Electronic Journal of Differential Equations, Vol. 2008(2008), No. 55, pp. 1–7.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
A NOTE ON LOCAL SMOOTHING EFFECTS FOR THE
UNITARY GROUP ASSOCIATED WITH THE KDV EQUATION
XAVIER CARVAJAL
Abstract. In this note we show interesting local smoothing effects for the
unitary group associated to Korteweg-de Vries type equation. Our main tools
are the Hardy-Littlewood-Sobolev and Hausdorff-Young inequalities. Using
our local smoothing effect and a dual version, we estimate the growth of the
norm of solutions of the complex modified KdV equation.
1. Introduction
In this note we describe some results on local smoothing effects for solutions of
the initial value problem (IVP)
∂t u + b∂x3 u = 0,
(1.1)
u(x, 0) = u0 (x).
We define the unitary group U (t)u0 as the solution of the linear initial-value problem
(1.1), in this way
3
U\
(t)u0 (ξ) = eit(bξ ) u
c0 (ξ).
(1.2)
Kenig et al. [3] (see also [1] and [4]) proved the following local smoothing effect
0
2 ≤ k∂x U (t )u0 kL∞ L2 ≤ cku0 kL2 .
k∂x U (t0 )u0 kL∞
x Lt
x
t
They also proved that
Z
2 t
∂x
U (t − t0 )f (t0 , x)dt0 L∞ L2 ≤ ckf kL1x L2t .
x
0
(1.3)
(1.4)
t
In this work we obtain a local smoothing effect (Theorem 1.1), more general than
local smoothing effect (1.3). We also consider the IVP for the complex modified
Korteweg-de Vries type equation:
∂t u + b∂x3 u + γ∂x (|u|2 u) = 0,
u(x, 0) = u0 (x),
(1.5)
where u is a complex valued function and b, γ are real parameters with bγ 6= 0.
Using our local smoothing effect we also proved an interesting result on growth
norms (Theorem 1.2).
2000 Mathematics Subject Classification. 35A07, 35Q53.
Key words and phrases. Modified KdV equation; KdV equation; local smoothing effect.
c
2008
Texas State University - San Marcos.
Submitted March 13, 2008. Published April 17, 2008.
1
2
X. CARVAJAL
EJDE-2008/55
The flow associated with (1.5) leads to the quantity
Z
I1 (u) =
|u(x, t)|2 dx,
(1.6)
R
which is conserved in time. Also, when b · γ 6= 0 we have the time invariant quantity
Z
Z
2
I2 (u) = k1 |∂x u(x, t)| dx + k2 |u(x, t)|4 dx,
(1.7)
R
R
where k1 = 3bγ and k2 = −3γ 2 /2. The main results in this work are stated as
follows.
Theorem 1.1. Let U (t)u0 be the solution of the linear problem associated to (1.1)
and let p ≥ 2 and 1/p + 1/q = 1.
If 2 < p < ∞ and 4/q − 2 < s < 1/q + 1 then
1/p
p ≤ c
k∂x U (t0 )u0 kL∞
kDs u0 kLq .
p,s (1 + t)
x Lt
If p = 2 and 0 ≤ s < 3/2, then
s/3
2 ≤ cs t
k∂x U (t0 )u0 kL∞
kDs u0 kL2 .
x Lt
(1.8)
If p = ∞ and 3/2 < s, then
∞ ≤ cs ku0 kH s .
k∂x U (t0 )u0 kL∞
x Lt
(1.9)
Theorem 1.2. Let u ∈ C(R, H 2 (R)) be solution of (1.5) and T > 0. Then for all
t ∈ (0, T ) there exist a function δ = δ(kukL2x L∞
, kukL∞ Ḣ 1/4 ) such that
T
T
ku(t)kḢ θ ≤ ku0 kḢ θ + δtku0 k3L2 ,
(1.10)
where 0 ≤ θ ≤ 1.
The notation used here is standard in partial differential equations. We will use
the Lebesgue space-time Lpx Lqτ endowed with the norm
p/q 1/p
Z Z τ
p
q
q
kf kLx Lτ = kf kLτ Lp =
|f (x, t)|q dt
dx
.
x
R
0
We will use the notation kf kLpx Lqt when the integration in the time variable is on
the whole real line. The notation kukLp is used when there is no doubt about the
variable of integration.
2. Smoothing Local Effects
In this section we prove new smoothing local effects for the unitary group associated with the Korteweg-de Vries equation (Theorem 1.1), which will be fundamental
in the proof of Theorem 1.2.
Linear Estimates. The next lemma is a preliminary result to be used in the proof
of Theorem 1.2.
Lemma 2.1. Let u(x, t0 ) = U (t0 )u0 (x) be the solution of (1.1). We have the
maximal function estimates
kU (t0 )u0 kL4x L∞
≤ ckD1/4 u0 kL2 ,
t
(2.1)
and for s > 3/4 and ρ > 3/4
kU (t0 )u0 kL2x L∞
≤ c(1 + t)ρ ku0 kH s .
t
(2.2)
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LOCAL SMOOTHING EFFECTS
and
2
∂x
Z
0
3
t
U (t − t0 )f (t0 , x)dt0 L∞ L2 ≤ ckf kL1x L2τ .
x
(2.3)
τ
Proof. The proof of (2.1) and (2.2) can be found in [3]. To prove (2.3), let τ > 0
and g(t0 , τ, x) = f (t0 , x)χ[0,τ ] (t0 ). Then
Z
Z
Z
2 1/2 τ 2 t
2 t
∞
∂x
∂x
U (t − t0 )g(t0 , τ, x)dt0 dt
U (t − t0 )f (t0 , x)dt0 L∞ L2 = Lx
x
τ
0
0
0
Z
Z
2 1/2 2 t
∞
∂x
U (t − t0 )g(t0 , τ, x)dt0 dt
≤
Lx
0
R
Z t
U (t − t0 )g(t0 , τ, x)dt0 L∞ L2 ,
= ∂x2
0
x
and by inequality (1.4) we obtain (2.3).
t
Proof of Theorem 1.1. Let ϕ ∈ C0∞ with ϕ(t0 ) = 1 in [−t, t], 0 ≤ ϕ(t0 ) ≤ 1 and
supp ϕ ⊂ [−2t, 2t], then
0
0
p ≤ kϕ(t )∂ U (t )u k ∞ p .
k∂x U (t0 )u0 kL∞
x
0 Lx Lt
x Lt
Using duality, we consider g ∈ Lq , kgkLq = 1 and the expression
Z
I(x, t) := g(t0 )ϕ(t0 )∂x U (t0 )u0 dt0 .
R
0
Now using the change of variable t = −t0 we can assume that
Z
I(x, t) := g(t0 )ϕ(t0 )∂x U (−t0 )u0 dt0 .
R
Fubinni Theorem and the definition of group U (t), shows that
Z
Z
3 0
I(x, t) = g(t0 )ϕ(t0 ) eixξ−iξ t iξc
u0 (ξ)dξdt0 R
ZR
Z
3 0
u0 (ξ)
g(t0 )ϕ(t0 )e−iξ t dt0 dξ = eixξ ξc
R
ZR
ixξ
= u
c0 (ξ)ξe ϕ
cg(ξ 3 )dξ ,
(2.4)
R
and by Plancherel’s equality, Hölder inequality and Hausdorff-Young inequality we
have
Z
ξeixξ
I(x, t) = |ξ|s u
c0 (ξ) s ϕ
cg(ξ 3 )dξ |ξ|
R
Z
ξeixξ
= Ds u0 (y)F
ϕ
cg(ξ 3 ) (y)dy s
|ξ|
R
(2.5)
ξeixξ
3
p
≤ kDs u0 kLq F
ϕ
c
g(ξ
)
(y)
L
|ξ|s
ξeixξ
≤ kDs u0 kLq s ϕ
cg(ξ 3 )Lq .
|ξ|
Now, we make the change of variable y = ξ 3 to obtain:
Z
ξeixξ
q
1
|c
ϕg(y)|q dy
3 cg(ξ ) =
,
(2.6)
s ϕ
|ξ|
3 R
|y|α
Lq
4
X. CARVAJAL
EJDE-2008/55
where α = (2 − (1 − s)q)/3. Note that if p = q = 2 and s = 0, then α = 0, therefore
in this case
I(x, t) ≤ cku0 kL2 kϕgkL2 ≤ cku0 kL2 kgkL2 = cku0 kL2 ,
and in this case we obtain (1.8).
If p = q = 2 and 0 < s < 3/2, then 0 < α = 2s/3 < 1, using properties of the
Fourier transform and the Hardy-Littlewood-Sobolev inequality it is not hard to
deduce the following string of inequalities
Z
Z
2
\
1
|c
ϕg(y)|2
dy
=
|c
ϕg(y)|2 1−s/3 (y) dy
2s/3
|x|
R
R |y|
2
1
≤ (ϕg) ∗ 1−s/3 L2
|x|
(2.7)
≤ cs kϕgk2L6/(3+2s)
≤ cs kϕk2L3/s kgk2L2
≤ cs t2s/3 kgk2L2 .
If p > 2 and 4/q − 2 < s < 1/q + 1, then 0 < α < 1 (observe that 4/q − 2 > 1 − 2/q),
we can write the integral in (2.6) as follows
Z
Z
Z
|c
ϕg(y)|q dy
|c
ϕg(y)|q dy
|c
ϕg(y)|q dy
=
+
:= I1q + I2q ,
|y|α
|y|α
|y|α
R
|y|≤1
|y|>1
hence
I1q ≤ cs,q kc
ϕgkqL∞ ≤ cs,q kϕgkqL1 ≤ cs,q kϕkqLp kgkqLq ≤ cs,q tq/p ,
note that s > 4/q − 2 implies αp/(p − q) > 1, therefore using Hölder inequality and
Hausdorf-Young inequality in I2q we obtain
1−q/p
Z
dy
q
q
I2 ≤ kc
≤ cs,q kϕgkqLq ≤ cs,q kgkqLq .
ϕgkLp
αp/(p−q)
|y|>1 |y|
If p = ∞ and s > 3/2, then (2.4) gives
I(x, t) ≤ kc
ϕgkL∞ kc
u0 (ξ)ξkL1 ≤ cs kgkL1 ku0 kH s .
Note that, for s > 1/2 using immersion we also have
0
∞ ≤ cs k∂x U (t )u0 kH s ≤ cs ku0 kH s+1 .
k∂x U (t0 )u0 kL∞
t Lx
Hence we have finished the proof of Theorem 1.1.
Corollary 2.2. Let 0 ≤ s ≤ 1 and u0 ∈ L2 . Then
(1−s)/3
2 ≤ cs t
kDxs U (t0 )u0 kL∞
ku0 kL2 .
x Lt
(2.8)
The proof of the above corollary follows from (1.8).
Corollary 2.3. Let f ∈ L1x L2t and U (t0 ) be as in (1.2). Then for 0 ≤ s ≤ 1 we
have
Z
s t
Dx
U (t − t0 )f (x, t0 )dt0 L2 ≤ cs t(1−s)/3 kf kL1x L2t .
(2.9)
0
x
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LOCAL SMOOTHING EFFECTS
5
Proof. Inequality (2.9) follows from (2.8) and a duality argument. In fact, by
Plancherel identity, definition of the group U (t) and (2.8), we have for kgkL2 = 1:
Z Z t
Z tZ
s
0
0
0
U (−t )f (x, t )dt g(x)dx =
Dx
f (x, t0 )Dxs U (t0 )g(x)dxdt0
R
0
0
R
2
≤ kf kL1x L2t kDxs U (t0 )g(x)kL∞
x Lt
≤ ct(1−s)/3 kf kL1x L2t kgkL2 .
Proof of Theorem 1.2. The next lemma is used in the proof.
Lemma 2.4. Let u ∈ C(R, H 2 ) be the solution of (1.5). Then
Z t
3/4+
3/4+
kukL2x L∞
≤
c(1
+
t)
ku(0)k
+
c(1
+
t)
(ku(t0 )kH 1/2+ ku(t0 )k2H 2
3/4+
H
t
0
+ ku(t
0
)k2H 1/2+ ku(t0 )kH 2 )dt0 .
(2.10)
Proof. To prove the first inequality we rely on the integral equation form
Z t
u(t) = U (t)u0 − γ
U (t − τ ) ∂x (|u|2 u) (τ ),
0
the linear estimate (2.2) show that if u(0) ∈ H 2 then for any t > 0,
kukL2x L∞
≤ c(1 + t)3/4+ ku(0)kH 3/4+
t
Z t
+ c(1 + t)3/4+
(k|u|2 u(t0 )kL2x + k∂x2 (|u|2 u)(t0 )kL2x )dt0 ,
(2.11)
0
using the immersions ku(t)kL∞
≤ cku(t)kH 1/2+ , ku(t)kL4x ≤ cku(t)kḢ 1/4 it follows
x
that
k|u|2 u(t0 )kL2x ≤ ku(t0 )kL∞
ku2 (t0 )kL2x ≤ cku(t0 )kH 1/2+ ku(t0 )k2L4x < ∞,
x
(2.12)
and using Leibniz rule, it is easy to see that
k∂x2 (|u|2 u)(t0 )kL2x ≤ ckuu2x (t0 )kL2x + cku2 uxx (t0 )kL2x
≤ cku(t0 )kH 1/2+ ku(t0 )k2H 2 + cku(t0 )k2H 1/2+ ku(t0 )kH 2 < ∞.
Hence combining this inequality and (2.11), we obtain (2.10).
Lemma 2.5. Let u ∈ C(R, H 2 (R)) be solution of (1.5) and 0 ≤ s ≤ 1. Then
kDxs u(t)kL2x ≤ kDs u0 kL2
1/2
2
2 + t
2 L∞ .
+ ct(1−s)/3 kuk2L2x L∞
ku
k
kuk
kuk
0
∞
L
L
1/4
x t
L Ḣ
t
(2.13)
t
Proof. Without loss of generality we restrict our attention to the real case u ∈ R.
The equivalent integral equation is
Z t
u(t) = U (t)u0 − γ
U (t − τ ) ∂x (u3 ) (τ )dτ =: U (t)u0 + z(t).
(2.14)
0
6
X. CARVAJAL
EJDE-2008/55
Let Γ(t) = kukL2x L∞
. From (2.14), Corollary 2.3 and Hölder inequality, we have
t
kDxs u(t)kL2x ≤ kDxs U (t)u0 kL2x + kDxs z(t)kL2x
≤ kDs u0 kL2 + ct(1−s)/3 ku2 ux kL1x L2t
s
(1−s)/3
≤ kD u0 kL2 + ct
(2.15)
2
2.
Γ(t) kux kL∞
x Lt
Using (1.3), (2.3) and Hölder inequality, we obtain
0
2 ≤ k∂x U (t )u0 kL∞ L2 + k∂x zkL∞ L2
k∂x ukL∞
x Lt
x
x
t
t
≤ cku0 kL2 + cku3 kL1x L2t
≤ cku0 kL2 + ckuk2L4 L4 Γ(t)
x
1/2
≤ cku0 kL2 + ct
(2.16)
t
kuk2L∞
4 Γ(t)
t Lx
≤ cku0 kL2 + ct1/2 kuk2L∞ Ḣ 1/4 Γ(t),
T
where in the last inequality we use immersion kukL4x ≤ kukḢ 1/4 . As a consequence
of (2.15) and (2.16) we have (2.13). Thus the proof is complete.
Proof of Theorem 1.2. Let T > 0. Then there is a δ0 = δ0 (T ) > 0 such that
kukL2x L∞ ([τ1 ,τ2 ]) < 2ku0 kL2 ,
for all τ1 , τ2 ∈ [0, T ], |τ1 − τ2 | ≤ δ0 .
(2.17)
To verify this we use contradiction, we suppose that for all n there exist τ1n , τ2n ∈
[0, T ], |τ1n − τ2n | < 1/n and
kukL2x L∞ ([τ1n ,τ2n ]) ≥ 2ku0 kL2 .
(2.18)
Since (τ1n ) and (τ2n ) are bounded sequences, we can suppose that there exist a
τ ∈ [0, T ] such that limn→∞ τ1n = limn→∞ τ2n = τ , using Lemma 2.4 and Lebesgue’s
Dominated Convergence Theorem, we have that
kukL2x L∞ ([τ1n ,τ2n ]) → ku(τ )kL2 = ku0 kL2
as n → ∞;
however, this contradicts the relation (2.18).
Let 0 ≤ tk ≤ t be a sequence with t0 = 0, tk+1 − tk = δ0 and let n ≈ t/δ0 such
that tn ≤ t < tn+1 . By Lemma 2.5 and (2.17), it follows that
(1−s)/3
kDxs u(tk )kL2x ≤ kDxs u(tk−1 )kL2 + cδ0
(1−s)/3+1/2
+ δ0
≤
kuk2L2x L∞ ([tk−1 ,tk ]) ku0 kL2
kuk2L∞ Ḣ 1/4 kuk3L2x L∞ ([tk−1 ,tk ])
kDxs u(tk−1 )kL2
T
(1−s)/3
+ cδ0
1/2
ku0 k3L2 (1 + δ0 kuk2L∞ Ḣ 1/4 ),
T
similarly we have
(1−s)/3
kDxs u(t)kL2x ≤ kDxs u(tn )kL2 + cδ0
1/2
ku0 k3L2 (1 + δ0 kuk2L∞ Ḣ 1/4 );
T
(2.19)
EJDE-2008/55
LOCAL SMOOTHING EFFECTS
7
therefore,
kDxs u(tn )kL2x − kDs u(0)kL2x =
≤
n
X
k=1
n
X
kDxs u(tk )kL2x − kDxs u(tk−1 )kL2
(1−s)/3
cδ0
1/2
ku0 k3L2 (1 + δ0 kuk2L∞ Ḣ 1/4 )
T
k=1
1/2
≤
ctku0 k3L2
(1 + δ0 kuk2L∞ Ḣ 1/4 )
T
(2+s)/3
,
δ0
so that we conclude
1/2
kDxs u(tn )kL2x
s
≤ kD u(0)kL2 +
ctku0 k3L2
(1 + δ0 kuk2L∞ Ḣ 1/4 )
T
(2+s)/3
δ0
,
(2.20)
combining (2.19) and (2.20) we obtain
kDxs u(t)kL2x ≤ kDs u(0)kL2 + ku0 k3L2
c(t + δ0 )
(2+s)/3
δ0
This completes the proof.
1/2
(1 + δ0 kuk2L∞ Ḣ 1/4 ).
T
References
[1] X. Carvajal and F. Linares; A higher order nonlinear Schrödinger equation with variable
coefficients, Differential and Integral Equations, 16 (2003), 1111-1130.
[2] T. Cazenave; An introduction to nonlinear Schrödinger equations, Textos de Métodos
Matemáticos 22 (Rio de Janeiro), 1989.
[3] C. E. Kenig, G. Ponce and L. Vega; Well-Posedness and Scattering Results for the Generalized Korteweg-de Vries Equation via the Contraction Principle, Comm. Pure and Applied
Math., 46 (1993), 527-620.
[4] G. Staffilani, On the Generalized Korteweg-de Vries-Type Equations, Differential and Integral
Equations 10 (1997), 777-796.
Xavier Carvajal
Instituto de Matemática - UFRJ Av. Horácio Macedo, Centro de Tecnologia Cidade
Universitária, Ilha do Fundão, Caixa Postal 68530 21941-972 Rio de Janeiro, RJ, Brasil
E-mail address: carvajal@im.ufrj.br