KORTEWEG-DE VRIES AND BENJAMIN-ONO EQUATIONS ON ZHIDKOV SPACES

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KORTEWEG-DE VRIES AND BENJAMIN-ONO EQUATIONS ON ZHIDKOV SPACES

Clément Gallo

To cite this version:

Clément Gallo. KORTEWEG-DE VRIES AND BENJAMIN-ONO EQUATIONS ON ZHIDKOV SPACES. Advances in Differential Equations, Khayyam Publishing, 2005, 10, pp.277 - 308. �hal- 01960979�

Korteweg-de Vries and Benjamin-Ono equations on Zhidkov spaces

Cl´ement Gallo

UMR de Math´ematiques, Bat. 425 Universit´e Paris-Sud

91405 Orsay, France.

Abstract. Motivated by the study of the Cauchy problem with bore-like ini- tial data, we show the “well-posedness” for Korteweg-de Vries and Benjamin- Ono equations with initial data in Zhidkov spacesX^s, with respectivelys >1 and s >5/4. Here, “well-posedness” includes local (global in some cases) ex- istence, uniqueness under a supplementary assumption and continuity with re- spect to the initial data.

Key Words. Korteweg-de Vries equation, Benjamin-Ono equation, Cauchy problem.

AMS Subject Classification. 35Q53, 35A07, 35A05.

1 Introduction

In [9], the authors considered the Cauchy problems for Korteweg-de Vries (KdV) and Benjamin-Ono (BO) equations with bore-like initial data, namely

∂tu+∂_x³u+u∂xu= 0, x∈R, t∈R

u(x,0) =g(x) (1)

and

∂tu+H∂_x²u+u∂xu= 0, x∈R, t∈R

u(x,0) =g(x) (2)

whereH denotes the Hilbert transform, and gsatisfies





i) g(x)→C± asx→ ±∞. ii) g^′∈H^s−1wheres>1 .

iii) g−C+ ∈L²([0,∞)), g−C− ∈L²((−∞,0]).

(3) They showed the local well-posedness of (1) and (2) with initial datagsatisfying (3) , under the assumption s > 3/2. Global well-posedness was obtained for s>2.

Our aim here is to improve this result by weakening the assumptions on the initial datag: we replace (3) by

g∈X^s (4)

where X^s denotes the Zhidkov space we introduced in [7] (see also [22]) for integer values ofs:

X^s:={f ∈ D^′(R), f ∈L^∞, f^′ ∈H^s−1(R)}. (5) Moreover (and that is probably the most interesting improvement), we assume only thats >1 in the KdV case, ands >5/4 in the BO case, instead ofs >3/2.

We first consider the KdV equation (1). Our strategy is as follows. Thanks to Lemma 2.1 below, for g ∈ X^s, there exists a function ψ ∈ C^∞(R) with ψ^′∈H^∞ such thatφ=g−ψ∈H^s (remark that it implies thatψis bounded, sinces>1). Similarly to [9], we write a solutionuof (1) as u=v+ψ, and we study the Cauchy problem associated withv, namely

∂tv+∂_x³v+v∂xv+∂x(vψ) =−(∂_x³ψ+ψψ^′), x∈R, t∈R

v(x,0) =φ(x) =g(x)−ψ(x) (6)

Our main result is as follows:

Theorem 1.1 Let ψ∈C_b^∞(R)such that ψ^′ ∈H^∞,s >1 andφ∈H^s(R).

Then there exists T =T(ψ, s,||φ||H^s)>0 and a unique v ∈C([−T, T], H^s)∩ C¹([−T, T], H^s−3)solving (6), and such thatvx∈L¹([−T, T], L^∞).

Moreover, for anyR >0, the mapφ→v is continuous from the ball of radius Rin H^s(R)toC([−T(R), T(R)], H^s).

From this local well-posedness result for (6) we deduce a local well-posedness result for (1):

Theorem 1.2 Let s >1,g∈X^s.

Then there existsT˜= ˜T(s,||g||X^s)>0and a unique solutionuof (1) such that u∈C([−T ,˜ T˜], X^s),u−g∈C([−T ,˜ T], H˜ ^s)andux∈L¹([−T ,˜ T], L˜ ^∞).

Moreover, for anyR >0, the map g→u is continuous from the ball of radius Rin X^s toC([−T˜(R),T˜(R)], X^s).

Equation (6) is just KdV equation perturbed by some terms. In the case of KdV, Kenig, Ponce and Vega ([16]) showed the local well-posedness inH^s, withs >−3/4, by the contraction principle. As it was mentionned in [9], this method fails here. Indeed, it does not seem to be possible to get an appropriate estimate on theH^snorm of the termψ∂xv, becauseψdoes not vanish at infinity.

Bourgain’s method (see [8]) fails for the same reason, as well as the method used by Kenig and Koenig in [12]. Iorio, Linares and Scialom ([9]) used a parabolic regularization and the Bona-Smith approximation. To improve their result, we use here the method that was employed by Koch and Tzetvkov in [17] to show the local well-posedness of BO in H^s, s > 5/4. Namely, we show Strichartz

estimates for a linearized version of (6). Next we derive the crucial non-linear estimate, using the Littlewood-Paley decomposition. To get an a priori estimate on theH^s norm of a solution of (6), a commutator lemma due to Kato ([10]) was used in [9]. This lemma fails fors 63/2. We use and prove here a new commutator lemma (Lemma 2.4 below), which is a variant of a lemma due to Kato and Ponce (see Lemma 2.3 below or [11]).

Our method gives similar results for the BO equation:

Theorem 1.3 Let ψ∈C_b^∞(R)such that ψ^′ ∈H^∞,s >5/4 andφ∈H^s(R).

Then there existsT =T(ψ, s,||φ||H^s)and an unique solutionv of ∂tv+H∂_x²v+v∂xv+∂x(vψ) =−(H∂_x²ψ+ψψ^′), x∈R, t∈R

v(x,0) =φ(x) =g(x)−ψ(x) , (7)

such thatv∈C([−T, T], H^s)∩C¹([−T, T], H^s−2)andvx∈L¹([−T, T], L^∞).

Moreover, for anyR >0, the mapφ→v is continuous from the ball of radius Rin H^s(R)toC([−T(R), T(R)], H^s).

Theorem 1.4 Let s >5/4,g∈X^s.

Then there existsT˜= ˜T(s,||g||X^s)>0and a unique solutionuof (2) such that u∈C([−T ,˜ T˜], X^s),u−g∈C([−T ,˜ T], H˜ ^s)andux∈L¹([−T ,˜ T], L˜ ^∞).

Moreover, for anyR >0, the map g→u is continuous from the ball of radius Rin X^s toC([−T˜(R),T˜(R)], X^s).

In fact, it also works for all the dispersions between BO and KdV. Namely, forα∈[1,2], the problem

∂tv−D^α∂xv+v∂xv+∂x(vψ) =D^α∂xψ−ψψ^′ , x∈R, t∈R

v(0) =φ∈H^s , (8)

whereD= (−∂_x²)^1/2 is locally well posed inH^sfors >3/2−α/4.

In [9], a global well-posedness result was obtained for (7) in H^s fors >2.

Since our local well-posedness result goes underneath 3/2, using the invariant of the Benjamin-Ono equation associated with theH^3/2 norm, we improve this result:

Theorem 1.5 Let φ ∈ H^s, s > 3/2. Then the solution of (7) obtained in Theorem 1.3 can be extended toR.

Corollary 1.1 Let g ∈ X^s, s > 3/2. Then the solution of (2) obtained in Theorem 1.3 can be extended toR.

Note that if Theorem 1.1 was true fors= 1, we would have a global well- posedness result. We failed to show this for general ψ ∈C_b^∞ with ψ^′ ∈ H^∞. However, ifψ ≡ais a constant, a change of variables shows thatv solves (6) if and only if w(x, t) := v(x+at, t) solves the classical KdV equation, which is known to be globally well-posed in H¹. For other dispersions, if we assume

thatψis constant, the results of Kenig and Koenig similarly ensure that (8) is locally well-posed inH^s,s >3/2−3α/8.

The proof of Theorem 1.3 (resp. 1.4) is similar and simpler to that of The- orem 1.1 (resp. 1.2), so that we will omit it.

This paper is organized as follows. In section 2, we state some prelimi- nary results, including the crucial commutator lemmas. In section 3, we prove Strichartz estimates for a linearized version of (6). In section 4, we prove a non-linear estimate. The proof is based on the Littlewood-Paley theory. In the last two sections, the main ideas are these of Koch and Tzvetkov explained in [17]. In section 5 and 6, we prove Theorem 1.1. In section 7, we derive Theorem 1.2. In section 8, we prove Theorem 1.5. Finally, we prove Lemma 2.4 in the appendix. The proof of this commutator lemma is inspired by that of Lemma 2.3 given in [11].

Notations. Throughout this paper, the notation A . B means that there exists an harmless constantc >0 such thatA6cB.

We denote by H^∞ the space H^∞ = ∩

s>0H^s, C_b^∞ the space of C^∞ bounded functions andS the space of Schwartz functions.

IfX is a Banach space, T a positive number and I⊂Ran interval, we define L^p_TX:=L^p([−T, T], X) andL^p_IX:=L^p(I, X) equipped with their natural norms.

We denoteL^∞_TH^∞:= ∩

s>0L^∞_T H^s.

Forσ>0, we denoteJ^σ:= (1−∂_x²)^σ/2,D^σ:= (−∂_x²)^σ/2.

The lettersλand µwill design dyadic integers. The notation P

λf(λ) should be understood asP∞

k=0f(2^k).

We call (q, p)∈R² an admissible pair if (q, p) = ( 6

θ(β+ 1), 2

1−θ), (θ, β)∈[0,1]×[0,1/2].

Note that to prove theorems 1.3 and 1.4 dealing with the BO equation, we should replace this definition by:

2 q +1

p =1

2, q∈(4,∞), p∈(2,∞). We recall that the solution of the initial value problem

∂tv+∂_x³v= 0, t, x∈R

v(x,0) =v0(x) (9)

is given by the unitary Airy group which will be denoted by {W(t)}t∈R, i.e.

v(., t) =W(t)v0=St⋆ v0 where fort >0

St=t^−1/3K(t^−1/3.) and

K(x) =c Z ∞

−∞

e^i(ξ3+xξ)dξ, x∈R.

2 Preliminary results

We now state a result that gives a decomposition of the initial datag, which is used to reduce the study of (1) (resp. (2)) to that of (6) (resp. (7)).

Lemma 2.1 Let g ∈ X^s, s >1. Then there exists ψ ∈ C^∞(R), φ ∈ H^s(R) such that

i) ψ^′∈H^∞(R).

ii) g=ψ+φ.

Moreover, the maps g → ψ and g →φ can be defined as linear maps such that for every s1 >1, g → ψ is continuous from X^s into X^s1 and g → φ is continuous fromX^sintoH^s.

Proof. Letk(x) = (4π)^−1/2e^−x2/4,ψ:=k ⋆ g.

Thenψ∈C_b^∞(R),ψ^′ =k ⋆ g^′∈H^∞, thereforeg−ψ∈L^∞⊂ S^′ and

\g−ψ(ξ) = (ˆg−ψ)(ξ) = (1ˆ −ˆk)ˆg(ξ) =1−e^−ξ2

| {z }ξ

∈L^∞

ξˆg(ξ)

| {z }

∈L²

.

Henceg−ψ∈L², and g−ψ∈H^s. Moreover,||ψ||L^∞ 6||g||L^∞ and

||ψ^′||²_Hs1−1 = Z

(1 +ξ²)^s1−1e^−2ξ2|ξˆg(ξ)|²dξ 6sup

ξ∈R

(1 +ξ²)^s1−1e^−2ξ2

||g^′||²_L2 , which shows the continuity ofg→ψfromX^sintoX^s1. Similarly,

||φ||²_Hs 6sup

ξ∈R



(1 +ξ²) 1−e^−ξ2 ξ

!2

||g^′||²_Hs−1

gives the continuity ofg→φfrom X^s intoH^s. Lemma 2.2 Let (aλ)λ, (dλ)λ two sequences indexed on dyadic integers λ = 2^j, j ∈N. Let s >0. Then

X

λ

λ^s X

µ>λ/8

aµdλ. X

λ

λ^2sa²_λ

!1/2

X

λ

d²_λ

!1/2

,

and hence by duality X

λ

λ^2s



 X

µ>λ/8

aµ





2

.X

λ

λ^2sa²_λ .

Proof. Using the Cauchy-Schwarz inequality, we get X

λ

λ^s X

µ>λ/8

aµdλ = X

λ

λ^s X∞ k=−3

2k λ>1

a2^kλdλ= X∞ k=−3

2^−ks X

λ>2^−k

(2^kλ)^sa2^kλdλ

6 X∞ k=−3

2^−ks



 X

λ>2^−k

(2^kλ)^sa2^kλ

²



1/2

 X

λ>2^−k

d²_λ





1/2

6 2^3s 1−2^−s

X

λ

λ^2sa²_λ

!1/2

X

λ

d²_λ

!1/2

.

Commutator and Bilinear Estimates

We will use in the sequel two commutator lemmas. The first one is due to Kato and Ponce and is proved in [11].

Lemma 2.3 If s >0,

||[J^s, f]g||L² .||∂xf||L^∞||J^s−1g||L²+||J^sf||L²||g||L^∞ . (10) The second commutator lemma is proved in the appendix.

Lemma 2.4 Let s >0, let s0>max(0,3−s). Then

||[J^s, f]g||L².||∂xf||L^∞||J^s−1g||L²+||J^s0+s−1∂xf||L²||g||L^∞ . (11) Next lemmas (see [21]) will be used in the proof of Theorem 1.5.

Lemma 2.5 Leta, b, c∈Rsuch thata>c,b>c,a+b>0anda+b−c > n/2.

Then the map(f, g)−→f gis a continuous bilinear form fromH^a(Rⁿ)×H^b(Rⁿ) intoH^c(Rⁿ).

Lemma 2.6 If s>1, then there is a constantC such that for all f ∈ S(Rⁿ), g∈H^s−1(Rⁿ),

||[J^s, f]g||L²6C||f^′||H^s||g||H^s−1 . We will need a generalized version of Lemma 2.6.

Lemma 2.7 Let n = 1. In Lemma 2.6 above, the assumption f ∈ S can be replaced byf ∈X^∞:= ∩

s>1X^s.

Proof. Let f ∈ X^∞. For ε >0, fε(x) :=e^−εx2f(x) belongs to S, therefore we can apply Lemma 2.6 tofε. Passing to the limit, we obtain Lemma 2.7.

In sections 3 to 6 below, we are interested in solving (6) where the functions ψandφare fixed.

3 The linear estimate

We first recall the Strichartz estimate with smoothing for the Airy group (see [15]).

Lemma 3.1 For any admissible pair (q, p) = (6/(θ(β + 1)),2/(1−θ)) with parameters(θ, β)∈[0,1]×[0,1/2], we have

||D^θβ2 W(t)u0||L^q_t(R,L^p).||u0||L²(R) .

As in [17], we deduce next from Lemma 3.1 a Strichartz inequality for a linearized version of (6).

Lemma 3.2 Let λ>1, T >0, σ >1/2. Letu: [−T, T]×R be a solution of

∂tu+∂_x³u+V1∂xu+V2∂xu+V3u=f (12) whereV1∈L^∞_TH^σ,V2∈L^∞_T L^∞,V3∈L^∞_T H^∞ andf ∈L¹_TL².

We assume moreover that there exists a constantC >0such that Supp(ˆu(., t))⊂C[−λ, λ], t∈[−T, T].

Then for every admissible pair (q, p) with parameter β = 1/2 fixed, for any intervalI⊂[−T, T]such that|I|.λ⁻¹,

||D^θ/4u||_L^q

IL^p.

1 +||J^σV1||L^∞_TL²+||V2||L^∞_TL^∞+||V3||_L¹

TL^∞

×

||u||L^∞_I L²+||f||_L¹_IL²

. (13) Moreover,

||D^θ/4u||_L^q_TL^p.(1 +T)^1/qλ^1/q

1 +||J^σV1||L^∞_TL²+||V2||L^∞_TL^∞+||V3||_L¹_TL^∞

×

||u||L^∞_TL²+||f||L¹_TL²

. (14)

Proof. The solutionv of the Cauchy problem ∂tv+∂_x³v=h∈L¹L²

v(0) =v0

is given by

v(t) =W(t)v0+ Z t

0

W(t−s)h(s)ds.

ApplyingD^θ/4 to this equation, we obtain that for anyt∈[−T, T],

||D^θ/4v(t)||_L^p_x 6||D^θ/4W(t)v0||_L^p_x+ Z T

−T

||D^θ/4W(t)W(−s)h(s)||_L^p_xds,

hence Lemma 3.1 yields

||D^θ/4v||L^q_TL^p.||v0||L²+||h||L¹_TL². We apply this tou:

||D^θ/4u||L^q_IL^p 6 ||u||L^∞_I L²+ (||V1||L^∞_I L^∞+||V2||L^∞_IL^∞)||∂xu||L¹_IL²

+||V3||L¹_IL^∞||u||L^∞_I L²+||f||L¹_IL². Now, as in [KT], using the Sobolev embedding, we have

||V1||L^∞_I L^∞ .||J^σV1||L^∞_I L²,

Next using the assumptions on the support of ˆu(., t) and on the length ofIand Plancherel’s theorem, we get

||∂xu||L¹_IL²=||ξˆu||L¹_IL² .λ|I|||ˆu||L^∞_IL².||u||L^∞_IL²,

which completes the proof of (13). The proof of (14) is the same that in [17]: we write [−T, T] = ∪ⁿ

k=1Ik where|Ik|6λ⁻¹. We may assume thatn < 1 + 2λT 6 2λ(1 +T), and hence using (13) applied toIk and summing overk, we get (14).

4 The nonlinear estimate

Notations. We use the same notations that in [17] about the Littlewood-Paley decomposition. Namely,

u=X

λ

uλ

whereuλ:= ∆λu, and the Fourier multiplier ∆λ is defined by

∆dλu(ξ) :=

φ(ξ/λ)ˆu(ξ) λ= 2^k, k>1 χ(ξ)ˆu(ξ) λ= 1 , whereχandφare nonnegative,C_c^∞ functions onRsatisfying

χ(ξ) +X

λ>1

φ(ξ/λ) = 1

and

φ(ξ) =

0 if|ξ|<5/8 or|ξ|>2 1 if 1<|ξ|<5/4 . For a dyadic integerλ, we also define

∆˜λ:=

∆λ/2+ ∆λ+ ∆2λ ifλ >1

∆1+ ∆2 ifλ= 1 .

Letube a regular solution of

∂tu+∂_x³u+u∂xu+ψ∂xu+ψ^′u=−(∂³_xψ+ψψ^′), x∈R, t∈R

u(0) =u0∈H^∞ . (15)

By “regular solution” we mean that u∈ ∩

s>0C(R, H^s). Theorem 1.2 in [9] en- sures that such a solution does exist (throughout [9], the assumption (3) can be replaced by “g∈X^s” for free).

Our aim in this section is to prove the following estimate onu.

Theorem 4.1 Let σ > 1/2, 0 < T 6 1, (q, p) an admissible pair with pa- rameters θ ∈ [0,1), β = 1/2, s = σ+ 1/q and u a regular solution of (15).

Then

||D^θ/4J^σu||_L^q_TL^p.(1+||J^σu||L^∞_TL²)(1+||ux||_L¹_TL^∞)^3/2(1+||J^su||²_L^∞

TL²)^1/2. (16) We split the proof in several lemmas.

Lemma 4.1 Let (q, p) be an admissible pair with parameter θ ∈ [0,1) (i.e.

p <∞), letσ >1/2. Then

||D^θ/4J^σu||L^q_TL^p. X

λ

λ^2σ||D^θ/4uλ||²_L^q

TL^p

!1/2

. (17)

Proof. A similar lemma was stated in [17]. We recall the proof. We define v:=D^θ/4u. We have

||J^σv||_L^q

TL^p = Z

t

||X

λ

J^σvλ(t)||^q_Lp xdt

!^1/q

6



Z

t

|| X

λ

|J^σvλ(t)|²

!1/2

||^q_L^p

xdt





1/q

(18)

=



 Z

t



Z

x

X

λ

|J^σvλ(t)|²

!p/2

dx





q/p

dt





1/q

6



Z

t

X

λ

Z

x

|J^σvλ(t)|^2.p/2dx

2/p!^q/2 dt





1/q

(19)

||J^σv||L^q_TL^p 6



Z

t

X

λ

||J^σvλ(t)||²_L^p_x

!q/2

dt





1/q

6 X

λ

Z

t

||J^σvλ(t)||^2.q/2_L^p

x dt

2/q!1/2

(20)

= X

λ

||J^σvλ(t)||²_L^q

tL^px

!1/2

. (21)

Here we have used the square functions theorem for the Littlewood-Paley de- composition (see [20]) to obtain (18) (1< p <∞), and the Minkowski inequality (see [18]) to get (19) and (20) (it works becausep/2>1 and q/2>1, respec- tively). Next, using the Mikhlin-Hrmander theorem (or more precisely Lemma 6.2.1 in [4]), we get, for allt,

||J^σvλ(t)||L^p .λ^σ||vλ(t)||L^p . (22)

(21) and (22) complete the proof of Lemma 4.1.

Lemma 4.2 There exists a constantC >0such that for allw∈L²andv∈C_b^∞ such thatvx∈H^∞,

||[∆λ, v∂x]w||L² 6C||vx||L^∞||w||L².

Proof. By density ofS inL², it suffices to show it forw∈ S. As in the proof of Lemma 2 in [17], we write forλ>2

[∆λ, v∂x]w(x) = Z ∞

−∞

K(x, y)w(y)dy , where

K(x, y) =c Z ∞

−∞

e^iλ(x−y)ηφ(η)

iλ²η(v(y)−v(x))−λvx(y) dη . Therefore, using the mean value theorem,

|K(x, y)|6cλ||vx||L^∞g(λ(y−x)) wheregis inL¹. Hence

sup

y

Z ∞

−∞

|K(x, y)|dx+ sup

x

Z ∞

−∞

|K(x, y)|dy.||vx||L^∞,

and the Schur lemma completes the proof of the lemma in the caseλ>2. The

proof is similar in the caseλ= 1.

Lemma 4.3 There exists a constant C > 0 such that for all v ∈ C_b^∞ and w∈L²,

||[∆λ, v]w||L² 6C||v||L^∞||w||L².