Boundary Smoothing Properties of the Korteweg-de Vries Equation in a Quarter Plane and Applications

Transcription

1 Dynamics of PDE, Vol., No., -69, 6 Boundary Smoothing Properties of the Korteweg-de Vries Equation in a Quarter Plane and Applications Jerry L. Bona, S. M. Sun, and Bing-Yu Zhang Communicated by Jerry Bona, received September 4, Mathematics Subject Classification. Primary 5; Secondary 76. Key words and phrases. KdV, boundary smoothing, well-posedness. JLB was partially supported by the National Science Foundation and by the W. M. Keck Foundation. SMS was partially supported by the National Science Foundation. BYZ was partially supported by the Taft Memorial Fund. c 6 International Press

2 Abstract. Reported here are results concerning the initial boundary value problem IBVP for the Korteweg-de Vries equation in a quarter plane, viz. 8 < u t + u x + uu + u xxx =, for x, t,. : ux, = φx, u, t = ht. The present study commences with a representation of solutions of. derived in our earlier paper [Trans. American Math. Soc. 54, 47 49]. The problem. arises naturally in the modeling of various types of wave phenomena, but the focus here will be on two mathematical points, namely a type of boundary smoothing and its impact upon the well-posedness of. in the L based Sobolev spaces H s R +. It has been known for some time that the KdV equation posed on the quarter plane possesses the Kato smoothing property just as do solutions on the whole plane of the pure initial value problem; that is to say s+ φ H s R + and h Hloc R+ implies u L, T ; H s+ loc R+ for any finite value of T for which the solution exists on [, T ]. It is shown here that the linear IBVP obtained by dropping the nonlinear term uu x in. has the following somewhat startling smoothing property: s+ if φ = and h Hloc R+, then the solution u of the linear version of. belongs to the space L, T ; H s+ R +. The linearized version of. with zero initial data, φ =, has another interesting property. The solution ux, t is the restriction to R + R + of a function wx, t defined on R R which is such that Z Z / + ξ s + τ ξ b ŵξ, τ dξdτ«c h H b+s / R + where b is any value in [, s if s <, b is any value in [, 5 6 s ] if < s < and C is a constant depending only on s and b. Aided by these boundary integral estimates, and after introduction of suitable versions of the Bourgain spaces whose underlying spatial-temporal domain is a quarter plane, we demonstrate that the full nonlinear IBVP. is unconditionally locally well-posed in the space H s R + for any s > 4. More precisely, it is shown that for a given compatible pair φ, h H s R + Hloc R+, there exists a T > such that the IBVP. admits a unique mild solution u C[, T ], H s R +, which depends continuously on the initial value φ and the boundary value h. Moreover, the IBVP. is shown to be unconditionally globally well- s+ s+ posed in H s R + Hloc R+ for s, while unconditional global wellposedness is shown to hold for s < in H s R + Hloc ɛ >. +s+ɛ R + for any Contents. Introduction. Linear Problems and Extension Formulas. Boundary smoothing properties 4. Well-posedness 4 5. Appendices 5 References 67

3 BOUNDARY SMOOTHING PROPERTIES. Introduction In this paper, we continue the study of the initial-boundary-value problem for the Korteweg-de Vries KdV equation posed in a quarter plane, namely u t + u x + uu x + u xxx =, for x, t,. ux, = φx, u, t = ht. As pointed out by several authors, see [] for an early commentary in the context of the BBM-equation, initial-boundary-value problems of the form. may serve as models for waves generated by a wave maker in a channel, or for waves approaching shallow water e.g. the shore from deep water. Similar problems arise in other physical contexts where KdV-type equations serve as models. Here, two mathematical issues connected to. will be addressed; boundary smoothing properties and the well-posedness of this initial-boundary value problem IBVP henceforth in the L based Sobolev spaces H s R +. The overall thrust of our theory is that stronger boundary smoothing properties than heretofore noticed allow the formulation of a sharper well-posedness theory. We begin with a review of existing theory which provides a setting in which to state precisely our results and put them into present day context. Recall that for the pure initial-value problem IVP henceforth for the KdV-equation. u t + uu x + u xxx =, ux, = φx, x, t R, written in traveling coordinates, it is well-known that there is no gain or loss of regularity in the Sobolev classes H s R. As Saut and Temam [8] pointed out, for any t R, u, t H s R if and only if φ H s R, at least for suitable values of s. There is, however, more subtle smoothing associated with the initial-value problem.. In the late 97 s, Kato [8, 9] discovered that for solutions of.,. φ H s R implies that u L, T ; H s+ loc R. This property, now known as Kato-smoothing, stimulated an extensive investigation of various smoothing properties associated with solving the KdV-equation and other dispersive wave equations see, for example, [,, 4,,,, 4, 9, 4] and the references contained therein. In particular, Kenig, Ponce and Vega [] demonstrated that, when φ H s R with s > 4, there is a unique solution u of. which belongs to the space CR; H s R and is such that.4 T sup <x<+ T x s+ ux, t dt C φ H s R,.5 and.6 T T + sup x u, t 4 dt <x<+ sup ux, t dx T t T 4 C φ Hs R C + T φ H s R

4 4 JERRY L. BONA, S. M. SUN, AND BING-YU ZHANG where the constants C on the right-hand sides depend only on s and on T when it appears. The inequality.4 is a sharp version of Kato smoothing and.5 is sometimes called global smoothing of Strichartz type, while.6 reveals a kind of global temporal smoothing. In the early 99 s, in attempting to establish the well-posedness of. in H s R for smaller values of s, Bourgain [] found a yet more subtle smoothing property for solutions of.. This property may be expressed as follows: for φ H s R with s,. admits a solution u C[, T ]; H s R which is the restriction to R, T of a function w on the whole plane R R such that /.7 + ξ s + τ ξ ŵξ, τ dξdτ C φ H s R where C depends only on s and ŵ is the Fourier transform of w with respect to both of the independent variables. Because of this smoothing property of solutions, Bourgain could show that. is conditionally well-posed in H s R for s. The distinction between well-posedness and conditional well-posedness will be drawn presently. Later, the smoothing property.7 was improved by Kenig, Ponce and Vega [, 4] to the stronger property.8 / N s,b w + ξ s + τ ξ b ŵξ, τ dξdτ C φ H s R for any φ H s R with s > 4, where < b < depends only on s and C depends on s and b. Note that if s R and U is the unitary group in H s R defined by Ut = exp itp D x, where P D x is the Fourier multiplier with symbol P ξ = ξ, then cf. [6] N s,b w U tw H s,b where H s,b Ht b R; HxR. s For given f H s,b, the larger the value of b, the smoother is f with respect to time t. In particular, if b >, Hs,b is continuously embedded into the space CR; HxR. s The inequality.8 allowed Kenig, Ponce and Vega to show that, locally in time, the IVP. is conditionally well-posed in H s R provided only that s > 4. This result was recently strengthened to include conditional global well-posedness in the same function classes, by Colliander, et al. in []. For the IBVP., the Kato smoothing property for. was established by Bona and Winther [8, 9], where they showed that solutions lie in L, T ; H n+ loc R+ if φ H n R + and h H n+ loc R+ for n. Smoothing properties analogous to.4-.6 were established by Bona, Sun and Zhang. These were derived in [4] in the following form. For s > 4, if φ Hs R + and h H s+ loc R+ satisfy certain compatibility conditions at x, t =,, then the IBVP. admits a unique solution u C, T ; H s R + L, T ; H s+ loc R+, which satisfies the additional properties.9 sup <x<+ T x s+ ux, t dt C φ H s R + + h H s+,,t

5 . and. T + BOUNDARY SMOOTHING PROPERTIES 5 sup x u, t 4 dt <x<+ sup ux, t dx t T 4 C φ H s R + h H s+,t C φ Hs R + h H s+,t where the constants depend only on s and T. As just stated, the results in [4] were local in time; corresponding global results were also established but were only optimal if s. Global results very nearly corresponding to the local theory in [4] are obtained in the recent paper [9] of Faminskii. Recently, Colliander and Kenig [] in a paper concerned with the IBVP for the generalized KdV-equation wherein uu x is replaced by u p u x, showed in the case p = that solutions of. also possess a Bourgain smoothing property which can be expressed precisely as follows. For φ L R + and h H R +,. admits a solution u C, T ; L R + which is the restriction of a function wx, t defined on the whole plane satisfying. Λα,b w C φ H s R + h H R + where Λ α s,b w = N s,b w + + τ α ŵξ, τ dξdτ with α > and where the constant b is required to be strictly less than in contrast to the theory for the IVP., where b > obtains. The discussion is now turned more directly to the contributions in the present essay. We commence with boundary smoothing properties. Note first that if the boundary value h in. vanishes identically, the solution u satisfies the energy identity d dt u x, tdx + u x, t = for all t. Thus, the L norm of the solution u is decreasing and is strictly so as long as u x, t. This suggests that some dissipative mechanism is introduced through imposition of the boundary condition at x =. An interesting question arises naturally in this situation: Can one quantify this boundary dissipative effect? It is well-known that a solution often becomes smoother under the influence of dissipative effects. Thus a further question presents itself: effect? Do solutions of. becomes smoother because of this boundary dissipative To address these issues, it is helpful to consider carefully the linear problem u t + u x + u xxx =, x, t R +,. ux, =, u, t = ht, x, t R +

6 6 JERRY L. BONA, S. M. SUN, AND BING-YU ZHANG associated to. and present some new boundary smoothing properties for its solutions. The Kato-smoothing result of [8], when extended to fractional-order spaces, states that the solution u of. belongs to the space CR + ; H s R + L R + ; H s+ loc R+ if h H s+ R +, at least if s is not too small. The space H rr+ is the closure of DR + in H r R +, as usual. It will be demonstrated in this paper that. possesses the following additional boundary smoothing properties. Theorem.. For a given pair b, s satisfying b < s if s, or.4 b < 5 6 s if < s <, there exists a constant C depending only on s and b such that for any h H s+ R +, the corresponding solution u of. is the restriction of a function wx, t defined on the whole plane satisfying.5 N s,b w C h H b+s / R +. Remark: Notice that this improves upon the just described result of Colliander and Kenig [] both as regards the range of b solutions are seen to be smoother in t and by allowing for negative values of s. As a corollary, there appears the following boundary smoothing property for solutions of the IBVP.. Theorem.. Let s and T > be given. There exists a constant C such that for any h H s+ R +, the corresponding solution u of. belongs to the space L, T ; H s+ R + and satisfies.6 u L,T ;H s+ R + C h H s+ R + for a constant C depending only on s and T. Remarks: i The smoothing property presented by.6 is global in the spatial variable x. ii This smoothing property only holds for.; it is not valid for the linear IBVP associated to. nor for the nonlinear problem.. iii For any T > and ɛ >, the following estimates were established by Faminskii [8] for the solution u of.;.7 u C,T ;L R + CT, ɛ h L 6+ɛ,T,.8 u x x, L t,t ;L x R+ C h H +ɛ R + and.9 u x LR + R + C h H 6 R +.

7 BOUNDARY SMOOTHING PROPERTIES 7 iv As a direct consequence of estimate.6, we have. u x x, L t,t ;L x R+ C h H +ɛ R + which is slightly stronger than.8. v The estimate.8 plays a key role in establishing sharper global wellposedness results for. in [9]. In addition, two improved versions of Bourgain smoothing are developed here for the nonlinear IBVP.. These take the following form. Theorem.. For given s in the interval 4 < s, there exists a constant b, ] depending on s such that for φ H s R + and h H b+s / R +, with φ = h in case s >, the IBVP. admits a solution u C, T ; H s R + which is the restriction to the domain R +, T of a function wx, t, defined on the whole plane, satisfying. Λα s,b w C φ H s R + h H b+s / R + for some α > s+. In particular, if h H R +, then. Λα s, w C φ H s R + h H s+. R + Remarks: i In case s =,. is a slightly stronger version of the estimate. due to Colliander and Kenig [] in that it allows b = instead of asking that b be strictly less than. ii As pointed out earlier, one needs that φ H s R + and h H s+ loc R+ to have the solution u of the IBVP. belonging to the space C, T ; H s R +. However, when b <, b + s / < + s. Estimate. thus reveals a boundary smoothing property for the nonlinear problem.. The second main issue addressed in this paper is the well-posedness of the IBVP.. Here and above, well-posedness means existence and uniqueness of solutions, and continuous dependence of solutions on auxiliary data. The following definition encapsulates the precise sense of well posedness enforced here. Definition.4 well-posedness. Let s, s R be given. The IBVP. is said to be locally well-posed in the space H s R + Hloc s R+ if for any r > there exists a T = T r > with T r as r such that for given φ H s R + and h Hloc s R+ satisfying suitable compatibility conditions, and if φ Hs R + + h H s,t r, then. admits a unique solution u = ux, t in the space C, T ; H s R +. Moreover, the solution depends continuously on its initial data φ and its boundary value h in the corresponding spaces. Remarks:

8 8 JERRY L. BONA, S. M. SUN, AND BING-YU ZHANG i The well-posedness described above is called local well-posedness since the T in the above definition may depend on r. If T may be taken to be independent of r, then. is said to be globally well-posed in the space H s R + Hloc s R+. ii By a standard scaling argument, the above definition of well-posedness is equivalent to the following statement: There exists a δ depending only on s and s, such that for given φ H s R + and h Hloc s R+ satisfying suitable compatibility conditions and having φ Hs R + + h H s, δ, then. admits a unique solution u = ux, t in the space C, ; H s R +. Moreover, the solution depends continuously on its initial data φ and its boundary value h in the corresponding spaces. iii There is a weaker notion discussed by Kato [] of conditional wellposedness in which solutions are only known to be unique if they satisfy additional auxiliary conditions. Solutions satisfying such conditions are often available via the contraction mapping principle applied to an associated integral equation, but they are not necessarily known to be unique in the broader class not respecting the extra conditions. This point will be further elaborated presently. The mathematical study of the IBVP. began with the work of Ton [4] in which, existence and uniqueness were established assuming that the initial data φ is smooth and the boundary data h. The first well-posedness result in the strict sense of Definition.4 for the IBVP. was presented by Bona and Winther [8, 9]. Theorem A The IBVP. is globally well-posed in the space H k+ R + H k+ loc R+ for k =,,. Faminskii, in a wide-ranging paper [6], deals with the IBVP. for a generalization of the KdV-equation somewhat like that appearing later in Craig, Kappeler and Strauss [5]. He puts forward a theory of well-posedness for generalized solutions set in weighted H Sobolev classes. Moreover, he obtains extra interior regularity in case the initial data decays suitably rapidly at +. In [4], Bona, Sun and Zhang obtained the following conditional well-posedness result for.. Theorem B The IBVP. is locally well-posed in the space H s R + H s+ loc R+ for s > /4 with the following auxiliary condition to ensure uniqueness;. the solution u satisfies the estimates.9,. and.. Remarks: Notice that the last result reveals the relationship s = s + in the notation of the definition of well-posedness. This turns out to be the natural consequence of the balance t x. It was not noticed in the early attacks [8,9,6-9,4] on..

9 BOUNDARY SMOOTHING PROPERTIES 9 In Theorem A, solutions are in fact classical, which is to say all the terms in the equation are bounded and continuous functions of x, t and the equation is satisfied identically. In Theorem B, the solutions are distributional, but of course have the further regularity attached to lying in C, T ; H s R + and satisfying.9,. and.. The following result for. was established recently by Colliander and Kenig []. Theorem C For any φ H s R + and h H s+ R + with s which satisfy the compatibility condition φ = h if s >, there exists a T = T φ, h > and a solution u C, T ; H s R + of the IBVP.. The map φ, h u is Lipschitz-continuous from H s R + H s+ R + to C, T ; H s R +. This is not a well-posedness result in the sense of Definition.4, since uniqueness is not discussed. Actually, a well-posedness result is established for an integral equation.4 w = HS φ, h + IHS ww x posed on the whole plane R R, where HS φ, h is an integral operator associated to the linear homogeneous problem v t + v x + v xxx =, x >, t, T, wx, = φx, w, t = ht, x >, t, T and IHS f is an integral operator associated with the linear inhomogeneous problem v t + v x + v xxx = f, x >, t, T, wx, =, w, t =, x >, t, T. The precise definitions of the integral operators HS and IHS are given in []. The relation between.4 and the IBVP. without the linear transport term u x in the equation is that a solution w of.4 on R R, when restricted to the domain R +, T, is a solution of.. For the integral equation.4, Colliander and Kenig established the following well-posedness result. Theorem D Let s be given with s. There exists a δ > such that, if φ, h H s R + H s+ R + satisfies φ, h Hs R + H s+ R + δ and φ = h when s >, then the integral equation.4 admits a unique solution u CR; L xr satisfying the auxiliary condition Λ α s,bw < for some α > and b in the range < b < see the text following. above. The well-posedness of. presented in Theorem B and Theorem D is conditional rather than in the sense of Definition.4 since auxiliary conditions are needed to ensure the uniqueness. By contrast, the well-posedness of. presented in Theorem A is in the strict sense of Definition.4 and is unconditional. This result has been extended recently by J. Holmer [5] to the case 4 < s <.

10 JERRY L. BONA, S. M. SUN, AND BING-YU ZHANG The issue of conditional well-posedness also appears in the works of Bourgain, and Kenig, Ponce and Vega for the IVP. where the uniqueness is established only for solutions in the space C T, T ; H s R satisfying certain auxiliary conditions. A basic question is are these auxiliary conditions really essential to the uniqueness? The reader is referred to [7, ] for further discussion of unconditional and conditional well-posedness for general classes of nonlinear evolution equations. The issue is more interesting than might appear at first glance. There are many ways to transform the IBVP. into an integral equation. Most of these admit an analysis something like that made in [] leading to Theorem C. The question is, when two such solutions are restricted to R +, T for some T >, are they equal to each other? For the linear problem, this is established in [], but the point is unresolved for the nonlinear problem. One of the main theorems proved in this paper is the following well-posedness result for., which also resolves the uniqueness issue for the nonlinear problem just mentioned. Theorem.9. Let s 4, ] and T > be given. For any φ H s R + and h H s+, T satisfying the compatibility condition φ = h when s >, there exists a T > depending only on φ, h Hs R + H s+,t such that the IBVP. admits a unique solution u C, T ; H s R + which is the restriction to R + [, T ] of a function w = wx, t satisfying the estimate.5 Λα s, w < for some α >. Remarks: i The solution given by this Theorem is smoother than that given by Theorem D of Colliander and Kenig, and by Holmer since b = rather than being strictly less than. ii The theorem still holds if we replace by some b < in.5. Thus the solutions given by Colliander and Kenig in Theorem D and the solution provided by Theorem.9 are the same when restricted to R +. Theorem.9 is also a conditional well-posedness result. It is natural to speculate whether or not the auxiliary condition.5 is removable. A way of resolving this issue is to introduce a concept of mild solution for the IBVP.. Definition. mild solution. Let s < and T > be given. For given φ H s R + and h H s+ loc R+, a function u C, T ; H s R + is said to be a mild solution of. on the time interval [, T ] if there exists a sequence {u n } n= in the space with such that C, T ; H R + C, T ; L R + φ n x = u n x,, h n t = u n, t, n =,,,

11 BOUNDARY SMOOTHING PROPERTIES i u n solves the equation in. in L R + for < t < T, which is to say, each term in the equation lies in C, T ; L R + and the equation is satisfies for each t, almost everywhere in space; ii lim n sup t T u n, t u, t H s R + = ; iii lim n φ n φ H s R + = and lim n h n h H s+,t =. Remark: A mild solution is a weak solution when s, but not necessarily vice versa. However, a mild solution might not on the face of it be a distributional solution when s < since u may not be a well-defined distribution. Classical energy arguments demonstrate uniqueness of solutions if s >. Hence, while larger values of s can be encompassed by demanding the sequence {u n } n= be drawn from even smoother function classes, there is no need for this in the present context. We will show that the following facts hold about mild solutions. Theorem. existence and uniqueness. a The weak solutions given by Theorem B, Theorem C and Theorem.9 are all mild solutions. b For given φ H s R + and h H s+ loc R+ with s > 4, the IBVP. admits at most one mild solution. An immediate consequence of this theorem is that the auxiliary conditions in Theorem B, Theorem C and Theorem.9 are not essential for the uniqueness and all of them can be removed. If the appellation solution in Definition.4 is understood as mild solution, then we have the following unconditional well-posedness results as one of the main theorems in this paper. Theorem. unconditional well-posedness. The IBVP. is unconditionally locally well-posed in the space H s R + H s+ loc R+ for s > 4. Its solution u has the additional properties: u satisfies the estimates.9-. if s > 4 ; u satisfies the estimates. if 4 < s <. Remark: As a model of real wave phenomena, the Korteweg-de Vries equation is not derived to take account of singularity formulation. Thus, one would hope that global well-posedness results obtain for.. Indeed this was shown to be the case in H s R + H s+ loc R+ in [4] for s. For s, the results of [4] only yielded conditional global well-posedness in the slightly smaller space H s R + H 7+s loc R +. Faminskii s recent work [9] showed that. is conditionally globally well-posed in the space H s R + H +s+ɛ loc R + when s <, but leaves open the question of whether the well-posedness is unconditional or not. Concerning global well-posedness, the following result follows readily from the local theory in Theorem.. Theorem. global well-posedness. The IBVP. is unconditionally globally well-posed in the space H s R + H +s+ɛ loc R + for s < and is unconditionally globally well-posed in the space H s R + H +s loc R+ for s.

12 JERRY L. BONA, S. M. SUN, AND BING-YU ZHANG The paper is organized as follows. In Section, explicit representation formulas are recalled for solutions of initial-boundary-value problems for the linear KdV equation. These were developed in our earlier paper [4] and will be used to establish the main theorems of this paper. We will demonstrate in this section how to convert the IBVP. posed in a quarter plane to an integral equation posed on the whole plane, which sets the stage for using powerful tools developed by Bourgain, Kenig, Ponce and Vega and others to study the well-posedness of the IVP. Three different types of extension of the boundary integral operator associated to the non-homogeneous linear boundary value problem. are provided, which are denoted by BI e t, BI m t and BI m t, respectively see Section for the precise definitions of those operators. Among those operators, BI e t is the simplest; it is basically an even extension with respect to the spatial variable x of the boundary integral operator, from the half line R + to the whole line R. Using this operator, we are able to prove Theorem. in case s. If s <, one only has the estimate: N s,b w C h b H R + instead of the inequality.5 featured in Theorem.. Because of this state of affairs, the more complicated boundary operators BI m t and BI m t are introduced. The operator BI m t is helpful when s <, whereas BI m t is effective when s <. In Section, attention is given to the non-homogeneous linear boundary value problem.. It is demonstrated there that the aforementioned global boundary smoothing properties obtain, as expounded in Theorem. and Theorem.. The boundary integral operators BI m t and BI m t play a crucial role in the analysis. In Section 4, it is shown that the IBVP. is locally unconditionally wellposed in the space H s R + H s+ loc R+ for s > 4 and is globally unconditionally well-posed in the same space for s and, for any ɛ >, in the space H s R + H s++ɛ loc R + when s <. The last section consists of two appendices. The proofs of some technical lemmas used in Section 4 are presented in Appendix I. The results are based on minor modification of arguments already in the literature. The proofs are sketched for the convenience of readers. Some discussion of the boundary integral operator BI e is provided in Appendix II. The outcome of the analysis is a result showing that estimates involving BI e t alone do not suffice to prove Theorem., thereby providing a reason for the introduction of the more complicated boundary operators BI m and BI m.. Linear Problems and Extension Formulas This section is divided into two subsections. In the first, some explicit representation formulas are recalled from [4] for solutions of initial-boundary-value problems for the linear KdV-equation. Then, a method is put forward to convert the IBVP. posed on a quarter plane to an integral equation posed on the whole plane. The second subsection features a discussion of the aforementioned three different extensions to R R of the boundary integral operator associated to the boundary value problem. given below. These extended boundary integral operators will play a central role in our analysis.

13 BOUNDARY SMOOTHING PROPERTIES. Explicit solution formulas for linear problems First, consider the non-homogeneous problem u t + u x + u xxx =, for x, t,. ux, =, u, t = ht. Its solution may be written in the form see [4]. ux, t = [W bdr th] x = [U b th] x + [U b th] x where, for x, t, [U b th] x = π e itµ µ e µ 4+iµ «x µ e iξµ µ hξdξdµ. Next, consider the same linear equation posed with zero boundary conditions, but non-trivial initial data, viz. u t + u x + u xxx =, for x, t,. ux, = φx, u, t =. By semigroup theory, its solution may be obtained in the form.4 ut = W c tφ where the spatial variable is suppressed and W c t is the C -semigroup in the space L R + generated by the operator Af = f f with the domain DA = {f H R + f = }. By d Alembert s formula, one may use the semigroup W c t to formally write the solution of the forced linear problem u t + u x + u xxx = f, for x, t,.5 ux, =, u, t =, in the form.6 u, t = t W c t τf, τdτ. The following helpful formula for W c tφ was established in [4]. As is apparent from.4 and.6, this gives an explicit representation for solutions of the inhomogeneous problems. and.5. and Proposition.. For any φ L R +, define U + tφx = π U + tφx = π U + tφx = πi e iµ t iµt iµ+ e iµ t iµt e e µt µt e µ i µ 4 µ +4 e iµx ξ φξdξdµ, «x «x e iµξ φξdξdµ e µξ φξdξdµ.

14 4 JERRY L. BONA, S. M. SUN, AND BING-YU ZHANG Then it follows that W c tφx = j= U + j tφx + U + j tφx. As a comparison, recall the explicit solution formula for the pure initial-value problem IVP for the linear KdV equation cf.. u t + u x + u xxx =, x, t R,.7 ux, = gx, x R, namely.8 ux, t = W R tgx = c e iξ ξt e ixξ e iyξ gydydξ. The formula for W R t is much simpler than that of W c t. We take advantage of this simplicity to give a related representation of W c t in terms of W R t and W bdr t. Let a function φ be defined on the half line R + and let φ be an extension of φ to the whole line R. The mapping φ φ can be organized so that it defines a bounded linear operator B from H s R + to H s R. Henceforth, φ = Bφ will refer to the result of such an extension operator applied to φ H s R +. Assume that v = vx, t is the solution of v t + v x + v xxx =, vx, = φ x for x R, t. If gt = v, t, then v g = v g x, t = W bdr tg is the corresponding solution of the non-homogeneous boundary-value problem. with boundary condition ht = gt for t. It is clear that for x > the function vx, t v g x, t solves the IBVP., and this leads directly to a representation of the semigroup W c t in terms of W bdr t and W R t. Proposition.. For a given s and φ H s R + with φ =, if φ is its extension to R as described above, then W c tφ may be written in the form W c tφ = W R tφ W bdr tg for any x, t >, where g is the trace of W R tφ at x =. Remark: This representation of W c t is less explicit than that presented in Proposition.. However, it enables us to use the well established theory for W R t to study W c t. It is worth emphasis, however, that W c, by its nature, does not depend upon the extension φ of φ. The representation in Proposition. does, of course, depend on the extension and this representation will be useful in deriving linear estimates. In a similar manner, one may derive an alternative representation of solutions of the inhomogeneous initial-boundary-value problem.5. Proposition.. If f, t = Bf, t is an extension of f from R + R + to R R +, say, then the solution u of.5 may be written in the form u, t = t W R t τf, τdτ W bdr tv This was suggested by one of the referees of our earlier paper [4].

15 BOUNDARY SMOOTHING PROPERTIES 5 for any x, t where v vt is the trace of t W Rt τf, τdτ at x =. Finally, consider the fully inhomogeneous initial-boundary-value problem u t + u x + u xxx = f, for x, t,.9 ux, = φx, u, t = ht, where φ and h are assumed to satisfy the compatibility condition h = φ. Let ux, t = zx, t + e x t h. If u solves.9, then zx, t solves z t + z x + z xxx = f + e x t h, for x, t, zx, = φx e x φ, Decompose z in the form z = w + v + y with and z, t = ht e t h. w t + w x + w xxx = f + e x t h, for x, t, wx, =, w, t =, v t + v x + v xxx =, for x, t, vx, = φx e x φ, v, t =, y t + y x + y xxx =, for x, t, yx, =, y, t = ht e t h. The following representation for the solution of.9 emerges from this decomposition together with the results of Propositions. and.4 and Duhamel s principle. Proposition.4. The solution ux, t of.9 may be realized in the form ux, t = W c t φx e x φ +. t W c t τ fx, τ + e x τ h dτ + + [ W bdr t ht e t h ] x + e x t h. In case s, the compatibility condition is not needed. φ = h =. In this situation,. becomes simply. ux, t = W c tφx + t One may choose W c t τfx, τdτ + [W bdr th] x. In any case, by Proposition. and Proposition.,. may be written as t ux, t = W R tφ x + W R t τ f x, τ + px, τ dτ +. for x R + and t, where [ ] + W bdr t ht e t h gt vt x + px, t φ x = φx e x φ, px, t = e x t h and gt and vt are the temporal traces of t W R tφ x and W R t τ f x, τ + px, τ dτ

16 6 JERRY L. BONA, S. M. SUN, AND BING-YU ZHANG at x =, respectively. Of course, if the right-hand side f already happens to be defined on R R, there is no need to apply the extension operator. The solution formula. holds only for x > and t >. It will be convenient to extend this formula in such a way that it holds for all x, t R. This will provide a context in which to establish the well-posedness of the nonlinear problem in the framework of Bourgain spaces. Note that the first two terms on the right side of. are naturally defined for x, t R. Only the third term, viewed as a function of x and t, needs to be extended from R + R + to R R. With an appropriate extension of the third term, to be denoted by [Wbdr t ht e t h gt vt] x, the function ux, t given by the formula. may be viewed as a function of x defined on the whole line R, which of course solves.9 when restricted to R + R +. If one replaces f in. by uu x and drops the extension operator, there appears the nonlinear integral equation ux, t = W R tφ x t t W R t τux, τu x x, τdτ + + W R t τpx, τdτ + ] [Wbdr t ht e t h gt vt x + px, t posed on the whole plane R R. It is clear from its construction that if this integral equation has a solution, then when the solution is restricted to the domain R +, T, it solves the IBVP.. It is also clear that if u C, T ; H R +, then it is a strong solution distributional solution for which all of the terms in the equation lie in C, T ; L R + and such that they sum to the zero function in this latter space.. Extensions of the boundary integral operator As pointed out in Section., to use the Bourgain spaces in a straightforward way to study well-posedness of the IBVP., one needs to extend the boundary integral operator W bdr t to an integral operator Wbdr t so that, for any given boundary value function ht, [Wbdr tht]x is, for each t R, a function of x defined on the whole line R. There are infinitely many ways to accomplish such an extension; among them the even extension is probably the simplest. However, as will be made clear presently, and especially in Appendix II, special extensions are needed if one intends to capture some of the more subtle smoothing properties induced via the imposition of a boundary condition at x =. Next are presented three different types of extensions of the boundary integral operator W bdr t. Rewrite W bdr t as [W bdr th] x = π Re e iµt iµt e µ 4+iµ x µ e iµ µξ hξdξdµ = 4 π Re e iµt iµt e µ 4+iµ φ x µ φ µ e iµ µξ hξdξdµ + π Re e iµ := π {I x, t + I x, t} t iµt e µ 4+iµ x µ φ µ e iµ µξ hξdξdµ

17 BOUNDARY SMOOTHING PROPERTIES 7 where φ µ and φ µ are nonnegative cut-off functions satisfying φ µ + φ µ = for any µ R + with supp φ, 4, supp φ, and φ x is a smooth function on R such that φ x = x for x, φ x = for x. Observe that µ 4 + iµ 4 µ + µ x = ix is purely imaginary for µ /. The integral I is naturally defined for all values of x and t and, viewed as a function defined on R R, is in fact C smooth there, with all its derivatives decreasing rapidly as x ±. Thus no complicated extension of I is required as the obvious one suffices. It is otherwise for I. To discuss I, it is convenient to let µλ denote the positive solution of µ µ = λ for λ and µ, while µλ = µ λ for λ <. Note that µλ is strictly increasing on [, and that values of µ correspond to values of λ. By a change of variables, the integral I can be rewritten in the form I x, t = Re = := Ex, t e iλt e µ λ 4+iµλ x e iλs φ µλhsdsdλ e x µ λ 4 cos λt s µλx φ µλhsdsdλ for x. Let the extension of Ex, t to x < be gx, t and write Ex, t, x, I x, t = gx, t, x <, where gx, t is to be defined. Note that F x [I ]ξ, t = = +i + Recall the identities I x, te ixξ dx = Ex, t cosxξ + i sinxξ dx Ex, t cosxξ + g x, t cosxξ dx Ex, t sinxξ g x, t sinxξ dx. sinxξ cosxη hηdηdx = gx, t cosxξ + i sinxξ dx ξ x hxdx

18 8 JERRY L. BONA, S. M. SUN, AND BING-YU ZHANG and hx = cosxξ cosξηhηdηdξ, π where hx is extended evenly to negative values of x. These relations yield Ex, t sinxξ g x, t sinxξ dx = π ξ η cosηxex, tdx In consequence, it transpires that [ ] F x,t [I ] = F t Ex, t cosxξ + g x, t cosxξ dx + i π [ ξ η F t cosηx Ex, tdx cosηx g x, tdx dη. ] cosηx g x, tdx dη. Note that different choices of gx, t give different extensions of Ex, t to x <. The following three choices of gx, t will be studied in this article. i Let g x, t = Ex, t for x R +. This is perhaps the simplest extension. It results in the formula ]. F x,t [I ] = F t [ Ex, t cosxξdx. The boundary integral operator corresponding to this extension of W bdr t is denoted by BI e t; the subscript e stands for even of course. ii For x >, choose g x, t such that [ ] [ F t g x, t cosxξdx τ = F t.4 ] Ex, t cosxξdx τθξ, τ [ ] +F t Ex, t cosxξdx τ Θξ, τ νξωτ where Θξ, τ = χ ξ δ τ / with δ > fixed, χξ everywhere, and, ξ <, χξ =, ξ >, whilst, ξ, νξ =, ξ <, and ωτ is a smooth and bounded function to be specified momentarily. It is easy to see that such a g is a combination of even and odd extension, viz. where Î ξ, τ = F t [ F x,t [I ] := Îξ, τ + Îξ, τ ] Ex, t cosxξdx τ Θξ, τ + νξωτ

19 BOUNDARY SMOOTHING PROPERTIES 9 and.5 Î ξ, τ = i π = i π [ Θη, τ + [ ] ξ η F t Ex, t cosxηdx τ ] Θη, τ νηωτ ξ η + [ ] F t Ex, t cosxηdx τ ξ + η [ ] Θη, τ + Θη, τ νηωτ dη. dη Because of the algebraic identity we may write Îξ, τ as Î ξ, τ = i πξ ξ η + ξ + η = ξ F t [ [ Θη, τ + + i πξ [ Θη, τ + + η ξ η, ] Ex, t cosxηdx ] Θη, τ νξωτ dη η/ξ [ ] η/ξ F t Ex, t cosxηdx ] Θη, τ νξωτ dη := Q ξ, τ + Q ξ, τ. Rewrite the integral Ex, t as follows; Ex, t = Re = = + + e iλt e µ λ 4+iµλ e iλt e µ λ 4+iµλ x x e iλt e µ λ 4 iµλ e iλt e µ λ 4+iµλ x e iλt e µ λ 4 iµλ e iλs φ µλhsdsdλ e iλs φ µλhsdsdλ x e iλs φ µλhsdsdλ e iλs φ µλhsdsdλ x e iλs φ µλhsdsdλ

20 JERRY L. BONA, S. M. SUN, AND BING-YU ZHANG where the fact that φ µλ = for λ < that follows since supp φ,. A direct computation reveals that and e µ λ 4+iµλ e µ λ 4 iµλ x x with µ λ 4.6 K η, λ =, µ λ 4 + η + µλ was used in the last step, a point cosxηdx = K η, λ+k η, λ+k η, λ+k 4 η, λ cosxηdx = K η, λ+k η, λ+k η, λ+k 4 η, λ and. µ λ 4 K η, λ =, µ λ 4 + η µλ µ λ 4 µλ K η, λ =, i µ λ 4 + η + µλ µ λ 4 + η µλ µ λ 4η µλ K 4 η, λ = i µ λ 4 + η + µλ µ λ 4 + η µλ K η, λ = K η, λ, K η, λ = K η, λ, K η, λ = K η, λ, K 4 η, λ = K 4 η, λ. Thus, F t [ Ex, t cosxηdx ] may be expressed in the form [ ] F t Ex, t cosxηdx τ. where = 4 K m η, τφ µτĥτ + m= ĥτ = 4 K m η, τφ µ τĥ τ m= e iτs hsds. Here, we note that from the definition of φ, for any given τ there is only one nonzero summand in. and the sum involving the terms K m η, τ is the same as the sum involving on the terms K m η, τ. Because φ µτ is a bounded C -function, and since, as direct calculation shows, 4 m= K m η, τ Θη, τ dη C >,

21 BOUNDARY SMOOTHING PROPERTIES where C is a fixed constant independent of τ, it follows that the formula,. + ωτ φ µτ 4 m= = φ µτ K m η, τ Θη, τ dη 4 K m η, τθη, τdη, defines the C R function ωτ in such a way that d k ω/dτ k is bounded on R, for k =,,,. It is clear that this choice of ωτ makes Hence, for ξ, m= Q ξ, τ, for all τ when ξ. Î ξ, τ = Q ξ, τ. where = i η [ ] πξ ξ η F t Ex, t cosxηdx τθ η, τdη Θ η, τ = Θη, τ + Θη, τ νηωτ. Moreover, when ξ and τ, Î ξ, τ = i [ 4 ] η.4 πξ ξ η K m η, τφ µτĥτ Θ η, τdη, whereas Î ξ, τ = i πξ m= [ 4 ] η ξ η K m η, τφ µ τĥ τ Θ η, τdη m= when ξ and τ <. The boundary integral operator corresponding to this extension of W bdr t is denoted by BI m t. iii For x >, choose g x, t such that [ ] [ F t g x, t cosxξdx τ = F t.5 In this case, +F t [ ] Ex, t cosxξdx τθξ, τνξωτ. ] Ex, t cosxξ τ Θξ, τ.6 F x,t [I ] := Î ξ, τ + Î ξ, τ where, if Θ η, τ = Θη, τ + Θη, τ νηωτ, then [ ] Îξ, τ = F t Ex, t cosxξdx τθξ, τ + νξωτ

22 JERRY L. BONA, S. M. SUN, AND BING-YU ZHANG and Î ξ, τ = i π = i π = i πξ + i πξ [ ξ η F t ξ η + ξ + η ] Ex, t cosxηdx τθ η, τdη F t [ [ ] F t Ex, t cosxηdx τθ η, τdη η/ξ η/ξ F t [ ] Ex, t cosxηdx τθ η, τdη ] Ex, t cosxηdx τθ η, τdη := Q ξ, τ + Q ξ, τ. Just as in case ii, one can choose an appropriate function ωτ such that Q ξ, τ = for ξ > and any τ R. The boundary integral operator corresponding to this extension of W bdr t is denoted by BI m t.. Boundary smoothing properties In this section, attention is focused upon the non-homogeneous boundaryvalue problem u t + u x + u xxx =, for x, t,. ux, =, u, t = ht. Our analysis turns around a detailed understanding of the boundary integral operators introduced in Section. For given s R, b, α > and any function w wx, t : R R R, define Λ s,b w = + τ ξ ξ b + ξ s ŵξ, τ dξdτ,. λ α w = ξ Consider first the operator BI m t. + τ α ŵξ, τ dξdτ. Theorem.. Let ψt be a given smooth function of t with compact support and assume that s and b are within the range b < + s <. Then there exists a constant C depending only on ψ such that. Λ s,b ψbi m h C h b s / H R + for any h H b s / R +.

23 BOUNDARY SMOOTHING PROPERTIES Proof: Recall that [BI m th] x = I x, t + I x, t where I x, t is a function defined on the whole plane R R and is, in fact, a C -smooth function of x and t. For any t R, I x, t L x R C µ φ µ e iµ µξ hξdξ C h L R +. L µ R This type of inequality is also valid for j x l t I for any j, l. Thus it is straightforward to see that if h L R +, then.4 Λ s,b ψi C h L R + for any given b and s R where the constant C depends only on ψ, b and s. To analyze I x, t, remember that where, for ξ, Î ξ, τ = F t [ F x,t [I ]ξ, τ = Îξ, τ + Îξ, τ ] Ex, t cosxξdx τ Θξ, τ + ωτ and Î ξ, τ = i π ξ η + [ ] F t Ex, t cosxηdx τθ η, τdη. ξ + η Since the relevant estimates in the region ξ < are straightforward, in what follows it is always assumed that ξ. First, consider the term + τ ξ ξ b s + ξ Î ξ, τ dξdτ. We have the following estimate for this term. Proposition.. Let s and < b < + s be given. There exists a constant C such that.5 + τ ξ ξ b + ξ s Î ξ, τ dξdτ C h H b s / R + for any h H b s / R +. Proof of Proposition.: According to., [ ] F t Ex, t cosx, ξdx τ = 4 K m ξ, τφ µτĥτ + m= 4 K m ξ, τφ µ τĥ τ. In the following, detailed analysis is given for terms containing K and K 4 ; the estimates for the other terms follow similar lines. Suppose that ξ. The case ξ < is entirely analogous. Write m= A m ξ, τ = K m ξ, τφ µτĥτ, m =,,, 4.

24 4 JERRY L. BONA, S. M. SUN, AND BING-YU ZHANG For given s and b >, we have C + τ ξ ξ b + ξ s A ξ, τ + ωτ Θξ, τ dξdτ φ µτ hse isτ ds B τdτ with B τ = + τ ξ ξ b s + ξ + ωτ Θξ, τ µ τ 4 φ µτ dξ. µ τ 4 + ξ µτ Claim: If b < + s, then, as τ, B τ = Oτ 6b s. To see the claim is true, note that in fact B τ = δ τ + τ ξ ξ µ τ 4 φ b µτ s + ξ µ τ 4 + ξ µτ dξ + ωτ Θξ, τ since Θξ, τ = when ξ < δ τ, where δ > is fixed, but arbitrary for the nonce see.4. Let ξ = ηζ be the real solution of the equation ξ ξ = ζ, for ζ < that connects continuously to the unique real root as ζ becomes large e.g. ζ >. Note that ηζ ζ as ζ.

25 BOUNDARY SMOOTHING PROPERTIES 5 For large τ, it is also the case that µτ τ. Thus, for τ > large enough, µ τ 4 + τ ζ b s B τ C + ζ δ τ µ τ 4 + ηζ µτ η ζ dζ C δ τ τ + τ ζ b + ζ s + τ + ηζ τ η ζ dζ τ = C δ τ + C τ τ + τ ζ b + ζ s + τ + ηζ τ τ + τ ζ b + ζ s + τ + ηζ τ := G τ + G τ. Continuing this sequence of inequalities, note further that τ G τ C + τ τ δ τ + τ ζ b + ζ s η ζ dζ η ζ dζ η ζ dζ C τ + τ b + τ τ δ τ + ζ +s dζ and Cτ 6b s G τ Cτ Cτ τ τ + τ ζ b dζ + τ + ζ + ζ s ζ ζ b ζ + s dζ Cτ 6b s if b < + s. The claim is thereby established. As a consequence, the following inequality emerges. For given s and b < + s, there exists a constant C such that.6 φ µτ hse iwτ dw B τdτ C φ µττ b s / C h H b s // R + for any h H b s / R +. hwe iwτ dw dτ

26 6 JERRY L. BONA, S. M. SUN, AND BING-YU ZHANG Next, consider the term A 4 ξ, τ = K 4 ξ, τφ µτĥτ involving K 4. For given b > and s R, + τ ξ ξ b + ξ s A 4 ξ, τ + ωτ dξdτ Θξ, τ C where B 4 τ is equal to φ µτ e isτ hsds B 4 τdτ 4ξ µ τ µτ + τ ξ ξ b + ξ s + ωτ Θξ, τ µ τ 4 + ξ + µτ µ τ 4 + ξ µτ dξ. As in the estimation of B τ, one shows that if b < + s, B 4 τ = Oτ 6b s as τ using again the fact that µτ τ for large positive value of τ. Consequently, for b < + s and s, there exists a constant C such that φ µτ hse isτ ds B 4 τdτ C h. H b s / R + The proof of Proposition. is complete. Next, attention is given to the term + τ ξ ξ b s + ξ Î ξ, τ dξdτ, for which we have the following estimate. Proposition.. Let s and b be given satisfying b < s < 4. Then there exists a constant C such that.7 + τ ξ ξ b s + ξ Î ξ, τ dξdτ < C h b s H R + for any h H b s R +. Proof of Proposition.: First, we note that to study Îξ, τ, we can use the form of [ Îξ, τ in.5 or.4. As before, details are given for only one term in F t Ex, t cosxηdx ], say A ξ, τ = K ξ, τφ µτĥτ. Notice that A ξ, τ = A ξ, τ. Hence, we may consider only the case wherein ξ. Denote by q the function q ξ, τ = µ τ 4 + ξ + µτ

27 BOUNDARY SMOOTHING PROPERTIES 7 and, for ξ, let D be given by D ξ, τ = for ξ δ µτ from.4 and η Θη, τ ξξ η q η, τdη + + ωτ η Θη, τ ξξ η q η, τdη ξθη, τ ξ Θη, τ D ξ, τ = ξ η q η, τdη + + ωτ ξ η q η, τdη for ξ δ µτ from.5, where δ > is a small constant. The relevance of these functions will become clear presently. First, note that A ξ, τ = q ξ, τφ µτĥτ µ τ 4. As for D, changing variables in the integrals of its definition shows it to have the form D ξ, τ = = a µ τ η ξξ η Θη, τq η, τdη η + + ωτ ξξ η Θη, τ q η, τdη + + ωτ µ τ η yy η Θµτη, τp η, τdη η a yy η Θµτη, τ p η, τdη.8 := D y, τ + D y, τ where a = δ τ +, a = δ τ µτ µτ, y = ξ/µτ δ, p η, τ = 4. µ τ + η + We have similar definitions for y δ. Remark that a is bounded independently of τ and so for y large enough, y η is bounded below for η [, a ]. Thus, where D y, τ = := a η y µ τ η/y Θµτη, τp η, τdη y µ τ D,τ, y D, τ, y C for all τ and y large.

28 8 JERRY L. BONA, S. M. SUN, AND BING-YU ZHANG Turning to D, note that Θµτη, τ = for η a, so yy η Θµτη, τ p η, τdη Of course, η a = η a yy η p η, τdη = y a y z z y p zy, τdz = z y a y z y p zy, τ /4 dz + 4y := D y, y, τ + D, y, τ. D, y, τ = 4 a y z dz = 4 since z dz = as a principal-value integral. It is therefore clear that a y z dz a y z dz D, y, τ C y for some constant C independent of τ when y is large, say y y. As for D, y, τ, note that η y p ηy, τ /4 = 4 y µ τy y 8η y 4 µ τy + η + y Rewrite D, y, τ as to obtain and := y p η, y, τ. D, y, τ = y + / / + a /y / + p η, y, τ η dη C / p η, y, τ a /y η dη C + ln y p η, y, τ η dη where C is independent of τ and y for µτ and y large. Thus, if y y, then.9 D µτy, τ C y + ln y µ τ

29 BOUNDARY SMOOTHING PROPERTIES 9 where C is independent of τ and y. The following calculation shows the relevance of D ; = = + τ ξ ξ b s + ξ π φ µτ ĥ τ ξ η A η, τθ η, τdη dξdτ µ τ 4 + τ ξ ξ b s + ξ ξ η q η, τθ η, τdη dξdτ π φ µτ ĥ τ µ τ 4 + τ ξ ξ b + ξ s D ξ, τ dξdτ. Appropriate bounds on D yield bounds on the left-hand side of the last formula. Consider the quantity E τ := = µ τ 4 δµτ µ τ 4 + τ ξ ξ b + ξ s D ξ, τ dξ + yµτ δ µτ + ξ s D ξ, τdξ + + τ ξ ξ b y µτ := E τ + E τ + E τ where δ is again a small constant. By the choice of ωτ, it transpires that for large τ, E τ Cτ µτ Cτ µτ 6b s Cτ 6b s y µτ y ξ 6b s 6 dξ ξ 6b s 6 dξ if 6b s 6 <, which is to say b < s For δ y y, say, D C µ + τ a y η Θµτη, τp η, τdη y + + ωτ Θµτη, τ p η, τdη a y η.

30 JERRY L. BONA, S. M. SUN, AND BING-YU ZHANG Note that if δ y a, then a y η Θµτη, τp a η, τdη Θµτη, τ p η, τ p y, τ dη y η a + p Θµτη, τ y, τ dη y η C. The same bound is valid if a y y. Thus,. D C µ τ and E τ Cτ Cτ Cτ Cτ Cτ τ δ µτ τ b τ yµτ δ µτ τ + + τ δ µτ + + τ ξ ξ b + ξ s dξ yµτ τ + τ ξ ξ b + ξ s dξ τ + ξ s dξ + + τ ξ ξ b ξ s dξ τ yµτ τ s b+ τ + τ 6b s+ + τ ξ 6b s dξ + τ ξ b τ s ξdξ + τ 6b s+ b+ s Cτ 6b s ξ since b. If y = µτ δ in the term D = D + D, then D C y µ τ y η Θµτη, τ p η, τdη C y µ τ and D = µ τ + a [ a a a + p η, τ p y, τ Θµτη, τdη y η p η, τ p y, τ Θµτη, τdη y + η y η p y, τ + y + η p y, τ := µ τ D + D + D. ] Θµτη, τdη

31 Recall that p η, τ = = = a a BOUNDARY SMOOTHING PROPERTIES 4 µ τ + η +, so that D y, τ + D y, τ = Θµτη, τ η + y + 4 µ τ + η + 4 µ τ + y + η y + 4 dη µ τ + y + [ Θµτη, τ 8yη + 4 µ τ + η + 4 µ τ + y + 4 µ τ + y + ] + y 4 µ τ + y dη. µ τ + y + It thus transpires that Also, D = p y, τ a = p y, τ a D + D C y. y η Θµτη, τdη + p y, τ y η dη + a +p y, τ a y η a y + η dη + = p y, τ ln a y + ln y = = a Θµτη, τdη a y + η + p y, τ a Θµτη, τdη ln a + y ln y Θµτη, τdη y + η a Θµτη, τdη a Θµτη, τdη +p y, τ y a η η + p y, τ y a η η + p y, τ + p y, τ ln a ln y + p y, τ ln y a +p y, τ ln + ya a + p y, τ a η + a +p y, τ y/η Θµτη, τdη a η y/η + p y, τ + p y, τ a ln a ln y + +p y, τ ln ya a + a +p y, τ ln + ya a a It follows that a η y/η Θµτη, τdη y/η Θµτη, τdη y Θµτη, τdη y ηη y Θµτη, τdη y + ηη D C y ln y +.

32 JERRY L. BONA, S. M. SUN, AND BING-YU ZHANG and which implies that D. D C y ln y + µ, τ C y ln y + µ τ Thus, it is apparent that δµτ E Cτ / + τ b + ξ s ξ τ + ln ξ µτ dξ δµτ Cτ 4/ + τ b + ξ s + ln ξ µτ ξ dξ δ Cτ 4/ + τ b + µτ ξ s µ τ + ln ξ ξ dξ δ Cτ s +b Cτ b s+/ ξ s + ln ξ dξ if s >. Combining these estimates, there obtains E τ Cτ b s+/ if s < and < b < + s. This in turn implies that + τ ξ ξ b + ξ s C Θη, τ + Θη, τ + ωτ τ b s+/ hwe iwτ dw Similar estimates for the other terms yield, in sum,. ξ η A η, τ dη dξdτ dτ C h. H b s 6 + τ ξ ξ b + ξ s Î ξ, τ dξdτ C h H b s 6 if s < / and b < s. This completes the proof of the Proposition... By combining above two propositions, we obtain the estimate.. The proof of Theorem. is complete. Now, attention is turned to the operator BI m t.

33 BOUNDARY SMOOTHING PROPERTIES Theorem.4. Let ψt be a given smooth function of t with compact support and assume that b < s with s <. Then there exists a constant C such that. Λ s,b ψbi m h C h H b s / R + for any h H b s / R +. Proof: As in the proof of Theorem., it suffices to prove the following two propositions. Proposition.5. Let s R and b be given. There exists a constant C such that. + τ ξ ξ b + ξ s Î ξ, τ dξdτ C h 9b s 5 H 9 C h H b if s, if s. Proposition.6. Let s and b be given satisfying b < s. There exists a constant C such that.4 + τ ξ ξ b s + ξ Îξ, τ dξdτ C h b s H R + for any h H b s R +. We only present a proof of Proposition.5. The proof of Proposition.6 follows the same line as that of Proposition. and is therefore omitted. As in the proof of Proposition., detailed analysis is given for the term containing K ; the estimates for the other terms are sufficiently similar that their proof does not require further elaboration. Suppose ξ and τ in what follows. The other cases are entirely analogous. Define For given s and b, we have C A m ξ, τ = K m ξ, τφ µτĥτ, m =,,. + τ ξ ξ b s + ξ A ξ, τ + ωτ φ µτ hse isτ ds B τdτ Θξ, τ dξdτ