We prove low regularity local well-posedness results in Bourgain- Klainerman-Machedon spaces for the Chern-Simons-Dirac system in the tem- poral gauge and the Coulomb gauge

Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 174, pp. 1–16.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

LOW REGULARITY SOLUTIONS FOR CHERN-SIMONS-DIRAC SYSTEMS IN THE TEMPORAL AND COULOMB GAUGE

HARTMUT PECHER

Abstract. We prove low regularity local well-posedness results in Bourgain- Klainerman-Machedon spaces for the Chern-Simons-Dirac system in the tem- poral gauge and the Coulomb gauge. Under slightly stronger assumptions on the data we also obtain “unconditional” uniqueness in the natural solution spaces.

1. Introduction and statement of main results Consider the Chern-Simons-Dirac system in two space dimensions

i∂_tψ+iα^j∂_jψ=mβψ−α^µA_µψ (1.1)

∂µAν−∂νAµ=−2µνλhψ, α^λψi (1.2) with initial data

ψ(0) =ψ₀, A_µ(0) =a_µ, (1.3) where we use the convention that repeated upper and lower indices are summed, Latin indices run over 1,2 and Greek indices over 0,1,2 with Minkowski metric of signature (+,−,−). Here ψ : R¹⁺² → C², Aν : R¹⁺² → R, m ∈R. α¹, α², β are hermitian (2×2)-matrices satisfying β² = (α¹)² = (α²)² = I, α^jβ+βα^j = 0, α^jα^k+α^kα^j = 2δ^jkI, α⁰ =I. h·,·idenotes the C²-scalar product. A particular representation is given by

α¹= 0 1

1 0

, α²=

0 −i i 0

, β =

1 0 0 −1

. µνλ is the totally skew-symmetric tensor with012= 1 .

This model was proposed by Cho, Kim and Park [5], and by Li and Bhaduri [11]. The equations are invariant under the gauge transformations

A_µ→A⁰_µ=A_µ+∂_µχ , ψ→ψ⁰=e^iχφ .

The most common gauges are the Coulomb gauge ∂^jA_j = 0, the Lorenz gauge

∂^µA_µ= 0 and the temporal gaugeA₀= 0.

Local well-posedness for data with minimal regularity assumptions was shown by Huh [7] in the Lorenz gauge for dataψ0∈H⁵⁸,aµ∈H^1/2using a null structure, in the Coulomb gauge forψ0∈H¹²⁺,ai∈L², and in temporal gauge forψ0∈H³⁴⁺,

2010Mathematics Subject Classification. 35Q40, 35L70.

Key words and phrases. Chern-Simons-Dirac;local well-posedness; Coulomb gauge;

temporal gauge.

c

2016 Texas State University.

Submitted November 10, 2015. Published July 6, 2016.

1

aj ∈H³⁴⁺+L², both without using a null structure. The result in Lorenz gauge was improved by Huh-Oh [8] where the regularity of the data was lowered down to ψ0∈H^s,aµ ∈H^s withs >1/4. Their proof relies also on a null structure in the nonlinear terms of the Dirac equation as well as the wave equation. They apply a Picard iteration in Bourgain-Klainerman-Machedon spaces X^s,b, which implies uniqueness in these spaces. Independently Okamoto [12] proved a similar result in Lorenz as well as Coulomb gauge also using a null structure of the system. The methods of Okamoto and Huh-Oh are different. Okamoto reduces the problem to a single Dirac equation with cubic nonlinearity forψ, which does not containAµ any longer. From a solutionψof this equation the potentialsAµcan be constructed by solving a wave equation in Lorenz gauge and an elliptic equation in Coulomb gauge.

Huh-Oh on the other hand directly solve a coupled system of a Dirac equation for ψ and a wave equation forAµ. Recently Bournaveas-Candy-Machihara [4] proved local well-posedness in Coulomb gauge under similar regularity assumptions without use of a null structure. Their proof relies on a bilinear Strichartz estimate given by Klainerman-Tataru [10].

A low regularity local well-posedness result in temporal gauge was given by Tao [14] for the Yang-Mills equations.

In the present paper we consider the temporal gauge as well as the Coulomb gauge. In temporal gauge we improve the result of Huh [7] to data ψ₀ ∈ H^s, aj ∈ H^s+18 with s > 3/8. We use Bourgain-Klainerman-Machedon spaces X^s,b adapted to the phase functionsτ± |ξ| on one hand andτ on the other hand. We decompose A_j into its divergence-free part A^df_j and its curl-free part A^cf_j . The main problem here is that there seems to be no null structure in the nonlinearity A^cf_j α^jψ in the Dirac equation whereas in Lorenz gauge A^cf_µ α^µψ has such a null structure. In fact all the other terms possess such a null structure. However we are not able to use it for an improvement of our result. We apply the bilinear estimates in wave-Sobolev spaces established in d’Ancona-Foschi-Selberg [2] which rely on Strichartz estimates. Morover we use a variant of an estimate for theL⁶_xL²_t - norm for the solution of the wave equation which goes back to Tataru and Tao.

When applying this estimate we partly follow Tao’s arguments in the case of the Yang-Mills equations [14]. We prove existence and uniqueness in X^s,b - spaces first (Theorem 1.1). Then we prove unconditional uniqueness under the stronger assumptions > ¹⁹₄₀ (Theorem 1.2) by using an idea of Zhou [16].

In Coulomb gauge we make the same regularity assumptions as Okamoto [12] and Bournaveas-Candy-Machihara [4], namelyψ₀∈H¹⁴⁺, and also reduce the problem to a single Dirac equation with cubic nonlinearity. We give a short (alternative) proof of local well-posedness inX^s,b - spaces without use of a null structure (The- orem 1.3) using d’Ancona-Foschi-Selberg [2] (cf. Proposition 1.5). We also prove unconditional uniqueness in the space ψ ∈ C⁰([0, T], H^s) under the assumption s >1/3 (Theorem 1.4).

We first give some notation. We denote the Fourier transform with respect to space and time byb. The operator |∇|^α is defined byF(|∇|^αf)(ξ) =|ξ|^α(Ff)(ξ), whereFis the Fourier transform, and similarlyh∇i^α, whereh·i:= (1+|·|²)^1/2. The inhomogeneous and homogeneous Sobolev spaces are denoted by H^s,p and ˙H^s,p, respectively. For p= 2 we simply denote them byH^sand ˙H^s. We repeatedly use the Sobolev embeddings ˙H^s,p ,→L^q for 1< p≤q <∞and ¹_q = ¹_p −^s₂, and also

H˙¹⁺∩H˙¹⁻ ,→L^∞ in two space dimensions. a+ :=a+ for a sufficiently small >0 , so thata < a+< a+ +, and similarlya− −< a−< a.

We define the standard spacesX_±^s,bof Bourgain-Klainerman-Machedon type be- longing to the half waves as the completion of the Schwarz spaceS(R³) with respect to the norm

kuk_Xs,b

± =khξi^shτ± |ξ|i^bu(τ, ξ)kb _L2 τ ξ. Similarly we define the wave-Sobolev spacesX_|τ|=|ξ|^s,b with norm

kuk_Xs,b

|τ|=|ξ|

=khξi^sh|τ| − |ξ|i^bu(τ, ξ)kb _L2 τ ξ

and alsoX_τ=0^s,b with norm kuk_Xs,b

τ=0

=khξi^shτi^bbu(τ, ξ)k_L²

τ ξ.

We also define X_±^s,b[0, T] as the space of the restrictions of functions in X_±^s,b to [0, T]×R²and similarlyX_|τ|=|ξ|^s,b [0, T] andX_τ=0^s,b [0, T]. We frequently use the obvious embeddingsX_|τ|=|ξ|^s,b ,→X_±^s,b forb≤0 andX_±^s,b ,→X_|τ|=|ξ|^s,b forb≥0.

We now formulate our main results in the case of the temporal gauge.

Theorem 1.1. Let > 0 and s > 3/8. The Chern-Simons-Dirac system (1.1), (1.2), (1.3) in temporal gauge A0 = 0 with data ψ0 ∈ H^s(R²), aj ∈ H^s+18(R²), satisfying the compatibility condition∂1a2−∂2a1=−2hψ0, ψ0i, has a local solution

ψ∈C⁰([0, T], H^s(R²)), |∇|Aj ∈C⁰([0, T], H^s+18−(R²)). More precisely ψ = ψ+ +ψ₋ with ψ_± ∈ X^s,

1 2+

± [0, T]. If A = A^df +A^cf is the decomposition into its divergence-free part and its “curl-free” part, where

A^df = (−∆)⁻¹(∂2(∂1A2−∂2A1), ∂1(∂2A1−∂1A2)),

A^cf =−(−∆)⁻¹(∂1(∂1A1+∂2A2), ∂2(∂1A1+∂2A2)) =−(−∆)⁻¹∇divA , one has

A^cf ∈X^s+

1 8,¹₂+

τ=0 [0, T], |∇|A^df ∈X^s+

3 8−,¹₂+

|τ|=|ξ| [0, T] and in these spaces uniqueness holds. Moreover we haveψ_±∈X_±^s,1[0, T].

Remark. The Chern-Simons-Dirac system is invariant under the scaling ψ^(λ)(t, x) =λψ(λt, λx), A^(λ)(t, x) =λAµ(λt, λx).

Thus in 2+1 dimensions the scaling critical Sobolev exponent is s = 0, i.e. ψ₀, a_µ ∈H^s=L². In Lorenz gauge Huh-Oh [8] remarked that their results >1/4 is probably optimal in view of Zhou [15], who proved that is the case for a system of nonlinear wave equations with nonlinearities, which fulfill a null condition. In our case of the temporal gauge however the system is reduced to a coupled system of a wave equation for ψand a transport equation for A^cf where null conditions seem to be not useful because they are only adapted for wave equations. Nevertheless it would be desirable to improve our result tos >1/4 forψ0 andaj .

Theorem 1.2. Let the assumptions of Theorem 1.1 be fulfilled. If s > 19/40, the solution of (1.1), (1.2), (1.3) is unique in the space ψ ∈ C⁰([0, T], H^s(R²)), A^cf ∈C⁰([0, T], H^s+18(R²)),|∇|A^df ∈C⁰([0, T], H^s+38−(R²)).

Consider now the Coulomb gauge condition ∂jA^j = 0. In this case one easily checks using (1.2) that the potentialsAµ satisfy the elliptic equations

A0= ∆⁻¹(∂2hψ, α¹ψi −∂1hψ, α²ψi),

A1= ∆⁻¹∂2hψ, ψi, A2=−∆⁻¹∂1hψ, ψi. (1.4) Inserting this into (1.1) we obtain

i∂tψ+iα^j∂jψ=mβψ+N(ψ, ψ, ψ), (1.5) where

N(ψ1, ψ2, ψ3)

= ∆⁻¹(∂1hψ1, α2ψ2i −∂2hψ1, α1ψ2i+∂2hψ1, ψ2iα1−∂1hψ1, ψ2iα2)ψ3. In the sequel we consider this nonlinear Dirac equation with initial condition

ψ(0) =ψ₀. (1.6)

Using an idea of d’Ancona - Foschi -Selberg [1] we simplify (1.5) by considering the projections onto the one-dimensional eigenspaces of the operator −iα· ∇ =

−iα^j∂j belonging to the eigenvalues ±|ξ|. These projections are given by Π_± = Π_±(D), whereD = ^∇_i and Π_±(ξ) = ¹₂(I±_|ξ|^ξ ·α). Then−iα· ∇=|D|Π+(D)−

|D|Π₋(D) and Π_±(ξ)β = βΠ_∓(ξ). Defining ψ_± := Π_±(D)ψ, the Dirac equation can be rewritten as

(−i∂t± |D|)ψ_±=mβψ_∓+ Π_±N(ψ++ψ₋, ψ++ψ₋, ψ++ψ₋). (1.7) The initial condition is transformed into

ψ_±(0) = Π_±ψ₀. (1.8)

We now formulate our results in the case of the Coulomb gauge.

Theorem 1.3. Assume ψ0 ∈ H^s(R²) with s > 1/4. Then (1.5),(1.6) is locally well-posed in H^s(R²). More precisely there exists T >0, such that there exists a unique solution ψ = ψ₊ +ψ₋ with ψ_± ∈ X_±^s,12+[0, T]. This solution belongs to C⁰([0, T], H^s(R²)).

The unconditional uniqueness result is the following.

Theorem 1.4. Assumeψ₀∈H^s(R²)withs >1/3. The solution of (1.5),(1.6)is unique inC⁰([0, T], H^s(R²)).

Fundamental for the proof of our theorems are the following bilinear estimates in wave-Sobolev spaces which were proven by d’Ancona, Foschi and Selberg in the two dimensional casen= 2 in [2] in a more general form which include many limit cases which we do not need.

Proposition 1.5. Let n= 2. The estimate kuvk_X^−s0,−b0

|τ|=|ξ| .kuk_Xs1,b1

|τ|=|ξ|

kvk_Xs2,b2

|τ|=|ξ|

holds, provided the following conditions hold:

b0+b1+b2>1

2, b0+b1≥0, b₀+b₂≥0, b₁+b₂≥0,

s₀+s₁+s₂>3

2 −(b₀+b₁+b₂), s0+s1+s2>1−min(b0+b1, b0+b2, b1+b2), s0+s1+s2> 1

2 −min(b0, b1, b2), s0+s1+s2> 3 4, (s₀+b₀) + 2s₁+ 2s₂>1, 2s₀+ (s₁+b₁) + 2s₂>1, 2s₀+ 2s₁+ (s₂+b₂)>1, s₁+s₂≥max(0,−b0),

s0+s2≥max(0,−b1), s0+s1≥max(0,−b2).

Another decisive tool are the estimates for the wave equation in the following proposition.

Proposition 1.6. The following estimates hold kuk_L6

xt .kuk

X

1 2,1

2+

|τ|=|ξ|

, (1.9)

kuk_L^p_xL²

t .kuk

X

1 2−2

p,1 2+

|τ|=|ξ|

for6≤p <∞, (1.10) especially kuk_L6

xL²_t .kuk

X

1 6,1

2+

|τ|=|ξ|

, (1.11)

kuk_L∞

xL²_t .kuk

X

1 2+,1

2+

|τ|=|ξ|

, (1.12)

kuk_L∞

xL²⁺_t .kuk

X

1 2+,1

2+

|τ|=|ξ|

, (1.13)

kuk_L6

xL²⁺_t .kuk

X

16+,1 2+

|τ|=|ξ|

, (1.14)

kuk_L4

xL²⁺_t .kuk

X

1 8+,3

8+

|τ|=|ξ|

, (1.15)

kuk_Lp

xL²⁺_t .kuk

X

1 2−2

p+,1 2+

|τ|=|ξ|

for6≤p <∞. (1.16) Proof. (1.9) is the standard Strichartz estimate combined with the transfer princi- ple. Concerning (1.10) we use [9] (appendix by D. Tataru) Thm. B2:

kFtukL²_τL⁶_x.ku0k

H˙

1 x6

,

ifu=e^it|∇|u0 , and F denotes the Fourier transform with respect to time. This implies by Plancherel, Minkowski’s inequality and Sobolev’s embedding theorem

kuk_L^p_xL2

t =kFtuk_L^p_xL2

τ .kFtuk_L2

τL^p_x .kFtuk

L²_τH

13−2 p,6 x

.ku0k

H

12−2 p,2 x

.

The transfer principle gives (1.10). (1.12) follows similarly using Hx^13+,6 ,→ L^∞_x . (1.14) is obtained by interpolation between (1.11) and (1.9), and (1.15) by inter- polation between (1.14) and the trivial identitykuk_L2

xt=kuk_X0,0

|τ|=|ξ|

. Moreover we obtain (1.13) and (1.16) by interpolation between (1.12) and (1.10), resp., and the estimatekukL^∞_xt.kuk

X^1+,

1 2+

|τ|=|ξ|

.

2. Reformulation of the problem in temporal gauge

Imposing the temporal gauge conditionA₀= 0 the system (1.1), (1.2) is equiv- alent to

i∂_tψ+iα^j∂_jψ=mβψ−α^jA_jψ (2.1)

∂tA1=−2hψ, α²ψi, ∂tA2= 2hψ, α¹ψi (2.2)

∂1A2−∂2A1=−2hψ, ψi. (2.3)

We first show that (2.3) is fulfilled for any solution of (2.1), (2.2), if it holds initially, i.e., if the following compatability condition holds:

∂1A2(0)−∂2A1(0) =−2hψ(0), ψ(0)i, (2.4) which we assume from now on. Indeed one easily calculates using (2.1):

∂thψ, ψi=−∂jhψ, α^jψi, (2.5) which implies by (2.2)

∂t(∂1A2−∂2A1) = 2∂jhψ, α^jψi=−2∂thψ, ψi,

so that (2.3) holds, if it holds initially. Thus we only have to solve (2.1) and (2.2).

We decomposeA= (A1, A2) into its divergence-free partA^df and its “curl-free”

partA^cf, namelyA=A^df+A^cf, where

A^df = (−∆)⁻¹(∂2(∂1A2−∂2A1), ∂1(∂2A1−∂1A2)),

A^cf =−(−∆)⁻¹(∂₁(∂₁A₁+∂₂A₂), ∂₂(∂₁A₁+∂₂A₂)) =−(−∆)⁻¹∇divA . Then (2.3) and (2.2) imply

A^df =−2(−∆)⁻¹(∂₂hψ, ψi,−∂1hψ, ψi), (2.6)

∂tA^cf_j =−2(−∆)⁻¹∂j(∂2hψ, α¹ψi −∂1hψ, α²ψi). (2.7) Reversely, definingA=A^df+A^cf, we show that our new system (2.1), (2.6), (2.7) implies (2.1), (2.2), (2.3), so that both systems are equivalent. It only remains to show that (2.2) holds. By (2.6), (2.7), (2.5) we obtain

∂_tA₁=∂_tA^df₁ +∂_tA^cf₁

=−2(−∆)⁻¹ ∂₂∂_thψ, ψi+∂₁(∂₂hψ, α¹ψi −∂₁hψ, α²ψi)

= 2(−∆)⁻¹ ∂2∂jhψ, α^jψi −∂1(∂2hψ, α¹ψi −∂1hψ, α²ψi)

= 2(−∆)⁻¹(∂₂²+∂₁²)hψ, α²ψi=−2hψ, α²ψi and similarly

∂tA2= 2hψ, α¹ψi.

In the same way in which we obtained (1.7) the Dirac equation (2.1) can be rewritten as

(−i∂t± |∇|)ψ±=−mβψ∓−Π±(α^jAjψ), (2.8) whereAj =A^df_j +A^cf_j , and in (2.6),(2.7) and (2.8) we replaceψbyψ++ψ₋.

3. Proof of Theorem 1.1

Taking the considerations of the previous section into account Theorem 1.1 re- duces to the following proposition and its corollary.

Proposition 3.1. Let > 0 and s > 3/8. Then there exists T > 0 such that system (2.6), (2.7), (2.8) has a unique local solution ψ± ∈ X^s,

1 2+

± [0, T], A^cf ∈ X^s+

1 8,¹₂+

τ=0 [0, T]. Also A^df satisfies|∇|A^df_j ∈X^s+

3 8−,¹₂+

|τ|=|ξ| [0, T]andψ±∈X_±^s,1[0, T].

Corollary 3.2. The solution satisfiesψ∈C⁰([0, T], H^s),A^cf ∈C⁰([0, T], H^s+18),

|∇|A^df ∈C⁰([0, T], H^s+38−).

Proof of Proposition 3.1. We want to apply a Picard iteration. For the Cauchy problem for the Dirac equation

(−i∂t± |∇|)ψ_±=F_±, ψ_±(0) =ψ_±0 we use the well-known estimate (cf. e.g. [6])

kψ_±k_Xs,b

± [0,T] .kψ_±0k_Hs+T^1+b0−bkF_±k

X_±^s,b0[0,T],

which holds for 0< T ≤1 , −¹₂ < b⁰ ≤0 ≤b ≤b⁰+ 1, s∈R. Thus by standard arguments it suffices to show the following estimates for the right hand side of the Dirac equation (2.8):

kA^cf_j α^jψk

X^s,−

1 2++

|τ|=|ξ|

.kA^cfk

X^{s+ 18,}

1 2+ τ=0

kψk

X^s,

1 2+

|τ|=|ξ|

, (3.1)

kA^df_j α^jψk

X^s,−

1 2++

|τ|=|ξ|

.k|∇|A^df_j k

X^{s+ 38−,}

1 2+

|τ|=|ξ|

kψk

X^s,

1 2+

|τ|=|ξ|

, (3.2)

k|∇|A^df_j k

X^{s+ 38−,}

1 2+

|τ|=|ξ|

.kψk²

X^s,

1 2++

|τ|=|ξ|

. (3.3)

Similarly, for the right hand side of (2.7) we need khψ, α^jψik

X^{s+ 18,−}

1 2++

τ=0

.kψk²

X^s,

1 2+

|τ|=|ξ|

. (3.4)

Proof of (3.1): We even prove the estimate withX^s,−

1 2++

|τ|=|ξ| replaced byX_|τ|=|ξ|^s,0 on the left hand side. It reduces to

Z

∗

ub1(τ1, ξ1) hξ₁i^s+18hτ₁i¹²⁺

ub2(τ2, ξ2)

hξ₂i^sh|τ₂| − |ξ₂|i¹²⁺hξ3i^sbu3(τ3, ξ3)dξdτ .

3

Y

i=1

kuik_L2 xt, where * denotes integration overξ= (ξ1, ξ2, ξ3),τ= (τ1, τ2, τ3) withξ1+ξ2+ξ3= 0 andτ1+τ2+τ3= 0. We assume here and in the following without loss of generality that the Fourier transforms are non-negative.

Case 1: |ξ1| ≥ |ξ2| ⇒ hξ3i^s.hξ1i^s. It suffices to show Z

∗

bu1(τ1, ξ1) hτ1i¹²⁺

bu2(τ2, ξ2)

hξ2i^s+18h|τ2| − |ξ2|i¹²⁺bu3(τ3, ξ3)dξdτ .

3

Y

i=1

kuik_L2 xt. This follows under the assumptions >3/8 from the estimate

Z

v1v2v3dxdt

.kv1k_L2

xL^∞_t kv2k_L∞

xL²_tkv3k_L2 xL²_t

.kv1k

X^0,

1 2+ τ=0

kv2k

X

1 2+,1

2+

|τ|=|ξ|

kv3k_X0,0

|τ|=|ξ|,

(3.5)

where we used (1.12).

Case 2: |ξ2| ≥ |ξ1| ⇒ hξ3i^s . hξ2i^s. In this case the desired estimate follows from

Z

∗

m(ξ₁, ξ₂, ξ₃, τ₁, τ₂, τ₃)bu₁(ξ₁, τ₁)bu₂(ξ₂, τ₂)bu₃(ξ₃, τ₃)dξdτ .

3

Y

i=1

ku_ik_L2

xt, (3.6)

where

m= 1

h|τ₂| − |ξ₂|i¹²⁺hξ₁i¹²⁺hτ₁i¹²⁺.

The following argument is closely related to the proof of a similar estimate in [14].

By two applications of the averaging principle [13, Prop. 5.1], we may replace m by

m⁰= χ_{||τ2|−|ξ2||∼1}χ_|τ1|∼1 hξ1i¹²⁺ .

Let nowτ2 be restricted to the regionτ2=T +O(1) for some integerT. Then τ3

is restricted to τ₃ =−T+O(1), because τ₁+τ₂+τ₃ = 0, and ξ₂ is restricted to

|ξ2| =|T|+O(1). The τ₃-regions are essentially disjoint forT ∈Z and similarly theτ₂-regions. Thus by Schur’s test [13, Lemma 3.11], we only have to show

sup

T∈Z

Z

∗

χ_{τ3=−T+O(1)}χ_τ2=T+O(1)χ_|τ1|∼1χ_{|ξ2|=|T|+O(1)} hξ1i¹²⁺

×bu1(ξ1, τ1)bu2(ξ2, τ2)bu3(ξ3, τ3)dξdτ .

3

Y

i=1

kuik_L2 xt. Theτ-behaviour of the integral is now trivial, thus we reduce to

sup

T∈N

Z

P3 i=1ξi=0

χ_|ξ2|=T+O(1)

hξ₁i¹²⁺ fb1(ξ1)fb2(ξ2)fb3(ξ3)dξ.

3

Y

i=1

kfik_L2

x. (3.7) An elementary calculation shows that

L.H.S. of(3.7).sup

T∈N

kχ_|ξ|=T+O(1)∗ hξi⁻¹⁻k^1/2_L∞(R²) 3

Y

i=1

kfikL²_x .

3

Y

i=1

kfikL²_x,

so that the desired estimate follows.

Proof of (3.4): This reduces to Z

∗

ub₁(τ₁, ξ₁) hξ1i^sh|τ1| − |ξ1|i¹²⁺

bu₂(τ₂, ξ₂) hξ2i^sh|τ2| − |ξ2|i¹²⁺

hξ₃i^s+18bu₃(τ₃, ξ₃) hτ3i¹²⁻ dξdτ .

3

Y

i=1

ku_ik_L2 xt. Assuming without loss of generality|ξ₁| ≤ |ξ₂|we have to show

Z

∗

bu1(τ1, ξ1) hξ1i^sh|τ1| − |ξ1|i¹²⁺

bu2(τ2, ξ2) h|τ2| − |ξ2|i¹²⁺

hξ3i^1/8bu3(τ3, ξ3) hτ3i¹²⁻ dξdτ .

3

Y

i=1

kuik_L2 xt. Case 1: |τ2| |ξ2|. We reduce to

Z

∗

ub1(τ1, ξ1) hξ1i^sh|τ1| − |ξ1|i¹²⁺

ub2(τ2, ξ2) hξ2i³⁸⁺

ub3(τ3, ξ3) hτ3i¹²⁻ dξdτ .

3

Y

i=1

ku_ik_L2 xt. This follows from

Z

v₁v₂v₃dxdt

.kv1k_L6

xL²⁺_t kv2k_L3

xL²_tkv3k_L2 xL^∞−_t

.kv1k

X

1 6+,1

2+

|τ|=|ξ|

kv2k

X

1 3,0

|τ|=|ξ|

kv3k

X^0,

1 2− τ=0

,

where we used (1.14) for the first factor and Sobolev for the others. Obviously here is some headroom left.

Case 2: |τ2|&|ξ2|. In this case we useτ1+τ2+τ3= 0 to estimate 1.hτ2i¹²⁻

hξ2i¹²⁻ .hτ1i¹²⁻

hξ2i¹²⁻ +hτ3i¹²⁻ hξ2i¹²⁻ .

2.1: If the second term on the right hand side is dominant we have to show, using alsohξ3i^1/8.hξ2i^1/8:

Z

∗

bu1(τ1, ξ1) hξ1i^sh|τ1| − |ξ1|i¹²⁺

ub2(τ2, ξ2)

hξ2i³⁸⁻h|τ2| − |ξ2|i¹²⁺bu3(τ3, ξ3)dξdτ .

3

Y

i=1

kuik_L2 xt, which follows fors >³₈ by Prop. 1.5.

2.2: If the first term on the right hand side is dominant we consider two subcases.

2.2.1: |τ1|.|ξ1|. We reduce to Z

∗

ub1(τ1, ξ1)hξ1i^12−s−

h|τ1| − |ξ1|i¹²⁺

ub2(τ2, ξ2) hξ2i³⁸⁻h|τ2| − |ξ2|i¹²⁺

ub3(τ3, ξ3) hτ3i¹²⁻ dξdτ .

3

Y

i=1

kuik_L2 xt. Using|ξ₂| ≥ |ξ₁|ands > ³₈ it suffices to show

Z

∗

bu1(τ1, ξ1) hξ1i¹⁸⁺h|τ1| − |ξ1|i¹²⁺

ub2(τ2, ξ2) hξ2i¹⁸⁺h|τ2| − |ξ2|i¹²⁺

bu3(τ3, ξ3) hτ3i¹²⁻ dξdτ .

3

Y

i=1

kuik_L2 xt. This follows from

Z

v₁v₂v₃dxdt

.kv1k_L4

xL²⁺_t kv2k_L4

xL²⁺_t kv3k_L2 xL^∞−_t

.kv1k

X

1 8+,3

8+

|τ|=|ξ|

kv2k

X

1 8+,3

8+

|τ|=|ξ|

kv3k

X^0,

1 2− τ=0

, where we used (1.15).

2.2.2: |τ1| |ξ1| ⇒ h|τ1| − |ξ1|i¹²⁻∼ hτ1i¹²⁻. We have to show Z

∗

bu1(τ1, ξ1) hξ1i^s

ub2(τ2, ξ2) hξ2i³⁸⁻h|τ2| − |ξ2|i¹²⁺

ub3(τ3, ξ3) hτ3i¹²⁻ dξdτ .

3

Y

i=1

kuik_L²

xt. This follows from

Z

v₁v₂v₃dxdt

.kv₁k_L3

xL²_tkv₂k_L6

xL²⁺_t kv₃k_L2 xL^∞−_t

.kv₁k

X

1 3,0

|τ|=|ξ|

kv₂k

X

1 6+,1

2+

|τ|=|ξ|

kv₃k

X^0,

1 2− τ=0

,

where we used Sobolev for the first and last factor and (1.14) for the second one.

This completes the proof of (3.4).

Proof of (3.3): We distinguish between low and high frequencies ofA^df_j . For high frequencies, i.e. , supp(FA^df_j ) ⊂ {|ξ| ≥ 1}, we obtain by (2.6) and Prop. 1.5 for s >3/8:

k|∇|A^df_j k

X^{s+ 38−,}

1 2+

|τ|=|ξ|

.khψ, ψik

X^s−

5 8,1

2+

|τ|=|ξ|

.kψk²

X^s,

1 2++

|τ|=|ξ|

. In the low frequency case|ξ3| ≤1, wherehξ1i ∼ hξ2i, it suffices to show

Z

∗

bu₁(τ₁, ξ₁) hξ1i^sh|τ1| − |ξ1|i¹²⁺⁺

ub₂(τ₂, ξ₂) hξ2i^sh|τ2| − |ξ2|i¹²⁺⁺

×bu₃(τ₃, ξ₃)h|τ3| − |ξ3|i¹²⁺hξ3i^s+38−χ_{|ξ3|≤1}

|ξ3|¹⁻ dξdτ .

3

Y

i=1

kuik_L²

xt.

Assuming without loss of generalityhτ2i ≤ hτ1i, we obtainh|τ3|−|ξ3|i¹²⁺∼ hτ3i¹²⁺. hτ1i¹²⁺+hτ2i¹²⁺.hτ1i¹²⁺.

If|τ1| |ξ1|or |τ1| |ξ1|, it suffices to show Z

∗

ub1(τ1, ξ1) hξ₁i^s

ub2(τ2, ξ2) hξ2i^sh|τ2| − |ξ2|i¹²⁺⁺

bu3(τ3, ξ3)χ_{|ξ3|≤1}

|ξ₃|¹⁻ dξdτ .

3

Y

i=1

kuik_L2 xt. This follows from

Z

v1v2v3dxdt

.kv1k_L2

xtkv2kL^∞_t L²_xkv3k_L2 tL^∞_x , which gives the desired result using ˙H_x¹⁻,→L^∞_x for low frequencies.

If|τ₁| ∼ |ξ₁|, we usehξ₁i ∼ hξ₂iand reduce to Z

∗ub1(τ1, ξ1) bu2(τ2, ξ2)

hξ₂i^2s−12−hh|τ₂| − |ξ₂|i¹²⁺⁺

bu3(τ3, ξ3)χ_{|ξ3|≤1}

|ξ3|¹⁻ .

3

Y

i=1

kuik_L2 xt, which can be shown as before. We remark that we only used s >1/4 in the low frequency case.

Proof of (3.2): We even prove the estimate withX^s,−

1 2++

|τ|=|ξ| replaced byX_|τ|=|ξ|^s,0 on the left hand side. For high frequencies ofA^df_j we have to show

kA^df_j α^jψk_Xs,0

|τ|=|ξ| .kA^dfk

X^{s+ 38,}

1 2+

|τ|=|ξ|

kψk

X^s,

1 2+

|τ|=|ξ|

,

which follows by Proposition 1.5. For the low frequency case ofA^df_j it suffices to show

Z

∗

ub1(τ1, ξ1)

hξ1i^sh|τ1| − |ξ1|i¹²⁺bu2(τ2, ξ2)hξ2i^s ub3(τ3, ξ3)χ_{|ξ3|≤1}

|ξ3|hkτ3| − |ξ3|i¹²⁺hξ3i^s+38−dξdτ .

3

Y

i=1

kuik_L2 xt.

Usinghξ1i ∼ hξ2iandhξ3i ∼1 and ˙H_x ,→L^∞_x for low frequencies this easily follows from the estimate

Z

v1v2v3dxdt

.kv1k_L^∞

t L²_xkv2k_L2

tL²_xkv3k_L2 tL^∞_x .

This completes the proof of (3.1)-(3.4). The property ψ_± ∈ X_±^s,1[0, T] follows

immediately from the proof of (3.1) and (3.2).

4. Proof of Theorem 1.2

Proof. Assumes >19/40 , says= ¹⁹₄₀+δ with 1δ >0. Letψ∈C⁰([0, T], H^s), A_j ∈C⁰([0, T], H^s+18).

Claim 1: ψ_±∈X

1 4+α,¹₂+

± [0, T], whereα= ₄₀¹ +³₂δ−. By Sobolev’s multiplication law we obtain

kAjα^jψ_±k

L²([0,T],H^2s−78).kAk

C⁰([0,T],H^{s+ 1}8)kψk_C0([0,T],H^s)T^1/2<∞.