In this paper, we are interested in the dynamics near self-similar solutions for the modified Korteweg-de Vries equation:

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SELF-SIMILAR DYNAMICS FOR THE MODIFIED KORTEWEG-DE VRIES EQUATION

SIMÃO CORREIA, RAPHAËL CÔTE AND LUIS VEGA

ABSTRACT. We prove a local well posedness result for the modified Korteweg-de Vries equation in a critical space designed so that is contains self-similar solutions. As a consequence, we can study the flow of this equation around self-similar solutions: in particular, we give an asymptotic description of small solutions ast→+∞and construct solutions with a prescribed blow up behavior ast→0.

1. I

NTRODUCTION

In this paper, we are interested in the dynamics near self-similar solutions for the modified Korteweg-de Vries equation:

∂

t

u + ∂

_{x x x}³

u + ε∂

x

( u

³

) = 0, u : R

t

× R

x

→ R . (mKdV)

The signum ε ∈ {±1} indicates whether the equation is focusing or defocusing. In our framework, ε will play no major role.

The (mKdV) equation enjoys a natural scaling: if u is a solution then u

_λ

( t , x ) : = λ

^1/3

u (λ t, λ

^1/3

x )

is also a solution to (mKdV). As a consequence, the self-similar solutions, which preserve their shape under scaling

S ( t, x ) = t

^−1/3

V ( t

^−1/3

x ) , are therefore of special interest.

Self-similar solutions play an important role for the (mKdV) flow: they exhibit an explicit blow up behavior, and are also related to the long time description of solutions. For smooth and decaying data, this problem (the so-called soliton resolution conjecture) can be studied via the inverse transform method ((mKdV) is integrable): for generic such initial data, a self similar solution appear in the self-similar region t

⁻¹^/³

x = O ( 1 ) ; (we refer to Deift and Zhou [6] for the defocusing case and the recent work by Chen and Liu [3] and the references therein for the focusing case, with a description involving solitons and breathers (which live outside the self-similar region).

Even if one considers small (and smooth, decaying) initial data – an assumption which rules out solitons and breathers – solutions to (mKdV) display a modified scattering where self- similar solutions appear: we refer to Hayashi and Naumkin [ 17, 16 ] , which was revisited by Germain, Pusateri and Rousset [ 11 ] and Harrop-Griffiths [ 14 ] .

2010Mathematics Subject Classification. 35Q53 (primary), 35C06, 35B40, 34E10.

Simão Correia is supported by the Fundação para a Ciência e Tecnologia project UID/MAT/04561/2019.

Raphaël Côte is supported by the Agence Nationale de la Recherche contract MAToS ANR-14-CE25-0009-0. Luis Vega is supported by an ERCEA Advanced Grant 2014 669689 - HADE, by the MEIC project MTM2014-53850-P and MEIC Severo Ochoa excellence accreditation SEV-2013-0323.

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Another example where self-similar solutions of the (mKdV) equation are relevant is in the long time asymptotics of the so-called Intermediate Long Wave (ILW) equation. This equation occurs in the propagation of waves in a one-dimensional stratified fluid in two limiting cases.

In the shallow water limit, the propagation reduces to the KdV equation, while in the deep water limit, it reduces to the so-called Benjamin-Ono equation. In a recent work, Bernal- Vilchis and Naumkin [ 2 ] study the large-time behavior of small solutions of the (modified) ILW, and they prove that in the so-called self-similar region the solutions tend at infinity to a self-similar solution of (mKdV).

Self-similar solutions and the (mKdV) flow are also related to some other simplified models in fluid dynamics. More precisely, Goldstein and Petrich [12] find a formal connection between the evolution of the boundary of a vortex patch in the plane under Euler equations and a hierarchy of completely integrable dispersive equations. The first element of this hierarchy is:

∂

t

z = −∂

sss

z + ∂

s

¯ z (∂

ss

z )

²

, |∂

s

z |

²

= 1, (1)

where z = z ( t, s ) is complex valued and parametrize by its arctlength s a plane curve which evolves in time t . A direct computation shows that its curvature solves the focusing (mKdV) (with ε = 1), and self-similar solutions with initial data

U (t) * cδ

0

+ α v.p.

1 x

as t → 0

⁺

, α, c ∈ R, (2)

correspond to logarithmic spirals making a corner, see [ 22 ] .

Finally, (mKdV) is a member of a two parameter family of geometric flows that appears as a model for the evolution of vortex filaments. In this case, the filaments are curves that propagate in 3d, and their curvature and torsion determined a complex valued function that satisfies a non-linear dispersive equation. This equation, that depends on the two free pa- rameters, is a combination of a cubic non-linear Schrödinger equation (NLS) and a complex modified KdV equation.

The particular case of cubic (NLS) has received plenty of attention. The corresponding geometric flow is known as either the binormal curvature flow or the Localized Induction Ap- proximation, name that is more widely used in the literature of fluid dynamics. In this setting, the relevant role played by the self-similar solutions, including also logarithmic spirals, has been largely studied. We refer the reader to the recent paper by Banica and Vega [ 1 ] and the references there in. Among other things, in this article the authors prove that the self- similar solutions have finite energy, when the latter is properly defined. Moreover, they give a well-posedness result in an appropriately chosen space of distributions that contains the self-similar solutions.

Our goal in this paper is to continue our work initiated in [ 4 ] , and to study the (mKdV) flow in spaces in which self-similar solutions naturally live. As we will see, the number of technical problems increases dramatically with respect to the case of (NLS). This is due to the higher dispersion, which makes the algebra rather more complicated, and to the presence of derivatives in the non-linear term. In this article, we give the first steps to deal with these issues.

In order to have a hint at one of the main challenge we face, let us recal the main result of [ 4 ] , which gives a description in Fourier space of self-similar solution.

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Theorem. Given c, α ∈ R small enough, there exists unique a ∈ R , A, B ∈ C and a self-similar solution S ( t, x ) = t

^−1/3

V ( t

^−1/3

x ) , where V satisfies

for p ⩾ 2, e

⁻^{i t p}³

V ˆ ( p ) = Ae

^ia^ln^|^p^|

+ B e

^3ia^ln^|p|−i⁸⁹^p³

p

³

+ z ( p ) , (3)

for |p| ⩽ 1, e

⁻^{i t p}³

V ˆ (p) = c + 3i α

2 π sgn(p) + z(p), (4)

where z ∈ W

^1,∞

(R) , z ( 0 ) = 0 and for any k <

⁴₇

, | z ( p )| + | pz

^′

( p )| = O (| p |

^−k

) as | p | → +∞ . One sees that there is no decay for high frequency: this is a standard feature for a self- similar solution. The jump at frequency 0 for α ̸= 0 (as in (2)) encodes a different geometric information: for example, in the context of the vortex patch (1), it is related to the angle of the curve at the corner.

We emphasize that self-similar solutions do not fit in any of the functional spaces considered for the use of the inverse scattering transform as in [6, 3] or in the setting derived from [ 16 ] : both due to the behavior at high frequency, and to the jump at frequency 0. In those works, the lack of decay for high frequency was not important because self-similar solutions were only relevant in a bounded region (for fixed time); and decaying initial data means that α = 0 so that there was no jump. Hence, the analysis therein does not suit our purposes: the study of the (mKdV) flow around self-similar solutions must be done in sharper spaces than in earlier works, and preferably in a critical space.

2. D

EFINITIONS AND MAIN RESULTS

2.1. Notations and functional setting. We start with some notations. ˆ u represents the Fourier transform of u (in its space variable x only, if u is a space time function), and we will often denote p the variable dual to x in the Fourier side. We denote by G ( t ) the linear KdV group:

Ø G (t )v(p) = e

^{i t p}³

ˆ v(p),

for any v ∈ S

^′

(R) . Given a (space-time) function u, we denote ˜ u, the profile of u, as the function defined by

˜

u ( t , p ) : = G Û (− t ) u ( t )( p ) = e

^{−i t p}³

u ˆ ( t, p ) . (5)

In all the following, C denotes various constants, which can change from one line to the next, but does not depend on the other variables which appear. As usual, we use the conventions a ≲ b and a = O ( b ) to abbreviate a ⩽ C b.

We will also use the Landau notation a = o

_n

(b) when a and b are two complex quantities (depending in particular of n) such that a / b → 0 as n → +∞ ; and mutatis mutandis a = o

_ϵ

( b) when a/ b → 0 as ϵ → 0.

We use often the japanese bracket 〈 y 〉 = p

1 + | y |

²

, and the (complex valued) Airy-Fock function

(6) Ai(z) := 1

π

∫

_+∞

0

e

^{i pz}⁺^{i p}³

d p.

3

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For v ∈ S

^′

(R) such that ˆ v ∈ L

^∞

∩ H ˙

¹

, and for t > 0, we define the norm (depending on t) with which we will mostly work:

∥ v ∥

_E(t)

: = ∥ G Û (− t ) v ∥

L^∞(R)

+ t

^−1/6

∥∂

p

G Û (− t ) v ∥

L²((0,+∞))

. (7)

Let us remark that we will only consider real-valued functions u, and so ˆ u ( t , − p ) = u ˆ ( t, p ) . As a consequence, the knowledge of frequencies p > 0 is enough to completely determine u ( t ) , and in the above definition, the purpose of considering L

²

(( 0, +∞)) is to allow a jump at 0 as discussed above.

Observe that the E ( t ) norm is scaling invariant, in the following sense:

∥ u

_λ

( t )∥

_E(t)

= ∥ u (λ t )∥

_E(λt)

, λ > 0.

In particular, self-similar solutions have constant E ( t ) norm for t ∈ ( 0, +∞) .

If u is a space-time function defined on a time interval I ⊂ ( 0, +∞) , we extend the above definition and denote

∥ u ∥

_E(I)

: = sup

t∈I

∥ u ( t )∥

_E(t)

= sup

t∈I

∥ u ˜ ( t )∥

L^∞(R)

+ t

⁻¹^/⁶

∥∂

p

u ˜ ( t )∥

L²((0,+∞))

.

In the same spirit, we define the functional space

E (1) := {u ∈ S

^′

(R) : ∥u∥

_E(1)

< +∞}, and for I ⊂ (0, +∞),

E (I) = {u : I → S

^′

(R) : ˜ u ∈ C (I , C

b

((0, +∞))), ∂

p

u ˜ ∈ L

^∞

(I, L

²

((0, +∞)))}, (8)

endowed with the norm ∥ · ∥

_E(I)

.

2.2. Main results. We can now state our results. Our main result is a local well-posedness result in the space E ( I ) , for initial data u

₁

∈ E ( 1 ) at time t = 1.

Theorem 1. Let u

₁

∈ E ( 1 ) . Then there exist T > 1 and a solution u ∈ E ([ 1 / T, T ]) to (mKdV) such that u ( 1 ) = u

₁

.

Furthermore, one has forward uniqueness. More precisely, let 0 < t

₁

< t

₂

and u and v be two solutions to (mKdV) such that u, v ∈ E ([t

1

, t

₂

]). If u(t

1

) = v(t

2

), then for all t ∈ [t

1

, t

₂

], u ( t ) = v ( t ) .

For small data in E ( 1 ) , the solution is actually defined for large times, and one can describe the asymptotic behavior. This is the content of our second result.

Theorem 2. There exists δ > 0 small enough such that the following holds.

If ∥ u

₁

∥

_E(1)

⩽ δ , the corresponding solution satisfies u ∈ E ([ 1, +∞)) . Furthermore, let S be the self-similar solution such that

S ˆ ( 1, 0

⁺

) = ˆ u

₁

( 0

⁺

) ∈ C .

Then ∥u(t ) − S(t)∥

L^∞

≲ ∥u

1

∥

_E(1)

t

⁻⁵^/⁶⁻

and there exists a profile U

_∞

∈ C

b

(R \ {0}, C), with

|U

_∞

(0

⁺

)| = lim

_p_→+∞

| S ˆ (1, p)| is well-defined, and

u(t ˜ , p) − U

_∞

(p) exp

− iε

4 π |U

_∞

(p)|

²

log t ≲ δ

〈 p

³

t 〉

¹²¹

∥u

1

∥

_E(1)

. As a consequence, one has the asymptotics in the physical space.

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Corollary 3. We use the notations of Theorem 2, and let y =

p − x / 3t, if x < 0, 0, if x > 0.

One has, for all t ⩾ 1 and x ∈ R ,

u ( t, x ) − 1 t

^1/3

Re Ai

x t

^1/3

U

_∞

( y ) exp

− i ε

6 | U

_∞

( y )|

²

log t

≲ δ t

^1/3

x / t

^1/3

_3/10

. (9)

2.3. Outline of the proofs, comments and complementary results. In proving Theorem 1 and 2, we use a framework derived from the work of Hayashi and Naumkin [ 16 ] , improved so that only critically invariant quantities are involved (see Section 3). In particular, we use very similar multiplier identities and vector field estimates. An important new difficulty though is that to perform such energy-type inequalities, the precise algebraic structure of the problem has to be respected (for example, in integration by parts): it seems that one cannot use a perturbative argument like a fixed point, as the method truly requires nonlinear solutions.

On the other hand, the rigorous derivation of such inequalities at our level of regularity is quite nontrivial.

This problem does not appear in [ 16 ] as the authors work in a (weighted) subspace of H

¹

, for which a nice local (and global) well-posedness result hold ((mKdV) is actually well-posed in H

^s

for s ⩾ 1 / 4, see Kenig, Ponce and Vega [ 19 ] ). However, no nontrivial self-similar solution belongs to these spaces, as it can be seen from the lack of decay for large p in (3). Let us also mention the work by Grünrock and Vega [ 13 ] , where local well-posedness is proved in

Ò H

^s_r

= {u ∈ S

^′

(R) : ∥ 〈p〉

^s

ˆ u∥

_L^r′

< +∞} for 1 < r ⩽ 2, s ⩾ 1 2 − 1

2r .

This framework is still not enough: self-similar belong to Ò H

₁⁰

but not better. When finding a remedy for this, let us emphasize again that, due to the jump at the zero frequency for self-similar solutions displayed in (4), one must take extra care on the choice of the functional setting. In particular, smooth functions are not dense in E spaces (and they can not approximate self-similar solutions).

In a nutshell, we face antagonist problems coming from low and high frequencies, and we were fortunate enough to be able to take care of both simultaneously.

An important effort of this paper is to solve first an amenable approximate problem (in Section 4), for which we will then derive uniform estimates in the ideology of [16]. This approximate problem is actually a variant of the Friedrichs scheme where we filter out high frequencies via a cut-off function χ

n

(in Fourier space). We solve it via a fixed point argument:

the cut-off takes care of the lack of decay for large frequencies, but again, smooth functions are not dense in the space X

_n

where the fixed point is found (X

_n

is a version of E where high frequencies are tamed, but the jump at frequency 0 remains).

In order to obtain uniform estimates, due to the absence of decay for large frequencies of self-similar solutions, boundary terms cannot be neglected – unless the cut-off function χ

n

is chosen in a very particular way.

At this point, we pass to the limit in n (Section 5), and a delicate but standard compactness argument allows to prove the existence part of Theorem 1 and Theorem 2. The description

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for large time (the second part of Theorem 2 and Corollary 3) is then a byproduct of the above analysis.

The forward uniqueness result given in Theorem 1 requires a different argument. We consider the variation of localized L

²

norm of the difference w of two solutions. Our solutions do not belong to L

²

, but we make use of an improved decay of functions in E ( I ) on the right (for x > 0): in this region, one has a decay of 〈 x 〉

⁻³^/⁴

and therefore they belong to L

²

([0, +∞)). The use of a cut-off φ which is zero for x ≪ −1 allows to make sense of the L

²

quantity.

When computing the derivative of this quantity, one bad term can not be controlled a priori.

Fortunately, if φ is furthermore chosen to be non decreasing, this bad term has a sign, and can be discarded as long as one works forward in time (which explains the one-sided result).

This is related to a monotonicity property first observed and used by Kato [ 18 ] , and a key feature in the study of the dynamics of solitons by Martel and Merle [ 21 ] . We can then conclude the uniqueness property via a Gronwall-type argument.

Using the forward uniqueness properties, we can improve the continuity properties of the solution u: the derivative of its Fourier transform is continuous to the right in L

²

, see Propo- sition 19 for the details.

Backward uniqueness for solutions in E remains an open problem. One can recover it under some extra decay information, namely that u

₁

∈ L

²

(R) (of course this is no longer a critical space). This is the content of our next result, proved in Section 6.

Proposition 4. Let u

₁

∈ E ( 1 ) ∩ L

²

(R) . Then the solution u ∈ E ([ 1 / T, T ]) to (mKdV) given by Theorem 1 is unique and furthermore, there is persistence of regularity: u ∈ C ([ 1 / T, T ] , L

²

(R)) . The stability of self-similar solutions at blow-up time t = 0, or more generally the behavior of solutions with initial data in E (1) near t = 0 is a challenging question. In this direction, we present one result regarding the stability of self-similar blow-up.

We point out that even the description of the effects of small and smooth perturbations of self-similar solutions for small time is not trivial. For example, consider the toy problem of the linearized equation

∂

t

v + ∂

x x x

v + ε∂

x

(K

²

v) = 0.

near the fundamental solution K(t, x ) = t

^−1/3

Ai(t

^−1/3

x ) of the linear Korteweg-de Vries equation (which is, in some sense, the self-similar solution to the linear problem). The most natural move is to use the estimates of Kenig, Ponce and Vega [20], which allows to recover the loss of a derivative:

∥ v ∥

L^∞_t L²_x

≲ ∥ v ( 0 )∥

L²_x

+ ∥ K

²

v ∥

L¹_xL²_t

. Now one can essentially only use Hölder estimate:

∥ K

²

v ∥

L¹_xL²_t

⩽ ∥ K ∥

²_L4

xL^∞_t

∥ v ∥

L²_x,t

,

but, due to the slow decay for x ≪ −1, K(t) ∈ / L

⁴_x

for any t , and the argument can not be closed.

We can however prove a stability result of self-similar solutions up to blow-up time, for low frequency perturbations. Given α > 0 and a sequence ( a

_k

)

_k∈N₀

⊂ R

⁺

satisfying

a

₀

, a

₁

= 1, and for all k ⩾ 0, a

_k

⩽ α a

_2k+1

,

6

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let us define the remainder space

(10) R

_α

=

u ∈ C

^∞

(R) : sup

k⩾0

a

_k

∥∂

_x^k

u∥

²_L2

< ∞

endowed with the norm

(11) ∥u∥

_R_α

=

sup

k⩾0

a

_k

∥∂

_x^k

u∥

²_L2

_1/2

.

Proposition 5 (Stability of the self-similar blow-up under R

_α

-perturbations). There exists δ > 0 sufficiently small such that, if w

₁

∈ R

_α

and S is a self-similar solution with

∥ w

₁

∥

²_R_α

+ α∥ S ( 1 )∥

²_E(₁₎

< δ ,

then the solution u of (mKdV) with initial data u

₁

= S ( 1 ) + w

₁

is defined on ( 0, 1 ] and sup

t∈(0,1)

∥u(t ) − S(t )∥

²_R_α

< 2δ.

Obviously, we shrank considerably the critical space by taking smooth perturbations of self- similar solutions, but the above result still shows some kind of stability of self-similar blow up; observe in particular that the blow up time is not affected by the perturbation. The study of R

_α

perturbations is done in Section 8.

3. P

RELIMINARY ESTIMATES

Throughout this section, I ⊂ ( 0, +∞) is an interval.

Lemma 6 (Decay estimates). Let u ∈ C ( I , S

^′

) such that ∥ u ∥

_E(I)

< +∞ . Then there hold u ∈ C ( I, L

^∞_loc

(R)) and more precisely, for t ∈ I and x ∈ R , one has

| u ( t , x )| ≲ 1 t

¹^/³

| x |/ t

¹^/³

1/4

∥ u ( t )∥

_E(t)

(12)

|∂

x

u ( t, x )| ≲ 1 t

²^/³

| x |/ t

^1/3

_1/4

∥ u ( t )∥

_E(t)

. (13)

Consequently,

∥ u ( t )∥

³_L6

≲ t

⁻⁵⁶

∥ u ( t )∥

³_E(_t₎

(14)

∥ u ( t )∂

x

u ( t )∥

L^∞

≲ t

⁻¹

∥ u ( t )∥

²_E(_t₎

. (15)

Moreover, for x > t

^1/3

,

| u ( t, x )| ≲ 1 t

^1/3

x /t

^1/3

_3/4

∥ u ( t )∥

_E(t)

, (16)

|∂

x

u(t, x )| ≲ 1 t

²^/³

x /t

¹^/³

1/4

∥u(t )∥

_E(t)

, (17)

and for x < − t

^1/3

, (18)

u ( t, x ) − 1 t

^1/3

Re Ai

x t

^1/3

˜ u

t,

v t | x | 3t

≲ 1 t

^1/3

| x |/ t

^1/3

_3/10

∥ u ( t )∥

_E(t)

.

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Proof. The statement and proof are very similar to Lemma 2.1 in [ 16 ] ; notice, however, that the norm ∥·∥

X

therein is stronger than ours, so that we in fact need to systematically improve their bounds. For the convenience of the reader, we provide a complete proof.

We recall that ˜ u ( t ) is not continuous at 0, and may (and will) have a jump (because we only control ∥∂

p

u∥

L²(0,+∞)

: in the following computations ˜ u(t , 0) will mean the limit ˜ u(t, 0

⁺

)).

Setting

z = x p

3

t , y = s

− x

t for x ⩽ 0, y = 0 for x > 0, we have the identity

u ( t, x ) = 1 π Re

∫

_∞

0

e

^{i p x+i p}³^t

u ˜ ( t, p ) d p, q = p p

³

t

= 1

π p

³

t Re

∫

_∞

0

e

^iqz⁺^iq³

˜

u ( t , y ) +

˜ u

t , q

p

3

t

− u ˜ ( t, y )

dq

= 1 p

3

t Re Ai

x p

3

t

˜

u ( t, y ) + R ( t, x ) . (19)

In the case x ⩾ 0, we integrate by parts in the remainder R:

R ( t, x ) = 1 π p

³

t Re

∫

_∞

0

∂

q

qe

^iqz+iq³

1 1 + iq ( 3q

²

+ z )

˜ u

t , q

p

3

t

− u ˜ ( t, y )

dq

= 1

π p

³

t Re

∫

_∞

0

e

^iqz+iq³

1 + iq(3q

²

+ z)

×

iq ( 6q

²

+ z ) 1 + iq ( 3q

²

+ z )

˜ u

t , q

p

3

t

− u(t, ˜ y )

− q p

3

t ∂

p

u ˜

t, q

p

3

t

dq.

Since

| u ˜ ( t , p ) − u ˜ ( t, 0 )| ⩽

∫

_p

0

∂

p

˜ u ( t , q ) dq

⩽ p p ∥∂

p

u ˜ ( t )∥

L²((0,+∞))

, we can estimate the remainder in the following way:

| R ( t, x )| ≲ 1 p

3

t

∫

_∞

0

1 1 + q ( 3q

²

+ z )

u ˜

t, q

p

3

t

− u ˜ ( t , 0 ) + q

p

3

t ∂

p

u ˜

t , q

p

3

t

dq

≲ 1

p t ∥∂

p

u ˜ ( t )∥

L²

∫

_∞

0

p qdq 1 + q(3q

²

+ z) + 1

p

3

t

²

∫

_∞

0

∂

p

u ˜

t, q

p

3

t

2

dq

¹₂

∫

_∞

0

q

²

dq (1 + q(3q

²

+ z))

²

¹₂

≲ 1 p t

1 + | x |

t

¹^/³

₋1/4

∥∂

p

u(t)∥ ˜

L²((0,+∞))

. In the case x < 0, we denote r = p

−z/3. Integrating by parts, we get R(t, x ) = 1

π p

³

t Re

∫

_∞

0

∂

q

(q − r )e

^iqz⁺^iq³

1 1 + 3i ( q − r )

²

( q + r )

˜ u

t, q

p

3

t

− ˜ u(t , y )

dq

8

(9)

= − 1 π p

³

t Re

∫

_∞

0

e

^iqz+iq³

1 + 3i ( q − r )

²

( q + r )

3i(q − r )

²

(3q + r) 1 + 3i ( q − r )

²

( q + r )

˜ u

t , q

p

3

t

− u(t, ˜ y )

+ q − r p

3

t ∂

p

u ˜

t , q

p

3

t

dq − r π p

³

t Re u ˜ ( t, 0 ) − u ˜ ( t, y ) 1 + 3i r

³

. Then we can estimate

| R ( t, x )| ≲ 1 p

3

t

∫

_∞

0

1 1 + (q − r)

²

(q + r ) u ˜

t , q

p

3

t

− u ˜ ( t, y )

+ |q − r|

p

3

t ∂

p

˜ u

t , q

p

3

t

dq

+ 1

p t 〈 r 〉 ∥∂

p

u ˜ ( t )∥

L²((0,+∞))

≲ 1 p t

∫

_∞

0

p |q − r |dq 1 + ( q − r )

²

( q + r ) +

∫

_∞

0

( q − r )

²

dq ( 1 + ( q − r )

²

( q + r ))

²

1/2

!

× ∥∂

p

u ˜ ( t )∥

L²((0,+∞))

+ 1

p t 〈 r 〉 ∥∂

p

˜ u ( t )∥

L²((0,+∞))

≲ 1 p t p

〈 r 〉 ∥∂

p

u ˜ ( t )∥

L²((0,+∞))

.

It now follows from the decay of the Airy-Fock function | Ai ( z )| ≲ 〈 z 〉

⁻¹⁴

that

| u ( t, x )| ≲ t

⁻¹^/³

D x

t

^1/3

E

₋1/4

∥ u ˜ ( t )∥

L^∞

+ t

⁻¹⁶

∥∂

p

u ˜ ( t )∥

L²((0,+∞))

≲ 1

p

3

t

x /t

^1/3

_1/4

∥ u ( t )∥

_E(t)

.

This concludes the proof of (12). For (13), we split once again between the cases x ⩾ 0 and x < 0. In the second case, we have as in (19)

∂

x

u ( t , x ) = 1 p

3

t

²

Re Ai

^′

x p

3

t

˜

u ( t , y ) + R ˜ ( t, x ) , with R ˜ ( t , x ) : = 1

π p

³

t

²

Re

∫

_∞

0

iqe

^iqz⁺^iq³

˜ u

t , q

p

3

t

− u ˜ ( t, y )

dq.

Analogous computations done for R yield

| R(t, ˜ x )| ≲ 1 p

3

t

²

∫

_∞

0

q

1 + ( q − r )

²

( q + r ) u ˜

t, q

p

3

t

− u(t ˜ , y )

+ |q − r|

p

3

t ∂

p

u ˜

t, q

p

3

t

dq

≲ ∥ u ˜ ( t )∥

L^∞

p

3

t

²

∫

_∞

0

qdq

1 + 3 ( q − r )

²

( q + r ) + ∥∂

p

˜ u(t )∥

L²((0,+∞))

p

6

t

⁵

∫

_∞

0

q

²

( q − r )

²

dq ( 1 + 3 ( q − r )

²

( q − r ))

²

¹₂

≲ 1 t

²^/³

1 + | x |

t

¹^/³

1/4

∥ u ˜ ( t )∥

L^∞

+ t

⁻¹⁶

∥∂

p

u ˜ ( t )∥

L²((0,+∞))

9

(10)

≲ 1 t

²^/³

D x t

¹^/³

E

_1/4

∥u(t )∥

_E(t)

,

and the bound for ∂

x

u follows from the bound on the Airy-Fock function | Ai

^′

( z )| ≲ 〈 z 〉

¹⁴

. For x ⩾ 0, we write

∂

x

u ( t, x ) = 1 π p

³

t

²

Re

∫

_∞

0

ie

^iqz⁺^iq³

q˜ u

t, q

p

3

t

dq

= 1

π p

³

t

²

Re

∫

_∞

0

ie

^iqz⁺^iq³

q

u(t, 0) + ˜

∫

q

0

∂

p

u ˜

t , r

p

3

t

d r p

3

t

dq

= 1 π t Re

∫

_∞

0

∫

_∞

r

ie

^iqz+iq³

qdq

∂

p

u ˜

t , r

p

3

t

d r.

Applying Cauchy-Schwarz, and as for z, r > 0, we have

∫

_∞

r

e

^iqz⁺^iq³

qdq ≲ 1 z + r

²

, we obtain

|∂

x

u ( t, x )| ≲ 1 p

6

t

⁵

∥∂

p

u ˜ ( t )∥

L²((0,+∞))

∫

_∞

r

e

^iqz⁺^iq³

qdq

L²((0,∞),d r)

≲ 1 p

3

t

²

〈 z 〉

⁻¹⁴

∥∂

p

u ˜ ( t )∥

L²((0,+∞))

.

Hence (13) follows. The estimate for ∂

x

u in (16) is also a consequence of the above estimate.

Now we prove the first estimate in (16). To that end, we integrate by parts the expression for u:

u(t, x ) = 1 π

∫

_∞

0

e

^{i p x}⁺^{i p}³^t

u( ˜ t, p)d p = ˜ u(t , 0) π x − 1

π

∫

_∞

0

e

^{i p x}⁺^{i p}³^t

∂

p

˜ u(t , p) x + 3p

²

t

d p

= u ˜ ( t , 0 ) π x − 1

π

∫

_∞

0

e

^{i p x+i p}³^t

∂

p

u( ˜ t, p) x + 3p

²

t d p +

∫

_∞

0

e

^{i p x+i p}³^t

6pt˜ u ( t , p ) ( x + 3p

²

t )

²

d p.

The first and third terms are bounded directly, while the second term is bounded using Cauchy-Schwarz:

∫

_∞

0

e

^{i p x+i p}³^t

∂

p

˜ u ( t , p ) x + 3p

²

t d p

≲ ∥∂

p

u ˜ ( t )∥

L²((0,+∞))

∫

_∞

0

d p (x + 3p

²

t)

²

1/2

≲ t

¹^/⁶

1 t

¹^/⁴

x

³^/⁴

∥∂

p

˜ u(t )∥

L²((0,+∞))

≲ 1 t

¹^/³

x / t

¹^/³

∥∂

p

˜ u(t )∥

L²((0,+∞))

.

Finally, estimate (18) follows from [ 11, Lemma 2.9 ] . For completeness, we present the proof:

define ℓ

0

∈ Z so that

2

^ℓ⁰

∼ t

⁻¹^/³

(| x |/ t

¹^/³

)

⁻¹^/⁵

. We split the estimate for

R(t, x ) = 1 π Re

∫

_∞

0

e

^{i t}^Φ(^p⁾

(˜ u(t , p) − u(t, ˜ y))d p, Φ(p) = x

t p + p

³

,

10

(11)

in three regions, using appropriate cut-off functions χ

A

+ χ

B

+ χ

C

= 1:

Region A: | p − y | ⩾ y / 2. Over this region, ∂

p

Φ( p ) ≳ max { y, p } . Then an integration by parts yields

∫

_∞

0

e

^{i t}^Φ(^p⁾

( u ˜ ( t , p ) − u ˜ ( t, y ))χ

A

( p ) d p

≲ 1

t

^1/3

(| x |/ t

^1/3

)

^3/4

∥ u ( t )∥

_E(t)

.

Region B: |p − y | ⩾ 2

^ℓ⁰

. If | p − y| ∼ 2

^l

, with l ⩾ ℓ

0

, then |∂

p

Φ(p)| ≳ 2

^l

y and the same integration by parts gives

contribution of | p − y | ∼ 2

^l

≲

1 t

⁵^/⁶

2

^l^/²

y + 1 t2

^l

y

∥ u ( t )∥

_E(t)

. Summing in l ⩾ ℓ

0

,

∫

_∞

0

e

^{i t}^Φ(p)

( u ˜ ( t, p ) − u ˜ ( t, y ))χ

B

( p ) d p ≲

1 t

⁵^/⁶

2

^ℓ⁰^/2

y + 1 t 2

^ℓ⁰

y

∥ u ( t )∥

_E(t)

≲ 1

t

¹^/³

(| x |/ t

¹^/³

)

³^/¹⁰

∥u(t )∥

_E(t)

. Region C: | p − y | ⩽ 2

^ℓ⁰

. We decompose the integral as

∫

_∞

0

e

^{i t}^Φ(^p⁾

( ˜ u(t , p) − u(t ˜ , y ))χ

C

(p)d p

=

∫

_∞

0

e

i tΦ(p)−i t(Φ(y)+3y(p−y)²)

( u ˜ ( t , p ) − u ˜ ( t, y ))χ

C

( p ) d p + e

^{i t}^Φ(^y⁾

∫

_∞

0

e

^{3i t y}⁽^p⁻^y⁾²

( u ˜ ( t, p ) − u ˜ ( t , y ))χ

C

( p ) d p

= I

₁

+ I

₂

.

Since | u( ˜ t, p) − u(t ˜ , y )| ≲ t

¹^/⁶

|p − y |

¹^/²

∥u(t )∥

_E(t)

, one easily bounds these integrals:

| I

₁

| ≲ t 2

⁴^ℓ⁰

∥ u ∥ ≲ t

⁻¹^/³

(| x |/ t

¹^/³

)

⁴^/⁵

∥ u ( t )∥

_E(t)

| I

₂

| ≲ t

^1/6

2

^3ℓ⁰^/2

∥ u ( t )∥

_E(t)

≲ t

^−1/3

(| x |/ t

^1/3

)

^3/10

∥ u ∥

_E(t)

. □ Let u ∈ C (I , S

^′

) be a solution to (mKdV) in the distributional sense. Taking the Fourier transform of

(∂

t

+ ∂

x x x

) u = −ε∂

x

( u

³

) using ˜ u(t , p) = e

^{i t p}³

u(t ˆ , p), one obtains

(20) ∂

t

˜ u ( t , p ) + iε p 4π

²

∫∫

p₁+p₂+p₃=p

e

^{i t}⁽^p³⁻^p³¹⁻^p²³⁻^p³³⁾

u ˜ ( t, p

₁

) u ˜ ( t, p

₂

) u ˜ ( t, p

₃

) d p

₁

d p

₂

= 0.

This leads us to define (with the change of variables p

_i

= pq

_i

) (21) N [ u ]( t, p ) = i p

³

∫∫

q₁+q₂+q₃=1

e

^{−i t p}³^(1−q³¹^−q³²^−q³³⁾

u ˜ ( t, pq

₁

) ˜ u ( t , pq

₂

) u ˜ ( t, pq

₃

) dq

₁

dq

₂

,

11

(12)

so that

∂

t

u ˜ ( t , p ) = − ε

4π

²

N [ u ]( t , p ) . (22)

The following result is a stationary phase lemma for N [u]. Similar statements may be found in [ 16, Lemma 2.4 ] and [ 11 ] .

Lemma 7 (Asymptotics of the nonlinearity on the Fourier side). Let u ∈ C ( I, S

^′

) such that

∥u∥

_E(I)

< +∞. One has the following asymptotic development for N [u]: for all t ∈ I and p > 0,

N [ u ]( t, p ) = π p

³

〈 p

³

t 〉

i | ˜ u ( t , p )|

²

u ˜ ( t , p ) − 1 p 3 e

⁻^{8i t p}

3 9

u ˜

³

t , p

3 + R [ u ]( t , p ) (23)

where the remainder R satisfies the bound

|R[u](t, p)| ≲ p

³

∥ u ( t )∥

³_E(t)

(p

³

t)

⁵^/⁶

〈p

³

t 〉

¹^/⁴

. (24)

Proof. This essentially relies on a stationary phase type argument. We must however emphasize that the computations and the estimations of the errors have to be performed very carefully, because our setting allows few integration by parts and functions have limited

spatial decay. We postpone the proof to Appendix A. □

Lemma 8. Let I ⊂ ( 0, +∞) be an interval and t

₁

∈ I . Let u ∈ C ( I, S

^′

) be a solution to (mKdV) in the distributional sense such that ∥ u ∥

_E(I)

< +∞ .

Then, for some universal constant C (independent of I ), and for all t ∈ I

∥ u ˜ ( t )∥

L^∞

⩽ ∥ u ˜ ( t

₁

)∥

L^∞

+ C (∥ u ∥

³_E(_I₎

+ ∥ u ∥

⁵_E(_I₎

) ,

∥∂

t

u ˜ ( t )∥

L^∞

≲ 1

t ∥ u ( t )∥

³_E(t)

. Furthermore, if we denote

E

_u

( t, p ) : = exp

− i ε

∫

t

t₁

p

³

〈p

³

s〉 | u ˜ ( s, p )|

²

ds

, (25)

one has, for all t, τ ∈ I

(26) |( uE ˜

_u

)( t, p ) − ( uE ˜

_u

)(τ , p )| ≲ (∥ u ∥

³_E(_I₎

+ ∥ u ∥

⁵_E(_I₎

) min

|t − τ|

〈p

³

〉 , 1

〈τ p

³

〉

¹²¹

. Proof. Since ˜ u ( t , − p ) = u ˜ ( t, p ) , it suffices to consider p > 0. Using Lemma 7,

∂

t

u ˜ ( s, p ) = − ε 4π

p

³

〈p

³

s〉

1 p 3 e

⁻^8isp

3

9

u ˜

³

( s, p / 3 ) − i | ˜ u ( s, p )|

²

u ˜ ( s, p )

(27)

+ O

p

³

∥u∥

³_E(_t₎

(p

³

t)

^5/6

〈p

³

t 〉

^1/4

. (28)

12

(13)

Denote v ( t , p ) = u ˜ ( t , p ) E

_u

( t , p ) . Then the integration in time of (27) on [τ , t ] ⊂ I yields v ( t , p ) = v (τ , p ) − ε

∫

t

τ

p

³

4 π p

3 〈 p

³

s 〉 e

⁻^8isp

3

9

E

_u

( s, p ) u ˜

³

s, p 3

ds (29)

+ O

p

³

∥ u ∥

³_E(_I₎

∫

t

τ

ds ( p

³

s )

^5/6

〈 p

³

s 〉

^1/4

. (30)

We claim that (31)

∫

_t

τ

p

³

〈 p

³

s 〉 e

⁻^8isp

3

9

E

_u

(s, p) ˜ u

³

s, p 3

ds

≲ (∥u∥

³_E(I)

+ ∥u∥

⁵_E(I)

) min

1 〈 p

³

τ〉 , | t − τ|

〈 p

³

〉

Indeed, integrating by parts,

∫

_t

τ

p

³

〈 p

³

s 〉 e

⁻^8isp

3

9

E

_u

(s, p) ˜ u

³

s, p 3

ds

=

∫

t

τ

∂

s

se

⁻^8isp

3 9

1 1 −

^8isp₉³

p

³

〈 p

³

s 〉 E

_u

(s, p)˜ u

³

s, p 3

ds

= −

∫

t

τ

E

_u

( s, p ) e

⁻^8isp

3 9

1 −

^8isp₉³

˜

u

³

( s, p / 3 ) O(p

⁶

s)

1 −

^8isp₉³

〈 p

³

s 〉 + 3sp

³

〈 p

³

s 〉 u ˜

²

(s, p/3) u ˜

_s

(s, p/3) + i p

³

sp

³

〈p

³

s〉

²

˜ u

³

(s, p/3)| u(s, ˜ p)|

²

ds

+



 E

_u

( s, p ) e

⁻^8isp

3

9

u ˜

³

( s, p / 3 ) 1 −

^8isp₉ ³





s=t

s=τ

. From (27), we have

|∂

t

˜ u ( s, p )| ≲ s

⁻¹

∥ u ˜ ( s )∥

³_L∞

≲ s

⁻¹

∥ u ∥

³_E(_s₎

. Taking absolute values in the above expression,

∫

t

τ

p

³

〈p

³

s〉 e

⁻^8isp

3

9

E

_u

( s, p ) u ˜

³

s, p 3

ds

≲ (∥ u ∥

³_E([τ_,t_])

+ ∥ u ∥

⁵_E([τ_,t_])

)

∫

t

τ

p

³

ds

〈 p

³

s 〉

²

≲ (∥u∥

³_E(I)

+ ∥u∥

⁵_E(I)

) min

1 〈 p

³

τ〉 , | t − τ|

〈 p

³

〉

as claimed. We plug this estimate with τ = t

₁

in (29),

∥˜ u(t)∥

L^∞

= ∥v(t)∥

_∞

⩽ ∥v(t

1

)∥

L^∞

+ C (∥u∥

³_E(I)

+ ∥u∥

⁵_E(I)

).

Estimate (26) follows from (31):

|v(t , p) − v(τ, p)| ≲

∫

_t

τ

p

³

4 π p

3 〈 p

³

s 〉 e

⁻^8isp

3

9

E

_u

(s, p)˜ u

³

s, p 3

ds

+ O

p

³

∥u∥

³_E(I)

∫

_t

τ

ds (p

³

s)

⁵^/⁶

〈p

³

s〉

¹^/⁴

13

In this paper, we are interested in the dynamics near self-similar solutions for the modified Korteweg-de Vries equation:

SELF-SIMILAR DYNAMICS FOR THE MODIFIED KORTEWEG-DE VRIES EQUATION

1. I