GLOBAL EXISTENCE FOR THE CRITICAL GENERALIZED KDV

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 6, Pages 1847–1855 S 0002-9939(02)06871-5 Article electronically published on November 6, 2002

GLOBAL EXISTENCE FOR THE CRITICAL GENERALIZED KDV EQUATION G. FONSECA, F. LINARES, AND G. PONCE (Communicated by David S. Tartakoff)

Abstract. We discuss results regarding global existence of solutions for the critical generalized Korteweg-de Vries equation, ut + uxxx + u4 ux = 0,

x, t ∈ R.

The theory established shows the existence of global solutions in Sobolev spaces with order below the one given by the energy space H 1 (R), i.e. solutions corresponding to data u0 ∈ H s (R), s > 3/4, with ku0 kL2 < kQkL2 , where Q is the solitary wave solution of the equation.

1. Introduction Consider the initial value problem ( ut + u4 ux + uxxx = 0, (1.1) u(x, 0) = u0 (x).

x ∈ R, t ∈ R,

We study the global existence of solutions for data in H s (R), s ∈ (0, 1). In [8] the local (subcritical) well-posedness of IVP (1.1) was shown. For data in H s (R), s > 0, i.e. for any u0 ∈ H s (R), there exist T (ku0ks , s) > 0 and a unique solution u of (1.1) in C([0, T ] : H s (R)). It was also proved that for any data u0 ∈ L2 (R) there exist T (u0 ) > 0 and a unique strong solution u in C([0, T ] : L2 (R)). Moreover, the solution is global for small data. The latter follows from the local theory in [8] and the fact that L2 is a critical space (see the argument below). Notice that scaling also works for complex-valued functions where the conservation of ku(t)kL2 does not hold. More precisely, if u solves (1.1), then, for λ > 0, so does uλ (x, t) = λ1/2 u(λx, λ3 t), with data uλ (x, 0) = λ1/2 u(λx, 0). It follows that kuλ (·, 0)kH˙ s = λs ku(·, 0)kH˙ s . Received by the editors January 30, 2002. 2000 Mathematics Subject Classification. Primary 35Q53. The first author was partially supported by DIB-Universidad Nacional de Colombia. The second author was partially supported by CNP-q Brazil. The third author was partially supported by an NSF grant. c

2002 American Mathematical Society

1847

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G. FONSECA, F. LINARES, AND G. PONCE

This suggests that the optimal Sobolev index s is s = 0. Indeed, this result was established in [1]. Hence, the results in [8] imply that complex solutions corresponding to data u0 ∈ L2 (R) with ku0 kL2 ≤ 0

(1.2)

are global in time. On the other hand, Weinstein [14] showed the following Gagliardo-Nirenberg √ estimate for v ∈ H 1 (R) and Q(x) = {3c sech2 (2 c x)}1/4 , the solitary wave solution of (1.1): R Z Z 1 v 2 2 1 6 R v ≤ vx2 . (1.3) 6 2 Q2 This estimate combined with the conservation laws for real solutions of (1.1), Z Z (1.4) u20 , u2 (t) = and (1.5)

1 2

Z u2x (t) −

1 6

Z u6 (t) =

1 2

Z

u00 − 2

1 6

Z u60 ,

gives an a priori estimate in H 1 (R) provided ku0 kL2 < kQkL2 and therefore the global existence of solutions in H 1 (R) for initial data satisfying that condition. From this it is clear that 0 < kQkL2 . Thus if we want to show global existence of solutions in H s (R), s ∈ (0, 1), we must assume that the initial data satisfy 0 < ku0 kL2 < kQkL2 . Several interesting results have been lately obtained for solutions of IVP (1.1). Merle [12] proved the existence of real-valued solutions of (1.1) in H 1 (R) corresponding to data u0 ∈ H 1 satisfying ku0 kL2 > kQkL2 that blow up. There are also various results concerning instability of solitary wave solutions as well as the structure of the blow-up formation obtained by Martel and Merle [10], [11]. In the last few years there have been a great deal of results regarding global existence of solutions for several nonlinear dispersive equations below the energy space. The first result in this direction was obtained by Bourgain [2] for the twodimensional Schr¨ odinger equation. His method was used to obtain such kind of results, for instance, for the mKdV equation [5] and the KdV equation [3]; see also [7], [13]. Recently, a new variant of Bourgain’s method was introduced to obtain sharp global results for the KdV and mKdV equations; see [4]. In [6] for the gKdV-3 equation a sharp local result was obtained which implies L2 global well-posedness. Our main result in this paper reads as follows: Theorem 1.1. Let s > 3/4. Then for any initial data u0 ∈ H s (R) satisfying 0 < ku0 kL2 < kQkL2 , see (1.2), the unique solution of the IVP (1.1) given by the local theory (see Theorem 2.1) extends to any time interval [0, T ]. The idea of the proof is similar the one we developed in [5]. That is a combination of sharp smoothing effects present in solutions of the linear problem associated to (1.1) and the iteration process introduced by Bourgain. There are two new elements that can bring some extra difficulty. First, we need to be careful controlling the L2 norm of the initial data. We must be in the conditions of the global well-posedness in odinger equation H 1 (R). This situation is similar to the one for the derivative Schr¨


CRITICAL GENERALIZED KDV

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treated by Takaoka in [9]. Second, in the local well-posedness result, the time of existence depends on the Sobolev space H s (see below). We organized this paper as follows: In section 2, we give the statement of the local result obtained in [8] and the main estimates used in the subsequent sections. To prove Theorem 1.1 we need to study two auxiliary IVP’s and the properties of their solutions; this will be done in section 3. Finally, the proof of Theorem 1.1 is given in section 4. 2. Preliminary results We begin this section with the local sharp result obtained in [8]. Theorem 2.1 ([8]). Let s > 0. Then for any u0 ∈ H s (R) there exist T = T (ku0ks,2 ) (with T (ρ, s) → ∞ as ρ → 0) and a unique strong solution u(·) of the IVP (1.1) satisfying u ∈ C([−T, T ] : H s (R)),

(2.1) (2.2)

s/3

+ kDxs ukL5x L10 + kDt kukL5xL10 T T

ukL5xL10 < ∞, T

and (2.3)

s/3

s 2 + kD ∂x ukL∞ L2 + kD k∂x ukL∞ x t x LT x T

2 < ∞. ∂x ukL∞ x LT

Given T 0 ∈ (0, T ) there exists a neighborhood V of u0 in H s (R) such that the map ˜(t) from V into the class defined by (2.1), (2.2) and (2.3) with T 0 instead of u ˜0 → u T is Lipschitz. Remark 2.2. From the proof of Theorem 2.1 it follows that the time of existence T satisfies . T ' c ku0 k−3/s s

(2.4)

This can also be deduced from the scaling argument. The proof of Theorem 2.1 uses as main tools (1) The smoothing effect of Kato’s type associated to the linear problem ( ∂t u + ∂x3 u = 0, (2.5) u(x, 0) = u0 (x).

(2.6)

More precisely, let U (t) be the linear group describing the solution of (2.5), then 1/2 Z ∞ |∂x U (t)u0 (x)|2 dt = ku0 k, for any x ∈ R. −∞

(2) Its dual version, that is, Z ∞ U (−τ )g(x, τ ) dτ k ≤ c kgkL1xL2t . (2.7) k∂x −∞

(3) The double smoothing effect, solutions of the non-homogeneous linear equation ( ∂t u + ∂x3 u = f, (2.8) u(x, 0) = 0,


1850


satisfy the estimate 2 ≤ c kf kL1 L2 . k∂x2 ukL∞ x LT x T

(2.9)

(4) The fractional derivative commutators. Theorem 2.3. Let α ∈ (0, 1), α1 , α2 ∈ [0, α], α1 + α2 = α. Suppose p, p1 , p2 , q, q1 , q2 ∈ (1, ∞) with 1p = p11 + p12 and 1q = q11 + q12 . Then (2.10)

kDxα (f g) − f Dxα g − gDxα f kLpx LqT ≤ c kDxα1 f kLpx1 LqT1 kDxα2 gkLpx2 LqT2 . Moreover, for α1 = 0 the value q1 = ∞ is allowed.

(5) The chain rule for fractional derivatives Theorem 2.4. Let α ∈ (0, 1), and p, p1 , p2 , q, q2 ∈ (1, ∞), q1 ∈ (1, ∞] such that p1 = p11 + p12 and 1q = q11 + q12 . Then kDxα F (f )kLpx LqT ≤ c kF 0 (f )kLpx1 LqT1 kDxα f kLpx2 LqT2 .

(2.11)

(6) Several interpolation theorems. These tools plus the contraction mapping principle give the result. 3. Local well-posedness for the auxiliary problems In this section we describe the properties of two auxiliary IVP’s we will use in the proof of Theorem 1.1. We first set some notation. Let N be a sufficiently large positive number to be chosen later. Define the norm ||| · |||s by s/3

(3.1)

s s + kuk 5 10 + kD uk 5 10 + kD |||u|||s = kukL∞ Lx LT Lx LT x t T H s 2 + kD ∂x ukL∞ L2 + + k∂x ukL∞ x x LT x T

s/3 kDt

ukL5xL10 T

2 . ∂x ukL∞ x LT

Proposition 3.1. Consider the IVP ( vt + v 4 vx + vxxx = 0, x ∈ R, t > 0, (3.2) v(x, 0) = v0 (x) ∈ H 1 (R). If v0 satisfies (3.3)

kv0 kL2 < kQkL2

and

kv0 kH 1 ' N 1−s ,

−3(1−s) , the existence time given by Theorem 2.1. then for ∆T ' kv0 k−3 H1 ' c N

(1) The solution v of (3.2) satisfies (3.4)

sup kv(t)kH 1 ≤ cN 1−s . [0,∆T ]

(2) For any ρ ∈ (0, 1), the solution v of (3.2) satisfies (3.5)

|||v|||ρ ' N ρ(1−s) .

Proof. To prove (3.4) we use (1.5) and (1.3). The estimate (3.5) is established by using Theorem 2.1 and kv0 kH ρ ≤ c N ρ(1−s) .


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Proposition 3.2. Let w0 ∈ H s (R), s > 0, and let v be the solution of the IVP (3.2). Then there exists a unique solution w of the IVP ( ∂t w + ∂x3 w + ∂x (v 4 w + 2v 3 w2 + 2v 2 w3 + w4 v + 15 w5 ) = 0, (3.6) w(x, 0) = w0 (x), defined in the same interval of existence of v, [0, ∆T ], such that w ∈ C([0, ∆T ] : H s (R)), s/3

+ kDxs wkL5x L10 + kDt kwkL5x L10 ∆T ∆T

wkL5x L10 < ∞, ∆T

and s/3

2 2 + kDxs ∂x wkL∞ + kDt k∂x wkL∞ x L∆T x L∆T

2 ∂x wkL∞ < ∞. x L∆T

Proof. The argument is the same as used in [8] to prove Theorem 2.1. For the sake of clearness we will sketch it. We use the integral equation form of (3.6), that is, Z t 1 U (t−t0 ) ∂x (v 4 w+2v 3 w2 +2v 2 w3 +w4 v+ w5 )(t0 ) dt0 . (3.7) w(t) = U (t)w0 (x)− 5 0 Defining Xa,To = {w ∈ C([0, To ] : H ρ (R)) : |||w|||ρ ≤ a},

(3.8)

where ||| · ||| is as in (3.1), and (3.9) Φ(w) = Φv,w0 (w)

Z

t

≡ U (t)w0 (x) −

U (t − t0 ) ∂x (v 4 w + 2v 3 w2 + 2v 2 w3 + w4 v +

0

1 5 0 0 w )(t ) dt . 5

Next we shall show that Φ is a contraction. We will work in detail the case kDxρ ΦkL2 . The other estimates follow similar arguments. Using the dual of the smoothing effect (2.7), the chain and Leibniz rules for fractional derivatives (2.10) and (2.11), we obtain (3.10) Z

t

kDxρ

U (t − t0 ) ∂x (v 4 w)(t0 ) dt0 k = k∂x

Z

0

t

U (t − t0 ) Dxρ (v 4 w)(t0 ) dt0 k

0

≤ ckDxρ (v 4 w)kL1x L2T

o

ρ ≤ ckDxρ (v 4 )wkL1x L2T + ckv 4 kL5/4 5/2 kDx wkL5 L10 x T x L o

4

o

To

ckvk4L5 L10 kDxρ wkL5x L10 To x To

≤

ckvk3L5 L10 kDxρ vkL5x L10 kwkL5x L10 To To x To

≤

ρ/3 cTo4ρ/3 kDt vk3L5 L10 x T

o

+

+ ckvk4L5 L10 kDxρ wkL5x L10 T x

To

ρ/3 kDxρ vkL5x L10 kDt wkL5x L10 To To

ρ/3 cTo4ρ/3 kDt vk4L5 L10 x T

o

≤

o

To

≤ ckD (v )kL5/4 5/2 kwkL5 L10 + x T x L ρ

kDxρ wkL5x L10 T

o

cTo4ρ/3 |||v|||4ρ |||w|||ρ ,


o

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(3.11)


Z

t

U (t − t0 ) ∂x (v 3 w2 )(t0 ) dt0 k ≤ ckDxρ (v 3 w2 )kL1x L2T

kDxρ 0

≤

ckDxρ v 3 kL5/3 10/3 x L

≤

ckvk2L5 L10 kDxρ vkL5x L10 To x To

To

o

ρ 2 kw kL5/2 + ckv kL5/3 10/3 kDx w k 5/2 5 5 Lx L x L x L 2

3

To

To

kwk2L5 L10 x To

+

To

ckvk3L5 L10 kDxρ wkL5x L10 k wkL5x L10 To To x To

≤ cTo4ρ/3 |||v|||3ρ |||w|||2ρ , (3.12)

Z

kDxρ 0

≤

t

U (t − t0 ) ∂x (v 2 w3 )(t0 ) dt0 k ≤ ckDxρ (v 2 w3 )kL1x L2T

kvkL5xL10 kwk3L5 L10 ckDxρ vkL5x L10 To To x T

+

o

o

ckvk2L5 L10 kDxρ wkL5x L10 kwk2L5 L10 To x To x To

≤ cTo4ρ/3 |||v|||2ρ |||w|||3ρ , Z (3.13)

t

U (t − t0 ) ∂x (vw4 )(t0 ) dt0 k ≤ ckDxρ (w4 v)kL1x L2T

kDxρ 0

≤

kwk4L5 L10 ckDxρ vkL5x L10 To x T

≤

cTo4ρ/3 |||v|||ρ |||w|||4ρ ,

and finally

Z kDxρ 0

(3.14)

+

o

≤

t

o

ckvkL5x L10 kDxρ wkL5x L10 kwk3L5 L10 To To x T

o

U (t − t0 ) ∂x (w5 )(t0 ) dt0 k ≤ ckDxρ (w4 v)kL1x L2T

ckDxρ (w4 )kL5/4 5/2 kwkL5 L10 x To x L To

+

o

ckwk4L5 L10 kDxρ wkL5x L10 To x To

≤ ckwk4L5 L10 kDxρ wkL5x L10 T x

o

To

≤ cTo4ρ/3 |||w|||5ρ . Combining (3.7) and (3.10)–(3.14) we have n kDxρ Φ(w)k ≤ ckDxρ w0 k + To4ρ/3 |||v|||4ρ + |||v|||3ρ |||w|||ρ (3.15)

o + |||v|||2ρ |||w|||2ρ + |||v|||ρ |||w|||3ρ + |||w|||4ρ |||w|||ρ n 3ρ ≤ ckDxρ w0 k + To4ρ/3 |||v|||4ρ 1 + |||v|||1 |||w|||ρ o ρ 2 3 4 + |||v|||2ρ 1 |||w|||ρ + |||v|||1 |||w|||ρ + |||w|||ρ |||w|||ρ .

Similarly we can obtain estimates for the other norms involved in (3.1) to get (3.16)

|||Φ(w)|||ρ ≤ kw0 kH ρ + To4ρ/3 F (|||v|||1 , |||w|||ρ )|||w|||ρ .

Choosing a = max{kv0 k1 , kw0 kρ }(= kv0 k1 ) and taking To = ∆T we have 1 . 2 Thus Φ is well defined since F (|||v|||1 , |||w|||ρ ) ≤ c a4 . The same argument applies to show that Φ is a contraction. Hence the contraction mapping principle gives the result. c (∆T )4ρ/3 a4