Dispersive analysis I: Setup and the main proposition

In this section we start the proof of Proposition 2.3. WithU=h∇ih+i|∇|^1/2φ, assume thatU is a solution of the equation

(∂_t+iΛ)U=N2+N3+N_>4 (7.1) on some time interval [0, T],T>1, whereN2is a quadratic non-linearity inU andU,N3

is a cubic non-linearity, andN_>4 is a higher-order non-linearity. Such an equation will be verified below (see§C.2) starting from the main system (2.1) and using the expansion of the Dirichlet–Neumann operator in§B.1. The non-linearityN₂is of the form

N2= X

µ,ν∈{+,−}

Nµν(Uµ,Uν), (F Nµν(f, g))(ξ) =

Z

R²

mµν(ξ, η) ˆf(ξ−η)ˆg(η)dη,

(7.2)

whereU+=U andU−=U. The cubic non-linearity is of the form N3= X

µ,ν,β∈{+,−}

Nµνβ(Uµ,Uν,Uβ), (F Nµνβ(f, g, h))(ξ) =

Z

R²×R²

nµνβ(ξ, η, σ) ˆf(ξ−η)ˆg(η−σ)ˆh(σ)dη dσ.

(7.3)

The multipliers m_µν and n_µνβ satisfy suitable symbol-type estimates. We define the profilesV_σ(t)=e^itΛσU_σ(t),σ∈{⁺,−}, as in (1.11). The Duhamel formula is

(∂_tV)(ξ, s) =b e^isΛ(ξ)Nb₂(ξ, s)+e^isΛ(ξ)Nb₃(ξ, s)+e^isΛ(ξ)Nb_>4(ξ, s), (7.4)

or, in integral form,

V(ξ, t) =b V(ξ,b 0)+cW2(ξ, t)+Wc3(ξ, t)+

Z t 0

e^isΛ(ξ)Nb_>4(ξ, s)ds, (7.5) where, with the definitions in (2.13),

Wc2(ξ, t) := X

µ,ν∈{+,−}

Z t 0

Z

R²

e^{isΦ+µν(ξ,η)}mµν(ξ, η)bVµ(ξ−η, s)bVν(η, s)dη ds, (7.6)

Wc3(ξ, t) := X

µ,ν,β∈{+,−}

Z t 0

Z

R²×R²

e^{isΦ˜+µνβ(ξ,η,σ)}nµνβ(ξ, η, σ)

×bVµ(ξ−η, s)bVν(η−σ, s)Vbβ(σ, s)dη dσ ds.

(7.7)

The vector field Ω acts on the quadratic part of the non-linearity according to the identity

ΩξNb2(ξ, s) = X

µ,ν∈{+,−}

Z

R²

(Ωξ+Ωη)(mµν(ξ, η)Ubµ(ξ−η, s)Ubν(η, s))dη.

A similar formula holds for ΩξNb3(ξ, s). Therefore, for 16a6N1, letting m^b_µν:= (Ω_ξ+Ω_η)^bm_µν and n^b_µνβ:= (Ω_ξ+Ω_η+Ω_σ)^bn_µνβ, we have

Ω^a_ξ(∂tV)(ξ, s) =b e^isΛ(ξ)Ω^a_ξNb2(ξ, s)+e^isΛ(ξ)Ω^a_ξNb3(ξ, s)+e^isΛ(ξ)Ω^a_ξNb_>4(ξ, s), (7.8) where

e^isΛ(ξ)Ω^a_ξNb2(ξ, s) = X

µ,ν∈{+,−}

X

a1+a2+b=a

Z

R²

e^{isΦ+µν(ξ,η)}m^b_µν(ξ, η)

×(Ω^a1Vb_µ)(ξ−η, s)(Ω^a2Vb_ν)(η, s)dη (7.9)

and

e^isΛ(ξ)Ω^a_ξNb3(ξ, s) = X

µ,ν,β∈{+,−}

X

a₁+a₂+a₃+b=a

Z

R²×R²

e^{isΦ˜+µνβ(ξ,η,σ)}n^b_µνβ(ξ, η, σ)

×(Ω^a1Vbµ)(ξ−η, s)(Ω^a2Vbν)(η−σ, s)(Ω^a3Vbβ)(σ, s)dη dσ.

(7.10)

To state our main proposition, we need to make suitable assumptions on the non-linearitiesN₂,N₃, andN_>4. Recall the class of symbolsS^∞ defined in (A.5).

• Concerning the multipliers definingN2, we assume that (Ωξ+Ωη)m(ξ, η)≡0 and km^k,k1,k2kS^∞.2^k2^min(k1,k2)/2,

kD^α_ηm^k,k1,k2kL^∞.|α|2(|α|+3/2) max(|k₁|,|k₂|), kD^α_ξm^k,k1,k2kL^∞.|α|2(|α|+3/2) max(|k|,|k1|,|k2|),

(7.11)

for anyk, k1, k2∈Zandm∈{mµν:µ, ν∈{⁺,⁻}}, where

m^k,k1,k2(ξ, η) :=m(ξ, η)ϕk(ξ)ϕk₁(ξ−η)ϕk₂(η).

• Concerning the multipliers definingN3, we assume that (Ω_ξ+Ω_η+Ω_σ)n(ξ, η, σ)≡0 and

kn^k,k1,k2,k3kS^∞.2^{min(k,k1,k2,k3)/2}2^{3 max(k,k1,k2,k3,0)},

kD_η,σ^α n^k,k1,k2,k3;lk_L^∞.|α|2^{|α|max(|k1|,|k2|,|k3|,|l|)}2(7/2) max(|k1|,|k2|,|k3|), kD^α_ξn^k,k1,k2,k3k_L^∞.|α|2(|α|+7/2) max(|k|,|k1|,|k2|,|k3|),

(7.12)

for anyk, k1, k2, k3, l∈Zandn∈{nµνβ:µ, ν∈{⁺,⁻}}, where

n^k,k1,k2,k3(ξ, η, σ) : =n(ξ, η, σ)ϕk(ξ)ϕk₁(ξ−η)ϕk₂(η−σ)ϕk₃(σ), n^k,k1,k2,k3;l(ξ, η, σ) : =n(ξ, η, σ)ϕk(ξ)ϕk₁(ξ−η)ϕk₂(η−σ)ϕk₃(σ)ϕl(η).

Our main result is the following.

Proposition 7.1. Assume that U is a solution of the equation

(∂t+iΛ)U=N2+N3+N_>4, (7.13) on some time interval[0, T],T>1,with initial data U0. Define,as before,V(t)=e^itΛU(t) and V0=U0. With δ as in Definition 2.5, assume that

kU0k_HN0∩H_Ω^N1,N3+kV0kZ6ε₀1 (7.14) and

(1+t)^−δ2kU(t)k_HN0∩H^N_Ω^1,N3+kV(t)kZ6ε11, (1+t)²kN_>4(t)k_HN0−N3

∩H^N_Ω^1,0+(1+t)^1+δ2ke^itΛN_>4(t)k_Z6ε²₁,

(7.15) for all t∈[0, T]. Moreover,assume that the non-linearitiesN2and N3satisfy (7.2)–(7.3) and (7.11)–(7.12). Then, for any t∈[0, T],

kV(t)kZ.ε0+ε²₁. (7.16) We will show in§C.2below how to use this proposition and a suitable expansion of the Dirichlet–Neumann operator to complete the proof of Proposition2.3.

7.2. Some lemmas

In this subsection we collect several important lemmas which are used often in the proofs in the next two sections. Let Φ=Φ_σµν be as in (2.13).

7.2.1. Integration by parts

In this subsection we state two lemmas that are used in the paper in integration-by-part arguments. We start with an oscillatory integral estimate. See [42, Lemma 5.4] for the proof of (i), and the proof of (ii) is similar.

Lemma7.2. (i) Assume that0<ε61/ε6K,N>1is an integer,and f, g∈C^N(R²).

provided that f is real-valued,

|∇xf|>1_suppg, and kD^α_xf·1suppgkL^∞.Nε^1−|α|, 26|α|6N+1. (7.18)

provided that f is real-valued,

|Ωf|>1suppg, and kΩ^mf·1suppgkL^∞.^N%^1−m, 26m6N+1. (7.20) We will need another result about integration by parts using the vector field Ω. This lemma is more subtle. It is needed many times in the next two sections to localize and then estimate bilinear expressions. The point is to be able to take advantage of the fact that our profiles are “almost radial” (due to the bootstrap assumption involving many copies of Ω), and prove that for such functions one has better localization properties than for general functions.

Fix ξ∈R² and let, for t∈[2^m−1,2^m+1], Ip(f, g) :=

Z

R²

e^itΦ(ξ,η)m(ξ, η)ϕp(ΩηΦ(ξ, η))ϕk(ξ)ϕk₁(ξ−η)ϕk₂(η)f(ξ−η)g(η)dη.

If 2^p6U2^|k1|/2+100 and A62^mU⁻²,then

|Ip(f, g)|.N(2^p+m)^−NU^2N[2^m/2+A2^p]^N+2^−10m. (7.23) In addition,assuming that 1+¹₄δ

ν>−m, the same bound holds when Ip is replaced by I˜_p(f, g) : =

Z

R²

e^itΦ(ξ,η)ϕ_ν(Φ(ξ, η))m(ξ, η)ϕ_p(Ω_ηΦ(ξ, η))

×ϕk(ξ)ϕ_k1(ξ−η)ϕk2(η)f(ξ−η)g(η)dη.

A slightly simpler version of this integration by parts lemma was used recently in [30]. The main interest of this lemma is that we have essentially no assumption ong and very mild assumptions onf.

Proof of Lemma 7.3. We decompose first

f=R₆m/10f+[I−R₆m/10]f and g=R₆m/10g+[I−R₆m/10]g, where the operatorsR₆L are defined in polar coordinates by

(R₆Lh)(rcosθ, rsinθ) :=X

n∈Z

ϕ₆L(n)hn(r)e^inθ if h(rcosθ, rsinθ) :=X

n∈Z

hn(r)e^inθ. (7.24) Since Ω corresponds tod/dθin polar coordinates, using (7.22) we have,

k[I−R_6m/10]fk_L2+k[I−R_6m/10]gk_L2.2^−10m. Therefore, using the H¨older inequality,

|Ip([I−R₆m/10]f, g)|+|Ip(R₆m/10f,[I−R₆m/10]g)|.2^−10m. It remains to prove a similar inequality for Ip:=Ip(f1, g1), where

f1:=ϕ_{[k1−2,k1+2]}R_6m/10f and g1:=ϕ_{[k2−2,k2+2]}R_6m/10g.

It follows from (7.22) and the definitions that

kΩ^ag₁k_L2.a2^am/10 and kΩ^aD^αf₁k_L2.a2^am/10A^|α|, (7.25)

for anya>0 and|α|6N. Integration by parts gives Ip=cϕk(ξ)

Z

R²

e^itΦ(ξ,η)Ωη

m(ξ, η)ϕk₁(ξ−η)ϕk₂(η)

tΩηΦ(ξ, η) ϕp(ΩηΦ(ξ, η))f1(ξ−η)g1(η)

dη.

IteratingN times, we obtain an integrand made of a linear combination of terms like e^itΦ(ξ,η)ϕ_k(ξ)

1 tΩηΦ(ξ, η)

N

Ω^a_η¹(m(ξ, η)ϕ_k1(ξ−η)ϕk2(η))

×Ω^a_η²f1(ξ−η)·Ω^a_η³g1(η)·Ω^a_η⁴ϕp(ΩηΦ(ξ, η))·Ω^a_η⁵⁺¹Φ

ΩηΦ ...Ω^aη^q+1Φ ΩηΦ , whereP

iai=N. The desired bound follows from the pointwise bounds

|Ω^a_η{m(ξ, η)ϕk₁(ξ−η)ϕk₂(η)}|.2^am/2,

|Ω^a_ηϕ_p(Ω_ηΦ(ξ, η))|+

Ω^a+1_η Φ Ω_ηΦ

.U^2a2^am/2,

(7.26)

which hold in the support of the integral, and theL²bounds kΩ^a_ηg1(η)k_L2.2^am/4, kΩ^a_ηf1(ξ−η)ϕk(ξ)ϕ_{[k2−2,k2+2]}(η)ϕ₆p+2(ΩηΦ(ξ, η))k_L2

η.U^2a(2^m/2+A2^p)^a.

(7.27)

The first bound in (7.26) is direct (see (7.21)). For the second bound we notice that Ωη(ξ·η^⊥) =−ξ·η, Ωη(ξ·η) =ξ·η^⊥, ΩηΦ(ξ, η) =λ⁰_µ(|ξ−η|)

|ξ−η| (ξ·η^⊥),

|Ω^a_ηΦ(ξ, η)|.λ(|ξ−η|)(|ξ−η|^−2a|ξ·η^⊥|^a+|ξ−η|^−aU^a).

(7.28)

Sinceλ⁰(|ξ−η|)≈2^|k1|/2 in the support of the integral, we have

|ξ−η|⁻²|ξ·η^⊥| ≈2^p2^{−k1−|k1|/2}.

The second bound in (7.26) follows, once we recall the assumptions in (7.21).

We now turn to the proof of (7.27). The first bound follows from the construction ofg1. For the second bound, if 2^p&2^|k1|/2+min(k,k2), then we have the simple bound

kΩ^a_ηf1(ξ−η)ϕk(ξ)ϕ_{[k2−2,k2+2]}(η)k_L2

η.(A2^min(k,k2)+2^m/10)^a,

which suffices. On the other hand, if 2^p2^|k1|/2+min(k,k2), then we may assume that ξ=(s,0), withs≈2^k. The identities (7.28) show thatϕ_6p+2(Ω_ηΦ(ξ, η))6=0 only if

|ξ·η^⊥|62^p+202^k1−|k1|/2,

which gives

The last claim follows using formula (7.30), as in Lemma7.4below.

7.2.2. Localization in modulation

Our lemma in this subsection shows that localization with respect to the phase is often a bounded operation.

where the constant in the inequality only depends on the function χ.

Proof. We may assume thatm>10 and use the Fourier transform to write χ(2^−pΦ(ξ, η)) =c

Z

R

e^{i%2−pΦ(ξ,η)}χ(%)b d%. (7.30) The left-hand side of (7.29) is dominated by

C

Using (A.2), the contribution of the integral over |%|62^δ2m is dominated by the first term in the right-hand side of (7.29). The contribution of the integral over|%|>2^δ2m is arbitrarily small and is dominated by the second term in the right-hand side of (7.29).

7.2.3. Linear estimates

We first note the straightforward estimates

kPkfkL².min(2^(1−50δ)k,2^{−N k})kfk_Z1∩HN forN >0. (7.31) We now prove several linear estimates for functions in Z1∩H_Ω^N. As in Lemma 7.3, it is important to take advantage of the fact that our functions are “almost radial”. The bounds we prove here are much stronger than the bounds one would normally expect for general functions with the same localization properties, and this is important in the next two sections.

Lemma7.5. Assume that N>10and kfkZ₁+ sup

Recall the operatorsAn,γ₀ defined in (2.17). If j6(1−δ²)m+¹₂|k|and|k|+D6¹2m,then we have the more precise bounds

ke^−itΛA₆0,γ₀fj,k,nkL^∞.

2^−m+2δ2m2−(j−n)(1/2−δ⁰)2^n(δ+1/2N), if n>1,

2^−m+2δ2m2^k2^{−(1/2−δ0)j}, if n= 0. (7.41) Moreover,for l>1,

ke^−itΛA_l,γ0f_j,k,0kL^∞.

2^−m+2δ2m2^δ0j2m/2−j/2−l/2−max(j,l)/2, if 2l+max(j, l)>m, 2^−m+2δ2m2^δ0j2^(l−j)/2, if 2l+max(j, l)6m.

(7.42) In particular, if j6(1−δ²)m+¹₂|k| and |k|+D6¹₂m,then

ke^−itΛA₆0,γ₀fj,kkL^∞.2^−m+2δ2m2^k2^j(δ+1/2N), X

l>1

ke^−itΛAl,γ₀fj,kkL^∞.2^−m+2δ2m2^δ0j2^(m−3j)/6. (7.43) For all k∈Z we have the bounds

ke^−itΛA₆0,γ₀PkfkL^∞.(2^k/2+2^2k)2^−m(2^51δm+2^m(2δ+1/2N)), ke^−itΛA_>1,γ₀PkfkL^∞.2^{−5m/6+2δ2m}.

(7.44)

Proof. (i) The hypothesis gives

kfj,k,nkL².2(1/2−49δ)n−(1−50δ)j and kΩ^Nf_j,k,nkL².kΩ^NP_kfkL².1. (7.45) The bounds (7.35) follow using the general interpolation inequality

sup

θ∈S¹

|h(rθ)|

_{L2(r dr)}.L^1/2khk_L2+L^1/2−NkΩ^Nhk_L2, (7.46) for anyh∈L²(R²) and L>1, which easily follows using the operators R₆L defined in (7.24).

Inequality (7.36) follows from (7.35). Indeed, the left-hand side is dominated by C(2^−k) sup

θ∈S¹

Z

R

|fˆ_j,k,n(rθ)|1_R(ξ0;,%)(rθ)r dr .sup

θ∈S¹

kfˆ_j,k,n(rθ)k_L2(r dr)(2^−k)(2^kmin(%,2^k−n))^1/2, which gives the desired result.

We now consider (7.37). For any fixedθ∈S¹ we have

kfˆ_j,k,n(rθ)k_L^∞.2^j/2kfˆ_j,k,n(rθ)k_L2(dr)+2^−j/2k(∂_rfˆ_j,k,n)(rθ)k_L2(dr)

.2^j/22^−k/2kfˆ_j,k,n(rθ)k_L2(r dr),

using the support property of Qjkf in the physical space. The desired bound follows using (7.35) and the observation that ˆfj,k,n=0 unlessn=0 ork∈[−10,10]. The bound (7.38) also follows since differentiation in the Fourier space essentially corresponds to multiplication by factors of 2^j, due to space localization.

(ii) The bound (7.39) follows directly from Hausdorff–Young and (7.45). To prove (7.40), if|k|>10 then the standard dispersion estimate

Z

R²

e^{−itλ(|ξ|)}ϕk(ξ)e^ix·ξdξ

.2^2k(1+|t|2^k+|k|/2)⁻¹ (7.47) gives

ke^−itΛfj,k,nkL^∞. 2^2k

1+|t|2^k/2kfj,k,nk_L1. 2^2k

1+|t|2^k/22^50δj. (7.48) The bound (7.40) follows (in casem610 andk>0, one can use (7.39)).

We now prove (7.41). The operatorA₆0,γ0 is important here, because the function λhas an inflection point atγ0; see (10.3). Using Lemma7.2(i) and the observation that

|(∇Λ)(ξ)|≈2^|k|/2 if|ξ|≈2^k, it is easy to see that

|(e^−itΛA_60,γ0f_j,k,n)(x)|.2^−10m unless |x| ≈2^m+|k|/2.

Also, letting f_j,k,n⁰ :=R_6m/5fj,k,n (see (7.24)), we have kfj,k,n−f_j,k,n⁰ k_L2.2^−mN/5, and thus

ke^−itΛA₆0,γ₀(fj,k,n−f_j,k,n⁰ )kL^∞.kfˆj,k,n−fb_j,k,n⁰ k_L1.2^−2m2^k. (7.49) On the other hand, if|x|≈2^m+|k|/2 then, using again Lemma7.2and (7.38),

(e^−itΛA₆0,γ₀f_j,k,n⁰ )(x)

=C Z

R²

e^iΨ(ξ)ϕ(r⁻¹∇ξΨ)ϕ(⁻¹θ ΩξΨ)fb_j,k,n⁰ (ξ)ϕ_>−100(|ξ|−γ0)dξ+O(2^−10m),

(7.50)

where

Ψ : =−tΛ(ξ)+x·ξ,

r: = 2^δ2m(2(m+|k|/2−k)/2+2^j), θ: = 2^δ2m2(m+k+|k|/2)/2.

(7.51)

We notice that the support of the integral in (7.50) is contained in a×%rectangle in the direction of the vectorx, where

%. r

2^m+|k|/2−k, . θ

2^m+|k|/2, and .%.

This is because the function λ⁰⁰ does not vanish in the support of the integral, and so λ⁰⁰(|ξ|)≈2^|k|/2−k. Therefore, we can estimate the contribution of the integral in (7.50) using either (7.36) or (7.37). More precisely, if j6¹2 m+¹₂|k|−k

, then we use (7.37), while if j>¹2 m+¹₂|k|−k

, then we use (7.36) (and estimate min(1,2ⁿ%2^−k)62ⁿ%2^−k);

in both cases the desired estimate follows.

We now prove (7.42). We may assume that |k|610 andm>D. As before, we may assume that|x|≈2^m and replacefj,k,0 byf_j,k,0⁰ . As in (7.50), we have

(e^−itΛA_l,γ0f_j,k,0)(x)

=C Z

R²

e^iΨ(ξ)ϕ(2^{−m/2−δ2m}ΩξΨ)fb_j,k,0⁰ (ξ)ϕ_−l−100(|ξ|−γ0)dξ+O(2^−2m),

(7.52)

where Ψ is as in (7.51). The support of the integral above is contained in a×%rectangle in the direction of the vectorx, where %.2^−l and.2^−m/2+δ2m. Since

|fb_j,k,0⁰ (ξ)|.2^−j/2+δ0j

in this rectangle (see (7.37)), the bound in the first line of (7.42) follows ifl>j. On the other hand, ifl6j then we use (7.36) to show that the absolute value of the integral in (7.52) is dominated byC2^−j+δ0j%^1/2, which gives again the bound in the first line of (7.42).

It remains to prove the stronger bound in the second line of (7.42) in the case 2l+max(j, l)6m. We notice that λ⁰⁰(|ξ|)≈2^−l in the support of the integral. Assume thatx=(x1,0), withx1≈2^m, and notice that we can insert an additional cutoff function of the form

ϕ[r⁻¹(x1−tλ⁰(|ξ1|) sgn (ξ1))], wherer:= 2^δ2m(2^(m−l)/2+2^j+2^l),

in the integral in (7.52), at the expense of an acceptable error. This can be verified using Lemma7.2(i). The support of the integral is then contained in a×%rectangle in the direction of the vectorx, where%.r2^−m2^l and .2^−m/2+δ2m. The desired estimate then follows as before, using the L^∞ bound (7.37) if 2j6m−l and the integral bound (7.36) if 2j>m−l.

The bounds in (7.43) follow from (7.41) and (7.42) by summation over n and l, respectively. Finally, the bounds in (7.44) follow by summation (use (7.39) ifj>(1−δ²)m or m64D, use (7.40) if j6(1−δ²)m and |k|>10, and use (7.43) if j6(1−δ²)m and

|k|610).

Remark 7.6. We notice that we also have the bound (with no loss of 2^2δ2m) ke^−itΛA_60,γ0f_j,k,0k_L^∞.2^−m2^k2^{−(1/2−δ0−δ)j}, (7.53)

provided that j6(1−δ²)m+¹₂|k| and |k|+D6¹2m. Indeed, this follows from (7.41) if j>10¹m. On the other hand, ifj610¹m, then we can decompose (compare with (7.50)),

(e^−itΛA_60,γ0f_j,k,0)(x) =X

p>0

C Z

R²

e^iΨ(ξ)ϕ^[0,∞)_p (⁻¹∇ξΨ) ˆf_j,k,0(ξ)ϕ_>−100(|ξ|−γ0)dξ, where:=2(m+|k|/2−k)/2. The contribution ofp=0 is estimated as before, using (7.37), while forp>1 we can first integrate by parts at most three times, and then estimate the integral in the same way.

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