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The Cauchy Problem for Higher-Order KP Equations
journal of differential equations 153, 196222 (1999) Article ID jdeq.1998.3534, available online at http:www.idealibrary.com on
The Cauchy Problem for Higher-Order KP Equations J. C. Saut and N. Tzvetkov Analyse numerique et EDP, Universite de Paris Sud, Ba^t. 425, 91405 Orsay Cedex, France Received April 13, 1998; revised July 7, 1998
We study the local well-posedness of higher-order KP equations. Our well-posedness results make an essential use of a global smoothing effect for the linearized equation established in Ben-Artzi and Saut (preprint, 1997), injected into the framework of Fourier transform restriction spaces introduced by Bourgain. Our illposedness results rely on the existence of solitary wave solutions and on scaling arguments. The method was first applied in the context of the KdV and Schrodinger equations (cf. Birnir et al., Ann. Inst. H. Poincare Anal. Non Lineaire 13 (1996), 529535; Birnir et al., J. London Math. Soc. 53 (1996), 551559). 1999 Academic Press
1. INTRODUCTION This paper is concerned with the Cauchy problem associated to KadomtsevPetviashvili (KP) equations having higher order dispersion in the main direction of propagation. Such equations occur naturally in the modeling of certain long dispersive waves (cf. [2, 15, 16]). The study of their solitary wave solutions was done in [6, 7]. Thus we consider the Cauchy problems in R d, d=2, 3
{
(u t +:u xxx +;u xxxxx +uu x ) x +u yy =0 u(0, x, y)=,(x, y)
(1)
in the two dimensional case and (u t +:u xxx +;u xxxxx +uu x ) x +u yy +u zz =0
{u(0, x, y, z)=,(x, y, z)
(2)
in the three dimensional case. The ``usual'' KP equations correspond to ;=0 and :=&1 (KP-I) or :=+1 (KP-II). We are interested in the local well-posedness of the Cauchy problems (1) and (2). Following [4] and [13] we introduce the notion for well-posedness which will be convenient for our purposes. 196 0022-039699 30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved.
CAUCHY PROBLEM FOR KP EQUATIONS
197
Definition 1. The Cauchy problem (1) (resp. (2)) is locally well-posed in the space X if for any , # X there exists T=T(&,& X )>0 (T is a nondecreasing continuous function such that lim \ Ä 0 T(\)= ) and a map F from X to C([0, T ]; X) such that u=F(,) solves the equation (1) (resp. (2)) and F is continuous in the sense that &F(, 1 )&F(, 2 )& L ([0, T ]; X ) M(&, 1 &, 2 & X , R) for some locally bounded function M from R +_R + to R + such that M(S, R) Ä 0 for fixed R when S Ä 0 and for , 1 , , 2 # X such that &, 1 & X +&, 2 & X R. We introduce the nonisotropic Sobolev spaces H s1 , s2 (R d ) equipped with the norm
|
&u& 2H s1 , s2 = ( !) 2s1 ( ') 2s2 |u^(!, ')| 2 d! d', where ( } ) =(1+ | } | 2 ) 12 and '=(' 1, ' 2 ) in the case of three space dimensions. By H4 s1 , s2 (R d ) we shall denote the homogeneous nonisotropic Sobolev spaces equipped with the norm
|
&u& 2H4 s1 , s2 = |!| 2s1 |'| 2s2 |u^(!, ')| 2 d! d'. Taking into account the specific structure of the KP-type equations (cf. [18]) we introduce the modified Sobolev space H s1 , s2 (R d ) equipped with the norm
|
&u& 2H s1, s2 = (!) 2s1 ( ') 2s2 (1+ |!| &1 ) 2 |u^(!, ')| 2 d! d'. Note that any u # H s1 , s2 (R d ) has (formally) a zero x mean value. We shall denote H 0, 0 by L 2. For b, s 1 , s 2 # R we define X b, s1 , s2 (R d+1 ) to be the completion of the functions of C 0 with zero x mean value with respect to the norm
|
&u& 2X b, s1 , s2 = ({+ p(!, ')) 2b ( !) 2s1 ( ') 2s2 (1+ |!| &1 ) 2 |u^({, !, ')| 2 d{ d! d', where p(!, ')=;! 5 &:! 3 +
|'| 2 . !
If : or ; vanishes then the equations (1) and (2) are scale invariant. If u(t, x, y) is a solution of (1) with ;=0 then so is u *(t, x, y)=* 2u(* 3t, *x, * 2y)
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and &u *(t, .)& H4 s1 , s2 =* s1 +2s2 +12 &u(* 3t, .)& H4 s1 , s2 . Hence one may expect local well-posedness (resp. ill-posedness) in H s1 , s2 of (1) for ;=0 when s 1 +2s 2 >&12 (resp. s 1 +2s 2 <&12). Similarly if u(t, x, y, z) is a solution of (2) with ;=0 then so is u *(t, x, y, z)=* 2u(* 3t, *x, * 2y, * 2z) and &u *(t, .)& H4 s1 , s2 =* s1 +2s2 &12 &u(* 3t, .)& H4 s1 , s2 . Hence one may expect local well-posedness (resp. ill-posedness) in H s1 , s2 of (2) for ;=0 when s 1 +2s 2 >12 (resp. s 1 +2s 2 <12). If u(t, x, y) is a solution of (1) with :=0 then so is u *(t, x, y)= * 4u(* 5t, *x, * 3y) and &u *(t, .)& H4 s1 , s2 =* s1 +3s2 +2 &u(* 5t, .)& H4 s1 , s2 . Hence one one may expect local well-posedness (resp. ill-posedness) in H s1 , s2 of (1) for :=0 when s 1 +3s 2 >&2 (resp. s 1 +3s 2 <&2). Similarly if u(t, x, y, z) is a solution of (2) with :=0 then so is u *(t, x, y, z)=* 4u(* 5t, *x, * 3y, * 3z) and &u *(t, .)& H4 s1 , s2 =* s1 +3s2 +12 &u(* 5t, .)& H4 s1 , s2 . Hence one may expect local well-posedness (resp. ill-posedness) in H s1 , s2 of (2) for :=0 when s 1 +3s 2 >&12 (resp. s 1 +3s 2 <&12). Now we state the well-posedness result. The proof uses the Fourier transform restriction norms introduced by Bourgain [810], in the context of Schrodinger, KdV or KP equations. In the nonlinear estimates we shall make an essential use of the smoothing effects for the linear group associated to (1) or (2) established in [3]. Theorem 1. Let d=2 and ;<0, : # R. Then for any s 1 &14, s 2 0 the Cauchy problem (1) is locally well-posed in H s1 , s2. Moreover there exists b>12 such that the solution u(t, x, y) satisfies u # X b, s1 , s2. Let d=3 and ;<0, : # R. Then for any s 1 &18, s 2 0 the Cauchy problem (2) is locally well-posed in H s1 , s2. Moreover there exists b>12 such that the solution u(t, x, y, z) satisfies u # X b, s1 , s2. If ;=0 and :>0 (the ``usual'' KP-II equation) local well-posedness of (1) in L 2 is established in [10]. The proof uses Fourier transform restriction norms, a dyadic decomposition related to the structure of the symbol of the linearized operator and can be performed for periodic initial data. Local well-posedness of KP-II in H s1 , s2, s 1 >&14, s 2 0 is established in [20]. The sign of ; is crucial in the proof of Theorem 1. We do not know of a similar result when ;>0 (KP-I type equations). In this case, however, it is easy to obtain global weak solutions by energy methods (cf. [19]). The uniqueness of such solutions is unknown. Note that L 2 norms of the local solutions of (1) and (2) are conserved. Lemma 1. If u is a local solution of (1) or (2) in the time interval [0, T ] then for any t # [0, T ] &u(t, .)& L2 =&,& L2 .
CAUCHY PROBLEM FOR KP EQUATIONS
Proof.
199
It suffices to multiply (1) or (2) with u and integrate over R d.
Using Theorem 1 with s 1 =s 2 =0 and Lemma 1 we obtain Theorem 2. Let ;<0, : # R. Then for any , # L 2(R 2 ) (resp. L 2(R 3 )) there exists a unique global solution of the Cauchy problem (1) (resp. (2)). In [4] it is shown that the solitary wave solutions can be used to construct examples proving local ill-posedness for the (generalized) KdV equation. One considers the limit of the solitary wave solutions as the propagation speed tends to infinity. In the favorable cases the initial data tends weakly to a nonzero distribution (for example the Dirac delta function for the KdV equation with cubic nonlinearity) while the solution at time t>0 tends weakly to zero. Using the idea of [4, 5] together with the properties of the solitary waves of KP-I type equations (cf. [6, 7]), we can prove the next Theorem. Theorem 3. Let d=2. If :<0 and ;=0 then (1) is locally ill-posed in H s, 0 for s<&12. Let d=3. If :<0 and ;=0 then (1) is locally ill-posed in H4 s1 , s2 for s 1 +2s 2 =12, s 1 0, s 2 0. If :=0 and ;>0 then (2) is locally ill-posed in H s, 0 for s<&12. Note that our well-posedness or ill-posedness results do not contradict with the scaling argument. In fact the region [(s 1 , s 2 ): s 1 &14, s 2 0] is a subset of the region suggested by the scaling argument [(s 1 , s 2 ): s 1 +3s 2 +2>0], where well-posedness is expected to hold in the case of two space dimensions when :=0, ;<0. The same holds for the region [(s 1 , s 2 ): s 1 &18, s 2 0] which is a subset of [(s 1 , s 2 ): s 1 +3s 2 +12>0] in the case of three space dimensions. If d=2, :<0 and ;=0 then ill-posedness holds for the pairs (s, 0), s< &12 which can achieve the ``critical'' line s 1 +2s 2 +12=0, suggested by the scaling argument. If d=3, :<0 and ;=0 then ill-posedness holds for the pairs [(s 1 , s 2 ): s 1 +2s 2 =12, s 1 0, s 2 0] which are a part of the line suggested by the scaling argument. If d=3, :=0 and ;>0 then the pairs (s, 0), s<&12 can achieve the line s 1 +3s 2 +12=0, suggested by the scaling argument. The method for obtaining local well-posedness works just for KP-II type equations (i.e., ;<0 or ;=0, :>0) since one can recuperate the derivative x in the non-linear term by the aid of an algebraic relation for the symbol of the linearized operator. (cf. (16) below). In the case of KP-I type equations (;>0 or ;=0, :<0) we can not gain the x derivative using the relation (16). However, using the parabolic regularisation method it is possible to prove local well-posedness of (1) in H s, s>2 and local wellposedness of (2) in H s, s>52 (cf. for instance [14] for an illustration of this method). The proof does not make use of the specific structure of the
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KP-type equations and could be performed for quite general evolution equations. The conditions for s are in order to control the L norm of the gradient of the solution and they seem to be very restrictive. Hence there is still no satisfactory theory for the local well-posedness of the Cauchy problem for KP-I type equations. On the other hand our ill-posedness results are only valid for KP-I type equations since the existence of solitary wave solutions is used in an essential way. In [6] using Pohojaev type identities it is shown that the KP-II type equations do not possess localized solitary wave solutions. Note also that we are forced to take : or ; zero in order to keep the scale invariance of the equation. The paper is organized as follows. In Section 2 we recall the global smoothing effects for (1) and (2) established in [3]. Then we inject these estimates into the framework of the Bourgain spaces associated to (1) and (2). In Section 3 we prove the crucial nonlinear estimate. In Section 4 we perform a fixed point argument to complete our well-posedness result. Section 5 is devoted to the ill-posedness of (1) and (2). In Section 6 we remark that the ill-posedness of the (generalized) KdV equations does not depend on the special form of the solitary waves. Finally in an appendix we give the proof of a decay estimate for a fractional derivative of the ``lump'' solitary wave of KP-I equation.
2. ESTIMATES FOR THE LINEAR EQUATION Let us consider the linear initial value problem associated to (1) or (2) i t u= p(D) u
{u(0, x, y)=,,
(3)
where D=(D 1 , D 2 ), when d=2, D=(D 1 , D 2 , D 3 ), when d=3 D 1 =&i x , D 2 =&i y and D 3 =&i z . We recall that p(!, ')=;! 5 &:! 3 +(|'| 2!). We shall denote by U(t)=exp(&ip(D)) the unitary group which generates the solutions of (3). We have the following representation for the solutions of (3), when d=3 u(t, x, y, z)=U(t) ,=G t V ,, where G t is the oscillatory integral G t(x, y, z)=c
|
exp(itp(!, ')+i(x!+ y' 1 +z' 2 )) d! d'. R3
CAUCHY PROBLEM FOR KP EQUATIONS
201
A similar representation for the solutions of (3) holds when d=2. We have in fact a global smoothing effect for the solutions of (3). Lemma 2 (cf. [3], Theorem 4.2). Let $(r)=d( 12 &(1r)). Then the following estimates hold &|D x | $(r)2 U(t),& L qt (L r(x, y) ) c &,& L2 ,
(d=2),
&|D x | $(r)6 U(t) ,& L qt (L r(x, y, z) ) c &,& L2 ,
(d=3),
provided 2 =$(r). q Next we shall prove an inequality which will be used in the proof of the nonlinear estimates. Lemma 3. Let d=2 and 0<% 1,
= 1 >0,
0 b 12+= 1 .
Then the following inequality holds &F &1( |!| * ( {+ p(!, ')) &b |u^({, !, ')| ); L qt(L r(x, y) )& c &u& L2 ,
(4)
where %b 2 =1& , q 12+= 1
$(r)=
(1&%) b , 12+= 1
*=
$(r) b . 2(12+= 1 )
Let d=3 and 0<% 1,
= 1 >0,
0 b 12+= 1 .
Then the following inequality holds &F &1( |!| * ({+ p(!, ')) &b |u^({, !, ')| ); L qt(L r(x, y, z) )& c &u& L2 ,
(5)
where %b 2 , =1& q 12+= 1 Proof.
$(r)=
(1&%) b , 12+= 1
*=
$(r) b . 6(12+= 1 )
Let d=3. Using Lemma 2 and [12], Lemma 3.3 we obtain &|D x | $(r)6 u; L qt(L r(x, y, z) )& c &u; X 12+=1 , 0, 0 &,
(6)
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SAUT AND TZVETKOV
provided 2 =$(r). q Interpolating between (6) and &u; L 2t (L 2(x, y, z) )& &u; X 0, 0, 0 &, we obtain &|D x | * u; L qt(L r(x, y) )& c &u; X b, 0, 0 &,
(7)
provided %b 2 , =1& q 12+= 1
$(r)=
(1&%) b , 12+= 1
*=
$(r) b . 6(12+= 1 )
But (7) is equivalent to (5) which completes the proof of Lemma 3 when d=3. If d=2 then the arguments are the same. Corollary 1. Let d=2. Then the following inequalities hold &F &1( |!| 14 ( {+ p(!, ')) &b |u^({, !, ')| ); L 4t (L 4(x, y) )& c &u& L2 ,
(8)
where b>12. &F &1(( {+ p(!, ')) &b |u^({, !, ')| ); L qt(L 2(x, y) )& c &u& L2 ,
(9)
where 2 q , b 0 and (bq(q&2))> 12 . Let d=3. Then the following inequalities hold 4 &F &1(|!| 18 ( {+ p(!, ')) &b |u^({, !, ')| ); L 83 t (L (x, y, z) )& c &u& L2 , (10)
where b>12. &F &1(( {+ p(!, ')) &b |u^({, !, ')| ); L qt(L 2(x, y, z) )& c &u& L2 ,
(11)
where 2 q , b 0 and (bq)(q&2)> 12 . Proof. The proof of (8) follows from Lemma 3 (d=2) applied with %=12 and b=12+= 1 . Lemma 3 (d=3) applied with %=14 and b=12 += 1 yields (10). To prove (9) and (11) we apply Lemma 3 with %=1.
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CAUCHY PROBLEM FOR KP EQUATIONS
3. AN ESTIMATE FOR THE NONLINEAR TERM Lemma 4. Let d=2 and s 1 &14, s 2 0. Then for sufficiently small = we have &uu x & X &12+3=, s1 , s2 c &u& 2X 12+2=, s 1 , s2 .
(12)
Let d=3 and s 1 &18, s 2 0. Then for sufficiently small = we have &uu x & X &12+3=, s1 , s2 c &u& 2X 12+2=, s1 , s2 .
(13)
Proof. We shall give the proof only when d=3. In the case of two space dimensions the arguments are similar. We set w^({, !, ')=( {+ p(!, ')) 12+2= ( !) s1 ( ') s2 u^({, !, '), _ :=_({, !, ')={+ p(!, '),
_ 1 :=_({ 1 , ! 1 , ' 1 ),
_ 2 :=_({&{ 1 , !&! 1 , '&' 1 ). Let `=(!, '). Then (13) is equivalent to
"( _)
&12+3=
( !) 1+s1 (') s2
|| K({, `, { , ` ) w^({ , ` ) w^({&{ , `&` ) d{
_
1
1
1
1
1
1
1
d` 1
"
2
L ({, `)
c[&w& 2L2 +&w& L2 &|D x | &1 w& L2 +&|D x | &1 w& 2L2 ],
(14)
where K({, `, { 1 , ` 1 )=
( ! 1 ) &s1 ( !&! 1 ) &s1 ( ' 1 ) &s2 ( '&' 1 ) &s2 . (_ 1 ) 12+2= ( _ 2 ) 12+2=
Hence by the self-duality of L 2 we obtain that (14) is equivalent to
}|| K ({, `, { , ` ) w^({ , ` ) w^({&{ , `&` ) v^({, `) d{ d` d{ d` } 1
1
1
1
1
1
1
1
1
c[&w& 2L2 +&w& L2 &|D x | &1 w& L2 +&|D x | &1 w& 2L2 ] &v& L2 ,
(15)
where K 1({, `, { 1 , ` 1 )=
( !) 1+s1 ( ! 1 ) &s1 ( !&! 1 ) &s1 ( ') s2 . 12+2= 12+2= 12&3= s2 ( _ 1) ( _2) ( _) ( ' 1 ) ( '&' 1 ) s2
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SAUT AND TZVETKOV
Since for s 2 0 ( ') s2 const, ( ' 1 ) s2 ( '&' 1 ) s2 we shall suppose that s 2 =0 from now on. Without loss of generality we can assume that w^ 0 and v^ 0. We have the following relation, where '=(' 1, ' 2 ) and ' 1 =(' 11 , ' 21 ) _ 1 +_ 2 &_=&5;! 1 !(!&! 1 )(! 2 &!! 1 +! 21 )+3:! 1 !(!&! 1 ) +
(! 1 ' 1 &!' 11 ) 2 (! 1 ' 2 &!' 21 ) 2 + . ! 1 !(!&! 1 ) ! 1 !(!&! 1 )
(16)
Lemma 5. There exists a positive constant c 0 such that max[ |_|, |_ 1 |, |_ 2 | ] c |! 1(!&! 1 ) ! 3 |, Proof.
for
|!| c 0 .
(17)
Since ;<0 we have that 2 3 max[|_|, |_ 1 |, |_ 2 | ] |! 1 !(!&! 1 )| (& 15 4 ;! +3:),
where we used the elementary inequality ! 2 &!! 1 +! 21 34 ! 2. If :>0 then 2 we obtain (17) with c 0 =0. If :<0 then for & 15 8 ;! +3: 0 we have 2 3 max[ |_|, |_ 1 |, |_ 2 | ] |! 1 !(!&! 1 )| (& 15 8 ;! ),
which completes the proof of Lemma 5. We now continue the proof of Lemma 4. We shall denote by J the integral in the left-hand side of (15). We shall divide the domain of integration taking into account which term dominates in the left-hand side of (17). By symmetry we can assume that |_ 1 | |_ 2 |. Case 1. |!| max(2, c 0 ). We denote by J 1 the restriction of J on this region. In this case we have that K 1({, `, { 1 , ` 1 )
( ! ) 18 ( !&! 1 ) 18 . ( _1) ( _ 2 ) 12+2= (_) 12&3= 1 12+2=
CAUCHY PROBLEM FOR KP EQUATIONS
205
Case 1.1. |! 1 | 1, |!&! 1 | 1. We denote by J 11 the restriction of J on this region. In this case we have that K 1({, `, { 1 , ` 1 )
|! 1 | 18 |!&! 1 | 18 . ( _ 1 ) 12+2= ( _ 2 ) 12+2= (_) 12&3=
Hence using Holder inequality (10) and (11) we obtain J 11 &F &1(( _) &12+3= v^({, `))& L 4t (L 2(x, y, z ) ) 4 _&F &1(|! 1 | 18 ( _ 1 ) &12&2= w^({ 1 , ` 1 ))& L83 t (L (x, y, z) ) 4 _&F &1(|!&! 1 | 18 ( _ 2 ) &12&2= w^({&{ 1 , `&` 1 ))& L 83 t (L (x, y, z) )
c &w& 2L2 &v& L2 . Case 1.2. |! 1 | 1. We denote by J 12 the restriction of J on this region. In this case we have that K 1({, `, { 1 , ` 1 )
|! 1 | 18 |!&! 1 | 18 . |! 1 | ( _ 1 ) 12+2= ( _ 2 ) 12+2= ( _) 12&3=
Hence using Holder inequality (10) and (11) we obtain J 12 &F &1(( _) &12+3= v^({, `))& L 4t (L 2(x, y, z) ) 4 _&F &1( |! 1 | 18 ( _ 1 ) &12&2= |! 1 | &1 w^({ 1 , ` 1 ))& L 83 t (L (x, y, z) ) 4 _&F &1( |!&! 1 | 18 ( _ 2 ) &12&2= w^({&{ 1 , `&` 1 ))& L 83 t (L (x, y, z) )
c &w& L2 &|D x | &1 w& L2 &v& L2 . Case 1.3.
|!&! 1 | 1. This case can be treated similarly to Case 1.2.
Case 2. |!| max(2, c 0 ), |_| |_ 1 |. We denote by J 2 the restriction of J on this region. In this case we have that |_| 14&4= c |!| 34&12= |! 1 | 14&4= |!&! 1 | 14&4=. Hence K 1({, `, { 1 , ` 1 )
( !) 14+s1 +12= ( ! 1 ) &s1 +4= ( !&! 1 ) &s1 +4= . |! 1 | |!&! 1 | 14 ( _ 1 ) 12+2= ( _ 2 ) 12+2= ( _) 14 += 14
Case 2.1. |! 1 | 1, |!&! 1 | 1. We denote by J 21 the restriction of J on this region. In this case we have that K 1({, `, { 1 , ` 1 )
( _1)
|! | 18 |!&! 1 | 18 . ( _ 2 ) 12+2= (_) 14 +=
1 12+2=
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SAUT AND TZVETKOV
Using (10), (11) and Holder inequality we obtain J 21 &F &1(( _) &14&= v^({, `))& L4t (L2(x, y, z) ) 4 _&F &1(|! 1 | 18 ( _ 1 ) &12&2= w^({ 1 , ` 1 ))& L 83 t (L(x, y, z) ) 4 _&F &1(|!&! 1 | 18 ( _ 2 ) &12&2= w^({&{ 1 , `&` 1 ))& L83 t (L(x, y, z) )
c &w& 2L2 &v& L2 . Case 2.2. |! 1 | 1. We denote by J 22 the restriction of J on this region. In this case we have that K 1({, `, { 1 , ` 1 )
|! 1 | 18 |!&! 1 | 18 . |! 1 | ( _ 1 ) 12+2= ( _ 2 ) 12+2= ( _) 14 +=
Using (10), (11) and Holder inequality we obtain J 22 &F &1(( _) &14&= v^({, `))& L4t (L 2(x, y, z) ) 4 _&F &1( |! 1 | 18 ( _ 1 ) &12&2= |! 1 | &1 w^({ 1 , ` 1 ))& L83 (L (x, y, z) ) t 4 _&F &1( |!&! 1 | 18 ( _ 2 ) &12&2= w^({&{ 1 , `&` 1 ))& L 83 t (L (x, y, z) )
c &w& L2 &|D x | &1 w& L2 &v& L2 . Case 2.3.
|!&! 1 | 1. This case can be treated similarly to Case 2.2.
Case 3. |!| max(2, c 0 ), |_ 1 | |_|. We denote by J 3 the restriction of J on this region. In this case we have that |_ 1 | 14&4= c |!| 34&12= |! 1 | 14&4= |!&! 1 | 14&4=. Hence K 1({, `, { 1 , ` 1 )
( !) 14+s1 +12= ( ! 1 ) &s1 +4= ( !&! 1 ) &s1 +4= . |! 1 | |!&! 1 | 14 ( _ 1 ) 14+6= ( _ 2 ) 12+2= ( _) 12&3= 14
Case 3.1. |! 1 | 1, |!&! 1 | 1. We denote by J 31 the restriction of J on this region. In this case we have that K 1({, `, { 1 , ` 1 )
( !) 12= ( ! 1 ) 4= ( !&! 1 ) 4= . ( _ 1 ) 14+6= ( _ 2 ) 12+2= (_) 12&3=
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CAUCHY PROBLEM FOR KP EQUATIONS
Using Lemma 3 and Holder inequality we obtain J 31 &F &1(|!| (12&3=)(12+=1 )($(r1 )6) ( _) &12+3= v^({, `))& Lqt (L r(x,1 y, z) ) t
_&F &1(|! 1 | (14+6=)(12+=1 )($(r2 )6) ( _ 1 ) &14&6= w^({ 1 , ` 1 ))& L qt 2 (L r(x,2 y, z) ) _&F &1(|!&! 1 | (12+2=)(12+=1 )($(r3 6) ( _ 2 ) &12&2= _w^({&{ 1 , `&` 1 ))& L q3 (L r(x,3 y, z) ) t
c &w&
2 L2
&v& L2 ,
provided 2 %(12&3=) =1& , q1 12+= 1
$(r 1 )=
(1&%)(12&3=) , 12+= 1
%(14+6=) 2 =1& , q2 12+= 1
$(r 2 )=
(1&%)(14+6=) , 12+= 1
%(12+2=) 2 =1& , q3 12+= 1
$(r 3 )=
(1&%)(12+2=) , 12+= 1
1 1 1 + + =1, q1 q2 q3
3 $(r 1 )+$(r 2 )+$(r 3 )= . 2
Now it is sufficient to take = 1 =2= and %=25 to ensure the restrictions of the Holder inequality. Case 3.2. |! 1 | 1. We denote by J 32 the restriction of J on this region. Using the arguments of Case 3.1, we similarly obtain J 32 c &w& L2 &&|D x | &1 w& L2 & &v& L2 . Case 3.2. |!&! 1 | 1. This case can be treated similarly to Case 3.2. This completes the proof of Lemma 4.
4. PROOF OF THEOREM 1 In this section we shall give the proof of Theorem 1. Note that the Eqs. (1) or (2) are equivalent to the integral equation u(t)=U(t) ,&
|
t
U(t&t$) u(t$) x u(t$) dt$. 0
(18)
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SAUT AND TZVETKOV
Let be a cut-off function such that # C 0 (R),
supp /[&2, 2],
=1 over the interval [&1, 1].
We truncate (18) u(t)=(t) U(t) u(0)&(tT )
|
t
U(t&t$) u(t$) x u(t$) dt$.
(19)
0
We shall solve (19) globally in time. To the solutions of (19) will correspond local solutions of (18) in the time interval [&T, T ]. We shall apply a fixed point argument to solve (19) in X 12+=, s1 , s2 for sufficiently small =. Now we state the estimates for the two terms in the left-hand side of (19). Lemma 6. The following estimates hold for sufficiently small =>0 &(t) U(t) ,& X 12+2=, s1 , s2 c &,& H s1 , s2 ,
(20)
t
"(tT ) | U(t&t$) u(t$) u(t$) dt$ " x
0
X 12+2=, s1 , s2
=
cT &uu x & X &12+3=, s1 , s2 . Proof.
(21)
To prove (20) it is sufficient to note that &u& X b .s1 , s2 =&U(&t) u& H b, s1 , s2 ,
where H b, s1 , s2 is equipped with the norm
|
&u& 2H b, s1 , s2 = ( {) b ( !) s1 ( ') s2 (1+ |!| &1 ) 2 |u^({, !, ')| 2 d{ d! d'. The proof of (21) is a direct consequence of [12], Lemma 3.2 applied with b=12+2= and b$=&12+3=. This completes the proof of Lemma 6. Now we define an operator L Lu(t) :=(t) U(t) u(0)&(tT )
|
t
U(t&t$) u(t$) x u(t$)) dt$.
0
We obtain from Lemmas 5 and 6 &Lu& X 12+2=, s1 , s2 c &,& H s +cT = &u& 2X 12+2=, s1 , s2 .
(22)
Similarly we obtain &Lu&Lv& X 12+2=, s1 , s2 cT = &u+v& X 12+2=, s1 , s2 &u&v& X 12+2=, s1 , s2 .
(23)
209
CAUCHY PROBLEM FOR KP EQUATIONS
Using (22) and (23) we can apply the contraction mapping principle for sufficiently small T which completes the proof of Theorem 1.
5. PROOF OF THEOREM 3 It is known (cf. [6]) that in some cases (1) (resp. (2)) possesses solitary wave solutions. We recall that a solitary wave is a ``localized'' solution of (1) (resp. (2)) of the form u(x&ct, y) (resp. u(x&ct, y, z)). More precisely we have the following theorem. Theorem 5.1. (cf. [6, 7]). Let ;>0 and : 0. Then (1) (resp. (2)) possesses a solitary wave solution u(x&ct, y) (resp. u(x&ct, y, z)) such that
u # L 2y(L 1x ) , H s, s=0
\
+
resp. u # L 2(y, z)(L 1x ) , H s . s=0
The same statement holds when ;=0 and :<0. We have (cf. [7]) that r $u # L 2(R d ), for $
(d=2),
u # L 2(y, z)(L 1x )
(d=3).
The rest of the proof of Theorem 5.1. is contained in [6, 7]. Note that when ;=0, :<0 an explicit form of a solitary wave can be derived by the inverse scattering method and we will use it below. In the 3D case we will use essentially scaling arguments and no explicit form of the solitary waves is needed. Actually, solitary waves for (1) or (2) (cf. [6, 7]) can be obtained by the aid of the concentration compactness principle for some pure power nonlinearities and the method for obtaining ill-posedness could be extended to these cases (see the end of the section). v Let d=2, ;=0 and :<0. Then (1) has a solitary wave solution of the form u c(t, x, y)=, c(x&ct, y),
c>0,
where u c(0, x, y)=, c(x, y)=c, 1(c 12x, cy) :=c,(c 12x, cy).
210
SAUT AND TZVETKOV
Here u c is the ``lump''solitary wave of KP-I equation. Recall that c c2 8c 1& (x&ct) 2 + y 2 3 3 u c(t, x, y)= 2 2 c c 1+ (x&ct) 2 + y 2 3 3
\
+
\
+
is a solitary wave solution of (1) with :=&1 and ;=0. We will need a decay estimate of a fractional derivative of , c . Obviously x, c does not belong to L 2(R 2 ), but Lemma 7 (cf. Appendix). Let =>0. Then |D x | = (x, c ) # L 2(R 2 ). We shall denote by Fx the partial Fourier transform with respect to x. One has Fx (, c )(!, y)=c 12Fx (,)
\c
! 12
+
, cy .
Further we have &, c & 2H s, 0 =
|
R2
}
( !) 2s Fx (,)
2
!
\c +} d! dy. ,y 12
Hence for s<&12 Lebesgue theorem yields lim &, c & 2H s, 0 =c s &Fx (,)(0, .)& 2L2 =c s &,& 2L 2y (L 1x ) ,
cÄ
where cs =
|
( !) 2s d!.
&
Similarly we obtain (, c1 , , c2 ) s =
c2 c1
\ +
12
|
R2
( !) 2s Fx (,)
!
\c , y+ F (,) \c 12 1
x
! 12 2
where (., .) s stays for the H s, 0 scalar product. Hence lim
c1 , c2 Ä , c1 c2 Ä 1
(, c1 , , c2 ) s =c s &F x (,)(0, .)& 2L2
,
c2 y d! dy, c1
+
211
CAUCHY PROBLEM FOR KP EQUATIONS
and lim
c1 , c2 Ä , c1 c2 Ä 1
&, c1 &, c2 & H s, 0 =0.
Now we have that Fx (u c )(t, !, y)=c 12 exp(&itc!) Fx (,)(!c 12, cy). The presence of the oscillatory term exp(&itc!) makes the expression (u c1 (t, .), u c2 (t, .)) s tend to zero as n Ä by taking c 1 =n 2 and c 2 =(n+1) 2. More precisely we have by integration by parts (u n 2 (t, .), u (n+1) 2 (t, .)) s =c(n, t)
|
R2
!(e it(2n+1) ! ) (!) 2s Fx (,)
_Fx (,)
\
=2sc(n, t)
|
_Fx (,)
! (n+1) 2 y d! dy , n+1 n2
+
R2
\
!
\ n , y+
e it(2n+1) ! ( !) 2s&2 !Fx (,)
! ,y n
\ +
! (n+1) 2 , y d! dy n+1 n2
c(n, t) n = + in
+
|
e it(2n+1) !
R2
_F x (|D x | = (x,)) c(n, t)(n+1) = + i(n+1)
( !) 2s |!| =
! ! (n+1) 2 , y Fx (,) , y d! dy n n+1 n2
\ +
|
e it(2n+1) !
R2
_F x (|D x | = (x,))
\
\
+
(!) 2s ! Fx (,) , y = |!| n
\ +
! (n+1) 2 y d! dy :=(1)+(2)+(3), , n+1 n2
+
where c(n, t)=
n+1 . in(2n+1) t
212
SAUT AND TZVETKOV
Using CauchySchwarz inequality and the estimate &,& L &,& L1 we obtain (n+1) 2 |(1)| 2s |c(n, t)| ( !) 2s&2 &,(., y)& L1 , ., y d! dy n2 R2 L1 cn |c(n, t)| &,& 2L 2y (L 1x ) . n+1
"\
|
+"
It is clear that the last expression tends to zero as n tends to infinity. Further we have |(2)|
n = |c(n, t)| n
( !) 2s ! Fx (|D x | = (x,)) , y = n R 2 |!|
}
|
n = |c(n, t)| n &,& L y2 (Lx1 ) n n+1
(n+1) 2 y n2
\ +} " \ +" ! {| }F (|D | (x,)) \n , y+} d! dy= , .,
2
d! dy L1
12
=
R2
x
x
n = |c(n, t)| n 32 &,& L 2y (Lx1 ) &|D x | = (x,)& L 2(x, y) , n n+1
where we used that |!| &= ( !) 2s # L 2! if =<12 and Lemma 7. Hence we obtain that for =<12, |(2)| tends to zero as n tends to infinity. The term |(3)| can be estimated in a similar fashion. Hence lim (u (n+1) 2 (t, .), u n 2 (t, .)) s =0.
nÄ
Moreover, for any t>0 lim &u (n+1) 2 (t, .)&u n 2 (t, .)& 2H s, 0 =2c s &Fx (,)(0, .)& 2L2 .
nÄ
Therefore we obtain that (1) is locally ill-posed in H s, 0(R 2 ) for ;=0 and :<0, when s<&12. v Let d=3, ;>0 and :=0. Then (2) has a solitary wave solution of the form u c(t, x, y, z)=, c(x&ct, y, z),
c>0,
where u c(0, x, y, z)=, c(x, y, z)=c, 1(c 14 x, c 34 y, c 34 z) :=c,(c 14 x, c 34 y, c 34 z). One obtains easily that F x (, c )(!, y, z)=c 34 F x (,)
\c
! 14
+
, c 34 y, c 34 z .
213
CAUCHY PROBLEM FOR KP EQUATIONS
Further we have &, c & 2H s, 0 =
|
}
( !) 2s F x (,)
R3
\c
2
!
, y, z 12
+} d! dy dz.
Hence for s<&12 Lebesgue theorem yields lim &, c & 2H s, 0 =c s &Fx (,)(0, .)& 2L2 =c s &,& 2L 2(y, z) (L1x ) .
cÄ
Similarly we obtain c2 c1
\+
(, c1 , , c2 ) s =
34
|
_Fx (,)
R2
\c
! 14 2
,
c2 c1
!
\c , y, z+ c y, z d! dy dz, c +
( !) 2s Fx (,)
14 1
2
1
where (., .) s stays for the H s, 0 scalar product. Hence lim
c1 , c2 Ä , c1 c2 Ä 1
(, c1 , , c2 ) s =c s &Fx (,)(0, .)& 2L2
and lim
c1 , c2 Ä , c1 c2 Ä 1
&, c1 &, c2 & H s, 0 =0.
Now we have that Fx (u c )(t, !, y, z)=c 34 exp(&itc!) Fx (,)(!c 14, cy, cz). The presence of the oscillatory term exp(&itc!) makes the expression (u c1 (t, .), u c2 (t, .)) s tend to zero as n Ä by taking c 1 =n 2 and c 2 =(n+1) 2. Let x$=( y, z). Then (u n 2 (t, .), u (n+1) 2 (t, .)) s =c(n, t)
|
R3
_F x(,)
!(e it(2n+1) ! ) (!) 2s Fx (,)
\
\
! , x$ n 12
(n+1) 2 ! , x$ d! dx$ 12 (n+1) n2
+
+
214
SAUT AND TZVETKOV
!
\n
|
e it(2n+1) ! ( !) 2s&2 !Fx (,)
_Fx (,)
\
! (n+1) 2 , x$ d! dx$ (n+1) 12 n2
c(n, t) in 12
|
=2sc(n, t)
R3
+
_F x (,) +
, x$
+
+
e it(2n+1) ! ( !) 2s Fx (x,)
R3
\
!
\n
, x$
12
+
! (n+1) 2 , x$ d! dx$ 12 (n+1) n2
+
c(n, t) i(n+1) 12
_Fx (x,)
12
\
|
R3
e it(2n+1) ! ( !) 2s Fx (,)
!
\n
12
, x$
+
(n+1) 2 ! , x$ d! dx$ (n+1) 12 n2
+
:=(1)+(2)+(3), where c(n, t)=
(n+1) 32 . in (2n+1)t 32
Using CauchySchwarz inequality and the estimate &,& L &,& L1 we obtain |(1)| 2s |c(n, t)| |c(n, t)|
|
R3
"\
( !) 2s&2 &,(., x$)& L1 , .,
(n+1) 2 x$ n2
+"
d! dx$ L1
cn 2 &,& 2L 2x$ (L1x) . (n+1) 2
It is clear that the last expression tends to zero as n tends to infinity. Further we have |(2)|
|c(n, t)| n 12
|
R3
}
( !) 2s Fx (x,)
! , x$ n 12
(n+1) 2 x$ n2
\ +} " \ +" d! dx$ ! {| }F (x,) \n , x$+} d! dx$= , .,
|c(n, t)| n 2 &,& L 2x$ (L1x ) n 12 (n+1) 2
2
|c(n, t)| n 14 n 2 &,& L2x$ (L1x ) &x,& L 2(x, x$) , n 12 (n+1) 2
R3
x
L1
12
12
(53)
215
CAUCHY PROBLEM FOR KP EQUATIONS
where we used that x, # L 2(R 3 ), which follows from the decay properties of the solitary waves. (See [7].) Hence we obtain that |(2)| tends to zero as n tends to infinity. The term |(3)| can be estimated in a similar fashion. Hence lim (u (n+1) 2 (t, .), u n 2 (t, .)) s =0.
nÄ
Moreover lim &u (n+1) 2 (t, .)&u n 2 (t, .)& 2H s, 0 =2c s &Fx (,)(0, .)& 2L2 .
nÄ
Therefore we obtain that (2) is locally ill-posed in H s, 0(R 3 ) for ;>0 and :=0, when s<&12. v Let d=3, ;=0 and :<0. Then (2) has a solitary wave solution of the form u c(t, x, y, z)=, c(x&ct, y, z), where u c(0, x, y, z)=, c(x, y)=c,(c 12x, cy, cz). We have that , c(!, ')=c &32,
!
' . c
\c + 12
,
Further we have &, c & 2H4 s1 , s2 =c s1 +2s2 &12
|
R3
|!| 2s1 |'| 2s2 |,(!, ')| 2 d! d'=c s1 +s2 &12 &,& 2H4 s1 , s2 .
Hence when s 1 +2s 2 =12, s 1 0, s 2 0 we have that &, c & H4 s1 , s2 =&,& H4 s1 , s2 . Similarly we obtain (, c1 , , c2 ) s1 , s2 =
c2 c1
\ +
&32
|
|!| 2s1 |'| 2s2 ,(!, ') , R3
c1 c2
\\ +
12
!,
c1 ' d! d', c2
+
where now (., .) s1 , s2 stays for the H4 s1 , s2 scalar product. Hence using Lebesgue theorem we obtain lim
c1 , c2 Ä , c1 c2 Ä 1
(, c1 , , c2 ) s1 , s2 =&,& 2H4 s1 , s2
216
SAUT AND TZVETKOV
and &, c1 &, c2 & H4 s1 , s2 =0.
lim
c1 , c2 Ä , c1 c2 Ä 1
Now we have that u^ c(t, !, ')=c &32 exp(&itc!) ,
!
'
\c , c+ , 12
and furthermore (u c1 (t, .), u c2 (t, .)) s1 , s2 =
c2 c1
&32
\ + | c _, \\c +
R3
exp(&it!(c 1 &c 2 )) |!| 2s1 |'| 2s2 ,(!, ')
12
1
2
!,
c1 ' d! d'. c2
+
Hence using RiemannLebesgue lemma arguments we obtain as above lim (u (n+1) 2 (t, .), u n 2 (t, .)) s1 , s2 =0.
nÄ
Moreover lim &u (n+1) 2 (t, .)&u n 2 (t, .)& 2H s1 , s2 =&,& 2H4 s1 , s2 .
nÄ
Therefore we obtain that (1) is locally ill-posed in H4 s1 , s2 (R 2 ) for ;=0 and :<0, when s 1 +2s 2 =12 and s 1 0, s 2 0. This completes the proof of Theorem 3. There are some difficulties to extend the results of Theorem 3 to the two dimensional case with higher order term. If d=2, ;>0 and :=0 then (1) has a solitary wave solution of the form u c(t, x, y)=, c(x&ct, y),
c>0,
where u c(0, x, y)=, c(x, y)=c, 1(c 14 x, c 34 y) :=c,(c 14 x, c 34 y). One has , c(!, ')=,
\c
! 14
,
' . c 34
+
CAUCHY PROBLEM FOR KP EQUATIONS
217
and furthermore &, c & 2H s1 , s2 =
|
} \c
( !) 2s1 ( ') 2s2 , R2
!
, 14
' c 34
2
+} d! d'.
(24)
It is easy to see (cf. [7]) that , does not belong to L 1(x, y) . Hence one can not define the trace of ,(!, ') at zero and therefore it is impossible to pass to the limit as c tends to infinity in (24). We conclude this section with some remarks for the generalized KP-I equation
{
(u t +:u xxx +;u xxxxx +u pu x ) x +u yy +u zz =0 u(0, x, y, z)=,(x, y, z).
(25)
If ;=0 and u(t, x, y, z) is a solution of (25) then so is u *(t, x, y, z)=* 2 pu(* 3 t, *x, * 2y, * 2z) and &u *(t, .)& H4 s1 , s2 =* s1 +2s2 &(5p&4)2p &u(* 3t, .)& H4 s1 , s2 . In [6] it is shown that for 1 p<43 and :<0 (25) possesses nontrivial localized solitary waves of the form u(t, x, y, z)=, c(x&ct, y, z),
c>0,
where , c(x, y, z)=c 1 p,(c 12x, cy, cz), Using the arguments of the proof of Theorem 3 one can prove that (25) is locally ill-posed in H4 s1 , s2 for s 1 +2s 2 &(5p&4)2p=0, s 1 0, s 2 0. If :=0 and u(t, x, y, z) is a solution of (25) then so is u *(t, x, y, z)=* 4 pu(* 5t, *x, * 3y, * 3z) and &u *(t, .)& H4 s1 , s2 =* s1 +3s2 +(8&7p)2p &u(* 5t, .)& H4 s1 , s2 . In [6] it is shown that for 1 p<83 and ;>0 (25) possesses nontrivial localized solitary waves of the form u(t, x, y, z)=, c(x&ct, y, z),
c>0,
where , c(x, y, z)=c 1 p,(c 14x, c 34y, c 34z), Using the arguments of the proof of Theorem 3 one can prove that (25) is locally ill-posed in H4 s1 , s2 for
218
SAUT AND TZVETKOV
s 1 +2s 2 +(8&7p)2p=0, s 1 0, s 2 0. In particular when p=87 (25) is locally ill-posed in L 2. Finally we note that when :<0 and ;>0, the existence of solitary waves is known for a suitable range of p's (cf. [6]) but the scaling argument leading to ill-posedness does not work.
6. A REMARK ON THE (GENERALIZED) KDV EQUATION In this section we shall show that the example providing ill-posedness for the (generalized) KdV equations uses just scaling properties and hence the special form of the solitary waves is not needed. Consider the (generalized) KdV equation u t +u xxx +u pu x =0.
(26)
The local ill-posedness of (26) is studied in [4, 5]. It is known that (26) possesses solitary wave solutions of the form u c, p(t, x)=, c, p(x&ct), where , p(x)=[ 12 ( p+2) sech 2( 12 px)] 1 p.
u c, p(0, x)=c 1p, p(c 12x),
Since the solitary waves of (26) decays exponentially at infinity one can define the trace of its Fourier transforms at zero and then prove local illposedness of (26) for p=2 in H s, s<&12 which is the result of [5]. Actually if p=2 then , c, 2(!)=, 2(!c 12 ) and one obtains that for s<&12 lim &, c, 2 & 2H s = lim
cÄ
cÄ
| ( !)
s
, 2(!c 12 ) d!=c s , 2(0)
and therefore one can easily obtain lim
(, c1 , , c2 ) s =c s , 2(0)
lim
&, c1 , 2 &, c2 , 2 & H s =0.
c1 , c2 Ä , c1 c2 Ä 1
and moreover c1 , c2 Ä , c1 c2 Ä 1
Further we have F(u c, 2 )(t, !)=exp(&itc!) , 2(!c 12 ).
219
CAUCHY PROBLEM FOR KP EQUATIONS
Due to the RiemannLebesgue lemma, the presence of the oscillatory term exp(&itc!) makes the expression (u c1 , 2(t, .), u c2 , 2(t, .)) s tend to zero as n Ä by taking c 1 =n 2 and c 2 =(n+1) 2. Therefore we obtain that (26) is locally ill-posed in H s for p=2 when s<&12. In fact one can easlily show that the sequence , c, 2 converge strongly in H s, s<&12 to - 2 ?$ as c tends to infinity. Here $ stays for Dirac delta function. Similarly we can see that u c, 2(t, .) converge weakly to zero in H s. Now the boundedness of &, c & H s , s<&12 provides the locall ill-posedness. Similarly when p=1 (the ``usual'' KdV equation) one can prove that , c, 1(x) converge weakly in H s, s<&32 to - 2 ? x $(x) as c tends to infinity. We also have that u c, 1(t, .) converge weakly to zero as c tends to infinity. We also have that &u c, 1(t, .)& H s =&, c, 1 & H s =c 12 &, 1 & H s . Since x $(x) # H s, s<&32 the last arguments strongly suggest local illposedness of (26) in H s, s<&32 when p=1. However, as was pointed out to us by the referee, J. Bourgain has shown in [11] that if one requires that the map data solution u 0 [ u(t) be smooth then the results of KenigPonceVega [17] for the KdV ( p=1, s>&34) or the modified KdV ( p=2, s 14) are optimal.
7. APPENDIX In this appendix we shall give the proof of Lemma 7. Proof of Lemma 7. By scaling it suffices to prove that |D x | = (x,) # L (R 2 ), where 2
,(x, y)=
1&x 2 + y 2 . (1+x 2 + y 2 ) 2
We have ,(x, y)=
1 2x 2 & :=, 1(x, y)+, 2(x, y). 2 2 1+x + y (1+x 2 + y 2 ) 2
A simple computation shows that 1 , 1(!, y) :=Fx (, 1 )(!, y)= (1+ y 2 ) 12 =
1 2 12 e &(1+ y ) |!|. (1+ y 2 ) 12
2 12 x!
e &i(1+ y ) 1+x 2 &
|
dx
220
SAUT AND TZVETKOV
Hence , 1 2 12 (!, y)=&sgn ! } e &(1+ y ) |!| ! and
|
|!| 2=
R2
}
, 1 (!, y) !
2
} d! dy= |
|!| 2= e &2(1+ y
2 ) 12 |!|
d! dy
R2
=
|X | 2= e &2 |X | dX dy< , 2 12+= R 2 (1+ y )
|
since =>0. Let now (x, y)=& 12 , 2(x, y). A simple calculation yields (!, y) :=Fx ()(!, y)=
1 (1+ y 2 ) 12
2 12 x!
x 2 e &i(1+ y ) (1+x 2 ) 2 R2
|
dx.
Write x 2(1+x 2 ) 2 =(11+x 2 )&(1(1+x 2 ) 2 ) and recall that (cf. [1]) F
1
\1+x + (!)=e
&|!|
2
and F
1
\(1+x ) + (!)=c |!| 2 2
32
K 32( |!| ),
where K 32 is the modified Bessel function of order 32 (cf. [1]) K 32(z)=
?
1
2z } e \1+ z+ . &z
Therefore (!, y)= =
1 1 1 2 12 e &(1+ y ) |!| + F ((1+ y 2 ) 12 !) (1+ y 2 ) 12 (1+ y 2 ) 12 (1+x 2 ) 2
\
+
1 2 12 e &(1+ y ) |!| (1+ y 2 ) 12 ? c 2 12 (1+ |!| (1+ y 2 ) 12 ) e &(1+ y ) |!| } 2 (1+ y 2 ) 12
& =
1&c - ?2 &(1+ y 2 ) 12 |!| e &c (1+ y 2 ) 12
:= 1(!, y)+ 2(!, y).
?
2 } |!| e
&(1+ y 2 ) 12 |!|
(61)
221
CAUCHY PROBLEM FOR KP EQUATIONS
The term 1(!, y) leads to a computation similar to that of , 1 . Now 2 ? 2 12 2 12 (!, y)=&c } sgn ! } (e &(1+ y ) |!| & |!| (1+ y 2 ) 12 e &(1+ y ) |!| ) ! 2 := 21(!, y)+ 22(!, y).
Now we have that
|
R2
| =c |
|!| 2= | 21(!, y)| 2 d! dy=c
R2
|!| 2= e &2(1+ y
2 ) 12 |!|
d! dy
|X | 2= e &2 |X | dX dy< , 2 12+= R 2 (1+ y )
since =>0 and furthermore
|
R2
|!| 2= | 22(!, y)| 2 d! dy=c
|
=c
|
|!| 1+2= (1+ y 2 ) 12 e &2(1+ y
2 ) 12 |!|
d! dy
R2
|X | 1+2= e &2 |X | dX dy< . 2 12+= R 2 (1+ y )
This completes the proof. Remark. We conjecture that the result of Lemma 7 is valid for any localized solitary wave of KP-I type equation.
REFERENCES 1. M. Abramowitz and I. A. Stegun, ``Handbook of Mathematical Functions,'' Dover, New York, 1965. 2. L. A. Abramyan and Y. A. Stepanyants, The structure of two-dimensional solitons in media with anomalously small dispersion, Sov. Phys. JETP 61 (1985), 963966. 3. M. Ben-Artzi and J.-C. Saut, Uniform decay estimates for a class of oscillatory integrals and applications, Diff. Int. Eq., to appear. 4. B. Birnir, C. Kenig, G. Ponce, N. Svanstedt, and L. Vega, On the ill-posedness of the IVP for the generalized Kortewegde Vries and nonlinear Schrodinger equations, Ann. IHP Anal. Non Lineaire 13 (1996), 529535. 5. B. Birnir, G. Ponce, and N. Svanstedt, The local ill-posedness of the modified KdV equation, J. London Math. Soc. 53 (1996), 551559. 6. A. de Bouard and J.-C. Saut, Solitary waves of generalized KadomtsevPetviashvili equations, Ann. IHP Anal. Non Lineaire 14 (1997), 211236. 7. A. de Bouard and J.-C. Saut, Symmetries and decay of the generalized KP solitary waves, SIAM J. Math. Anal. 28 (1997), 10641085. 8. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrodinger equations, GAFA 3 (1993), 107156.
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9. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations II. The KdV equation, GAFA 3 (1993), 209262. 10. J. Bourgain, On the Cauchy problem for the KadomtsevPetviashvili equation, GAFA 3 (1993), 315341. 11. J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Selecta Math. 3 (1997), 115159. 12. J. Ginibre, Le probleme de Cauchy pour des EDP semi-lineaires periodiques en variables d'espace (d'apres Bourgain), Seminaire Bourbaki n. 796, Asterique 237 (1996), 163187. 13. J. Ginibre, Y. Tsutsumi, and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal. 151 (1997), 384436. 14. R. J. Iorio Jr. and W. V. L. Nunes, On equations of KP-type, Proc. Roy. Soc. Edinburgh 128A (1998), 725743. 15. V. I. Karpman and V. Yu. Belashov, Dynamics of two dimensional solitons in weakly dispersive media, Phys. Lett. A 154 (1991), 131139. 16. V. I. Karpman and V. Yu, Belashov, Evolution of three-dimensional nonlinear pulses in weakly dispersive media, Phys. Lett. A 154 (1991), 140144. 17. C. Kenig, G. Ponce, and L. Vega, Well-posedness and scattering results for the generalized KdV equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), 527560. 18. J.-C. Saut, Remarks on the generalized KadomtsevPetviashvili equations, Indiana Univ. Math. J. 42 (1993), 10171029. 19. M. M. Tom, On a generalized KadomtsevPetviashvili equation, Contem. Math. 200 (1996), 193210. 20. N. Tzvetkov, On the Cauchy problem for KadomtsevPetviashvili equation, preprint, Orsay, 1998.
The Cauchy Problem for Higher-Order KP Equations J. C. Saut and N. Tzvetkov Analyse numerique et EDP, Universite de Paris Sud, Ba^t. 425, 91405 Orsay Cedex, France Received April 13, 1998; revised July 7, 1998
We study the local well-posedness of higher-order KP equations. Our well-posedness results make an essential use of a global smoothing effect for the linearized equation established in Ben-Artzi and Saut (preprint, 1997), injected into the framework of Fourier transform restriction spaces introduced by Bourgain. Our illposedness results rely on the existence of solitary wave solutions and on scaling arguments. The method was first applied in the context of the KdV and Schrodinger equations (cf. Birnir et al., Ann. Inst. H. Poincare Anal. Non Lineaire 13 (1996), 529535; Birnir et al., J. London Math. Soc. 53 (1996), 551559). 1999 Academic Press
1. INTRODUCTION This paper is concerned with the Cauchy problem associated to KadomtsevPetviashvili (KP) equations having higher order dispersion in the main direction of propagation. Such equations occur naturally in the modeling of certain long dispersive waves (cf. [2, 15, 16]). The study of their solitary wave solutions was done in [6, 7]. Thus we consider the Cauchy problems in R d, d=2, 3
{
(u t +:u xxx +;u xxxxx +uu x ) x +u yy =0 u(0, x, y)=,(x, y)
(1)
in the two dimensional case and (u t +:u xxx +;u xxxxx +uu x ) x +u yy +u zz =0
{u(0, x, y, z)=,(x, y, z)
(2)
in the three dimensional case. The ``usual'' KP equations correspond to ;=0 and :=&1 (KP-I) or :=+1 (KP-II). We are interested in the local well-posedness of the Cauchy problems (1) and (2). Following [4] and [13] we introduce the notion for well-posedness which will be convenient for our purposes. 196 0022-039699 30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved.
CAUCHY PROBLEM FOR KP EQUATIONS
197
Definition 1. The Cauchy problem (1) (resp. (2)) is locally well-posed in the space X if for any , # X there exists T=T(&,& X )>0 (T is a nondecreasing continuous function such that lim \ Ä 0 T(\)= ) and a map F from X to C([0, T ]; X) such that u=F(,) solves the equation (1) (resp. (2)) and F is continuous in the sense that &F(, 1 )&F(, 2 )& L ([0, T ]; X ) M(&, 1 &, 2 & X , R) for some locally bounded function M from R +_R + to R + such that M(S, R) Ä 0 for fixed R when S Ä 0 and for , 1 , , 2 # X such that &, 1 & X +&, 2 & X R. We introduce the nonisotropic Sobolev spaces H s1 , s2 (R d ) equipped with the norm
|
&u& 2H s1 , s2 = ( !) 2s1 ( ') 2s2 |u^(!, ')| 2 d! d', where ( } ) =(1+ | } | 2 ) 12 and '=(' 1, ' 2 ) in the case of three space dimensions. By H4 s1 , s2 (R d ) we shall denote the homogeneous nonisotropic Sobolev spaces equipped with the norm
|
&u& 2H4 s1 , s2 = |!| 2s1 |'| 2s2 |u^(!, ')| 2 d! d'. Taking into account the specific structure of the KP-type equations (cf. [18]) we introduce the modified Sobolev space H s1 , s2 (R d ) equipped with the norm
|
&u& 2H s1, s2 = (!) 2s1 ( ') 2s2 (1+ |!| &1 ) 2 |u^(!, ')| 2 d! d'. Note that any u # H s1 , s2 (R d ) has (formally) a zero x mean value. We shall denote H 0, 0 by L 2. For b, s 1 , s 2 # R we define X b, s1 , s2 (R d+1 ) to be the completion of the functions of C 0 with zero x mean value with respect to the norm
|
&u& 2X b, s1 , s2 = ({+ p(!, ')) 2b ( !) 2s1 ( ') 2s2 (1+ |!| &1 ) 2 |u^({, !, ')| 2 d{ d! d', where p(!, ')=;! 5 &:! 3 +
|'| 2 . !
If : or ; vanishes then the equations (1) and (2) are scale invariant. If u(t, x, y) is a solution of (1) with ;=0 then so is u *(t, x, y)=* 2u(* 3t, *x, * 2y)
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and &u *(t, .)& H4 s1 , s2 =* s1 +2s2 +12 &u(* 3t, .)& H4 s1 , s2 . Hence one may expect local well-posedness (resp. ill-posedness) in H s1 , s2 of (1) for ;=0 when s 1 +2s 2 >&12 (resp. s 1 +2s 2 <&12). Similarly if u(t, x, y, z) is a solution of (2) with ;=0 then so is u *(t, x, y, z)=* 2u(* 3t, *x, * 2y, * 2z) and &u *(t, .)& H4 s1 , s2 =* s1 +2s2 &12 &u(* 3t, .)& H4 s1 , s2 . Hence one may expect local well-posedness (resp. ill-posedness) in H s1 , s2 of (2) for ;=0 when s 1 +2s 2 >12 (resp. s 1 +2s 2 <12). If u(t, x, y) is a solution of (1) with :=0 then so is u *(t, x, y)= * 4u(* 5t, *x, * 3y) and &u *(t, .)& H4 s1 , s2 =* s1 +3s2 +2 &u(* 5t, .)& H4 s1 , s2 . Hence one one may expect local well-posedness (resp. ill-posedness) in H s1 , s2 of (1) for :=0 when s 1 +3s 2 >&2 (resp. s 1 +3s 2 <&2). Similarly if u(t, x, y, z) is a solution of (2) with :=0 then so is u *(t, x, y, z)=* 4u(* 5t, *x, * 3y, * 3z) and &u *(t, .)& H4 s1 , s2 =* s1 +3s2 +12 &u(* 5t, .)& H4 s1 , s2 . Hence one may expect local well-posedness (resp. ill-posedness) in H s1 , s2 of (2) for :=0 when s 1 +3s 2 >&12 (resp. s 1 +3s 2 <&12). Now we state the well-posedness result. The proof uses the Fourier transform restriction norms introduced by Bourgain [810], in the context of Schrodinger, KdV or KP equations. In the nonlinear estimates we shall make an essential use of the smoothing effects for the linear group associated to (1) or (2) established in [3]. Theorem 1. Let d=2 and ;<0, : # R. Then for any s 1 &14, s 2 0 the Cauchy problem (1) is locally well-posed in H s1 , s2. Moreover there exists b>12 such that the solution u(t, x, y) satisfies u # X b, s1 , s2. Let d=3 and ;<0, : # R. Then for any s 1 &18, s 2 0 the Cauchy problem (2) is locally well-posed in H s1 , s2. Moreover there exists b>12 such that the solution u(t, x, y, z) satisfies u # X b, s1 , s2. If ;=0 and :>0 (the ``usual'' KP-II equation) local well-posedness of (1) in L 2 is established in [10]. The proof uses Fourier transform restriction norms, a dyadic decomposition related to the structure of the symbol of the linearized operator and can be performed for periodic initial data. Local well-posedness of KP-II in H s1 , s2, s 1 >&14, s 2 0 is established in [20]. The sign of ; is crucial in the proof of Theorem 1. We do not know of a similar result when ;>0 (KP-I type equations). In this case, however, it is easy to obtain global weak solutions by energy methods (cf. [19]). The uniqueness of such solutions is unknown. Note that L 2 norms of the local solutions of (1) and (2) are conserved. Lemma 1. If u is a local solution of (1) or (2) in the time interval [0, T ] then for any t # [0, T ] &u(t, .)& L2 =&,& L2 .
CAUCHY PROBLEM FOR KP EQUATIONS
Proof.
199
It suffices to multiply (1) or (2) with u and integrate over R d.
Using Theorem 1 with s 1 =s 2 =0 and Lemma 1 we obtain Theorem 2. Let ;<0, : # R. Then for any , # L 2(R 2 ) (resp. L 2(R 3 )) there exists a unique global solution of the Cauchy problem (1) (resp. (2)). In [4] it is shown that the solitary wave solutions can be used to construct examples proving local ill-posedness for the (generalized) KdV equation. One considers the limit of the solitary wave solutions as the propagation speed tends to infinity. In the favorable cases the initial data tends weakly to a nonzero distribution (for example the Dirac delta function for the KdV equation with cubic nonlinearity) while the solution at time t>0 tends weakly to zero. Using the idea of [4, 5] together with the properties of the solitary waves of KP-I type equations (cf. [6, 7]), we can prove the next Theorem. Theorem 3. Let d=2. If :<0 and ;=0 then (1) is locally ill-posed in H s, 0 for s<&12. Let d=3. If :<0 and ;=0 then (1) is locally ill-posed in H4 s1 , s2 for s 1 +2s 2 =12, s 1 0, s 2 0. If :=0 and ;>0 then (2) is locally ill-posed in H s, 0 for s<&12. Note that our well-posedness or ill-posedness results do not contradict with the scaling argument. In fact the region [(s 1 , s 2 ): s 1 &14, s 2 0] is a subset of the region suggested by the scaling argument [(s 1 , s 2 ): s 1 +3s 2 +2>0], where well-posedness is expected to hold in the case of two space dimensions when :=0, ;<0. The same holds for the region [(s 1 , s 2 ): s 1 &18, s 2 0] which is a subset of [(s 1 , s 2 ): s 1 +3s 2 +12>0] in the case of three space dimensions. If d=2, :<0 and ;=0 then ill-posedness holds for the pairs (s, 0), s< &12 which can achieve the ``critical'' line s 1 +2s 2 +12=0, suggested by the scaling argument. If d=3, :<0 and ;=0 then ill-posedness holds for the pairs [(s 1 , s 2 ): s 1 +2s 2 =12, s 1 0, s 2 0] which are a part of the line suggested by the scaling argument. If d=3, :=0 and ;>0 then the pairs (s, 0), s<&12 can achieve the line s 1 +3s 2 +12=0, suggested by the scaling argument. The method for obtaining local well-posedness works just for KP-II type equations (i.e., ;<0 or ;=0, :>0) since one can recuperate the derivative x in the non-linear term by the aid of an algebraic relation for the symbol of the linearized operator. (cf. (16) below). In the case of KP-I type equations (;>0 or ;=0, :<0) we can not gain the x derivative using the relation (16). However, using the parabolic regularisation method it is possible to prove local well-posedness of (1) in H s, s>2 and local wellposedness of (2) in H s, s>52 (cf. for instance [14] for an illustration of this method). The proof does not make use of the specific structure of the
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KP-type equations and could be performed for quite general evolution equations. The conditions for s are in order to control the L norm of the gradient of the solution and they seem to be very restrictive. Hence there is still no satisfactory theory for the local well-posedness of the Cauchy problem for KP-I type equations. On the other hand our ill-posedness results are only valid for KP-I type equations since the existence of solitary wave solutions is used in an essential way. In [6] using Pohojaev type identities it is shown that the KP-II type equations do not possess localized solitary wave solutions. Note also that we are forced to take : or ; zero in order to keep the scale invariance of the equation. The paper is organized as follows. In Section 2 we recall the global smoothing effects for (1) and (2) established in [3]. Then we inject these estimates into the framework of the Bourgain spaces associated to (1) and (2). In Section 3 we prove the crucial nonlinear estimate. In Section 4 we perform a fixed point argument to complete our well-posedness result. Section 5 is devoted to the ill-posedness of (1) and (2). In Section 6 we remark that the ill-posedness of the (generalized) KdV equations does not depend on the special form of the solitary waves. Finally in an appendix we give the proof of a decay estimate for a fractional derivative of the ``lump'' solitary wave of KP-I equation.
2. ESTIMATES FOR THE LINEAR EQUATION Let us consider the linear initial value problem associated to (1) or (2) i t u= p(D) u
{u(0, x, y)=,,
(3)
where D=(D 1 , D 2 ), when d=2, D=(D 1 , D 2 , D 3 ), when d=3 D 1 =&i x , D 2 =&i y and D 3 =&i z . We recall that p(!, ')=;! 5 &:! 3 +(|'| 2!). We shall denote by U(t)=exp(&ip(D)) the unitary group which generates the solutions of (3). We have the following representation for the solutions of (3), when d=3 u(t, x, y, z)=U(t) ,=G t V ,, where G t is the oscillatory integral G t(x, y, z)=c
|
exp(itp(!, ')+i(x!+ y' 1 +z' 2 )) d! d'. R3
CAUCHY PROBLEM FOR KP EQUATIONS
201
A similar representation for the solutions of (3) holds when d=2. We have in fact a global smoothing effect for the solutions of (3). Lemma 2 (cf. [3], Theorem 4.2). Let $(r)=d( 12 &(1r)). Then the following estimates hold &|D x | $(r)2 U(t),& L qt (L r(x, y) ) c &,& L2 ,
(d=2),
&|D x | $(r)6 U(t) ,& L qt (L r(x, y, z) ) c &,& L2 ,
(d=3),
provided 2 =$(r). q Next we shall prove an inequality which will be used in the proof of the nonlinear estimates. Lemma 3. Let d=2 and 0<% 1,
= 1 >0,
0 b 12+= 1 .
Then the following inequality holds &F &1( |!| * ( {+ p(!, ')) &b |u^({, !, ')| ); L qt(L r(x, y) )& c &u& L2 ,
(4)
where %b 2 =1& , q 12+= 1
$(r)=
(1&%) b , 12+= 1
*=
$(r) b . 2(12+= 1 )
Let d=3 and 0<% 1,
= 1 >0,
0 b 12+= 1 .
Then the following inequality holds &F &1( |!| * ({+ p(!, ')) &b |u^({, !, ')| ); L qt(L r(x, y, z) )& c &u& L2 ,
(5)
where %b 2 , =1& q 12+= 1 Proof.
$(r)=
(1&%) b , 12+= 1
*=
$(r) b . 6(12+= 1 )
Let d=3. Using Lemma 2 and [12], Lemma 3.3 we obtain &|D x | $(r)6 u; L qt(L r(x, y, z) )& c &u; X 12+=1 , 0, 0 &,
(6)
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provided 2 =$(r). q Interpolating between (6) and &u; L 2t (L 2(x, y, z) )& &u; X 0, 0, 0 &, we obtain &|D x | * u; L qt(L r(x, y) )& c &u; X b, 0, 0 &,
(7)
provided %b 2 , =1& q 12+= 1
$(r)=
(1&%) b , 12+= 1
*=
$(r) b . 6(12+= 1 )
But (7) is equivalent to (5) which completes the proof of Lemma 3 when d=3. If d=2 then the arguments are the same. Corollary 1. Let d=2. Then the following inequalities hold &F &1( |!| 14 ( {+ p(!, ')) &b |u^({, !, ')| ); L 4t (L 4(x, y) )& c &u& L2 ,
(8)
where b>12. &F &1(( {+ p(!, ')) &b |u^({, !, ')| ); L qt(L 2(x, y) )& c &u& L2 ,
(9)
where 2 q , b 0 and (bq(q&2))> 12 . Let d=3. Then the following inequalities hold 4 &F &1(|!| 18 ( {+ p(!, ')) &b |u^({, !, ')| ); L 83 t (L (x, y, z) )& c &u& L2 , (10)
where b>12. &F &1(( {+ p(!, ')) &b |u^({, !, ')| ); L qt(L 2(x, y, z) )& c &u& L2 ,
(11)
where 2 q , b 0 and (bq)(q&2)> 12 . Proof. The proof of (8) follows from Lemma 3 (d=2) applied with %=12 and b=12+= 1 . Lemma 3 (d=3) applied with %=14 and b=12 += 1 yields (10). To prove (9) and (11) we apply Lemma 3 with %=1.
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CAUCHY PROBLEM FOR KP EQUATIONS
3. AN ESTIMATE FOR THE NONLINEAR TERM Lemma 4. Let d=2 and s 1 &14, s 2 0. Then for sufficiently small = we have &uu x & X &12+3=, s1 , s2 c &u& 2X 12+2=, s 1 , s2 .
(12)
Let d=3 and s 1 &18, s 2 0. Then for sufficiently small = we have &uu x & X &12+3=, s1 , s2 c &u& 2X 12+2=, s1 , s2 .
(13)
Proof. We shall give the proof only when d=3. In the case of two space dimensions the arguments are similar. We set w^({, !, ')=( {+ p(!, ')) 12+2= ( !) s1 ( ') s2 u^({, !, '), _ :=_({, !, ')={+ p(!, '),
_ 1 :=_({ 1 , ! 1 , ' 1 ),
_ 2 :=_({&{ 1 , !&! 1 , '&' 1 ). Let `=(!, '). Then (13) is equivalent to
"( _)
&12+3=
( !) 1+s1 (') s2
|| K({, `, { , ` ) w^({ , ` ) w^({&{ , `&` ) d{
_
1
1
1
1
1
1
1
d` 1
"
2
L ({, `)
c[&w& 2L2 +&w& L2 &|D x | &1 w& L2 +&|D x | &1 w& 2L2 ],
(14)
where K({, `, { 1 , ` 1 )=
( ! 1 ) &s1 ( !&! 1 ) &s1 ( ' 1 ) &s2 ( '&' 1 ) &s2 . (_ 1 ) 12+2= ( _ 2 ) 12+2=
Hence by the self-duality of L 2 we obtain that (14) is equivalent to
}|| K ({, `, { , ` ) w^({ , ` ) w^({&{ , `&` ) v^({, `) d{ d` d{ d` } 1
1
1
1
1
1
1
1
1
c[&w& 2L2 +&w& L2 &|D x | &1 w& L2 +&|D x | &1 w& 2L2 ] &v& L2 ,
(15)
where K 1({, `, { 1 , ` 1 )=
( !) 1+s1 ( ! 1 ) &s1 ( !&! 1 ) &s1 ( ') s2 . 12+2= 12+2= 12&3= s2 ( _ 1) ( _2) ( _) ( ' 1 ) ( '&' 1 ) s2
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SAUT AND TZVETKOV
Since for s 2 0 ( ') s2 const, ( ' 1 ) s2 ( '&' 1 ) s2 we shall suppose that s 2 =0 from now on. Without loss of generality we can assume that w^ 0 and v^ 0. We have the following relation, where '=(' 1, ' 2 ) and ' 1 =(' 11 , ' 21 ) _ 1 +_ 2 &_=&5;! 1 !(!&! 1 )(! 2 &!! 1 +! 21 )+3:! 1 !(!&! 1 ) +
(! 1 ' 1 &!' 11 ) 2 (! 1 ' 2 &!' 21 ) 2 + . ! 1 !(!&! 1 ) ! 1 !(!&! 1 )
(16)
Lemma 5. There exists a positive constant c 0 such that max[ |_|, |_ 1 |, |_ 2 | ] c |! 1(!&! 1 ) ! 3 |, Proof.
for
|!| c 0 .
(17)
Since ;<0 we have that 2 3 max[|_|, |_ 1 |, |_ 2 | ] |! 1 !(!&! 1 )| (& 15 4 ;! +3:),
where we used the elementary inequality ! 2 &!! 1 +! 21 34 ! 2. If :>0 then 2 we obtain (17) with c 0 =0. If :<0 then for & 15 8 ;! +3: 0 we have 2 3 max[ |_|, |_ 1 |, |_ 2 | ] |! 1 !(!&! 1 )| (& 15 8 ;! ),
which completes the proof of Lemma 5. We now continue the proof of Lemma 4. We shall denote by J the integral in the left-hand side of (15). We shall divide the domain of integration taking into account which term dominates in the left-hand side of (17). By symmetry we can assume that |_ 1 | |_ 2 |. Case 1. |!| max(2, c 0 ). We denote by J 1 the restriction of J on this region. In this case we have that K 1({, `, { 1 , ` 1 )
( ! ) 18 ( !&! 1 ) 18 . ( _1) ( _ 2 ) 12+2= (_) 12&3= 1 12+2=
CAUCHY PROBLEM FOR KP EQUATIONS
205
Case 1.1. |! 1 | 1, |!&! 1 | 1. We denote by J 11 the restriction of J on this region. In this case we have that K 1({, `, { 1 , ` 1 )
|! 1 | 18 |!&! 1 | 18 . ( _ 1 ) 12+2= ( _ 2 ) 12+2= (_) 12&3=
Hence using Holder inequality (10) and (11) we obtain J 11 &F &1(( _) &12+3= v^({, `))& L 4t (L 2(x, y, z ) ) 4 _&F &1(|! 1 | 18 ( _ 1 ) &12&2= w^({ 1 , ` 1 ))& L83 t (L (x, y, z) ) 4 _&F &1(|!&! 1 | 18 ( _ 2 ) &12&2= w^({&{ 1 , `&` 1 ))& L 83 t (L (x, y, z) )
c &w& 2L2 &v& L2 . Case 1.2. |! 1 | 1. We denote by J 12 the restriction of J on this region. In this case we have that K 1({, `, { 1 , ` 1 )
|! 1 | 18 |!&! 1 | 18 . |! 1 | ( _ 1 ) 12+2= ( _ 2 ) 12+2= ( _) 12&3=
Hence using Holder inequality (10) and (11) we obtain J 12 &F &1(( _) &12+3= v^({, `))& L 4t (L 2(x, y, z) ) 4 _&F &1( |! 1 | 18 ( _ 1 ) &12&2= |! 1 | &1 w^({ 1 , ` 1 ))& L 83 t (L (x, y, z) ) 4 _&F &1( |!&! 1 | 18 ( _ 2 ) &12&2= w^({&{ 1 , `&` 1 ))& L 83 t (L (x, y, z) )
c &w& L2 &|D x | &1 w& L2 &v& L2 . Case 1.3.
|!&! 1 | 1. This case can be treated similarly to Case 1.2.
Case 2. |!| max(2, c 0 ), |_| |_ 1 |. We denote by J 2 the restriction of J on this region. In this case we have that |_| 14&4= c |!| 34&12= |! 1 | 14&4= |!&! 1 | 14&4=. Hence K 1({, `, { 1 , ` 1 )
( !) 14+s1 +12= ( ! 1 ) &s1 +4= ( !&! 1 ) &s1 +4= . |! 1 | |!&! 1 | 14 ( _ 1 ) 12+2= ( _ 2 ) 12+2= ( _) 14 += 14
Case 2.1. |! 1 | 1, |!&! 1 | 1. We denote by J 21 the restriction of J on this region. In this case we have that K 1({, `, { 1 , ` 1 )
( _1)
|! | 18 |!&! 1 | 18 . ( _ 2 ) 12+2= (_) 14 +=
1 12+2=
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Using (10), (11) and Holder inequality we obtain J 21 &F &1(( _) &14&= v^({, `))& L4t (L2(x, y, z) ) 4 _&F &1(|! 1 | 18 ( _ 1 ) &12&2= w^({ 1 , ` 1 ))& L 83 t (L(x, y, z) ) 4 _&F &1(|!&! 1 | 18 ( _ 2 ) &12&2= w^({&{ 1 , `&` 1 ))& L83 t (L(x, y, z) )
c &w& 2L2 &v& L2 . Case 2.2. |! 1 | 1. We denote by J 22 the restriction of J on this region. In this case we have that K 1({, `, { 1 , ` 1 )
|! 1 | 18 |!&! 1 | 18 . |! 1 | ( _ 1 ) 12+2= ( _ 2 ) 12+2= ( _) 14 +=
Using (10), (11) and Holder inequality we obtain J 22 &F &1(( _) &14&= v^({, `))& L4t (L 2(x, y, z) ) 4 _&F &1( |! 1 | 18 ( _ 1 ) &12&2= |! 1 | &1 w^({ 1 , ` 1 ))& L83 (L (x, y, z) ) t 4 _&F &1( |!&! 1 | 18 ( _ 2 ) &12&2= w^({&{ 1 , `&` 1 ))& L 83 t (L (x, y, z) )
c &w& L2 &|D x | &1 w& L2 &v& L2 . Case 2.3.
|!&! 1 | 1. This case can be treated similarly to Case 2.2.
Case 3. |!| max(2, c 0 ), |_ 1 | |_|. We denote by J 3 the restriction of J on this region. In this case we have that |_ 1 | 14&4= c |!| 34&12= |! 1 | 14&4= |!&! 1 | 14&4=. Hence K 1({, `, { 1 , ` 1 )
( !) 14+s1 +12= ( ! 1 ) &s1 +4= ( !&! 1 ) &s1 +4= . |! 1 | |!&! 1 | 14 ( _ 1 ) 14+6= ( _ 2 ) 12+2= ( _) 12&3= 14
Case 3.1. |! 1 | 1, |!&! 1 | 1. We denote by J 31 the restriction of J on this region. In this case we have that K 1({, `, { 1 , ` 1 )
( !) 12= ( ! 1 ) 4= ( !&! 1 ) 4= . ( _ 1 ) 14+6= ( _ 2 ) 12+2= (_) 12&3=
207
CAUCHY PROBLEM FOR KP EQUATIONS
Using Lemma 3 and Holder inequality we obtain J 31 &F &1(|!| (12&3=)(12+=1 )($(r1 )6) ( _) &12+3= v^({, `))& Lqt (L r(x,1 y, z) ) t
_&F &1(|! 1 | (14+6=)(12+=1 )($(r2 )6) ( _ 1 ) &14&6= w^({ 1 , ` 1 ))& L qt 2 (L r(x,2 y, z) ) _&F &1(|!&! 1 | (12+2=)(12+=1 )($(r3 6) ( _ 2 ) &12&2= _w^({&{ 1 , `&` 1 ))& L q3 (L r(x,3 y, z) ) t
c &w&
2 L2
&v& L2 ,
provided 2 %(12&3=) =1& , q1 12+= 1
$(r 1 )=
(1&%)(12&3=) , 12+= 1
%(14+6=) 2 =1& , q2 12+= 1
$(r 2 )=
(1&%)(14+6=) , 12+= 1
%(12+2=) 2 =1& , q3 12+= 1
$(r 3 )=
(1&%)(12+2=) , 12+= 1
1 1 1 + + =1, q1 q2 q3
3 $(r 1 )+$(r 2 )+$(r 3 )= . 2
Now it is sufficient to take = 1 =2= and %=25 to ensure the restrictions of the Holder inequality. Case 3.2. |! 1 | 1. We denote by J 32 the restriction of J on this region. Using the arguments of Case 3.1, we similarly obtain J 32 c &w& L2 &&|D x | &1 w& L2 & &v& L2 . Case 3.2. |!&! 1 | 1. This case can be treated similarly to Case 3.2. This completes the proof of Lemma 4.
4. PROOF OF THEOREM 1 In this section we shall give the proof of Theorem 1. Note that the Eqs. (1) or (2) are equivalent to the integral equation u(t)=U(t) ,&
|
t
U(t&t$) u(t$) x u(t$) dt$. 0
(18)
208
SAUT AND TZVETKOV
Let be a cut-off function such that # C 0 (R),
supp /[&2, 2],
=1 over the interval [&1, 1].
We truncate (18) u(t)=(t) U(t) u(0)&(tT )
|
t
U(t&t$) u(t$) x u(t$) dt$.
(19)
0
We shall solve (19) globally in time. To the solutions of (19) will correspond local solutions of (18) in the time interval [&T, T ]. We shall apply a fixed point argument to solve (19) in X 12+=, s1 , s2 for sufficiently small =. Now we state the estimates for the two terms in the left-hand side of (19). Lemma 6. The following estimates hold for sufficiently small =>0 &(t) U(t) ,& X 12+2=, s1 , s2 c &,& H s1 , s2 ,
(20)
t
"(tT ) | U(t&t$) u(t$) u(t$) dt$ " x
0
X 12+2=, s1 , s2
=
cT &uu x & X &12+3=, s1 , s2 . Proof.
(21)
To prove (20) it is sufficient to note that &u& X b .s1 , s2 =&U(&t) u& H b, s1 , s2 ,
where H b, s1 , s2 is equipped with the norm
|
&u& 2H b, s1 , s2 = ( {) b ( !) s1 ( ') s2 (1+ |!| &1 ) 2 |u^({, !, ')| 2 d{ d! d'. The proof of (21) is a direct consequence of [12], Lemma 3.2 applied with b=12+2= and b$=&12+3=. This completes the proof of Lemma 6. Now we define an operator L Lu(t) :=(t) U(t) u(0)&(tT )
|
t
U(t&t$) u(t$) x u(t$)) dt$.
0
We obtain from Lemmas 5 and 6 &Lu& X 12+2=, s1 , s2 c &,& H s +cT = &u& 2X 12+2=, s1 , s2 .
(22)
Similarly we obtain &Lu&Lv& X 12+2=, s1 , s2 cT = &u+v& X 12+2=, s1 , s2 &u&v& X 12+2=, s1 , s2 .
(23)
209
CAUCHY PROBLEM FOR KP EQUATIONS
Using (22) and (23) we can apply the contraction mapping principle for sufficiently small T which completes the proof of Theorem 1.
5. PROOF OF THEOREM 3 It is known (cf. [6]) that in some cases (1) (resp. (2)) possesses solitary wave solutions. We recall that a solitary wave is a ``localized'' solution of (1) (resp. (2)) of the form u(x&ct, y) (resp. u(x&ct, y, z)). More precisely we have the following theorem. Theorem 5.1. (cf. [6, 7]). Let ;>0 and : 0. Then (1) (resp. (2)) possesses a solitary wave solution u(x&ct, y) (resp. u(x&ct, y, z)) such that
u # L 2y(L 1x ) , H s, s=0
\
+
resp. u # L 2(y, z)(L 1x ) , H s . s=0
The same statement holds when ;=0 and :<0. We have (cf. [7]) that r $u # L 2(R d ), for $
(d=2),
u # L 2(y, z)(L 1x )
(d=3).
The rest of the proof of Theorem 5.1. is contained in [6, 7]. Note that when ;=0, :<0 an explicit form of a solitary wave can be derived by the inverse scattering method and we will use it below. In the 3D case we will use essentially scaling arguments and no explicit form of the solitary waves is needed. Actually, solitary waves for (1) or (2) (cf. [6, 7]) can be obtained by the aid of the concentration compactness principle for some pure power nonlinearities and the method for obtaining ill-posedness could be extended to these cases (see the end of the section). v Let d=2, ;=0 and :<0. Then (1) has a solitary wave solution of the form u c(t, x, y)=, c(x&ct, y),
c>0,
where u c(0, x, y)=, c(x, y)=c, 1(c 12x, cy) :=c,(c 12x, cy).
210
SAUT AND TZVETKOV
Here u c is the ``lump''solitary wave of KP-I equation. Recall that c c2 8c 1& (x&ct) 2 + y 2 3 3 u c(t, x, y)= 2 2 c c 1+ (x&ct) 2 + y 2 3 3
\
+
\
+
is a solitary wave solution of (1) with :=&1 and ;=0. We will need a decay estimate of a fractional derivative of , c . Obviously x, c does not belong to L 2(R 2 ), but Lemma 7 (cf. Appendix). Let =>0. Then |D x | = (x, c ) # L 2(R 2 ). We shall denote by Fx the partial Fourier transform with respect to x. One has Fx (, c )(!, y)=c 12Fx (,)
\c
! 12
+
, cy .
Further we have &, c & 2H s, 0 =
|
R2
}
( !) 2s Fx (,)
2
!
\c +} d! dy. ,y 12
Hence for s<&12 Lebesgue theorem yields lim &, c & 2H s, 0 =c s &Fx (,)(0, .)& 2L2 =c s &,& 2L 2y (L 1x ) ,
cÄ
where cs =
|
( !) 2s d!.
&
Similarly we obtain (, c1 , , c2 ) s =
c2 c1
\ +
12
|
R2
( !) 2s Fx (,)
!
\c , y+ F (,) \c 12 1
x
! 12 2
where (., .) s stays for the H s, 0 scalar product. Hence lim
c1 , c2 Ä , c1 c2 Ä 1
(, c1 , , c2 ) s =c s &F x (,)(0, .)& 2L2
,
c2 y d! dy, c1
+
211
CAUCHY PROBLEM FOR KP EQUATIONS
and lim
c1 , c2 Ä , c1 c2 Ä 1
&, c1 &, c2 & H s, 0 =0.
Now we have that Fx (u c )(t, !, y)=c 12 exp(&itc!) Fx (,)(!c 12, cy). The presence of the oscillatory term exp(&itc!) makes the expression (u c1 (t, .), u c2 (t, .)) s tend to zero as n Ä by taking c 1 =n 2 and c 2 =(n+1) 2. More precisely we have by integration by parts (u n 2 (t, .), u (n+1) 2 (t, .)) s =c(n, t)
|
R2
!(e it(2n+1) ! ) (!) 2s Fx (,)
_Fx (,)
\
=2sc(n, t)
|
_Fx (,)
! (n+1) 2 y d! dy , n+1 n2
+
R2
\
!
\ n , y+
e it(2n+1) ! ( !) 2s&2 !Fx (,)
! ,y n
\ +
! (n+1) 2 , y d! dy n+1 n2
c(n, t) n = + in
+
|
e it(2n+1) !
R2
_F x (|D x | = (x,)) c(n, t)(n+1) = + i(n+1)
( !) 2s |!| =
! ! (n+1) 2 , y Fx (,) , y d! dy n n+1 n2
\ +
|
e it(2n+1) !
R2
_F x (|D x | = (x,))
\
\
+
(!) 2s ! Fx (,) , y = |!| n
\ +
! (n+1) 2 y d! dy :=(1)+(2)+(3), , n+1 n2
+
where c(n, t)=
n+1 . in(2n+1) t
212
SAUT AND TZVETKOV
Using CauchySchwarz inequality and the estimate &,& L &,& L1 we obtain (n+1) 2 |(1)| 2s |c(n, t)| ( !) 2s&2 &,(., y)& L1 , ., y d! dy n2 R2 L1 cn |c(n, t)| &,& 2L 2y (L 1x ) . n+1
"\
|
+"
It is clear that the last expression tends to zero as n tends to infinity. Further we have |(2)|
n = |c(n, t)| n
( !) 2s ! Fx (|D x | = (x,)) , y = n R 2 |!|
}
|
n = |c(n, t)| n &,& L y2 (Lx1 ) n n+1
(n+1) 2 y n2
\ +} " \ +" ! {| }F (|D | (x,)) \n , y+} d! dy= , .,
2
d! dy L1
12
=
R2
x
x
n = |c(n, t)| n 32 &,& L 2y (Lx1 ) &|D x | = (x,)& L 2(x, y) , n n+1
where we used that |!| &= ( !) 2s # L 2! if =<12 and Lemma 7. Hence we obtain that for =<12, |(2)| tends to zero as n tends to infinity. The term |(3)| can be estimated in a similar fashion. Hence lim (u (n+1) 2 (t, .), u n 2 (t, .)) s =0.
nÄ
Moreover, for any t>0 lim &u (n+1) 2 (t, .)&u n 2 (t, .)& 2H s, 0 =2c s &Fx (,)(0, .)& 2L2 .
nÄ
Therefore we obtain that (1) is locally ill-posed in H s, 0(R 2 ) for ;=0 and :<0, when s<&12. v Let d=3, ;>0 and :=0. Then (2) has a solitary wave solution of the form u c(t, x, y, z)=, c(x&ct, y, z),
c>0,
where u c(0, x, y, z)=, c(x, y, z)=c, 1(c 14 x, c 34 y, c 34 z) :=c,(c 14 x, c 34 y, c 34 z). One obtains easily that F x (, c )(!, y, z)=c 34 F x (,)
\c
! 14
+
, c 34 y, c 34 z .
213
CAUCHY PROBLEM FOR KP EQUATIONS
Further we have &, c & 2H s, 0 =
|
}
( !) 2s F x (,)
R3
\c
2
!
, y, z 12
+} d! dy dz.
Hence for s<&12 Lebesgue theorem yields lim &, c & 2H s, 0 =c s &Fx (,)(0, .)& 2L2 =c s &,& 2L 2(y, z) (L1x ) .
cÄ
Similarly we obtain c2 c1
\+
(, c1 , , c2 ) s =
34
|
_Fx (,)
R2
\c
! 14 2
,
c2 c1
!
\c , y, z+ c y, z d! dy dz, c +
( !) 2s Fx (,)
14 1
2
1
where (., .) s stays for the H s, 0 scalar product. Hence lim
c1 , c2 Ä , c1 c2 Ä 1
(, c1 , , c2 ) s =c s &Fx (,)(0, .)& 2L2
and lim
c1 , c2 Ä , c1 c2 Ä 1
&, c1 &, c2 & H s, 0 =0.
Now we have that Fx (u c )(t, !, y, z)=c 34 exp(&itc!) Fx (,)(!c 14, cy, cz). The presence of the oscillatory term exp(&itc!) makes the expression (u c1 (t, .), u c2 (t, .)) s tend to zero as n Ä by taking c 1 =n 2 and c 2 =(n+1) 2. Let x$=( y, z). Then (u n 2 (t, .), u (n+1) 2 (t, .)) s =c(n, t)
|
R3
_F x(,)
!(e it(2n+1) ! ) (!) 2s Fx (,)
\
\
! , x$ n 12
(n+1) 2 ! , x$ d! dx$ 12 (n+1) n2
+
+
214
SAUT AND TZVETKOV
!
\n
|
e it(2n+1) ! ( !) 2s&2 !Fx (,)
_Fx (,)
\
! (n+1) 2 , x$ d! dx$ (n+1) 12 n2
c(n, t) in 12
|
=2sc(n, t)
R3
+
_F x (,) +
, x$
+
+
e it(2n+1) ! ( !) 2s Fx (x,)
R3
\
!
\n
, x$
12
+
! (n+1) 2 , x$ d! dx$ 12 (n+1) n2
+
c(n, t) i(n+1) 12
_Fx (x,)
12
\
|
R3
e it(2n+1) ! ( !) 2s Fx (,)
!
\n
12
, x$
+
(n+1) 2 ! , x$ d! dx$ (n+1) 12 n2
+
:=(1)+(2)+(3), where c(n, t)=
(n+1) 32 . in (2n+1)t 32
Using CauchySchwarz inequality and the estimate &,& L &,& L1 we obtain |(1)| 2s |c(n, t)| |c(n, t)|
|
R3
"\
( !) 2s&2 &,(., x$)& L1 , .,
(n+1) 2 x$ n2
+"
d! dx$ L1
cn 2 &,& 2L 2x$ (L1x) . (n+1) 2
It is clear that the last expression tends to zero as n tends to infinity. Further we have |(2)|
|c(n, t)| n 12
|
R3
}
( !) 2s Fx (x,)
! , x$ n 12
(n+1) 2 x$ n2
\ +} " \ +" d! dx$ ! {| }F (x,) \n , x$+} d! dx$= , .,
|c(n, t)| n 2 &,& L 2x$ (L1x ) n 12 (n+1) 2
2
|c(n, t)| n 14 n 2 &,& L2x$ (L1x ) &x,& L 2(x, x$) , n 12 (n+1) 2
R3
x
L1
12
12
(53)
215
CAUCHY PROBLEM FOR KP EQUATIONS
where we used that x, # L 2(R 3 ), which follows from the decay properties of the solitary waves. (See [7].) Hence we obtain that |(2)| tends to zero as n tends to infinity. The term |(3)| can be estimated in a similar fashion. Hence lim (u (n+1) 2 (t, .), u n 2 (t, .)) s =0.
nÄ
Moreover lim &u (n+1) 2 (t, .)&u n 2 (t, .)& 2H s, 0 =2c s &Fx (,)(0, .)& 2L2 .
nÄ
Therefore we obtain that (2) is locally ill-posed in H s, 0(R 3 ) for ;>0 and :=0, when s<&12. v Let d=3, ;=0 and :<0. Then (2) has a solitary wave solution of the form u c(t, x, y, z)=, c(x&ct, y, z), where u c(0, x, y, z)=, c(x, y)=c,(c 12x, cy, cz). We have that , c(!, ')=c &32,
!
' . c
\c + 12
,
Further we have &, c & 2H4 s1 , s2 =c s1 +2s2 &12
|
R3
|!| 2s1 |'| 2s2 |,(!, ')| 2 d! d'=c s1 +s2 &12 &,& 2H4 s1 , s2 .
Hence when s 1 +2s 2 =12, s 1 0, s 2 0 we have that &, c & H4 s1 , s2 =&,& H4 s1 , s2 . Similarly we obtain (, c1 , , c2 ) s1 , s2 =
c2 c1
\ +
&32
|
|!| 2s1 |'| 2s2 ,(!, ') , R3
c1 c2
\\ +
12
!,
c1 ' d! d', c2
+
where now (., .) s1 , s2 stays for the H4 s1 , s2 scalar product. Hence using Lebesgue theorem we obtain lim
c1 , c2 Ä , c1 c2 Ä 1
(, c1 , , c2 ) s1 , s2 =&,& 2H4 s1 , s2
216
SAUT AND TZVETKOV
and &, c1 &, c2 & H4 s1 , s2 =0.
lim
c1 , c2 Ä , c1 c2 Ä 1
Now we have that u^ c(t, !, ')=c &32 exp(&itc!) ,
!
'
\c , c+ , 12
and furthermore (u c1 (t, .), u c2 (t, .)) s1 , s2 =
c2 c1
&32
\ + | c _, \\c +
R3
exp(&it!(c 1 &c 2 )) |!| 2s1 |'| 2s2 ,(!, ')
12
1
2
!,
c1 ' d! d'. c2
+
Hence using RiemannLebesgue lemma arguments we obtain as above lim (u (n+1) 2 (t, .), u n 2 (t, .)) s1 , s2 =0.
nÄ
Moreover lim &u (n+1) 2 (t, .)&u n 2 (t, .)& 2H s1 , s2 =&,& 2H4 s1 , s2 .
nÄ
Therefore we obtain that (1) is locally ill-posed in H4 s1 , s2 (R 2 ) for ;=0 and :<0, when s 1 +2s 2 =12 and s 1 0, s 2 0. This completes the proof of Theorem 3. There are some difficulties to extend the results of Theorem 3 to the two dimensional case with higher order term. If d=2, ;>0 and :=0 then (1) has a solitary wave solution of the form u c(t, x, y)=, c(x&ct, y),
c>0,
where u c(0, x, y)=, c(x, y)=c, 1(c 14 x, c 34 y) :=c,(c 14 x, c 34 y). One has , c(!, ')=,
\c
! 14
,
' . c 34
+
CAUCHY PROBLEM FOR KP EQUATIONS
217
and furthermore &, c & 2H s1 , s2 =
|
} \c
( !) 2s1 ( ') 2s2 , R2
!
, 14
' c 34
2
+} d! d'.
(24)
It is easy to see (cf. [7]) that , does not belong to L 1(x, y) . Hence one can not define the trace of ,(!, ') at zero and therefore it is impossible to pass to the limit as c tends to infinity in (24). We conclude this section with some remarks for the generalized KP-I equation
{
(u t +:u xxx +;u xxxxx +u pu x ) x +u yy +u zz =0 u(0, x, y, z)=,(x, y, z).
(25)
If ;=0 and u(t, x, y, z) is a solution of (25) then so is u *(t, x, y, z)=* 2 pu(* 3 t, *x, * 2y, * 2z) and &u *(t, .)& H4 s1 , s2 =* s1 +2s2 &(5p&4)2p &u(* 3t, .)& H4 s1 , s2 . In [6] it is shown that for 1 p<43 and :<0 (25) possesses nontrivial localized solitary waves of the form u(t, x, y, z)=, c(x&ct, y, z),
c>0,
where , c(x, y, z)=c 1 p,(c 12x, cy, cz), Using the arguments of the proof of Theorem 3 one can prove that (25) is locally ill-posed in H4 s1 , s2 for s 1 +2s 2 &(5p&4)2p=0, s 1 0, s 2 0. If :=0 and u(t, x, y, z) is a solution of (25) then so is u *(t, x, y, z)=* 4 pu(* 5t, *x, * 3y, * 3z) and &u *(t, .)& H4 s1 , s2 =* s1 +3s2 +(8&7p)2p &u(* 5t, .)& H4 s1 , s2 . In [6] it is shown that for 1 p<83 and ;>0 (25) possesses nontrivial localized solitary waves of the form u(t, x, y, z)=, c(x&ct, y, z),
c>0,
where , c(x, y, z)=c 1 p,(c 14x, c 34y, c 34z), Using the arguments of the proof of Theorem 3 one can prove that (25) is locally ill-posed in H4 s1 , s2 for
218
SAUT AND TZVETKOV
s 1 +2s 2 +(8&7p)2p=0, s 1 0, s 2 0. In particular when p=87 (25) is locally ill-posed in L 2. Finally we note that when :<0 and ;>0, the existence of solitary waves is known for a suitable range of p's (cf. [6]) but the scaling argument leading to ill-posedness does not work.
6. A REMARK ON THE (GENERALIZED) KDV EQUATION In this section we shall show that the example providing ill-posedness for the (generalized) KdV equations uses just scaling properties and hence the special form of the solitary waves is not needed. Consider the (generalized) KdV equation u t +u xxx +u pu x =0.
(26)
The local ill-posedness of (26) is studied in [4, 5]. It is known that (26) possesses solitary wave solutions of the form u c, p(t, x)=, c, p(x&ct), where , p(x)=[ 12 ( p+2) sech 2( 12 px)] 1 p.
u c, p(0, x)=c 1p, p(c 12x),
Since the solitary waves of (26) decays exponentially at infinity one can define the trace of its Fourier transforms at zero and then prove local illposedness of (26) for p=2 in H s, s<&12 which is the result of [5]. Actually if p=2 then , c, 2(!)=, 2(!c 12 ) and one obtains that for s<&12 lim &, c, 2 & 2H s = lim
cÄ
cÄ
| ( !)
s
, 2(!c 12 ) d!=c s , 2(0)
and therefore one can easily obtain lim
(, c1 , , c2 ) s =c s , 2(0)
lim
&, c1 , 2 &, c2 , 2 & H s =0.
c1 , c2 Ä , c1 c2 Ä 1
and moreover c1 , c2 Ä , c1 c2 Ä 1
Further we have F(u c, 2 )(t, !)=exp(&itc!) , 2(!c 12 ).
219
CAUCHY PROBLEM FOR KP EQUATIONS
Due to the RiemannLebesgue lemma, the presence of the oscillatory term exp(&itc!) makes the expression (u c1 , 2(t, .), u c2 , 2(t, .)) s tend to zero as n Ä by taking c 1 =n 2 and c 2 =(n+1) 2. Therefore we obtain that (26) is locally ill-posed in H s for p=2 when s<&12. In fact one can easlily show that the sequence , c, 2 converge strongly in H s, s<&12 to - 2 ?$ as c tends to infinity. Here $ stays for Dirac delta function. Similarly we can see that u c, 2(t, .) converge weakly to zero in H s. Now the boundedness of &, c & H s , s<&12 provides the locall ill-posedness. Similarly when p=1 (the ``usual'' KdV equation) one can prove that , c, 1(x) converge weakly in H s, s<&32 to - 2 ? x $(x) as c tends to infinity. We also have that u c, 1(t, .) converge weakly to zero as c tends to infinity. We also have that &u c, 1(t, .)& H s =&, c, 1 & H s =c 12 &, 1 & H s . Since x $(x) # H s, s<&32 the last arguments strongly suggest local illposedness of (26) in H s, s<&32 when p=1. However, as was pointed out to us by the referee, J. Bourgain has shown in [11] that if one requires that the map data solution u 0 [ u(t) be smooth then the results of KenigPonceVega [17] for the KdV ( p=1, s>&34) or the modified KdV ( p=2, s 14) are optimal.
7. APPENDIX In this appendix we shall give the proof of Lemma 7. Proof of Lemma 7. By scaling it suffices to prove that |D x | = (x,) # L (R 2 ), where 2
,(x, y)=
1&x 2 + y 2 . (1+x 2 + y 2 ) 2
We have ,(x, y)=
1 2x 2 & :=, 1(x, y)+, 2(x, y). 2 2 1+x + y (1+x 2 + y 2 ) 2
A simple computation shows that 1 , 1(!, y) :=Fx (, 1 )(!, y)= (1+ y 2 ) 12 =
1 2 12 e &(1+ y ) |!|. (1+ y 2 ) 12
2 12 x!
e &i(1+ y ) 1+x 2 &
|
dx
220
SAUT AND TZVETKOV
Hence , 1 2 12 (!, y)=&sgn ! } e &(1+ y ) |!| ! and
|
|!| 2=
R2
}
, 1 (!, y) !
2
} d! dy= |
|!| 2= e &2(1+ y
2 ) 12 |!|
d! dy
R2
=
|X | 2= e &2 |X | dX dy< , 2 12+= R 2 (1+ y )
|
since =>0. Let now (x, y)=& 12 , 2(x, y). A simple calculation yields (!, y) :=Fx ()(!, y)=
1 (1+ y 2 ) 12
2 12 x!
x 2 e &i(1+ y ) (1+x 2 ) 2 R2
|
dx.
Write x 2(1+x 2 ) 2 =(11+x 2 )&(1(1+x 2 ) 2 ) and recall that (cf. [1]) F
1
\1+x + (!)=e
&|!|
2
and F
1
\(1+x ) + (!)=c |!| 2 2
32
K 32( |!| ),
where K 32 is the modified Bessel function of order 32 (cf. [1]) K 32(z)=
?
1
2z } e \1+ z+ . &z
Therefore (!, y)= =
1 1 1 2 12 e &(1+ y ) |!| + F ((1+ y 2 ) 12 !) (1+ y 2 ) 12 (1+ y 2 ) 12 (1+x 2 ) 2
\
+
1 2 12 e &(1+ y ) |!| (1+ y 2 ) 12 ? c 2 12 (1+ |!| (1+ y 2 ) 12 ) e &(1+ y ) |!| } 2 (1+ y 2 ) 12
& =
1&c - ?2 &(1+ y 2 ) 12 |!| e &c (1+ y 2 ) 12
:= 1(!, y)+ 2(!, y).
?
2 } |!| e
&(1+ y 2 ) 12 |!|
(61)
221
CAUCHY PROBLEM FOR KP EQUATIONS
The term 1(!, y) leads to a computation similar to that of , 1 . Now 2 ? 2 12 2 12 (!, y)=&c } sgn ! } (e &(1+ y ) |!| & |!| (1+ y 2 ) 12 e &(1+ y ) |!| ) ! 2 := 21(!, y)+ 22(!, y).
Now we have that
|
R2
| =c |
|!| 2= | 21(!, y)| 2 d! dy=c
R2
|!| 2= e &2(1+ y
2 ) 12 |!|
d! dy
|X | 2= e &2 |X | dX dy< , 2 12+= R 2 (1+ y )
since =>0 and furthermore
|
R2
|!| 2= | 22(!, y)| 2 d! dy=c
|
=c
|
|!| 1+2= (1+ y 2 ) 12 e &2(1+ y
2 ) 12 |!|
d! dy
R2
|X | 1+2= e &2 |X | dX dy< . 2 12+= R 2 (1+ y )
This completes the proof. Remark. We conjecture that the result of Lemma 7 is valid for any localized solitary wave of KP-I type equation.
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SAUT AND TZVETKOV
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