km1(η,−τ )φ2

μ(−τ )h(τ )ˆ

2Θ(η, τ )+

1−Θ(η, τ )

1+ω(τ ) dη

when |ξ|1 and τ <0. The boundary integral operator corresponding to this extension of Wbdr(t ) is denoted byBI1(t ).

(ii) Chooseg(−x, t )such that Ft

_∞

0

g(−x, t )cos(xξ ) dx

(τ )= −Ft

_∞

0

E(x, t )cos(xξ )

(τ )

1−Θ(ξ, τ )

+Ft

∞ 0

(τ )Θ(ξ, τ )ν(ξ )ω(τ ) (2.13)

whereΘis as before. In this case, we have Fx,t[I₂] := ˆI₂₁^∗(ξ, τ )+ ˆI₂₂^∗(ξ, τ ) with

Iˆ₂₁^∗(ξ, τ )=Ft

∞ 0

Θ(ξ, τ )

1+ν(ξ )ω(τ ) and