dispersion estimates for third order equations in two...

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS

Vol. 28, Nos. 11 & 12, pp. 1943–1974, 2003

Dispersion Estimates for Third Order Equations

in Two Dimensions

Matania Ben-Artzi,1,* Herbert Koch,2 and Jean-Claude Saut3

1Institute of Mathematics, Hebrew University, Jerusalem, Israel2Fachbereich Mathematik, Universitat Dortmund,

Dortmund, Germany3UMR de Mathematiques, Bat, Universite de Paris-Sud,

Orsay, France

ABSTRACT

Two-dimensional deep water waves and some problems in nonlinear optics can be

described by various third order dispersive equations, modifying and generalizing

the KdV as well as nonlinear Schrodinger equations. We classify all third order

polynomials up to certain transformations and study the pointwise decay for the

fundamental solutions,ZR

2eitpð�Þþix�� d�

for all third order polynomials p in two variables. We deduce the corresponding

Strichartz estimates. These estimates imply global existence and uniqueness for

the Shrira system.

*Correspondence: Matania Ben-Artzi, Institute of Mathematics, Hebrew University,

Jerusalem 91904, Israel; E-mail: [email protected].

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120025491_PDE28_11-12_R1_082903

1943

DOI: 10.1081/PDE-120025491 0360-5302 (Print); 1532-4133 (Online)

Copyright & 2003 by Marcel Dekker, Inc. www.dekker.com

+ [29.8.2003–10:16am] [1943–1974] [Page No. 1943] F:/MDI/Pde/28(11&12)/120025491_PDE_028_011-012_R1.3d Communications in Partial Differential Equations (PDE)

1. INTRODUCTION

In has been known since 1968 (Zakhrov, 1968) that the stability of finite ampli-tude gravity waves of deep water is governed by a nonlinear Schrodinger equation(NLS). This is found by a perturbation analysis to O("3) in the wave-steepness"¼ ka� 1, where a is a typical wave amplitude and k the modulus of the meanwave number.

Taking perturbation analysis one step further to O("4), Dysthe (1979) hasderived a system which improves significantly the results relating the stability offinite amplitude waves. One of the dominant new effects is the wave-induced meanflow with potential �. Solving the equation for � in terms of the complex amplitudeof the wave packet allows to put the Dysthe system (in dimensionless variables) inthe following form, see Ghidaglia and Saut (1993):

2i@A

@tþ1

2

@A

@x

� �þ1

2

@2A

@y2�1

4

@2A

@x2� AjAj2

¼i

8

@3

@x3� 6

@3

@x@y2

!Aþ

i

2A A

@ �AA

@x� �AA

@A

@x

� ��5i

2jAj2

@A

@xþ AR1

@

@xjAj2 ð1:1Þ

where R1 is the Riesz transform in R2, that is

FðR1 Þ ¼ i�1j�j :

Here F denotes the Fourier transform.The usual NLS is obtained by neglecting the right hand side of Eq. (1.1), which

is of order "4 in the dimensional variables.A similar derivation of the fourth order (in ") evolution equations for the

amplitude of a train of nonlinear gravity-capillary waves on the surface of an idealfluid of infinite depth was performed by Hogan (1985). The equation reads

2i@A

@tþ cg

@A

@x

� �þ p

@2A

@x2þ q

@2A

@y2� �jAj2A

¼ �is@3A

@x@y2� ir

@3A

@x3� iujAj2

@ �AA

@xþ ivjAj2

@A

@xþ AR1

@

@xjAj2 ð1:2Þ

where cg is the group velocity and �, p, q, s, r, u, and v are real parameters dependingon the surface tension parameter. Note that q and s are strictly positive, while p canachieve both signs (in particular it is negative for purely gravity waves as in theDysthe system and positive for purely capillary waves).

A similar system has been obtained by Lo and Mei (1985) and Hara and Mei(1994) for finite depth gravity waves. Another example of a nonlinear Schrodingertype equation involving third order derivatives has been proposed by Shrira (1981)to describe the evolution of a three dimensional packet of weakly nonlinear internalgravity waves propagating obliquely at an arbitrary angle � to the vertical. If thedependence of the wave packet amplitude A on the transversal coordinate y is much

1944 Ben-Artzi, Koch, and Saut

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slower than that on the x and z directions, one obtains the equation

i@A

@tþ!k

2

@2A

@x2þ!nn

2

@A

@z2þ !nk

@2A

@x@z

� i!kkk

6

@3A

@x3þ!kkn

2

@3A

@x2@zþ!knn

2

@3A

@x@z2þ!nnn

6

@3A

@z3

" #

þ i�A A@ �AA

@s� �AA

@A

@s

� �¼ 0 ð1:3Þ

Here @=@s ¼ cðsin�ð@=@xÞ � cos �ð@=@zÞÞ. The coefficients of the linear terms inEq. (1.3) can be computed explicitly as function of �. For nonzero constants �and � we have

!kkn ¼ � sin� cos�ð3� 5 cos2 �Þ

!nnn ¼ � sin� cos�ð3� 5 sin2 �Þ

!nnk ¼ ��ð5 sin2 � cos2 �� 2=3Þ

!kkk ¼ � sin2 �ð4� 5 sin2 �Þ

!kk ¼ �3� sin2 �

!kn ¼ � tan�ð2� 3 sin2 �Þ

!nn ¼ ��ð3 sin2 �� 1Þ

see Shrira (1981) and Ghidaglia and Saut (1993, Chap. 3). In particular !kkn and !knn

cannot vanish simultaneously.Similar problems occur in nonlinear optics (see for instance Zozulya (1999)), in

particular in the modeling of the dynamics of femtosecond laser pulses in a nonlinearmedia with normal dispersion. The evolution of the complex envelope E(x, y, z, t) ofthe field is described by the third order NLS

i@

@zE þ ð1� i"1@tÞ�?E �

@2E

@t2� i"2

@3E

@t3þ ð1þ i"1@tÞgðjEj

2ÞE ¼ 0 ð1:4Þ

where �? ¼ @2=@x2 þ @2=@y2 is the transverse Laplacian and "2 2 R, "1>0. As usualin nonlinear optics the evolution variable (which plays mathematically the role oftime) is z. The transverse Laplacian accounts for diffraction, while the second andthird time derivatives describe group velocity and third over dispersion.

Very little is known concerning the Cauchy problem for Eqs. (1.1)–(1.4). Theonly results available are local existence for analytic Cauchy data for Eqs. (1.1) and(1.2) by de Bouard (1993) and global existence for a very special case of Eq. (1.3), seeGhidaglia and Saut (1993).

The aim of the present article, as a first step towards solving the Cauchy problemin Sobolev classes, is to derive dispersion estimates for the linear group associated tothe linearized equation at 0. Those estimates have an independent interest and seem

Dispersion Estimates for Third Order Equations 1945

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120025491_PDE28_11-12_R1_082903

to be new for this class of third order symbols in 2 dimensions. On the other hand itis well known that Strichartz estimates are an important tool to solve the Cauchyproblem by a fixed point argument.

Thus we study linear third order equations, more precisely the decay ofthe fundamental solutions in x and t. This requires first a classification, second acalculation of several Fourier transforms, and third several applications of themethod of stationary phases.

The short time behavior is dominated by the homogeneous part (unlessit is strongly nongeneric) whereas the long time behavior is determined by thedegeneracy at the strongest singularity. For generic two-dimensional third-orderpolynomials the long time decay is t�3/4 and the generic short time bound is ct�2/3.

Stationary phase relates the map �! rpð�Þ with the decay of the fundamentalsolution. As side product we analyze the pointwise decay of the unique singularity(up to transformations) of codimension two (the singularity of codimension one hasnormal form �31, where the counting of the codimension is different from the one insingularity theory since linear terms do not change the decay). The decay estimatesimply Strichartz estimates, which in turn allow to prove global well-posedness for theCauchy problem for the Shrira system (1.3).

The article is organized as follows. In Sec. 2 we introduce notation. Section 3 isdevoted to a classification of third degree polynomials up to affine transformationsof coordinates and the addition of affine terms. This extends the work of Dzhuraevand Popelek, (Dzhuraev and Popelek, 1989 and 1991).

In Sec. 4 we give the inverse Fourier transform of eitp(�) for those polynomials forwhich the inverse Fourier transform can be computed in terms of exponentials,trigonometric functions and the Airy functions. The next section, Sec. 5, introducesthe Pearcey integral, which occurs for one of the phase functions, and studies onemore oscillatory integral, where a change of the contour of integration allows toobtain good estimates.

All this relies on the fact that we deal with very specific phase functions.The last two oscillatory integrals require natural but much deeper tools: We

have to study the contribution from general singularities of fold type and cusp type.In Sec. 6 we study local changes of coordinates and local normal forms for

nondegenerate points, folds, and cusps. Here Mather’s theory of stable mapsenters crucially. The reduction to normal forms allows to establish decay estimatesfor compactly supported amplitudes, which is done in Sec. 7. The next section, Sec. 8applies these results to obtain decay estimates for the inverse Fourier transforms ofeitp(�) for the remaining two polynomials.

Section 9 is standard: the decay estimates imply Strichartz estimates, some ofthem with smoothing. Well-posedness for the Shrira system is an easy consequence.We plan to return to the study of the other systems.

It should be noted that there are at least two motivations for the problems athand. First we establish sharp estimates for cusps—here the Pearcey integral plays asimilar role as the Airy function for holds. This is a natural step in the study ofoscillatory integrals. Even though the tools are available we are not aware of similarestimates in the literature, despite their importance for degenerate dispersive inte-grals. Secondly systems like the Shrira system have a strong motivation from physics,which makes a good analytical understanding highly desirable.


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120025491_PDE28_11-12_R1_082903

2. NOTATION

We denote the Fourier transform by F and its inverse by F�1:

Ff ð�Þ ¼ ff ð�Þ ¼ ð2�Þ�n=2

Ze�ix�f ðxÞ dx

and

F�1ð�Þx ¼ ��ðxÞ ¼ 2ð�Þ�n=2

Zein��ð�Þ dx:

For x in Rn we set ~xx ¼ ðx1, . . . , xn�1Þ. If A � R

n is measurable we denote its volumeby |A|. The Lebesgue spaces are denoted by Lp.

We define Lpw by the quasinorm

k f kLpw¼ sup

�>0�jfx : j f ðxÞj > �gj1=p:

These spaces are Banach spaces if 1 < p <1. They occur in the context of theweak Young inequality (see Stein and Weiss (1971)):

Lemma 2.1. We have

k f � gkLr � ck f kLpkgkLqw

provided 1 < p, q, r <1 and

1

pþ1

q¼ 1þ

1

r:

In what follows, we shall derive space-time estimates for the oscillatory integral

Iðx, tÞ ¼

Zeitpð�Þþix��d�:

The real polynomial p (in � 2 R2) is a general cubic polynomial. However, for

the estimates the constants clearly play no role while the linear terms can bedispensed with by a suitable shift x to xþ at. Thus, we shall assume that p containsonly third and second order homogeneous parts. In the following section we shallreduce it to one of a class of normal forms (modulo linear terms), by suitable lineartransformations of �.

3. NORMAL FORMS

Homogeneous second degree polynomials are quadratic forms. There are onlytwo invariants: rank and signature. There are three equivalence classes, representedby �21, �

21 þ �

22, and �1�2.


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120025491_PDE28_11-12_R1_082903

Note let p be homogeneous of degree three. Assuming p(1, 0) 6¼ 0 (otherwise werotate our coordinates) we can scale the �1 direction to obtain

pð�1, �2Þ ¼ �31 þ C�21�2 þ A�1�22 þ B�32:

We will keep this normalization for some time. We change coordinates to~��1 ¼ �1 þ ðC=3Þ�2 so that

pð�1, �2Þ ¼ ~�� 31 þ ~AA ~��1�22 þ ~BB�32 ð3:1Þ

with

~AA ¼ A�C2

3and ~BB ¼ B�

AC

3þ2C3

27:

We drop the tilde in the sequel and suppose that p is a polynomial of the form (3.1).Given a regular matrix D ¼ d11 d12

d21 d22

� �we consider

pDð�Þ ¼ pðD�Þ ¼ �1�31 þ �2�1�

22 þ �3�

21�2 þ �4�

32

where �i¼ �i(D). This defines a smooth map from the space of regular matrices toð�iÞ1�i�4 2 R

4. An immediate computation gives

pDð�Þ ¼ ðd311 þ Ad11d

221 þ Bd3

21Þ�31 þ ð3d2

11d12 þ Aðd12d221 þ 2d11d21d22Þ

þ 3Bd221d22Þ�

21�2 þ ð3d11d

212 þ Aðd11d

222 þ 2d12d21d22Þ þ 3Bd21d

222Þ�1�

22

� ðd312 þ Ad12d

222 þ Bd3

22Þ�32: ð3:2Þ

The derivative at the identity is gives by the matrix

3 0 0 00 3 2A 0A 0 3B 2A0 A 0 3B

0BB@

1CCA ð3:3Þ

which has determinant �ðA,BÞ :¼ 3ð27B2þ 4A3

Þ. This is zero if and only ifA/3¼�(B/2)2/3. Suppose that �(A,B) 6¼ 0. Using the implicit function theorem wecan find a nonsingular matrix if ð �AA, �BBÞ is close to (A,B) such that

pDð�Þ ¼ �31 þ �AA�1�22 þ

�BB�32:

A simple covering argument shows that the same is true for each ð �AA, �BBÞ in thepathcomponent of �(A,B) 6¼ 0. Obviously there are two path connected opencomponents ��(A,B)>0.

We choose two representative polynomials one for each component, therebychanging coefficient of �31 : ð1=6Þ�

31 � ð1=2Þ�1�

22 and ð1=6Þð�31 þ �

32Þ.

Now we consider the case A/3¼�(B/2)2/3. If A¼ 0 then the polynomial is �31.If not we may scale �2 so that A¼ 3. Then B¼�2 and changing �2 to ��2 if necessarywe arrive at

pð�Þ ¼ �31 � 3�1�22 þ 2�32: ð3:4Þ


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120025491_PDE28_11-12_R1_082903

In summary, we have four representatives for the homogeneous third degreepolynomial. A short check shows that they are classified by the shape of the zero setof the determinant of the Hessian �! detD2pð�Þ, which is either the whole space, aline, the union of two transversal lines or a point. The shape is invariant under theoperations we used above. This consideration allows us to use the following set ofrepresentatives:

1

6�31,

1

2�1�

22,

1

6ð�31 þ �

32Þ and

1

6�31 �

1

2�1�

22: ð3:5Þ

Note that the second of those four representatives is replacing the one in Eq. (3.4).For notational simplicity we introduced coefficients different from 1. In what followswe refer to the four cases in Eq. (3.5) by Cases I–IV.

We now want to classify all polynomials with cubic and quadratic terms and tofind normal forms for each class. We shall use affine changes and the addition oflinear and constant terms, as long as they leave the cubic terms in Eq. (3.5) invariant.As has been observed earlier, the transformed polynomials are considered moduloaffine terms.

Thus, let p be a polynomial with cubic and quadratic terms. We set

d ¼ degree detðD2pÞ and S ¼ f� : detðD2pð�ÞÞ ¼ 0g

and consider the four cases separately.

Case I. If the coefficient of �22 is nonzero we may suppose that it is one. Then we havethe polynomial

1

6�31 þ b�21 þ a�1�2 þ �

22 ¼

1

6�31 þ ða�1=2þ �2Þ

2þ ðb� a2=4Þ�21: ð3:6Þ

We set �¼ �2þ a�1/2, plug it into the formula and rename � to �2. Then

pð�Þ ¼1

6�31 þ �

22 þ c�21 ð3:7Þ

and we may assume that c¼ 0 since the term can be eliminated by a shift in �1.This creates affine terms, which we neglect in our classification. It is easily seenthat d¼ 1.

Now we consider the case that the coefficient of �22 is zero and that of �1�2 is not.Then we may assume that it is one and we are left with ð1=6Þ�31 þ �1�2. Then d¼ 0 anddet(D2p) is constant but not zero. If both coefficients are zero then det(D2p)¼ 0.We conclude that Case I leads to three normal forms:

1

6�31,

1

6�31 þ

1

2�22, and

1

6�31 þ �1�2:

Case II. We shift �1 so that the coefficient of �22 is zero. Shifting �2 we achieve thesame for the coefficient of �1�2. We can normalize the coefficient of �21 and henceobtain the two representative polynomials ð1=2Þ�1�

22 and ð1=2Þ�1�

22 þ ð1=2Þ�21.


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120025491_PDE28_11-12_R1_082903

Then d¼ 2 and S is a line or a parabola. The second case provides a normal form fora cusp. The notion of a cusp here is motivated by the fact that the image of S underthe map �! rpð�Þ is a geometric cusp. The inverse Fourier transform can beexpressed by the Pearcey integral (5.1).

Case III. The same arguments show that there are only two classes with normalpolynomials

pð�Þ ¼1

6ð�31 þ �

32Þ and pð�Þ ¼

1

6ð�31 þ �

32Þ þ �1�2:

Here d¼ 2 and S consists of a hyperbola or two lines.

Case IV. There are two classes with normal forms

1

6�31 �

1

2�1�

22 þ

1

2ð�21 þ �

22Þ, and

1

6�31 �

1

2�1�

22:

Here d¼ 2 and S is either a circle or a point. There cannot be any further reductionwhich can be seen by checking the zero sets of the determinant of the Hessian.

This completes the classification: The third order polynomials:

1

6�31 �

1

2�1�

22 þ

1

2�21 þ

1

2�22, ðIVÞ

1

6�31 þ

1

6�32 þ �1�2 ðIIIÞ, ð3:8Þ

will be discussed in Sec. 8. Their study is the most demanding and the most impor-tant part of this article. We recall that S is a circle (the first polynomial) resp. ahyperbola. The leading part of the asymptotic expansion of the oscillatory integral

Iðx, tÞ ¼

ZR

2eitpð�Þþix� d�

is expressed through the Airy function near points (x, t) for which there aredegenerate critical �c satisfying

xþ trpð�cÞ ¼ 0

for which the null space of D2p(�c) is transversal to S—this corresponds to values ofx/t lying in the image of S 3! �rpð�Þ, where the image is smooth. There exist,however, three cusps �0, �1 and �2 resp. one cusp �0. The leading part of theTaylorexpansion of tp(�) þ x� at these values is, in suitable coordinates,ðt=2Þð�21�2 þ �

22Þ þ x � �, which is the second polynomial in the list below. The leading

part of the expansion of I(x, t) close to directions

x=t ¼ �rpð�jÞ

is expressed using the Pearcey integral. Proving this fact requires deeper notionsand arguments from singularity theory. Even though the technique for getting theestimates is more or less standard it requires considerable work to carry it throughfor the specific case of the cusp.


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120025491_PDE28_11-12_R1_082903

The polynomials

1

6�31 �

1

2�1�

22, ðIVÞ and

1

2ð�21 þ �1�

22Þ ðIIÞ ð3:9Þ

are discussed in Sec. 5. The oscillatory integral Ix,t with the second polynomial istransformed to the Pearcey integral. It is the model for the cusp singularity.

The inverse Fourier transforms of eitp with p from the lists

1

6�31 þ

1

6�32 ðIIIÞ,

1

2�1�

22 ðIIÞ,

1

6�31 þ �1�2 ðIÞ, ð3:10Þ

1

6�31 þ

1

2�22,

1

6�31 ðIÞ, �1�2

1

2ð�21 þ �

22Þ and

1

2�21 ð3:11Þ

will be given in the next section.

4. SOME FOURIER TRANSFORMS

Here we give the (inverse) Fourier transforms of several functions. For simplicitywe always assume t>0.

(1) Quadratic phase functions: Letffiffi�

pdenote the square root in the slit domain

C=R� and x, � 2 R. Then F�1ðeðit=2Þ�

2

ÞðxÞ ¼ ð1=ffiffiffiit

pÞe�ðx2=2tÞi. We define

KðxÞ ¼1ffiffiffiffiffiffiffi2�i

p eði=2Þx2

:

(2) The Airy function: The Airy function is defined by

AiðxÞ ¼ ð2�Þ�1=2F�1ðei�

3=3ÞðxÞ:

It is the unique bounded solution to

Ai00ðxÞ � xAiðxÞ ¼ 0 Aið0Þ ¼ 3�1=6 �ð1=3Þ

2�:

We have the estimates

jeffiffiffiffiffi�x

p 3

j ð1þ jxj1=4ÞjAiðxÞj þ ð1þ jxjÞ�1=4jAi0ðxÞj

� �� c

which can be seen by the WKB method or standard calculations.(3) Let x, � 2 R

2. Then

F�1ðexpðit�31=6ÞÞðxÞ ¼ ðt=2Þ�1=30ðx2ÞAiðx1=ðt=2Þ

1=3Þ:

(4)

F�1ðexpðitð�31=6þ �

22=2ÞÞÞðxÞ ¼

ffiffiffiffiffiffiffi2�i

pð2tÞ�5=6Aiðx1=ðt=2

1=3ÞÞKðx=

ffiffit

pÞ:

(5) F�1ðexpðitð�31 þ �1�2ÞÞ ¼ t�1eið�xð3=2Þ=t2�x1x2=tÞ, hence

F�1 exp it1

6�31 þ �1�2

� �� ¼

ffiffiffiffiffi6t

p �1eið�

ffiffi6

px32=t

2�6x1x2=tÞ:


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120025491_PDE28_11-12_R1_082903

(6) The inverse Fourier transform of expðitð1=2Þ�1�22Þ is

F�1 exp it1

2�1�

22

� �� ¼

0 if x1 01

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�tx1=2

p cosðx2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2x1=tÞ

pif x1 < 0

8<:

See also Fedoryuk (1977).(7) F�1

ðexpðit16ð�31 þ �

32ÞÞÞ ¼ 2�ðt=2Þ�2=3Aiðx1=ðt=2Þ

1=3ÞAiðx2=ðt=2Þ

1=3Þ:

5. TWO OTHER CASES

We recall the lemma of van der Corput in the form of Stein (1993).

Proposition 5.1. Suppose that k 2, f 2 Ck and f (k) 1 on R. Let 2 C10ðRÞ. ThenZ

R

eitf ð�Þ ð�Þ d�

�� ct�1=k

k 0kL1 :

This can be extended to noncompactly supported functions. In that case the integralhas to be understood as suitable limit of truncated integrals.

We begin with the prototype of a phase function where we have to contentourselves with estimates for large x. Let

Bð y, zÞ :¼

ZR

eið1=24s4þ1=2ys2þzsÞ ds ð5:1Þ

for y, z 2 R. This integral was introduced by Pearcey (1946). The level lines of itsmodulus are shown in Arnold et al. (1999). In contrast to the previous cases there isan explicit formula for the Fourier transform. We collect estimates for B below:

Lemma 5.2. We have

jBð y, zÞj � cð1þ jzj2 þ j yj3Þ�1=18½1þ ð1þ jzj2 þ j yj3Þ�5=9

jð3zÞ2 þ ð2yÞ3j�1=4:

Proof. Set

pðsÞ ¼1

24s4 þ

1

2ys2 þ zs:

Since p(4)¼ 1 the van der Corput lemma implies |B( y, z)|�C. It suffices there-fore to take

� :¼ ð3zÞ2 þ ð2j yjÞ3 1:

We normalize the integral by

~yy ¼ ��1=3y, ~zz ¼ ��1=2z; and ~ss ¼ ��1=6s,


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120025491_PDE28_11-12_R1_082903

so that, with

~ppð~ssÞ ¼1

24~ss 4 þ

1

2~yy~ss2 þ ~zz ~ss,

Bð y, zÞ ¼ �1=6ZR

ei�2=3 ~ppð ~ss Þ d ~ss

¼ �1=6Z�ð~ssÞei�

2=3 ~ppð~ss Þ d ~ssþ

Zð1� �ð~ss Þei�

2=3 ~ppð~ssÞ d ~ss

¼ �1=6½B1ð ~yy, ~zzÞ þ B2ð ~yy, ~zz Þ

where � 2 C10 is identically 1 for all arguments j~ssj � 10. Then, since the derivative of

~pp is bounded from below outside [�5, 5],

jB2ð ~yy, ~zzÞj � xk��k forall k 2 N uniformly in ~yy, ~zz:

The cubic equation

~pp0ð~ss Þ ¼1

6~ss3 þ ~yy~ssþ ~zz ¼ 0

has (for given ð ~yy, ~zzÞ 6¼ ð0, 0Þ) either one, two or three roots, depending on whetherthe discriminant

D ¼ ð3 ~zzÞ2 þ ð2 ~yyÞ3

is positive, zero, or negative. If D then,

j ~pp0ð~ss Þj þ j ~pp00ð~ss Þj cðÞ

and, by the method of stationary phase

jB1ð ~yy, ~zz Þj � c��1=3:

The third derivative of ~ff is uniformly bounded from below at zeroes of ~pp00. Hence

jB1ð ~yy, ~zzÞj � c��2=9

for all y and z. We need however a more precise estimate if D< . Arguing as forfold singularities (see Hormander (1983) and Sec. 7.5 below) we obtain

jB1ð ~yy, ~zzÞj � c��1=18ð1þ �4=9jDjÞ

�1=4

hence

jBð y, zÞj � ��1=18½1þ ��5=9

ðj3zÞ2 þ ð2yÞ3jÞ�1=4:

Remark 5.3. It is not hard to prove the estimates of Lemma 5.2 by elementarycalculation and the van der Corput lemma. The key is the algebraic estimate

j ~pp0j þ j ~pp00j cjDj1=2, ð5:2Þ


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120025491_PDE28_11-12_R1_082903

which implies the estimate by the van der Corput lemma. It suffices to verify Eq. (5.2)at critical points. There the second derivative of ~pp is of size of the square root of thediscriminant, which implies Eq. (5.2)

The cusp singularity. Here we consider pð�Þ ¼ 1=2ð�1�22 þ �

21Þ. The phase function

tpð�Þ þ x�

has degenerate critical points for x/t in a cusp. Figure 1 (a) shows the setS¼ {�|rkD2p(�)<2} together with the kernel of the Hessian D2

�p and Fig. 1 (b)shows its image and the image of close by level curves of det(D2p) under the map� ! �rpð�Þ.

Let

ItðxÞ ¼ ð2�Þ�1

ZR

2eiðt=2Þð�1�

22þ�

21Þþi�x d�:

Then

Itðx1, x2Þ ¼ t�3=4I1ðx1t�1=2, x2t

�1=4Þ ð5:3Þ

and, by performing the integration with respect to �1,ZR

2ei1=2ð�

21þ�1�

22Þþix� d� ¼ 2��1=2

ZR

ei1=2ð1=2�22þx1Þ

2þix2�2 d�2

¼ ceix21=2Bð3�1=2x1, 3

�1=4x2Þ

where B is the Pearcey integral (5.1). The estimate

kItkL1 � ct�3=4

follows from Lemma 5.2.

The homogeneous integral. pð�Þ ¼1

6�31 �

1

2�1�

22. Let

ItðxÞ ¼

Zeitpð�Þþix� d�: ð5:4Þ

where pð�Þ ¼ 1=6�31 � 1=2�1�22. Then scaling shows

ItðxÞ ¼ t�2=3I1ðxt�1=3

Þ

which motivates the definition

CðxÞ ¼ I1ðxÞ:

Proposition 5.4. The following estimates hold:

jCðxÞj � cð1þ jxjÞ�1=2

jrxCðxÞj � c:

F1


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120025491_PDE28_11-12_R1_082903

−0.5 0 0.5 1 1.5 2 2.5 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

xi

XI

Figure 1ðaÞ. The set S and the kernel of D2�p.

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5−3

−2

−1

0

1

2

3

x

y

Figure 1ðbÞ. The image of S 3 �! �rpð�Þ.


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120025491_PDE28_11-12_R1_082903

The following corollary is an immediate consequence.

Corollary 5.5. We have

jItðxÞ � ct�2=3ð1þ jxjt�1=3

Þ�1=2

and

jrxItðxÞj � ct�1

Proof of Proposition 5.4. There are many ways to prove the estimate. We choose achange of the contour of integration, because this argument carries over to theinhomogeneous phase function and large x/t with only minor changes. For thesame reason we do not make use of scaling here. Let

�ð�Þ ¼1

2j�jð�21 � �

22Þ, �

1

j�j�1�2

� �:

Then j�j� ¼ rp and

j�j2 ¼1

j�j21

4ð�21 � �

22Þ

2þ �21�

22

� �¼

1

4j�j2

and

Im pð� þ i�ð�ÞÞ ¼1

j�jjrpj2 � pð�Þ

1

4j�j3 �

1

6j�j3

1

5j�j3:

We change the contour of integration:

CðxÞ ¼

Zexpðipð� þ i�Þ þ ixð� þ i�ÞÞ d�

hence for all x

jCðxÞj

Ze�1=5j�j3þ1=2jxk�jd� <1:

We want to show that C(x) decays when x is large. Let x 6¼ 0. We identify R2 with C

and define

�xð�Þ :¼�1 � i�2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðx1 þ ix2Þ

p� ��1 � i�2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðx1 þ ix2Þ

p� �2

ffiffiffiffiffiffijxj

pþ j�j

� � ¼r�ð pð�Þ � x�Þffiffiffiffiffiffi

jxjp

þ j�j:

Let �c be one of the two critical points of �! p(�)þ x�. Then we have j�cj ¼ 2ffiffiffiffiffiffijxj

p

and

Im pð� þ i�Þ j�2 � �2c j

2

8ðffiffiffiffiffiffijxj

pþ j�jÞ

ffiffiffiffiffiffijxj

pminfj� � �cj

2, j� þ �cj2g:

Moving the domain of integration as above gives

jCðxÞj � cð1þ jxjÞ�1=2:


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120025491_PDE28_11-12_R1_082903

The estimate for the gradient is straightforward: there is an additional factor 2ffiffiffiffiffiffijxj

p.

We obtain the following result when we combine Lemma 2.1 with the previousestimates to u(t)¼S(t)u0 where S is the group defined by the differential equation(where we plug in i@xj for D)

@tu� ipðDÞu ¼ 0:

Theorem 5.6. Consider the integral (5.4), where p is the polynomial 1=6�31 � 1=2�1�22.

1. Let 1� r� q�1. The following estimates hold:

kuðtÞkLq � ct�2=3ð1=r�1=qÞku0kLr

if

1

r�1

q

3

4:

If the difference is 3=4 we require that 1< r<q<1.2. Moreover

krukL1 � cjtj�1kuð0ÞkL1 :

6. SOME STABLE MAPS

It remains to study the two polynomials p1ð�Þ ¼ 1=6�31 � 1=2�1�22þ 1=2ð�21 þ �

22Þ

and p2ð�Þ ¼ 1=6�31 þ 1=6�32 þ �1�2. Let Jið�Þ ¼ detD2�pið�Þ, i¼ 1.2. Then

J1ð�Þ ¼ 1� j�j2, J2ð�Þ ¼ �1�2 � 1

which vanish respectively on a circle or on a hyperbola. Let Si be the set of pointsx 2 R

2 where thee is a degenerate critical point of �! pið�Þ þ x�. The set Si is onedimensional. It has three cusps if i¼ 1 and it consists of two unbounded curves ifi¼ 2, one of which contains one cusp. The leading part in suitable coordinates isgiven by the cusp singularity discussed above. It will turn out that the asymptoticbehavior of the Pearcey integral (5.1) determines the asymptotic behavior of theFourier cotransforms of eitp(�), t>0, but this requires some singularity theory aswell as general results about oscillatory integrals.

More generally we will be concerned with the problem of bounding oscillatoryintegrals of the type

It, ðxÞ ¼

ZR

n ð�Þeitð f ð�Þþx��Þ d�

where 2 C10 ðR

nÞ and where f is a smooth function. Suppose that

f ð�Þ þ x � � ¼ f0ð y, �Þ þ �ð yÞ ð6:1Þ


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120025491_PDE28_11-12_R1_082903

where x¼ x( y) and �¼ �( y, �) in the support of . Then

It, ðxÞ ¼ eit�ð yÞZR

n ð�ð y, �ÞÞdet

@�

@�

� �eitf0ð y, �Þ d�:

This observation decomposes the study of oscillatory integrals into two parts:a classification of the relevant normal forms for the phase function, and anestimation of oscillatory integrals with these phase functions. In this section weclassify the phase functions which are relevant. We follow the work ofDuistermaat (1974) and Mather (1968, 1969). From the discussion above it isclear that we should (and do) replace f(�)þ x�� by more general functions f(x, �).

Definition 6.1. A function f0(x, �) is called stable at (x0, �0) if there exists ", k, and such that for every smooth function f with

k f � f0kCkðBðx0, �0ÞÞ� "

there exist local diffeomorphisms ( y, n) – (x( y), �( y, �)) (which, together with itslocal inverse, are defined in B(x0)�B(�0) and � 2 C1

ðBðx0ÞÞ satisfying (6.1) in aneighborhood of (x0, �0).

Definition 6.2. We call f0 infinitesimally stable at (x0, �0) if for every functionf 2 C1

ðRn� R

nÞ there exist smooth functions gj, hj, and � with

f ðx, �Þ ¼Xnj¼1

gjðx, �Þ@�j f0 þXnj¼1

hjðxÞ@xj f0 þ �ðxÞ

for x and � in a neighborhood of (x0, �0).Given a function f we define the set

Sf0 ¼ fðx, �Þj@�j f ðx, �Þ ¼ 0 for 1 � j � ng

Proposition 6.3. (Theorem 2.1.5 of Duistermaat (1974)). Suppose f0 is smooth near(x0, �0) and that the projection S� 3 ðx, �Þ ! x is proper. Then f0 is stable at (x0, �0) ifand only if it is infinitesimally stable at (x0, �0).

Proof. The implication stable implies infinitesimally stable is obvious. The otherdirection has been proven (in much greter generality) by Mather (1968 and 1969),and closer in the form needed here, by Duistermaat (1974). œ

It is instructive to look at examples. Let A be a symmetric nondegenerate realmatrix. Then

f0ðx, �Þ ¼1

2�tA� þ x � �

is stable for every point (x, �). To see this we assume that f is close to f0 and we haveto find g¼ (g1, . . . , gn), h¼ (h1, . . . ,hn) and �

f ðx, �Þ ¼ gtðx, �ÞðA� þ xÞ þ ht� þ �ðxÞ


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120025491_PDE28_11-12_R1_082903

in a neighborhood of (x0, �0). This is trivial if

A�0 þ x0 6¼ 0

and we restrict ourselves to A�0¼�x0 in the sequel. Let �¼ �� 0 and y¼ x� x0.Then the problem reduces to the same problem near (0, 0). For simplicity we assumethat A is the identity (otherwise we first change coordinates so that A is a diagonalwith entries � 1 in the diagonal). Let

~ff ðx, �Þ ¼ f ðx, �� xÞ:

Then it suffices to fine ~ggðx, �Þ and � such that

~ff ðx, �Þ ¼ ~ggtðx, �Þ � �þ �ðxÞ:

We set

�ðxÞ ¼ ~ff ðx, 0Þ

and

~ggðx, �Þ ¼

Z 1

0

r� f ðx, t�Þ dt:

The same arguments show that f ð�Þ þ x � � is stable at (x0, �0) if D2f(�0))ij isnondegenerate.

Now let A be a symmetric nonsingular real (n� 1)� (n � 1) matrix. Let ~�� ¼ð�1, . . . , �n�1Þ. We claim that the normal form for a fold

f0ðx, �Þ ¼1

6�3n þ

1

2~��tA ~�� þ x � �

is stable at (0, 0). We again restrict ourselves to the case A ¼ idR

n�1 .We want to verify infinitesimal stability which amounts to finding g, h, and � so

that (with f a given function)

f ðx, �Þ ¼Xn�1

j¼1

gjðx, �Þð�j þ xjÞ þ gnðx, �Þ1

2�2n þ xn

� �þ htðxÞ � � þ �ðxÞ:

We begin with the one-dimensional case. Given ~ff we want to find ~gg, ~hh, and ~�� with

~ff ðs, tÞ ¼ ~ggðs, tÞ1

2t2 þ s

� �þ ~hhðsÞtþ ~��ðsÞ

near (0, 0). This cane be done by the Malgrange preparation theorem, which is theappropriate tool in general. Here we may also use a more elementary construction.It is not hard to see that

~ff �1

2t2, t

� �¼ ~hh �

1

2t2

� �tþ ~��

1

2t2

� �

hence

~�� 1

2t2

� �¼

1

2~ff �

1

2t2, t

� �þ ~ff �

1

2t2, � t

� ��


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120025491_PDE28_11-12_R1_082903

and

~hh �1

2t2

� �¼

1

2t~ff �

1

2t2, t

� �� ~ff �

1

2t2, � t

� �� :

We extend both functions smoothly to positive arguments. The definition of ~gg isstraight forward.

We can do the same construction in the higher dimensional case as long aswe restrict ourselves to the subspace where �1. . . �n�1¼ 0. Then we repeat theconstruction for quadratic phase functions above to obtain the remaining functions.

It is clear that the same arguments imply stability for (with 3< k� nþ 1)

f0ðx, �Þ ¼1

2~��tA ~�� þ �kn þ x � � þ

Xk�3

j¼1

xj�jþ1n :

This is a normal form for a cusp if k¼ 4.Moreover, if f is stable at (x0, �0) and we add a nondegenerate phase function in

additional variables, then we obtain a stable singularity.Finally we consider

f0ðx, �Þ ¼1

2�1�

22 þ

1

2�21 þ x�:

We claim that f0 is stable at (0, 0). To see this we change variables: let

�1 ¼ �1 þ1

2�22 and �2 ¼ �2:

Then

f0ðx, �Þ ¼ ~ff ðx, �Þ ¼1

2�21 �

1

8�42 þ x1�1 �

1

2x1�

22 þ x2�2:

This function is infinitesimally stable near (0, 0). The properness assumption isclearly satisfied. Hence f0 is stable.

We collect the results:

Lemma 6.4. The following phase functions are stable at (0, 0):

(1) f ðx, �Þ ¼ �ta� þ x � � (if A is symmetric and invertible),(2) f ðx, �Þ ¼ 1=6�31 þ ��tA� þ x � � (where A is a(n� 1)� (n� 1) matrix,

symmetric and invertible),(3) f ðx, �Þ ¼ �41 � �

22 þ x2�

21 þ x � � and

(4) f ðx, �Þ ¼ 1=2�21�2 þ 1=2�22 þ x � �.

7. ESTIMATES FOR OSCILLATORY INTEGRALS

In this section we prove estimates for oscillatory integrals with degenerate phasefunctions (not necessarily polynomials).


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120025491_PDE28_11-12_R1_082903

We introduce some notation. Let U � Rn be open and f 2 C1

ðUÞ.

Definition 7.1. A (Nondegenerate). We say that f is nondegenerate at �0 if D2f isnondegenerate at �0, or, equivalently, if the map

� ! rf ð�Þ

is nondegenerate near �0. Let

J ¼ detðD2f ð�ÞÞ:

Then f is nondegenerate at �0 iff J(�0) 6¼ 0.At a degenerate point �0 we assume that J has a simple zero. As a consequence,

its zero level-set is a smooth hypersurface S, and the null-space of D2f is one-dimen-sional at all points � 2 S. Let X 2 kerðD2f Þ be a nontrivial vectorfield, defined alongS i.e., at every point in S, but not necessarily in TS.

B (Fold). We say that f has a fold at the degenerate point �0 if X is transversal to S at�0., i.e., if X � rJ 6¼ 0 at �0.

C (Cusp). We say that f has a cusp at the degenerate point �0 if

(1) X � rJ has a simple zero at �0 (as a function on S ) and:(2) Let S1 be the zero level-set of X � rJ in S. Then X is transversal to S1.

D (Global). We say that f is nondegenerate if it nondegenerate at every point. It hasat most folds if at every point it is either nondegenerate or it has a fold. It has at mostcusps if at every point it is either nondegenerate, or it has a fold or a cusp.

Remark 7.2. Nondegeneracy at �0 is a condition on the Taylor expansion up tosecond order, existence of a fold at �0 is decided by the Taylor expansion up toorder 3 and the conditions for a cusp require a Taylor expansions up to order 4.Nondegeneracy implies that the map

� ! rf ð�Þ

is invertible in a neighborhood of �0. If �0 is a fold then

S 3 �! rf ð�Þ

parameterizes a smooth hypersurface (since the derivative of (rf )|S has full rank)and the image of

� ! rf ð�Þ

lies locally on one side of S—which is a consequence of the results below. If �0 is acusp then

S1 3 � ! rf ð�Þ

parameterizes a smooth submanifold of codimension two and the image ofS 3 � ! rf ð�Þ is a geometric cusp near rf(�0), which is again a consequence ofthe considerations below.

A fold corresponds to the singularity A2 and a cusp to A3 in the notation ofArnol’d et al. (1985), Sec. 11.1.


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120025491_PDE28_11-12_R1_082903

Remark 7.3. We will apply the estimates below only to functions with two dimen-sional � variable. In that case S1 will consist of isolated point.

Remark 7.4. We shall see that the functions above are closely related to the onesdiscussed in the last section.

Theorem 7.5. Let 2 C10 ðUÞ and

It, ðxÞ ¼

ZU

ð�Þeitf ð�Þþix� d�:

1. Suppose that f is non-degenerate in the support of . Then

kIt, kL1ðRnÞ � ct�n=2 and kIt, kL1ðRnÞ � ctn=2:

2. Suppose that f has at most folds in the support of . Then

kIt, kL1ðRnÞ � ct�n=2þ1=6, kIt, kL4wðR

nÞ � ct�n=4, and kIt, kL1ðRnÞ � ctn=2:

3. Suppose that f has at most cusps in the support of . Then

kIt, kL1ðRnÞ � ct�n=2þ1=4, kIt, kL4wðR

nÞ � ct�n=4, and kIt, kL1ðRnÞ � ctn=2:

Remark 7.6. Estimates for other values of p follow by interpolation. Then for p<4the estimate for kIt, kLp are the same in all cases.

Proof. We begin with the observation that we have for every integer

jIt, ðxÞj � cNð1þ jtj þ jxjÞ�Nð7:1Þ

for x with dist(�x/t,D) 1 where D ¼ frf ð�Þ : � 2 supp g.A standard application of stationary phase shows that for the nondegenerate

phase function

jIt, ðxÞj � ct�n=2:

This implies the first statement.Using a suitable partition of unity it suffices to prove the statements for ampli-

tudes supported in a small neighbourhood of a given point. The proof of the secondstatement will be achieved by a (\,finite) sequence of lemmas.

Step 2 folds. Let �0 be a point where f has a fold. We change coordinates so that�0¼ 0, S¼ {� : J(�)¼ 0} is tangential to R

n�1 and ð0R

n�1 , 1Þ is in the null space ofD2f(0). We may and do assume that is supported in B"(0) for a small number "and f(0)¼�f(0)¼ 0. Then, as above,

jIt, ðxÞj � cN, "ð1þ jtj þ jxjÞ�N

if |x| 2"|t|.


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120025491_PDE28_11-12_R1_082903

After a linear change of coordinates there exists a symmetric nondegenerate(n� 1)� (n� 1) matrix A such that

f ð�Þ ¼1

2~��tA ~�� þ

1

6�3n þOðj ~��j2j�j þ j�nj

4Þ:

Here ~�� denotes that first n� 1 components of the vector. Now we define�� ¼ ð�1=2 ~��, �1=3�nÞ and f�(�)¼ �

�1f�(��). Then

k f�ð�Þ �1

2~��tA ~��

1

6�3nkCN ðB1ð0ÞÞ

� c�1=3:

for all N 2 N. The function f0 ¼ 1=2 ~��tA ~�� þ 1=6�3n þ x� is stable. Thus, if � is small wecan change coordinates according to the last section so that we have the phasefunction f0 and an amplitude depending on x/t and �. Then estimate follows fromLemma 7.7 below.

Lemma 7.7. Let A be a nondegenerate symmetric (n� 1)� (n� 1) matrix and let 2 C1

0 ðB1ð0Þ � B"ð0ÞÞ be a smooth compactly supported function. Then

It, ðxÞ : ¼

ZR

n ðx=t, �Þeitð1=2

~��tA ~��þ1=6�3nÞþix� d�

¼ t�n=2þ1=6 ðx=t, � ½A�1 ~xx=t, 0Þeð ~xxtA�1 ~xxÞ=2ti

ffiffiffiffiffiffiffiffiffiffiffiffidet iA

p �1Aiðxn=ð2tÞ

1=3Þ

þOðt�n=2Þ:

where ½A�1 ~xx=t, 0 denotes the obvious vector in Rn.

Proof. We may assume that x/t is small. We haveZR

n�1 ðx=t, �Þeitð1=2Þ

~��tA ~��þi ~xx� ~�� d ~��

¼ ð4�tÞð1�nÞ=2ðdetðiAÞÞ�1=2 ðx=t, � ~xx=t, �nÞe

ið ~xxtA�1 ~xxÞ=2t

þ

Z½ ðx=t, �Þ � ðx=t, � ~xx=t, �nÞe

it=ð1=2Þ ~��tA ~��þi ~xx ~��d ~��:

The second term in the right hand side is O(t�n/2). This calculation reduces theproblem to the one dimensional one:

Lemma 7.8.ZR

ðs, Þeitð1=6s3þsÞ ds ¼ ð0, 0Þt�1=3Aiðt2=3Þ þOðt�2=3

Þ

if 0 and, for <0,

ZR

ðs, Þeitð1=6s3þsÞ ds ¼

ðffiffiffiffiffiffiffiffiffiffiffiffi�=3

p, 0þ ð�

ffiffiffiffiffiffiffiffiffiffiffiffi�=3

p, 0Þ

2t�1=3Aiðt2=3Þ þOðt�2=3

Þ:


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120025491_PDE28_11-12_R1_082903

The proof of this estimate is straight forward. See also Theorem 7.7.18(Hormander, 1983) for related estimates. œ

Now, if ct�n/2� �� ct�n/2þ1/6

fx : jIt, ðxÞj > �g � fx : jxj � ct, jxnj � c��4t1�2ng

hence, since |It, (x)|� ct�n/2þ1/6 and since the estimate is obvious for smaller �,

jfx : jIt, ðxÞj > �gj � c��4t�n

and

kIt, k4L4w¼ sup �4jfx : jIt, ðxÞj > �gj � ct�n:

This is the L4w estimate. The L1 estimate is much simpler and we omit it.

Step 3, cusps. We suppose that f has a cusp at �0. We shift �0 to zero. Without loss ofgenerality we assume that f ð0Þ ¼ rf ð0Þ ¼ 0. By assumption J has a simple zero at 0.We make a linear change of coordinates so that X¼ en at �¼ 0 and rJð0Þ is amultiple of e1. Thus the tangent space of S at �¼ 0 is orthogonal to e1. LetA¼D2f(0) with components aij and ~AA ¼ ðaijÞ1�i, j�n�1. The Taylor expansion is

f ð�Þ ¼1

2aij�i�j þ

1

6cijk�i�j�k þ

1

24dijkl�i�j�k�l þOðj�j5Þ

where cijk and d i jkl are symmetric in all coefficients, the summation convention isused, and, by assumption, ain¼ ani¼ 0 for 1� i� n and ~AA is invertible. Then

J ¼ cnni�i det ~AAþ oðj�jÞ

and hence cnni¼ 0 for 2� i� n and cnni 6¼ 0. After scaling �1 we may and do assumethat cnn1¼ 1. Then

ðD2f Þij ¼ aij þ cijk�k þ1

2dijkl�k�l þOðj�j3Þ

and, since by definition on S

0 ¼ ðD2f ÞijXj ¼ aijXj þ cijk�kXj þOðj�j2Þ

and since we may set Xn¼ 1 we have (on S) for 1� i� n� 1

Xi ¼ � ~AA�1ij cnjk�k þOðj�j2Þ:

We compute

J ¼ detð ~AAÞ �1 þ1

2d mnij�i�j � ~AA�1

ij cnik�kcnjm�m þ ~AA�1

ij �k�1

� �þOðj�j3Þ

where the sum runs from 1 to n� 1 if ~AA is involved. Thus

rJ � X ¼ detð ~AAÞðd nnni�i � 3 ~AA�11j c

njk�kÞ þOðj�j2Þ:


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120025491_PDE28_11-12_R1_082903

The singularity is a cusp provided

d nnnn6¼ 3 ~AA�1

11 :

We set

�i ¼ �i þ1

2~AA�11i �

2n, �n ¼ �n

which gives

f ð�Þ ¼1

2aij�i�j þ

1

24ðd nnnn

� 3 ~AA�111 Þ�

4n þOðj�j5 þ j ~��j2j�jÞ

and

f ð�Þ þx

t� � ¼

1

2aij�i�j þ

1

24ðd nnnn

� 3 ~AA�111 Þ�

4n þ

x

t� ��

1

4�2n ~AA�1

1i

xitþOðj�j5 þ j ~��j2j�jÞ:

For notational simplicity we restrict to ~AA ¼ 1R

n�1 and ðd nnnn� 3A�1

11 Þ ¼ 1. The lead-ing part is stable by Lemma 6.4. Scaling ~��i ¼ �1=2�i for 1� i� n� 1 and ~��n ¼ �1=4�nwe see as above that, at the expense of having an amplitude � dependingon y ¼ �ðx=tÞ as well, we may suppose that we have the phase function

f ð�Þ þ x=t � � ¼1

2j ~��j2 þ

1

24�4n þ

1

2y1�

2n þ y � �:

We integrate over ~�� and obtain for a suitable function �

It, ðxÞ ¼ ðitÞ�ðn�1Þ=2eitj ~yyj2=2

ZR

�ð y, ½ � ~yy, sÞeitð1=24s4þ1=2y1s

2þynsÞ dsþOðt�n=2

Þ:

We continue with the proof of Theorem 7.5, Step 3. Let

Mt, � ¼ fx : jIt, j > �g:

By Eq. (7.1) we have for n 2, 1� p� 8, and some c>0

Z ðctÞ�1

�p�1jMt, � \ ðR

nnBctð0Þjc� � cNt

�N :

where |.| denotes the volume. Clearly Mt,� is empty if �>> t�n/2þ1/4.The L4

w estimate follows from

jMt, � \ Btð0Þj � ct�n��4: ð7:2Þ

This estimate is trivial for �� t�n/2 and it remains verify Eq. (7.2) for

c�1t�n=2� � � ct�n=2þ1=4: ð7:3Þ

According to Duistermaat (1974) there exists a smooth function � with compactsupport and new coordinates y ¼ �ðx=tÞ with

det@y

@x

� �� t�n


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120025491_PDE28_11-12_R1_082903

such that locally

It, ðxÞ ¼ t�n=2þ1=4�ð ~xx=tÞeit ~yyt ~AA ~yyBðt1=2y1, t

3=4ynÞ þOðt�n=2Þ:

where B is the Pearcey integral (5.1). The O(t�n/2) term is easily dominated.After localization and a change of coordinates it suffices to show for

~MMt, � ¼ f y 2 R2k yj � 1, jBðt1=2y1, t

3=4y2jÞ > t2n�1=4�g:

with

t�n=2� � � t�n=2þ1=4

the estimate

j ~MMt, �j � ct�2n��4: ð7:4Þ

We set R¼ z2þ |y|3 and T¼ z2þ y3. By Lemma 5.2 it suffices to prove

kfð y, zÞ 2 B1ð0ÞjR�1=18

½t�2=3þ R�5=9

jT j�1=4 > �gj � c��4:

for

1 << � � t1=4:

This set is contained in

A1 ¼ fð y, zÞkz2 þ y3j � ��4R1=3g, and in A2 ¼ fð y, zÞjðz2 þ j yj3Þ < ��24=5

g:

The second set has size

jA2j � ��4

and, if ð y, zÞ 2 A1, for fixed |z| 1/2��2, then y lies in an interval of size

�y � ��4z�2=3:

This implies Eq. (7.4), and hence the L4w estimate. The L1 and L8 estimate are simple

consequences. œ

8. THE REMAINING CASES

The classification of third order polynomials has led to the list (3.8)–(3.11). Wehave seen that there are explicit formulas for the oscillatory integrals of Sec. 2 for pas in Eqs. (3.10) and (3.11). The polynomial in list (3.9) have been studied above.It remains to study the oscillatory integrals with p in the list (3.8).

A. The case pð�Þ ¼ 1=6�31 � 1=2�1�22 þ 1=2ð�21 þ �

22Þ: This is a second order

perturbation of Eq. (5.4). The second order term changes the long term asymptotics.


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120025491_PDE28_11-12_R1_082903

The degenerate singularity of the previous example is broke up into three cusps,see Fig. 1 (c) for S and the null space D2

�p and (d) for the image of �rp|s andcloseby lines.

We begin with a discussion of the mapping properties of � ! rp and of degen-erate points. We have detðD2

�pÞ ¼ 1� j�j2, which vanishes on the circle of radius 1.There are three points where the kernal of D2

�p is tangent to that curve. Those are themost degenerate sets. They are mapped to three points (�1, 0), ð1=2,

ffiffiffi3

p=2Þ, and

ð1=2, �ffiffiffi3

p=2Þby � ! �rp in the x space, which are connected through the image

of the arcs of the circle. Every point outside has two preimages (by mapping degreearguments) and every point inside has four preimages. The map of the circle to thatcurve is a homeomorphism. This is discussed in detail in Arnol’d et al. (1985).

Suppose that |x|/t� 3. Let � 2 C10 ðR

2Þ, �ð�Þ ¼ 1 for |�|� 10 and

It,�ðxÞ ¼

Z�ð�Þeitpð�Þþx�� d�:

Then by Theorem 7.5

kIt,�kL1 � cð1þ tÞ�3=4

and

kIt,�kL4!� cð1þ tÞ�1=2:

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

xi

XI

Figure 1ðcÞ. The set S and the kernel of D2�p:


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120025491_PDE28_11-12_R1_082903

We claim that for the points x under considerationZð1� �ð�ÞÞeitpð�Þþx�� d�

�� ct�N :

Indeed, let (�) be supported in B(0, 20) and (�)¼ 1 for |�|� 10. We define

� ¼ ð1� ð�ÞÞrpð�Þ

1þ j�j

and deform the contour of integration. We decompose the integral into one where wemultiply the integrand by (./2) and one where we multiply it with 1� (./2). Thereis no critical point of the phase function in the support of the amplitude. Hence thisintegral decays fast. The other integrand is bounded by

ce�tðj�jþ1Þ

which decays exponentially in t.If |x|/ 3 we use the same arguments as in the previous section, replacing

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix1 þ ix2

pby the roots of rp(�)¼�x/t. Here we mix real and complex notation.

We collect all estimates:


jItðxÞj � cminft�3=4, t�2=3g,

jrxItðxÞj � cmax½t�1, t�3=4g:

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5

−2

−1.5

1

−0.5

0

0.5

1

1.5

2

x

y

Figure 1ðd Þ. The image under S 3 �! �rpð�Þ:


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120025491_PDE28_11-12_R1_082903

If |x|/t 3 then

jItðxÞj � ct�1=2jxj�1=2:

We have

kItkL4w� ct�ð1=2Þ:

Theorem 8.2. Part 1 of Theorem 5.6 holds here. Moreover

kruðtÞkL1 � cðjtj�1þ jtj�3=4

Þkuð0ÞkL1 :

B. The polynomial ð1=6Þð�31 þ �32Þ þ �1�2. This is a second order perturbation of

ð1=6Þ�31 þ ð1=6Þ�22. The second order terms lead to a cusp, see Fig. 1(e) and 1(f ).

ItðxÞ ¼1

2�

Zeitðð1=6Þ�

31þð1=6Þ�32þ�1�2Þþix� d�:

The map � ! rpð�Þ is degenerate at the hyperbola, �1�2¼ 1. The kernel of D2�p is

tangential to the hyperbola only at (1, 1). The hyperbola is mapped to two curves inthe x space with a singularity at (�1.5,�1.5). We denote these curves by ��.

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5

xi

XI

Figure 1ðeÞ. The set S and the kernel of D2�p.


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120025491_PDE28_11-12_R1_082903


kItkL1 � cminft�2=3, t�3=4g:

Moreover, for all p>4

kItkLp � ct�ð2=3Þþ2=3p:

Proof. We consider first the case |x|/t� 5. Then

It,�ðxÞ ¼1

2�

Z�ð�Þeitðð1=6Þ�

31þð1=6Þ�32þ�1�2Þþix� d�

is controlled by Theorem 7.5. Now

1

2�

Zð1� �ð�ÞÞeitðð1=6Þ�

31þð1=6Þ�32þ�1�2Þþix�d�

¼ �1

2�t

Zeitðð1=6Þ�

31þð1=6Þ�32þ�1�2Þþix�

r �ð1� �ð�ÞÞð�21, �

22Þ

12�41 þ

12�42 þ �1�

22 ¼ �2�

21 þ x1�

11 þ x2�

22

d�:

It is not hard to see that the integrand is integrable, which yields the decay t�1.The argument could be iterated to yield cNt

�N. For t� 1 we apply a similarargument, but with cutoff function

�ð�Þ ¼ �ð�t1=3Þ:

−6 −4 −2 0 2 4

−6

−4

−2

0

2

4

x

y

Figure 1ðf Þ. The image under S 3 �! �rpð�Þ.


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120025491_PDE28_11-12_R1_082903

Then the argument gives

jI j � ct�2=3:

It remains to study the decay if |x| 5t. We may assume that |x1| |x2|.Let R¼ |x|/t, y¼ x/R, � ¼ �=

ffiffiffiffiR

p. Then it suffices to consider

It,�ðxÞ ¼ R

Z�ð�ÞeitR

3=2ð�31þ�

22þR�1=2�1�2þy��Þ d�

since the integral with amplitude 1� �ð�Þ can be estimated by cjxj�1=2t�1=2 by thesame arguments as above. The estimate follows now from Theorem 7.5 by a tediouscomputation. œ

Theorem 8.4. The estimate

kuðtÞkLq � ct�ð2=3Þð1=p�1=qÞkuð0ÞkLp

holds for p and q satisfying the strict inequalities of Theorem 5.6.

9. STRICHARTZ ESTIMATES AND CONSEQUENCES

The (local) Strichartz estimates follow by standard arguments (complex inter-polation, TT * arguments and the Hardy-Littlewood-Sobolev inequality) from thelocal estimate

jItðxÞj � ct�� for x 2 R2, t 2 ð0,T : ð9:1Þ

A pair (q, r) is admissible if (for Rn)

2 � r <1,1

q¼ �

1

2�1

r

� �:

If � denotes the operator

�ð f ÞðtÞ ¼

Z t

0

Sðt� sÞ f ðsÞ ds for 0 < t � T

then for any admissible pairs (q, r) and ð �qq, �rrÞ one has

sup0<t�T

k�f ðtÞkL2 þ k�f kLqð½0,T ;LrðR2ÞÞ � ck f kL �qq0 ð½0,T ;L�rr0 ðR2ÞÞ

where �qq0 and �rr0 are the Holder conjugate exponents of �qq and �rr, and

kSð�Þ�kLqð½0,T ;LrðR2ÞÞÞ � ck�kL2ðR2Þ:

For instance estimate (9.1) holds with �¼ 2/3 if p(�) is one of the followingpolynomials

1

6�31 �

1

2�1�

22,

1

6ð�31 þ �

32Þ,

1

6�31 �

1

2�1�

22 þ

1

2ð�21 þ �

22Þ,

1

6ð�31 þ �

32Þ þ �1�2: ð9:2Þ


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120025491_PDE28_11-12_R1_082903

It holds with �¼ 3/4 if pð�Þ ¼ ð1=2Þð�1�22 þ �

21Þ, with �¼ 5/6 if pð�Þ ¼ �31=6þ �

22 and

with �¼ 1 if pð�Þ ¼ ð1=6Þ�31 þ �1�2. For these polynomials we obtain the admissiblepairs and the corresponding Strichartz estimates by the procedure described above.

We now give a simple application of the Strichartz estimates to the Cauchyproblem for the Shrira system (1.3). A partial result has been obtained in Ghidagliaand Saut (1993). The idea was to use the conservation laws (see Proposition 3.6in Ghidaglia and Saut (1993))

d

dt

ZR

2jAj2 dx dz ¼ 0 and

d

dt

ZR

2jAsj

2 dx dz ¼ 0

together with the Strichartz estimates.In Ghidaglia and Saut (1993) only two very special cases were considered,

corresponding to

pð�Þ ¼ �31 þ �32, and pð�Þ ¼

1

6�31 þ

1

2�22:

We obtain

Theorem 9.1. Let

L ¼!kk

2

@2

@x2þ !nn

@2

@z2þ!nk

2

@2

@x@z

� i!kkk

6

@3

@x3þ!kkn

2

@3

@x2@zþ!knn

2

@3

@x@z2þ!nnn

6

@3

@z3

" #

and assume that the symbol is equivalent to a phase p with � 2/3.Then, if A0 2 L2

ðR2Þ and @sA0 2 L2

ðR2Þ there exists a unique solution of Eq. (1.3)

with initial data A0 linear part L such that

A 2 CðRþ;L2ðR

2ÞÞ \ L�locðR

þ,L6ðR

2ÞÞ;

@A

@s2 CðRþ;L

2ðR

2ÞÞ,

where

� ¼9

2if � ¼

2

3, � ¼

18

5if � ¼

5

6, � ¼ 4 if � ¼

3

4, � ¼ 3 if � ¼ 1: ð9:3Þ

Proof. It results immediately from the proof of Theorem 3.8 in Ghidaglia and Saut(1993) by using the Strichartz estimates explained above. œ

Remark 9.2. A straight forward calculation shows that the second order part of J is

ð!kkk!knn � !2kknÞ�

21 þ ð!kkk!nnn � !kkn!knnÞ�1�2 þ ð!knn!nnn � !

2knnÞ�

22:


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120025491_PDE28_11-12_R1_082903

This is a quadratic form with determinant

D ¼ ð!kkk!knn � !2kknÞð!knn!nnn � !

2knnÞ � ð!kkk!nnn � !kkn!knnÞ:

The number D is a function of �. If D 6¼ 0 then the polynomial is equivalent toone of the four polynomial with �¼ 2/3. Numerically one sees that there are twoangles (<�/2) where D vanishes: � ¼ 0, and � � 7:405. In both cases the normalform for the polynomial is

1

2ð�21 þ �1�

22Þ:

Thus we obtain global existence for all cases of the Shrira system.

ACKNOWLEDGMENT

The work of M. Ben-Artzi was partially supported by a grant from the IsraelScience Foundation – Basic research.

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Received July 2002Revised May 2003


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120025491_PDE28_11-12_R1_082903

dispersion estimates for third order equations in two...

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