On the local and global well-posedness theory for the KP-I equation

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On the local and global well-posedness theory for the KP-I equation

Sur le caractère bien posé local et global pour l’équation KP-I

Carlos E. Kenig

aSchool of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA bDepartment of Mathematics, University of Chicago, Chicago, IL 60637, USA

Received 28 October 2003; accepted 8 December 2003 Available online 29 July 2004

Abstract

In this paper we obtain new local and global well-posedness results for the KP-I equation.

2004 Elsevier SAS. All rights reserved.

Résumé

Dans cet article nous obtenons de nouveaux résultats sur le caractère bien posé local et global de l’équation KP-I.

2004 Elsevier SAS. All rights reserved.

Keywords: Local and global well-posedness; KP-I equation

Introduction

In this note we prove a new local well-posedness theorem for the KP-I equation (1.2), and use the well-known conservation laws to establish a new global well-posedness result as a consequence. These, our main results, are Theorems 3.1 and 3.4.

As has been shown in [13–15], the KP-I equation is badly behaved with respect to Picard iterative methods in the standard Sobolev spaces and the natural energy space, since the flow map fails to beC²at the origin in these spaces. Thus, one is left with two choices: one either considers function spaces that are different from the standard Sobolev spaces, and then shows that the Picard iteration scheme does apply for them (this is the strategy

E-mail address: cek@math.uchicago.edu (C.E. Kenig).

1 Supported in part by NSF, and at IAS by The von Neumann Fund, The Weyl Fund, The Oswald Veblen Fund and the Bell Companies Fellowship.

0294-1449/$ – see front matter 2004 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpc.2003.12.002

of [4–6], where weighted spaces are used), or one decides to abandon the Picard iteration scheme and tries a different approach to well-posedness theory, which is the approach taken in the present paper. The KP-I equation arises in water wave theory, as modeling capillary gravity waves, in the presence of strong surface tension effects.

Its companion equation, the KP-II equation (where the sign in front of the∂_x⁻¹∂_y²u term in (1.2) is+ instead of−) is also of importance in water wave theory, and from the point of view of well-posedness theory it is much better understood, since the above mentioned difficulty with Picard iterative methods does not occur for it. In fact (see [2]) KP-II is known to be both locally and globally well-posed inL²(R²), “by Picard iteration”, and even in some Sobolev spaces of negative index [18–20]. In [13], the authors use the local well-posedness theorem of [7], obtained by a version of the classical energy method, and they couple it with some of the known (formal) conserved quantities, and an ingenious use of the dispersive estimates of Strichartz type for KP-I (see [17]), to show global well-posedness for KP-I in the spaceZ= {φ∈L²(R²):φ_L²+ ∂_x³φ_L²+ ∂_yφ_L²+ ∂_x∂_yφ_L²+ ∂_x⁻¹∂_yφL² + ∂_x⁻²∂_y²φL^∞<∞}. Our contribution comes from improving on the local well-posedness result given by the classical energy estimate, by showing local well-posedness in the spaceY_s = {φ∈L²(R²): φL²+ J_x^sφL²+ ∂_x⁻¹∂yφL²<∞}fors >3/2, whereJ_x^sf (ξ, η)=(1+ |ξ|²)^s/2f (ξ, η). We first observe that, in thisˆ context, it is still enough to control∂xu_L¹_tL∞xy (Lemma 1.3). We then adapt the version of the argument first introduced by Koch and Tzvetkov [12] in the context of the Benjamin–Ono equation, given in [10], also for the Benjamin–Ono equation. This uses the ‘interpolation inequality’ of Lemma 1.7, which is the main new ingredient of our proof. Another important ingredient of the proof is a ‘product theory’ Leibniz rule for fractional derivatives, given in Lemma 1.8(ii), which follows from [16]. Once our local well-posedness result is established, we use the results in [13], to establish global well-posedness in

Z₀=

φ∈L²(R²): φL²+ ∂_x⁻¹∂_yφL²+ ∂_x²φL²+ ∂_x⁻²∂_y²φL²<∞ .

The question of the local and global well-posedness of KP-I in the energy spaceY = {φ∈L²(R²): φL²+ ∂xφL²+ ∂_x⁻¹∂yφL2<∞}remains a challenging open problem.

1. Preliminary estimates

We introduce the space, fors∈R, H₋^s₁(R²)=

u∈S(R²): uH₋₁^s (R²)<∞ , where

uH₋₁^s (R²)=

1+ |ξ|⁻¹2

1+ |ξ|²+ |η|²s

·u(ξ, η)ˆ ²dξ dη

1/2

andH₋^∞₁(R²)=

s0H₋^s₁(R²). We recall a local well-posedness result of Iorio and Nunes [7]. Foru∈H₋^s₁(R²), s2 we define∂_x⁻¹∂_y²u(ξ, η)= −η²/ξu(ξ, η).ˆ

Lemma 1.1. The Cauchy problem ∂tu+∂_x³u−∂_x⁻¹∂_y²u+u∂xu=0,

u|t=0=u₀ (1.2)

is locally well-posed inH₋^s₁(R²), fors >2.

Also, recall that, from [14], if, in addition,u₀Z<∞, whereZ= {φ∈L²(R²): φZ<∞}, and φZ= φ_L2+ ∂_x³φL²+ ∂_yφL²+ ∂_xy² φL²+ ∂_x⁻¹∂_yφL²+ ∂_x⁻²∂_y²φL²,

thenuin fact extends for all time and remains also inZfor all time.

Lemma 1.3. Letube a solution to (1.2), withu₀∈H₋^∞₁(R²)∩Z. LetJ_x^sf (ξ, η)=(1+ |ξ|²)^s/2f (ξ, η). Then, forˆ s0, we have, for anyT >0:

sup

0<t <T

J_x^su_L²_xy+ ∂_x⁻¹∂_yu_L²_xyC_sexp

C_s T

0

∂_xuL^∞_xy

J_x^su_0L²_xy+ ∂_x⁻¹∂_yu_0L²_xy .

Proof. We applyJ_x^s to (1.2), multiply byJ_x^su, and integrate in(x, y). We obtain

∂t

R²

J_x^su 2

2

−

R²

∂_x⁻¹∂_y²J_x^suJ_x^su+

R²

∂_x³J_x^suJ_x^su= − J_x^s(u∂xu)J_x^s(u).

The last two terms in the left-hand side of the equality are 0. To bound the right-hand side, we use the Kato and Ponce [9] commutator estimate, in the form

J^s(f g)−f J^s(g)

L²(R)C_s.

∂_xfL^∞(R)J^s−1gL²(R)+ J^sfL²(R)gL^∞(R)

. (1.4)

We thus see that the right-hand side above equals (withf=u,g=∂_xu), integrating first inx, and then iny

= −

uJ_x^s∂xu.J_x^su+O

∂xuL^∞(R²)J_x^su²_L2(R²)

.

Thus, d dt

|J_x^su|²dx dyCs∂xuL^∞(R²)J_x^su²_L2(R²),

and the desired estimate for J_x^suL²(R²) follows. To estimate ∂_x⁻¹∂_yu, we apply∂_x⁻¹∂_y to (1.2), multiply by (∂_x⁻¹∂_yu)and integrate. The right-hand side will be, this time,

− ∂_y u²

2 ·∂_x⁻¹∂_yu= − u∂_yu·∂_x⁻¹∂_yu= − u∂_x∂_x⁻¹∂_yu·∂_x⁻¹∂_yu= ∂xu∂_x⁻¹∂yu∂_x⁻¹∂yu

2 ,

and the desired estimate follows. 2

We next recall the Strichartz estimates associated to the free KP-I evolution.

Remark. For future use, forf ∈S(R²)we defineD_x^af (ξ, η)= |ξ|^af (ξ, η),ˆ D_y^bf (ξ, η)= |η|^bf (ξ, η).ˆ

Lemma 1.5. LetU (t)=exp(−t (∂_x³−∂_x⁻¹∂_y²))be the unitary group defining the free KP-I evolution. Then, the following estimates hold

U (t)φ_L^q

TL^r_xyφL²,

t

0

U (t−t)F (t) dt L^q_TL^r_xy

F_Lq¯ TL^r_xy^¯ , where_q¹+¹_r =_q¹_¯+¹_r¯−1=¹₂, 2r <+∞, 1<r¯2.

Note that the constant in the two inequalities of Lemma 1.5 is independent ofT. The proof of Lemma 1.5 is given in [17]. Combining Lemma 1.5 with a Sobolev-embedding, we obtain

Corollary 1.6. For eachT >0,ε >0, we have U (t)φ

L²_TL^∞_xyC_ε,T

φL²+ D_x^εφL²+ D^ε_yφL²

.

Our main new linear estimate is contained in the next lemma. It is an adaptation to the present setting of Proposition 2.8 in [10], which in turn is a reformulation of the main estimate in [12]. See also [15] for another use of the [12] idea in the context of KP-I.

Lemma 1.7. Letδ >0,T ∈(0,1]. Assume thatw∈C([0, T];H₋³₁(R²))is a solution to the linear equation

∂tw+∂_x³w−∂_x⁻¹∂_y²w=F.

Then,

∂xw_L¹

TL^∞_xyCδ,T

sup

0<t <T

J

3 2+2δ

x wL²+ sup

0<t <T

J

3 2+δ

x D^δ_ywL²

+ T

0

J_x^1/2+2δF_L²+ J_x^1/2+δD^δ_yF_L²

.

Proof. We use a Littlewood–Paley decomposition ofwin theξ variable. More precisely, ifη∈C₀^∞(¹₂<|ξ|<2), χ∈C₀^∞(|ξ|<2), are such that 1=_∞

k=1η(2^−kξ )+χ (ξ ), and forλ=2^k,k1, we definew_λ=Q_k(w), where Qˆ_kw(ξ, η)=η(2^−kξ )w(ξ, η),ˆ w₀=Q₀(w), and Q₀w(ξ, η)=χ (ξ )w(ξ, η). We first estimateˆ ∂_xw_λL¹

TL^∞_xy: let us assume, for simplicity, thatT =1. Split[0,1] =

jIj, whereIj = [aj, bj]andbj−aj=1/λ,j =1, . . . , λ.

Then,

∂xwλ_L¹

TL^∞_xy

j

∂xwλ_L¹

IjL^∞_xyλ

j

wλ_L¹

IjL^∞_xy,

where we have used thatξ η(2^−kξ )has inverse Fourier transform whoseL¹norm inx is bounded byCλ. We now use Cauchy–Schwartz inIj, and continue with

λ^1/2

j

wλ_L²

IjL^∞_xy.

We now apply Duhamel’s formula, in eachIj, to obtain, fort∈Ij, w_λ(t)=U (t−a_j)w_λ(−, a_j)+

t

aj

U (t−t)F_λ(t) dt. We use Corollary 1.6, and continue with

λ^1/2

j

J_x^δw_λ(−, a_j)

L²+D_y^δw_λ(−, a_j)

L²

+λ^1/2

j

Ij

J_x^δF_λL²+ D_y^δF_λL²

sup

t∈[0,1]

J_x^3/2+δw_λ(−, t)

L²+J_x^3/2D_y^δw_λ(−, t)

L²+ t

0

J_x^1/2+δF_λL²+ J_x^1/2D^δ_yF_λL².

Write noww1=

k1Qk(w), so thatw=w1+w0, andw1=

λwλ. Then,

∂xw₁

λ

∂xwλ_L¹

TL^∞_xy

k1

sup

t

J_x^3/2+δQk(w)

L²+J_x^3/2D^δ_yQk(w)

L²

+ T

0

J

1 2+δ

x Q_k(F )

L²+J_x^1/2D_y^δQ_k(F )

L²

k1

2^−kδ

sup

t

J_x^3/2+2δQ_k(w)

L²+J_x^3/2+δD^δ_yQ_k(w)

L²

+ T

0

J_x^12+2δQ_k(F )

L²+J_x^1/2+δD^δ_yQ_k(F )

L²

,

and the desired estimate forw₁follows. The proof of the estimate forw₀is simpler, and Lemma 1.7 follows. 2 Lemma 1.8 (Leibniz rule for fractional differentiation).

(i) For 0< α <1, D_x^α(f g)

L²(R)C

D_x^αfL^p1(R)gL^q1(R)+ fL^r1(R)D_x^αgL^s1(R)

with ¹₂=_p¹₁ +_q¹₁ =_r¹₁ +_s¹₁, 1< p1, q1, r1, s1∞. (See, for example, [11].) (ii) For 0< α,β <1, we have

D_x^αD_y^β(f.g)

L²_xyC f_L^p1

xyD^α_xD_y^βg_L^q1

xy + g_L^p2

xyD^α_xD_y^βf_L^q2

xy+ D_x^αg_L^p3

xyD_y^βf_L^q3 xy

+ D^β_yg_L^p4

xyD_xαf_L^q4 xy

,

where ¹₂=_p¹

i +_q¹

i, 1< p_i∞, 1< q_i∞.

The proof of (ii) reduces, using an argument similar to the one given in [9] to a (now two parameter) version of the multilinear theorem of Coifman and Meyer [3]. The proof of (ii) is given in [16] (see also [8]).

2. A priori estimates

In this section we prove:

Lemma 2.1. Letube a solution to (1.2), withu₀∈H₋^∞₁(R²)∩Z, defined, for allT. Let u₀Ys = J_x^su₀L²_xy+(∂_x⁻¹∂_yu₀)

L²_xy.

Then, for anys >3/2, there existsT =T_s, depending only onu₀Ys,s, and a constantC_T, depending only on u₀Ys,s, so that

T

0

∂xuL^∞_xydt+ T

0

uL^∞_xydtCT. (2.2)

Proof. Lets=³₂+δ₀, and chooseδ >0 so that 2δδ₀. Let us define

f (T )= ∂_xu_L¹

TL^∞_xy+ u_L¹

TL^∞_xy= T

0

∂_xuL^∞_xy+ uL^∞_xy

dt. (2.3)

We will prove the estimate f (T )Cu₀Ys

exp Cf (T )

+1

, (2.4)

for a fixed, universal constantC.

Let us take (2.4) temporarily for granted, and let us conclude the proof of (2.2). We first note that there exists ε₀>0 such that, ifu₀Ys ε₀, then (2.4) implies thatf (1)M, for a universal constantM. This follows from a standard continuity argument. (For instance, letF (X, ε)=X−Cεexp(Cx)−Cε. Note thatF (0,0)=0, and that

∂F

∂X(0,0)=1, ^∂F_∂ε(0,0)= −2C. The implicit function theorem now guarantees that for 0< ε < ε₀, there exists a smooth functionA(ε), which is increasing inε, so thatF (A(ε), ε)=0. IfM=A(u₀Ys), andu₀Ys ε₀, it is easy to see thatf (1)M.) Next, note thatuis a solution of (1.2), with initial datau₀, if and only ifu_λ(x, y, t)= λ²u(λx, λ²y, λ³t)is a solution of (1.2) with initial datau_0,λ(x, y)=λ²u₀(λx, λ²y). Sinceu_0,λL²=λ^1/2u_0L², D^s_xu_0,λL² =λ^12+sD_x^su₀L² and∂_x⁻¹∂_yu_0,λL²=λ^1+1/2∂_x⁻¹∂_yu₀L², we can first chooseλ=λ(u₀Y^s)so thatu_0,λYsε₀, and apply the above conclusion tou_λ, to obtain (2.2).

We now turn to the proof of (2.4): we will use Lemma 1.7, and write (1.2) as

∂tu+∂_x³u−∂_x⁻¹∂_y²u= −u∂xu.

We then obtain:

∂_xu_L¹

TL^∞_xyC_δ,T sup

0<t <T

J_x^3/2+2δuL²+ sup

0<t <T

J_x^3/2+δD_y^δuL²

+C_δ,T T

0

J_x^1/2+2δ(u∂_xu)

L²+C_δ,T T

0

J_x^12+δD^δ_y(u∂_xu)

L²

=I+II+III+IV.

To bound I, we use Lemma 1.3 to see that

IC_sexp

C_s T

0

∂_xuL^∞_xy

u₀Ys.

To bound II, we use Plancherel’s theorem, and the fact thatab^a_q^q +^b_p^p, _p¹+_q¹=1, to see that 1+ |ξ|³₂₊δ

.|η|^δ=

1+ |ξ|³₂₊δ

.|ξ|^δ. |η|

|ξ|

δ

1+ |ξ|(³₂+2δ)|η|

|ξ|

δ

1+ |ξ|(³₂+2δ)/(1−δ)

+|η|

|ξ|, where we have chosenq=1/δ,p=1/(1−δ). If nowδis chosen so small that(³₂+2δ)/(1−δ)³₂+δ0, we see that IIsup0<t <TuY_s, and in view of Lemma 1.3,

IICsexp

Cs

T

0

∂xuL^∞_xy

u0Y_s.

In order to estimate III, we use thatJ_x^1/2+2δ(f )L² fL² + D_x^1/2+2δ(f )L², to bound III by III₁+III₂. For III1 we see that it is bounded by sup0<t <TuL².T

0 ∂xuL^∞, and hence it has the desired bound because Lemma 1.3 gives sup0<t <TuL²Cu0Y_s.expCs

T

0 ∂xuL^∞. For III2, we use Lemma 1.8(i), to obtain:

III2 T

0

D

1 2+2δ

x ∂_xuL²uL^∞+ ∂_xuL^∞D

1 2+2δ x uL²

(in view of Lemma 1.3)Csu0Y_sexp

Cs

T

0

∂xuL^∞

× T

0

uL^∞+C_su₀Ys

T

0

∂_xuL^∞exp

C_s T

0

∂_xuL^∞

C_su₀Ysexp

C_sf (T ) . Finally, for IV, we bound it first by_T

0 D_y^δ(u∂_xu)_L²+_T

0 D

1 2+δ

x D_y^δ(u∂_xu)_L²=IV₁+IV₂. To bound IV₁, we use Lemma 1.8(i), in theyvariable, to see that

IV1 T

0

D_y^δ(u)

L²∂xuL^∞+ uL^∞D^δ_y∂xuL² =IV1,1+IV1,2.

To bound IV_1,1, we will use Plancherel’s theorem for the first factor, and the inequalities

|η|^δ= |ξ|^δ |η|

|ξ|

δ

1+ |ξ|_δ|η|

|ξ|

δ

1+ |ξ|_δ/1−δ +|η|

|ξ|,

whereδis chosen so thatδ/(1−δ)3/2+δ0. This shows that (using Lemma 1.3) IV_1,1C_su₀YsexpC_s

T

0

∂_xuL^∞.

For IV1,2we proceed similarly, using that

|η|^δ|ξ| = |ξ|^1+δ |η|

|ξ|

δ

1+ |ξ|1+δ

|η|

|ξ|

δ

1+ |ξ|(1+δ)/(1−δ)

+|η|

|ξ|,

and choosingδso that(1+δ)/(1−δ)3/2+δ0. This gives the bound IV1,2Csu0Y_sexp(Csf (T )). We next turn to IV2, and use Lemma 1.8(ii) to bound it by

T

0

∂_xuL^∞D

1 2+δ

x D^δ_yuL²+ T

0

∂_xD

1 2+δ

x D^δ_yuL²uL^∞

+ T

0

D

1 2+δ

x uL^∞D_y^δ∂xuL²+ T

0

D^δ_yuL^p1D^1/2_x ^+δ∂xuL^q1,

where_p¹

1+_q¹₁ =¹₂are to be determined. We call each one of these four terms IV2,i. To bound IV2,1, we again use Plancherel, and

|ξ|^12+δ|η|^δ

1+ |ξ|¹

2+2δ

|η|

|ξ|

δ

1+ |ξ|(¹₂+2δ)/(1−δ)

+|η|

|ξ|, and if(¹₂+2δ)/(1−δ)³₂+δ₀, Lemma 1.3 shows that

IV2,1Csu0Y_sexp

Cs

T

0

∂xuL∞

. For IV2,2we proceed similarly, using that

|ξ| |ξ|^12+2δ |η|

|ξ|

δ

1+ |ξ|₍³

2+2δ)/(1−δ)

+|η|

|ξ|,

and the fact that(³₂+2δ)/(1−δ)³₂+δ0. We obtain, using Lemma 1.3, that IV_2,2C_su₀Ysexp

C_sf (T ) .

In order to bound IV2,3, we will use the following inequality(δ <¹₂) D

1 2+δ

x uL^∞_xyuL^∞_xy+ ∂_xuL^∞_xy. (2.5)

To establish (2.5), let us use a Littlewood–Paley decomposition, as in the proof of Lemma 1.7, and writeu= u₀+u₁, as in that proof. In order to estimateD

1 2+δ

x u₀, note that, iff (ξ )ˆ = |ξ|^1/2+δχ (ξ ), thenf∈L¹(R). Hence, D

1 2+δ

x u₀L^∞_xyuL^∞_xy. Moreover, in order to estimateD

1 2+δ

x u₁=

k1D

1 2+δ

x Q_ku, note that D

1 2+δ

x Q_ku(ξ, η)= |ξ|^12+δη(2^−kξ )u(ξ, η)ˆ

=|ξ|^12+δ

ξ ξ η(2^−kξ )u(ξ, η)ˆ =2^{−k(12−δ)}|2^−kξ|^12+δ

(2^−kξ ) η(2^−kξ )ξu(ξ, η),ˆ and that ifg(ξ )ˆ =(|ξ|^12+δ/ξ )η(ξ ), theng∈L¹(R). Thus,D

1 2+δ

x QkuL∞xy 2^{−k(21−δ)}∂xuL∞xy, and (2.5) follows.

Because of (2.5), IV2,3is bounded by

sup

0<t <T

D^δ_y∂xuL²

T 0

uL^∞+ ∂xuL^∞

. Using Plancherel, and

|η|^δ|ξ| |η|

|ξ|

δ

1+ |ξ|1+δ

|η|

|ξ|+

1+ |ξ|1+δ/(1−δ)

,

choosingδso that(1+δ)/(1−δ)³₂+δ₀, and using Lemma 1.3, we see that IV_2,3C_su₀Ysexp

C_sf (T ) .

To bound IV_2,4, we need to prove first a couple of auxiliary estimates:

D^1/2_x ^+δ∂_xu_L^s1

TL^q_xy¹∂_xu^θ_L1

TL^∞_xy· J_x^3/2+δ0u¹_L⁻_∞^θ

TL²_xy, (2.6)

D^δ_yu_L^r1 TL^p_xy¹

u_L¹

TL^∞_xy

η

D^1/2_y uL^∞_TL²_xy+ uL^∞_TL²_xy

1−η

, (2.7)