[3] (see also [1] and [4]) proved the following local smoothing effect k∂xU(t0)u0kL∞ xL2t ≤ k∂xU(t0)u0kL∞ xL2t ≤cku0kL2

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)

∂_tu+b∂³_xu= 0,

u(x,0) =u0(x). (1.1)

We define the unitary groupU(t)u0as the solution of the linear initial-value problem (1.1), in this way

U\(t)u₀(ξ) =e^it(bξ3)cu₀(ξ). (1.2) Kenig et al. [3] (see also [1] and [4]) proved the following local smoothing effect

k∂xU(t⁰)u0k_L∞

xL²_t ≤ k∂xU(t⁰)u0k_L∞

xL²_t ≤cku0k_L2. (1.3) They also proved that

∂_x² Z t

0

U(t−t⁰)f(t⁰, x)dt⁰ _L∞

xL²_t ≤ckfk_L1

xL²_t. (1.4) 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:

∂tu+b∂³_xu+γ∂x(|u|²u) = 0,

u(x,0) =u₀(x), (1.5)

whereuis a complex valued function andb, γ are real parameters withbγ6= 0.

Using our local smoothing effect we also proved an interesting result on growth norms (Theorem 1.2).

2000Mathematics 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

The flow associated with (1.5) leads to the quantity I₁(u) =

Z

R

|u(x, t)|²dx, (1.6)

which is conserved in time. Also, whenb·γ6= 0 we have the time invariant quantity I2(u) =k1

Z

R

|∂xu(x, t)|²dx+k2

Z

R

|u(x, t)|⁴dx, (1.7) where k1 = 3bγ and k2 = −3γ²/2. 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 letp≥2 and1/p+ 1/q= 1.

If 2< p <∞and4/q−2< s <1/q+ 1 then k∂xU(t⁰)u₀k_L^∞

xL^p_t ≤c_p,s(1 +t)^1/pkD^su₀kL^q. If p= 2 and0≤s <3/2, then

k∂xU(t⁰)u₀k_L^∞

xL²_t ≤c_st^s/3kD^su₀kL². (1.8) If p=∞and3/2< s, then

k∂_xU(t⁰)u₀k_L^∞

xL^∞_t ≤c_sku₀k_Hs. (1.9) Theorem 1.2. Let u∈ C(R, H²(R))be solution of (1.5)andT >0. Then for all t∈(0, T) there exist a functionδ=δ(kukL²_xL^∞_T ,kuk_L∞

TH˙^1/4)such that

ku(t)kH˙^θ ≤ ku0kH˙^θ+δtku0k³_L2, (1.10) where0≤θ≤1.

The notation used here is standard in partial differential equations. We will use the Lebesgue space-timeL^p_xL^q_τ endowed with the norm

kfk_L^p_xL^q_τ = kfk_L^q_τ

_Lp x=Z

R

Z τ 0

|f(x, t)|^qdt^p/q dx^1/p

.

We will use the notation kfk_L^p_xL^q

t when the integration in the time variable is on the whole real line. The notationkukL^p 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 associ- ated 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)u₀k_L4

xL^∞_t ≤ckD^1/4u₀k_L2, (2.1) and fors >3/4 andρ >3/4

kU(t⁰)u₀kL²_xL^∞_t ≤c(1 +t)^ρku0kH^s. (2.2)

and

∂_x² Z t

0

U(t−t⁰)f(t⁰, x)dt⁰ _L∞

xL²_τ ≤ckfk_L1

xL²_τ. (2.3) Proof. The proof of (2.1) and (2.2) can be found in [3]. To prove (2.3), letτ >0 andg(t⁰, τ, x) =f(t⁰, x)χ_[0,τ](t⁰). Then

∂_x² Z t

0

U(t−t⁰)f(t⁰, x)dt⁰ _L∞

xL²_τ =

Z τ 0

∂_x² Z t

0

U(t−t⁰)g(t⁰, τ, x)dt⁰

2dt^1/2 _L∞

x

≤

Z

R

∂_x² Z t

0

U(t−t⁰)g(t⁰, τ, x)dt⁰

2dt1/2

_L∞

x

=

∂_x² Z t

0

U(t−t⁰)g(t⁰, τ, x)dt⁰ _L∞

xL²_t,

and by inequality (1.4) we obtain (2.3).

Proof of Theorem 1.1. Let ϕ ∈ C₀^∞ with ϕ(t⁰) = 1 in [−t, t], 0 ≤ ϕ(t⁰) ≤ 1 and suppϕ⊂[−2t,2t], then

k∂xU(t⁰)u0k_L∞

xL^p_t ≤ kϕ(t⁰)∂xU(t⁰)u0k_L∞ xL^p_t. Using duality, we considerg∈L^q,kgk_Lq = 1 and the expression

I(x, t) :=

Z

R

g(t⁰)ϕ(t⁰)∂_xU(t⁰)u₀dt⁰ .

Now using the change of variablet⁰=−t⁰ we can assume that I(x, t) :=

Z

R

g(t⁰)ϕ(t⁰)∂xU(−t⁰)u0dt⁰ .

Fubinni Theorem and the definition of groupU(t), shows that I(x, t) =

Z

R

g(t⁰)ϕ(t⁰) Z

R

e^{ixξ−iξ3t0}iξcu0(ξ)dξdt⁰

= Z

R

e^ixξξcu0(ξ)Z

R

g(t⁰)ϕ(t⁰)e^−iξ3t0dt⁰ dξ

= Z

R

uc0(ξ)ξe^ixξϕg(ξc ³)dξ ,

(2.4)

and by Plancherel’s equality, H¨older inequality and Hausdorff-Young inequality we have

I(x, t) = Z

R

|ξ|^scu0(ξ)ξe^ixξ

|ξ|^s ϕg(ξc ³)dξ

= Z

R

D^su₀(y)Fξe^ixξ

|ξ|^s ϕg(ξc ³) (y)dy

≤ kD^su₀k_Lq

Fξe^ixξ

|ξ|^s ϕg(ξc ³) (y)

_Lp

≤ kD^su0kL^q

ξe^ixξ

|ξ|^s ϕg(ξc ³) _Lq.

(2.5)

Now, we make the change of variabley=ξ³ to obtain:

ξe^ixξ

|ξ|^s ϕg(ξc ³)

q L^q

= 1 3 Z

R

|ϕg(y)|c ^qdy

|y|^α , (2.6)

whereα= (2−(1−s)q)/3. Note that ifp=q= 2 ands= 0, thenα= 0, therefore in this case

I(x, t)≤cku₀k_L2kϕgk_L2≤cku₀k_L2kgk_L2 =cku₀k_L2, and in this case we obtain (1.8).

Ifp=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

R

|ϕg(y)|c ²

|y|^2s/3 dy= Z

R

|ϕg(y)|c ²

\1

|x|^1−s/3(y)

2dy

≤

(ϕg)∗ 1

|x|^1−s/3

2 L²

≤cskϕgk²_L6/(3+2s)

≤cskϕk²_L3/skgk²_L2

≤c_st^2s/3kgk²_L2.

(2.7)

Ifp >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

R

|ϕg(y)|c ^qdy

|y|^α = Z

|y|≤1

|ϕg(y)|c ^qdy

|y|^α + Z

|y|>1

|ϕg(y)|c ^qdy

|y|^α :=I₁^q+I₂^q, hence

I₁^q≤c_s,qkϕgkc ^q_L∞ ≤c_s,qkϕgk^q_L1 ≤c_s,qkϕk^q_Lpkgk^q_Lq ≤c_s,qt^q/p,

note thats >4/q−2 impliesαp/(p−q)>1, therefore using H¨older inequality and Hausdorf-Young inequality inI₂^q we obtain

I₂^q ≤ kϕgkc ^q_Lp

Z

|y|>1

dy

|y|^αp/(p−q) 1−q/p

≤c_s,qkϕgk^q_Lq ≤c_s,qkgk^q_Lq.

Ifp=∞ands >3/2, then (2.4) gives

I(x, t)≤ kϕgkc L^∞kcu0(ξ)ξkL¹ ≤cskgkL¹ku0kH^s. Note that, fors >1/2 using immersion we also have

k∂xU(t⁰)u0kL^∞_t L^∞_x ≤csk∂xU(t⁰)u0kH^s ≤csku0k_Hs+1.

Hence we have finished the proof of Theorem 1.1.

Corollary 2.2. Let 0≤s≤1 andu0∈L². Then kD^s_xU(t⁰)u0k_L∞

xL²_t ≤cst^(1−s)/3ku0k_L2. (2.8) The proof of the above corollary follows from (1.8).

Corollary 2.3. Let f ∈L¹_xL²_t and U(t⁰) be as in (1.2). Then for 0 ≤ s≤ 1 we have

D^s_x Z t

0

U(t−t⁰)f(x, t⁰)dt⁰ _L2

x

≤cst^(1−s)/3kfk_L1

xL²_t. (2.9)

Proof. Inequality (2.9) follows from (2.8) and a duality argument. In fact, by Plancherel identity, definition of the groupU(t) and (2.8), we have forkgkL² = 1:

Z

R

D^s_x

Z t 0

U(−t⁰)f(x, t⁰)dt⁰

g(x)dx= Z t

0

Z

R

f(x, t⁰)D_x^sU(t⁰)g(x)dxdt⁰

≤ kfk_L1

xL²_tkD_x^sU(t⁰)g(x)k_L∞ xL²_t

≤ct^(1−s)/3kfk_L1

xL²_tkgk_L2.

Proof of Theorem 1.2. The next lemma is used in the proof.

Lemma 2.4. Let u∈ C(R, H²)be the solution of (1.5). Then kuk_L2

xL^∞_t ≤c(1 +t)^3/4+ku(0)k_H3/4++c(1 +t)^3/4+

Z t 0

(ku(t⁰)k_H1/2+ku(t⁰)k²_H2

+ku(t⁰)k²_H1/2+ku(t⁰)kH²)dt⁰.

(2.10) Proof. To prove the first inequality we rely on the integral equation form

u(t) =U(t)u0−γ Z t

0

U(t−τ) ∂x(|u|²u) (τ), the linear estimate (2.2) show that ifu(0)∈H²then for anyt >0,

kukL²_xL^∞_t ≤c(1 +t)^3/4+ku(0)k_H3/4+

+c(1 +t)^3/4+

Z t 0

(k|u|²u(t⁰)kL²_x+k∂_x²(|u|²u)(t⁰)kL²_x)dt⁰,

(2.11)

using the immersions ku(t)kL^∞_x ≤cku(t)k_H1/2+, ku(t)k_L4

x ≤cku(t)kH˙^1/4 it follows that

k|u|²u(t⁰)k_L2

x≤ ku(t⁰)kL^∞_x ku²(t⁰)k_L2

x ≤cku(t⁰)k_H1/2+ku(t⁰)k²_L4

x <∞, (2.12) and using Leibniz rule, it is easy to see that

k∂_x²(|u|²u)(t⁰)k_L2

x ≤ckuu²_x(t⁰)k_L2

x+cku²u_xx(t⁰)k_L2 x

≤cku(t⁰)k_H1/2+ku(t⁰)k²_H2+cku(t⁰)k²_H1/2+ku(t⁰)k_H2 <∞.

Hence combining this inequality and (2.11), we obtain (2.10).

Lemma 2.5. Let u∈ C(R, H²(R))be solution of (1.5)and0≤s≤1. Then kD_x^su(t)kL²_x ≤ kD^su0kL²

+ct^(1−s)/3kuk²_L2 xL^∞_t

ku0k_L2+t^1/2kuk²_L∞

t H˙^1/4kuk_L2 xL^∞_t

. (2.13) Proof. Without loss of generality we restrict our attention to the real caseu∈R. The equivalent integral equation is

u(t) =U(t)u0−γ Z t

0

U(t−τ) ∂x(u³)

(τ)dτ =:U(t)u0+z(t). (2.14)

Let Γ(t) =kukL²_xL^∞_t . From (2.14), Corollary 2.3 and H¨older inequality, we have kD_x^su(t)k_L2

x ≤ kD_x^sU(t)u0k_L2

x+kD_x^sz(t)k_L2 x

≤ kD^su0k_L2+ct^(1−s)/3ku²uxk_L1 xL²_t

≤ kD^su₀k_L2+ct^(1−s)/3Γ(t)²ku_xk_L∞ xL²_t.

(2.15)

Using (1.3), (2.3) and H¨older inequality, we obtain k∂_xuk_L∞

xL²_t ≤ k∂_xU(t⁰)u₀k_L∞

xL²_t +k∂_xzk_L∞ xL²_t

≤cku0kL²+cku³k_L1 xL²_t

≤cku0kL²+ckuk²_L4 xL⁴_tΓ(t)

≤cku₀k_L2+ct^1/2kuk²_L∞ t L⁴_xΓ(t)

≤cku₀k_L2+ct^1/2kuk²

L^∞_TH˙^1/4Γ(t),

(2.16)

where in the last inequality we use immersionkuk_L4

x ≤ kukH˙^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. LetT >0. Then there is aδ0=δ0(T)>0 such that kuk_L2

xL^∞([τ1,τ2])<2ku0k_L2, for allτ1, τ2∈[0, T], |τ1−τ2| ≤δ0. (2.17) To verify this we use contradiction, we suppose that for all nthere exist τ₁ⁿ, τ₂ⁿ ∈ [0, T],|τ₁ⁿ−τ₂ⁿ|<1/nand

kuk_L2

xL^∞([τ₁ⁿ,τ₂ⁿ])≥2ku₀k_L2. (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

kuk_L2

xL^∞([τ₁ⁿ,τ₂ⁿ])→ ku(τ)k_L2 =ku₀k_L2 asn→ ∞;

however, this contradicts the relation (2.18).

Let 0≤tk ≤t be a sequence witht0 = 0,tk+1−tk =δ0 and letn≈t/δ0 such thattn≤t < tn+1. By Lemma 2.5 and (2.17), it follows that

kD_x^su(t_k)kL²_x ≤ kD^s_xu(t_k−1)kL²+cδ₀^(1−s)/3kuk²_L2

xL^∞([tk−1,tk])ku0kL²

+δ(1−s)/3+1/2

0 kuk²_L∞

TH˙^1/4kuk³_L2

xL^∞([tk−1,t_k])

≤ kD^s_xu(t_k−1)k_L2+cδ₀^(1−s)/3ku0k³_L2(1 +δ^1/2₀ kuk²_L∞ TH˙^1/4), similarly we have

kD_x^su(t)k_L2

x≤ kD^s_xu(tn)k_L2+cδ^(1−s)/3₀ ku0k³_L2(1 +δ^1/2₀ kuk²_L∞

TH˙^1/4); (2.19)

therefore,

kD_x^su(tn)k_L2

x− kD^su(0)k_L2 x=

n

X

k=1

kD^s_xu(tk)k_L2

x− kD^s_xu(t_k−1)k_L2

≤

n

X

k=1

cδ₀^(1−s)/3ku0k³_L2(1 +δ₀^1/2kuk²

L^∞_TH˙^1/4)

≤ctku₀k³_L2

(1 +δ^1/2₀ kuk²

L^∞_TH˙^1/4) δ₀^(2+s)/3

,

so that we conclude

kD_x^su(tn)kL²_x ≤ kD^su(0)kL²+ctku0k³_L2

(1 +δ₀^1/2kuk²

L^∞_TH˙^1/4) δ₀^(2+s)/3

, (2.20)

combining (2.19) and (2.20) we obtain kD^s_xu(t)kL²_x ≤ kD^su(0)kL²+ku0k³_L2

c(t+δ₀) δ₀^(2+s)/3

(1 +δ₀^1/2kuk²_L∞ TH˙^1/4).

This completes the proof.

References

[1] X. Carvajal and F. Linares; A higher order nonlinear Schr¨odinger equation with variable coefficients, Differential and Integral Equations,16(2003), 1111-1130.

[2] T. Cazenave; An introduction to nonlinear Schr¨odinger equations, Textos de M´etodos Matem´aticos 22 (Rio de Janeiro), 1989.

[3] C. E. Kenig, G. Ponce and L. Vega;Well-Posedness and Scattering Results for the Gener- alized 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 Equations10(1997), 777-796.

Xavier Carvajal

Instituto de Matem´atica - UFRJ Av. Hor´acio Macedo, Centro de Tecnologia Cidade Universit´aria, Ilha do Fund˜ao, Caixa Postal 68530 21941-972 Rio de Janeiro, RJ, Brasil

E-mail address:[email protected]