Local well-posedness of the Maxwell–Chern–Simons–Higgs system in the temporal gauge

Nonlinear Analysis 99 (2014) 128–135

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Local well-posedness of the Maxwell–Chern–Simons–Higgs system in the temporal gauge Jianjun Yuan Department of Basic Science, Yancheng Institute of Technology, Yancheng 224051, China

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Article history: Received 2 September 2013 Accepted 27 December 2013 Communicated by Enzo Mitidieri

In this paper, we investigate the local well-posedness of the Maxwell–Chern–Simons–Higgs system in the temporal gauge. By using the X s,b type spaces, we prove that the system is locally well-posed in H s × H s−1 (s ≥ 1). © 2014 Elsevier Ltd. All rights reserved.

MSC: 35Q55 35A15 35B30 Keywords: Maxwell–Chern–Simons–Higgs Maxwell–Higgs Temporal gauge

1. Introduction The Lagrangian density of the (2 + 1)-dimensional Maxwell–Chern–Simons–Higgs theory is given by 1

κ

4

4

L = − F µν Fµν +

1

1

2

2

ϵ µνρ Fµν Aρ + Dµ φ Dµ φ + ∂µ N ∂ µ N − (e|φ|2 + κ N − ev 2 )2 − e2 N 2 |φ|2 ,

(1.1)

where Aµ ∈ R is the gauge field, φ is a complex scalar field, N is a real scalar field, Fµν = ∂µ Aν − ∂ν Aµ is the curvature, Dµ = ∂µ − ieAµ is the covariant derivative, e is the charge of the electron, κ > 0 is the Chern–Simons constant, v is a nonzero constant, ϵ µνρ is the totally skew-symmetric tensor with ϵ 012 = 1, indices are raised and lowered with respect to the Minkowski metric gµν = diag(1, −1, −1). We use the convention that the Greek indices such as µ, ν run through {0, 1, 2}, the Latin indices such as j, k run through {1, 2}, and repeated indices are summed. This model was proposed in [1] to investigate the self-dual system when there are both Maxwell and Chern–Simons terms. The corresponding Euler–Lagrange equations are

∂λ F λρ +

κ 2

ϵ µνρ Fµν + 2eIm(φ Dρ φ) = 0,

Dµ Dµ φ + Uφ (|φ|2 , N ) = 0, µ

∂µ ∂ N + UN = 0,

E-mail address: [email protected]. 0362-546X/$ – see front matter © 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.12.018

(1.2)

J. Yuan / Nonlinear Analysis 99 (2014) 128–135

where U (|φ|2 , N ) =

1 2

129

(e|φ|2 + κ N − ev 2 )2 + e2 N 2 |φ|2 , and Uφ , UN are formal derivative of U (|φ|2 , N ) with respect to φ, N:

Uφ (|φ|2 , N ) = (e|φ|2 + κ N − ev 2 )φ + e2 N 2 φ, UN (|φ|2 , N ) = κ(e|φ|2 + κ N − ev 2 ) + 2e2 N |φ|2 . Setting ρ = 0 in the first equation of (1.2), we obtain the Gauss-Law constraint

∂j Fj0 − κ F12 − 2eIm(φ D0 φ) = 0.

(1.3)

The energy of (1.2) is conserved, E (t ) =

   2 1 R2

+

2 i=1 2 

F0i2 (x, t ) +

1 2

2 F12 (x, t ) +

2 

|Dµ φ(x, t )|2

µ=0

 |∂µ N (x, t )| + U (|φ| , N )(x, t ) dx = E (0), 2

2

t ≥ 0.

(1.4)

µ=0

There are two possible boundary conditions to make the energy finite: either (φ, N , A) → (0, evκ , 0) as |x| → ∞ or (|φ|2 , N , A) → (v 2 , 0, 0) as |x| → ∞. The former is called the nontopological boundary condition, and the latter is called the topological boundary condition. Eqs. (1.2), (1.3) are invariant under the gauge transformations 2

Aµ → A′µ = Aµ + ∂µ χ , φ → φ ′ = eieχ φ, Dµ → D′µ = ∂µ − ieA′µ . (1.5) Hence one may impose an additional gauge condition on A. Usually there are three gauge conditions to choose, Coulomb gauge ∂i Ai = 0; temporal gauge A0 = 0; Lorentz gauge ∂ µ Aµ = 0. In this paper we will mainly focus on the temporal gauge. Now we review some results on the Cauchy problem of (1.2), (1.3) in the literature. In [2], the authors show that the system is globally well-posed in the Lorenz gauge in H 2 × H 1 , and this was extended to H 1 × L2 regularity in [3]. In [4], the authors show that the system is globally well-posed in the temporal gauge in H 2 × H 1 . There are also the existence results on the Cauchy problem for other gauge fields equations in the temporal gauge, e.g. [5] on the Maxwell–Klein–Gordon system, and [6–8] on the Yang–Mills system. The Maxwell–Chern–Simons–Higgs system (1.2) under the temporal gauge does not have the scaling invariance, and a natural function space to choose to investigate the well-posedness of the system is to make the energy (1.4) finite. In this paper, we prove that the system is locally well posed in the energy regularity and above. A noteworthy fact in two space dimension is that we have the estimate

∥u ⟨∇⟩ v∥H 0,b−1+ϵ . ∥u∥H 1,b |τ |=|ξ |

|τ |=|ξ |

∥v∥H 1,b

(1.6)

|τ |=|ξ |

for b = 12 + and sufficiently small positive ϵ , and by using this we prove that (1.2) and (1.3) under Lorenz gauge are locally well-posed in the energy space. While in the temporal gauge, we will decompose A into divergence-free part Adf and curlfree part Acf , and Acf will satisfy a different type equation, so we need to put it in a different function space, and except using the estimate (1.6), we will need more estimates, see (3.1)–(3.10). One may also consider the global existence part of (1.2), (1.3) under the temporal gauge. Note that the energy E controls most of the norms of (A, φ, N ), except one norm ∥∇ Acf ∥L2 , which I do not know how to control at the moment. Also one can consider the system (1.2), (1.3) in the Coulomb gauge, by separating the gauge potential into A0 and (A1 , A2 ), which results in to estimate some terms containing A0 , which I did not resolved yet. I expect to see some results about these in the near future. Some notations: H s (s ∈ R) are Sobolev spaces with respect to the norms ∥f ∥H s = ∥⟨ξ ⟩s fˆ ∥L2 , where fˆ (ξ ) = Ff (ξ ) is the 1

Fourier transform of f (x) and we use the shorthand ⟨ξ ⟩ = (1 + |ξ |2 ) 2 . We use the shorthand X . Y for X ≤ CY , where C ≫ 1 is a constant which may depend on the quantities which are considered fixed. X ∼ Y means X . Y . X . We use b+ to denote b + ϵ , for a sufficiently small positive ϵ , and  := ∂tt − ∆. In Section 2, we will make some preliminaries and reductions. In Section 3, we will state our local well-posedness result, and prove it. 2. Preliminaries and reductions Under the temporal gauge, we rewrite the Euler–Lagrange equations (1.2) in terms of (A, φ, N ) as follows

¯ = 0, ∂t (divA) + κ(∂1 A2 − ∂2 A1 ) + 2eIm(φ∂t φ) ¯ A1 + ∂1 (divA) + κ∂t A2 + 2eIm(φ∂1 φ) + 2e2 A1 |φ|2 = 0, ¯ + 2e2 A2 |φ|2 = 0, A2 + ∂2 (divA) − κ∂t A1 + 2eIm(φ∂2 φ) φ = −2ieAj ∂j φ − ie∂j Aj φ − e2 A2j φ − (e|φ|2 + κ N − ev 2 )φ − e2 N 2 φ, N = −κ(e|φ|2 + κ N − ev 2 ) − 2e2 N |φ|2 ,

(2.1)

130

J. Yuan / Nonlinear Analysis 99 (2014) 128–135

with given initial data Ai (0, x), φ(0, x), N (0, x), ∂0 Ai (0, x), ∂0 φ(0, x), ∂0 N (0, x) satisfying the constraint

¯ 0, x) = 0. ∂j ∂0 Aj (0, x) + κ F12 (0, x) + 2eIm(φ∂t φ)(

(2.2)

For the nontopological boundary condition, we introduce  N satisfying  N + ev /k = N. Then we have (φ,  N , A1 , A2 ) → 0 as |x| → ∞. In this case, UN in the system (2.4) changes to U respectively. For the topological case, we will discuss a subcase N of this case, we assume lim|x|→∞ φ = λ for a fixed complex scalar λ with |λ| = v , i.e., φ tends to be constant at the infinity, this assumption is very natural. We introduce ϕ satisfying ϕ + λ = φ . Then we also have (ϕ, N , A1 , A2 ) → 0 as |x| → ∞. For u = (A, φ,  N ) or (A, ϕ, N ), where A = (A1 , A2 ), we denote 2

∥u(t )∥H s = ∥A(t )∥H s + ∥φ(t )∥H s + ∥ N (t )∥H s , in the nontopological case, and

∥u(t )∥H s = ∥A(t )∥H s + ∥ϕ(t )∥H s + ∥N (t )∥H s , in the topological case. We also denote

∥u(t )∥H s ×H s−1 = ∥u(t )∥H s + ∥∂0 u(t )∥H s−1 , and u0 = u(0, ·). Note that the term ∂i (divA) in the equation for Ai in (2.1) makes the equation not a wave equation. To eliminate this, we df cf decompose A = (A1 , A2 ) into divergence free part Adf = (Adf = (Acf1 , Acf2 ), such that A = Adf + 1 , A2 ) and curl free part A cf df cf A , divA = 0, and curlA = 0. Remember that for u = (u1 (x, y), u2 (x, y)), divu(x, y) = u1,x (x, y)+ u2,y (x, y), curlu(x, y) = ∂A

∂A

∂A

∂A

u2,x (x, y) − u1,y (x, y). And by calculating, we have Adf = (−∂2 ∆−1 ( ∂ x2 − ∂ y1 ), ∂1 ∆−1 ( ∂ x2 − ∂ y1 )), Acf = ∆−1 (∇ divA).

¯ ∂t Acf = −∆−1 ∇[κ(∂1 A2 − ∂2 A1 ) + 2eIm(φ∂t φ)], df −1 −1 ¯ A1 = −κ ∆ ∂12 ∂t A1 − κ ∆ ∂22 ∂t A2 + 2e∆−1 ∂12 Im(φ∂2 φ) −1 2 −1 2 2 −1 ¯ − 2e∆ ∂22 Im(φ∂1 φ) + 2e ∆ ∂12 (A2 |φ| ) − 2e ∆ ∂22 (A1 |φ|2 ), −1 ¯ Adf ∂11 ∂t A1 + κ ∆−1 ∂21 ∂t A2 − 2e∆−1 ∂11 Im(φ∂2 φ) 2 = κ∆ + 2e∆−1 ∂12 Im(φ∂1 φ) − 2e2 ∆−1 ∂11 (A2 |φ|2 ) + 2e2 ∆−1 ∂12 (A1 |φ|2 ),

φ = −2ieAj ∂j φ − 2ie∂j Aj φ − e2 A2j φ − (e|φ|2 + κ N − ev 2 )φ − e2 N 2 φ, N = −κ(e|φ|2 + κ N − ev 2 ) − 2e2 N |φ|2 .

(2.3)

We can also assume Acf (0) = 0.

(2.4)

This can be established by using the transformation (1.5), and let χ = −∆−1 divA(0). We prove Theorem 3.1 in the topological case only, the nontopological case is similar. Let df df Adf ϕ = ϕ+ + ϕ− , N = N+ + N− , i = Ai,+ + Ai,− , 1 df 1 (Ai ± i−1 ⟨∇⟩−1 ∂t Adf Adf i = 1, 2, ϕ± = (ϕ ± i−1 ⟨∇⟩−1 ∂t ϕ), i,± = i ), 2 2 1 −1 −1 N± = (N ± i ⟨∇⟩ ∂t N ), 2 Then (2.1), (2.2) transform to df ∂t Acf = −∆−1 ∇(κ(∂1 Adf 2 − ∂2 A1 ) + 2eIm(−i(ϕ+ + ϕ− + λ)⟨∇⟩(ϕ+ − ϕ− ))),

−1 −1 ⟨∇⟩−1 (Adf (i∂t ± ⟨∇⟩)Adf ∂12 ⟨∇⟩(A1,+ − A1,− ) − κ ∆−1 ∂22 ⟨∇⟩(A2,+ − A2,− ) 1,± = ±2 1 − iκ ∆ −1 + 2e∆ ∂12 Im((ϕ + λ)∂2 ϕ) ¯ − 2e∆−1 ∂22 Im((ϕ + λ)∂1 ϕ) ¯ 2 −1 2 2 −1 + 2e ∆ ∂12 ((A2,+ + A2,− )|ϕ + λ| ) − 2e ∆ ∂22 ((A1,+ + A1,− )|ϕ + λ|2 )), −1 −1 ⟨∇⟩−1 (Adf (i∂t ± ⟨∇⟩)Adf ∂11 ⟨∇⟩(A1,+ − A1,− ) + κ ∆−1 ∂21 ⟨∇⟩(A2,+ − A2,− ) 2,± = ±2 2 + iκ ∆ − 2e∆−1 ∂11 Im((ϕ + λ)∂2 ϕ) ¯ + 2e∆−1 ∂12 Im((ϕ + λ)∂1 ϕ) ¯ 2 −1 − 2e ∆ ∂11 ((A2,+ + A2,− )|ϕ + λ|2 ) + 2e2 ∆−1 ∂12 ((A1,+ + A1,− )|ϕ + λ|2 )),

(i∂t ± ⟨∇⟩)ϕ± = ±2−1 ⟨∇⟩−1 (ϕ − 2ieAj ∂j ϕ − ie∂j Acfj φ − e2 A2j (ϕ + λ) − (e|ϕ|2 + eϕλ + eϕλ + κ N )(ϕ + λ) − e2 N 2 ϕ + λ), (i∂t ± ⟨∇⟩)N± = ±2−1 ⟨∇⟩−1 (N − κ(e|ϕ|2 + eϕλ + eϕλ + κ N ) − 2e2 N |ϕ + λ|2 ),

(2.5)

with initial data Ai (0, x), φ(0, x), N (0, x), ∂0 Ai (0, x), ∂0 φ(0, x), ∂0 N (0, x) satisfying df df df ∂j ∂t Acfj (0, x) + κ(∂1 (Adf 2,+ + A2,− ) − ∂2 (A1,+ + A1,− ))(0, x) − 2eIm(i(ϕ+ + ϕ− + λ)⟨∇⟩(ϕ+ − ϕ− ))(0, x) = 0. (2.6)

J. Yuan / Nonlinear Analysis 99 (2014) 128–135

131

Here we construct our solutions in the X s,b type spaces. We recall some definitions and some basic properties of X s,b and H spaces. s,b

s,b

Definition 2.1. For s, b ∈ R, let X|τ |=|ξ |,± be the completion of the Schwarz space S (R1+2 ) with respect to the norm

∥u∥X s,b

|τ |=|ξ |,±

= ∥ ⟨ξ ⟩s ⟨−τ ± |ξ |⟩b uˆ (τ , ξ )∥L2 , τ ,ξ

s,b

where uˆ (τ , ξ ) denotes the space–time Fourier transformation of u(t , x). Let H|τ |=|ξ | be the completion of the Schwarz space S (R1+2 ) with respect to the norm

∥u∥H s,b

|τ |=|ξ |

= ∥ ⟨ξ ⟩s ⟨|τ | − |ξ |⟩b uˆ (τ , ξ )∥L2 . τ ,ξ

Clearly, we have

∥u∥H s,b

≤ ∥u∥X s,b

,

for b ≥ 0,

(2.7)

∥u∥H s,b

≥ ∥u∥X s,b

,

for b ≤ 0.

(2.8)

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |,±

|τ |=|ξ |,±

s ,b

For ST = (0, T ) × R2 , the restriction space X|τ |=|ξ |,± (ST ) is a Banach space with respect to the norm

∥u∥X s,b

|τ |=|ξ |,± (ST )

= inf{∥ vX s,b

|τ |=|ξ |,±

: v ∈ X|τs,b|=|ξ |,± and v = u on ST }.

(2.9)

s ,b

The restriction space H|τ |=|ξ | (ST ) is defined analogously. Now we consider the following linear Cauchy problem

(−i∂t ± ⟨∇⟩)u = F ,

u|t =0 = u0 .

(2.10) s,b−1+δ

Lemma 2.2. Let 1/2 < b ≤ 1, s ∈ R, 0 < T ≤ 1, Also, let 0 ≤ δ ≤ 1 − b, Then for F ∈ X|τ |=|ξ |,± (ST ), u0 ∈ H s , the Cauchy s,b

problem has a unique solution u ∈ X|τ |=|ξ |,± (ST ), satisfying the first equation in the sense of D (ST ). Moreover,

∥u∥X s,b

|τ |=|ξ |,± (ST )

′

≤ C (∥u0 ∥H s + T δ ∥F ∥X s,b−1+δ (S ) ), T

|τ |=|ξ |,±

where C only depends on b. s ,b

Definition 2.3. For s, b ∈ R, let Xτ =0 be the completion of the Schwarz space S (R1+2 ) with respect to the norm

∥u∥X s,b = ∥ ⟨ξ ⟩s ⟨τ ⟩b uˆ (τ , ξ )∥L2 , τ ,ξ

τ =0

where uˆ (τ , ξ ) denotes the space–time Fourier transformation of u(t , x). s,b s,b For ST = (0, T ) × R2 , the restriction space ∥u∥X s,b (S ) is defined similar to X|τ |=|ξ |,± (ST ) and H|τ |=|ξ | (ST ). τ =0

T

Now we consider the following linear Cauchy problem

∂t u = F ,

u|t =0 = u0 .

(2.11) s,b−1+δ

Lemma 2.4. Let 1/2 < b ≤ 1, s ∈ R, 0 < T ≤ 1, Also, let 0 ≤ δ ≤ 1 − b, Then for F ∈ Xτ =0 (ST ), u0 ∈ H s , the Cauchy s,b problem has a unique solution u ∈ Xτ =0 (ST ), satisfying the first equation in the sense of D′ (ST ). Moreover,

∥u∥X s,b

τ =0 (ST )

≤ C (∥u0 ∥H s + T δ ∥F ∥X s,b−1+δ (S ) ), τ =0

T

where C only depends on b. We also recall the product law in [9]. Theorem 2.5. Let s0 , s1 , s2 , b0 , b1 , b2 ∈ R. The product estimate in 1 + 2 dimension

∥uv∥H −s0 ,−b0 ≤ C ∥u∥H s1 ,b1 ∥v∥H s2 ,b2 |τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

holds for all u, v ∈ S (R1+2 ) if the following conditions are satisfied: b0 + b1 + b2 > b0 + b1 ≥ 0 , b0 + b2 ≥ 0 , b1 + b2 ≥ 0,

1 2

,

(2.12)

132

J. Yuan / Nonlinear Analysis 99 (2014) 128–135

s0 + s1 + s2 >

3 2

− (b0 + b1 + b2 ),

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

1 2 3 4

− min(b0 , b1 , b2 ), ,

(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 ). This product law is optimal up to endpoint cases. A more precise statement, including many endpoint cases, can be found in [9]. We say that the pair (q, r ) is wave admissible if 2 ≤ q ≤ ∞,

2 ≤ r < ∞,

2 q

≤

1 2

1

− . r

Theorem 2.6. The embedding 1− 2r − 1q ,θ

(ST ) ↩→ Lqt Lrx (ST )

H|τ |=|ξ |

holds whenever (q, r ) is wave admissible and θ >

1 . 2

3. Main results Theorem 3.1. The Maxwell–Chern–Simons–Higgs Cauchy problem (2.1), (2.2) is locally well-posed in H s × H s−1 , s ≥ 1, either in nontopological or in topological boundary conditions (in the topological case, we assume lim|x|→∞ φ = λ for a fixed complex scalar λ with |λ| = v ). To be precise, there exists a time T depending on the initial data norm ∥u0 ∥H s ×H s−1 , and a solution u of (2.1), (2.2) on [−T , T ] × R2 with the regularity u ∈ C ([−T , T ]; H s ),

∂t u ∈ C ([−T , T ]; H s−1 ).

The solution is unique in a certain subset of this regularity class. Moreover, the solution depends continuously on the data, and higher regularity persists. In particular, if the data are smooth, then so is the solution. We will only consider the topological boundary condition, the nontopological condition case is similar. As discussed in Section 2, the proof of Theorem 3.1 is reduced to prove the corresponding local existence for (2.5), (2.6). Also as in [8], we will exploit more regularity for Acf . Theorem 3.2. Given any initial data condition df s (ϕ+ , ϕ− , Adf i,+ , Ai,− , N+ , N− )|t =0 ∈ H ,

s ≥ 1,

df df df cf satisfying the constraint div(Adf i,+ + Ai,− )|t =0 = 0, div⟨∇⟩(Ai,+ − Ai,− )|t =0 = 0, A |t =0 = 0 and (2.6), there exists T depending on the initial data norm df s N0 = ∥(ϕ+ , ϕ− , Adf i,+ , Ai,− , N+ , N− )|t =0 ∥H , cf and a solution (ϕ+ , ϕ− , Adf i,± , Ai , N+ , N− ) of (2.5), (2.6). The solution has the regularity s,b s ϕ± , N± , Adf i,± ∈ X|τ |=|ξ | ,± (ST ) ⊂ C ([−T , T ], H ), s+α Acf ), i ∈ Xτ =0 (ST ) ⊂ C ([−T , T ], H s+α,b

for some b > class.

1 , 2

s ≥ 1,

and some sufficiently small α > 0 and δ > 0 such that α ≤ 1 − b − δ . The solution is unique in this regularity

J. Yuan / Nonlinear Analysis 99 (2014) 128–135

133

Proof. By (2.7) and (2.8), Theorem 3.2 is reduced to the following estimates:

∥ − ∆−1 ∇∂i Adf j ∥X s+α,b−1+δ . ∥Aj ∥H s,b τ =0

|τ |=|ξ |

∥φ⟨∇⟩ψ∥X s+α−1,b−1+δ . ∥φ∥H s,b τ =0

|τ |=|ξ |

,

i ̸= j,

(3.1)

,

(3.2)

∥ψ∥H s,b

|τ |=|ξ |

∥Ai ∥H s,b−1+δ . min{∥Ai ∥H s,b

, ∥Ai ∥X s+α,b },

∥φ∂i ψ∥H s−1,b−1+δ . ∥φ∥H s,b

∥ψ∥H s,b

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

df ∥Adf i φψ∥H s−1,b−1+δ . ∥Ai ∥H s,b

|τ |=|ξ |

|τ |=|ξ |

(3.3)

τ =0

,

(3.4)

∥φ∥H s,b

|τ |=|ξ |

∥ψ∥H s,b

|τ |=|ξ |

,

(3.5)

∥Acfi φψ∥H s−1,b−1+δ . ∥Acfi ∥X s+α,b ∥φ∥H s,b

∥ψ∥H s,b

∥Acfi ∂j φ∥H s−1,b−1+δ . ∥Acfi ∥X s+α,b ∥φ∥H s,b

,

(3.7)

∥∂j Acfj φ∥H s−1,b−1+δ . ∥Acfj ∥X s+α,b ∥φ∥H s,b

,

(3.8)

τ =0

|τ |=|ξ |

|τ |=|ξ |

τ =0

|τ |=|ξ |

|τ |=|ξ |

τ =0

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

∥Acfj Acfj φ∥H s−1,b−1+δ . ∥Acfj ∥X s+α,b ∥Acfj ∥X s+α,b ∥φ∥H s,b τ =0

|τ |=|ξ |

∥φψ∥H s−1,b−1+δ . ∥φ∥H s,b

|τ |=|ξ |

|τ |=|ξ |

∥φ∥H s−1,b−1+δ . ∥φ∥H s,b |τ |=|ξ |

|τ |=|ξ |

τ =0

∥ψ∥H s,b

|τ |=|ξ |

,

|τ |=|ξ |

(3.6)

,

(3.9)

,

(3.10)

.

(3.11)

For (3.1), we need to show ⟨ξ ⟩α ⟨τ ⟩b−1+δ . ⟨|τ | − |ξ |⟩b , which is equivalent to ⟨ξ ⟩α . ⟨τ ⟩1−b−δ ⟨|τ | − |ξ |⟩b , and this can be seen from |τ | ≤ |ξ | + ∥τ | − |ξ ∥ and α ≤ 1 − b − δ . For (3.2), it suffices to show

∥φψ∥X s+α−1,b−1+δ . ∥φ∥H s,b τ =0

|τ |=|ξ |

∥ψ∥H s−1,b .

(3.12)

|τ |=|ξ |

For this it suffices to show

∥φψ∥X α,b−1+δ . ∥φ∥H 1,b

∥ψ∥H s−1,b ,

(3.13)

∥φψ∥X α,b−1+δ . ∥φ∥H s,b

∥ψ∥H 0,b

.

(3.14)

τ =0

|τ |=|ξ |

τ =0

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

By using ⟨ξ ⟩α ⟨τ ⟩b−1+δ . ⟨|τ | − |ξ |⟩α , we see ∥φψ∥X α,b−1+δ . ∥φψ∥H 0,α τ =0

, and it suffices to prove

|τ |=|ξ |

∥φψ∥H 0,α

. ∥φ∥H 1,b

∥ψ∥H s−1,b ,

(3.15)

∥φψ∥H 0,α

. ∥φ∥H s,b

∥ψ∥H 0,b

.

(3.16)

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

(3.15) and (3.16) can be seen from Theorem 2.5. (3.3) and (3.11) are trivial. For (3.4), it suffices to show

∥φψ∥H s−1,b−1+δ . ∥φ∥H s,b

|τ |=|ξ |

|τ |=|ξ |

∥ψ∥H s−1,b ,

(3.17)

|τ |=|ξ |

and which can be seen from

∥φψ∥H 0,b−1+δ . ∥φ∥H 1,b

∥ψ∥H s−1,b ,

(3.18)

∥φψ∥H 0,b−1+δ . ∥φ∥H s,b

∥ψ∥H 0,b

.

(3.19)

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

(3.18) and (3.19) can be seen from Theorem 2.5. (3.10) can be seen from (3.17). For (3.5), we use the inequality df ∥Adf i φψ∥H s−1,b−1+δ . ∥Ai ∥H s,b

|τ |=|ξ |

|τ |=|ξ |

. ∥

Adf s,b i H|τ |=|ξ |

∥

∥φψ∥H s,0

|τ |=|ξ |

∥φ∥H s,b

which can be seen by using Theorem 2.5.

|τ |=|ξ |

∥ψ∥H s,b

|τ |=|ξ |

,

(3.20)

134

J. Yuan / Nonlinear Analysis 99 (2014) 128–135

For (3.6), it suffices to show

∥Acfi φψ∥H 0,b−1+δ . ∥Acfi ∥X 1+α,b ∥φ∥H s,b

∥ψ∥H s,b

,

(3.21)

∥Acfi φψ∥H 0,b−1+δ . ∥Acfi ∥X s+α,b ∥φ∥H 1,b

∥ψ∥H s,b

,

(3.22)

∥Acfi φψ∥H 0,b−1+δ . ∥Acfi ∥X s+α,b ∥φ∥H s,b

∥ψ∥H 1,b

.

(3.23)

∥ψ∥H 1,b

.

(3.24)

τ =0

|τ |=|ξ |

|τ |=|ξ |

τ =0

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

τ =0

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

And it suffices to show the s = 1 case,

∥Acfi φψ∥H 0,b−1+δ . ∥Acfi ∥X 1+α,b ∥φ∥H 1,b τ =0

|τ |=|ξ |

|τ |=|ξ |

|τ |=|ξ |

By duality, it can be seen from

     fghw dxdt  . ∥f ∥ ∞− 2+ ∥g ∥ 4+ ∞− ∥h∥ 4+ ∞− ∥w∥ 2 2 Lt Lx Lt Lx Lt Lx Lx Lt   . ∥f ∥X 1+α,b ∥g ∥H 1,b τ =0

∥h∥H 1,b

|τ |=|ξ |

|τ |=|ξ |

∥w∥H 0,1−b−δ .

(3.25)

|τ |=|ξ |

For the second inequality in the above, we use Theorem 2.6 and Sobolev embedding. For (3.7), it suffices to show

∥Acfi φ∥H s−1,b−1+δ . ∥Acfi ∥X s+α,b ∥φ∥H s−1,b τ =0

|τ |=|ξ |

(3.26)

|τ |=|ξ |

and it reduces to

∥Acfi φ∥H 0,b−1+δ . ∥Acfi ∥X 1+α,b ∥φ∥H s−1,b ,

(3.27)

∥Acfi φ∥H 0,b−1+δ . ∥Acfi ∥X s+α,b ∥φ∥H 0,b

.

(3.28)

.

(3.29)

τ =0

|τ |=|ξ |

|τ |=|ξ |

τ =0

|τ |=|ξ |

|τ |=|ξ |

And it suffices to prove the s = 1 case,

∥Acfi φ∥H 0,b−1+δ . ∥Acfi ∥X 1+α,b ∥φ∥H 0,b τ =0

|τ |=|ξ |

|τ |=|ξ |

This can be seen by duality, since

     fghdxdt  . ∥f ∥ 2 ∞ ∥g ∥ ∞ 2 ∥h∥ 2 2 Lt Lx Lt Lx Lt Lx   . ∥f ∥X 1+α,b ∥g ∥H 0,b τ =0

|τ |=|ξ |

∥h∥H 0,1−b−δ .

(3.30)

|τ |=|ξ |

For the second inequality in the above, we use Theorem 2.6 and Sobolev embedding. For (3.8), it suffices to show

∥Acfj φ∥H s−1,b−1+δ . ∥Acfj ∥X s+α−1,b ∥φ∥H s,b τ =0

|τ |=|ξ |

|τ |=|ξ |

,

(3.31)

and this reduces to show

∥Acfj φ∥H 0,b−1+δ . ∥Acfj ∥X α,b ∥φ∥H s,b τ =0

|τ |=|ξ |

|τ |=|ξ |

,

∥Acfj φ∥H 0,b−1+δ . ∥Acfj ∥X s+α−1,b ∥φ∥H 1,b τ =0

|τ |=|ξ |

|τ |=|ξ |

(3.32)

.

(3.33)

And it suffices to prove the s = 1 case,

∥Acfj φ∥H 0,b−1+δ . ∥Acfj ∥X α,b ∥φ∥H 1,b τ =0

|τ |=|ξ |

|τ |=|ξ |

.

(3.34)

This can be seen from

∥Acfj φ∥H 0,b−1+δ . ∥Acfj φ∥L2 L2x . ∥Acfj ∥L4− L2x + ∥φ∥L4+ L∞− x t

|τ |=|ξ |

. ∥

Acf j X α,b τ =0

∥

t

∥φ∥H 1,b

|τ |=|ξ |

t

.

For the third inequality in the above, we use Theorem 2.6 and Sobolev embedding.

(3.35)

J. Yuan / Nonlinear Analysis 99 (2014) 128–135

135

For (3.9), it suffices to show

∥Acfj Acfj φ∥H 0,b−1+δ . ∥Acfj ∥X 1+α,b ∥Acfj ∥X s+α,b ∥φ∥H s,b

,

(3.36)

∥Acfj Acfj φ∥H 0,b−1+δ . ∥Acfj ∥X s+α,b ∥Acfj ∥X s+α,b ∥φ∥H 1,b

.

(3.37)

.

(3.38)

τ =0

τ =0

|τ |=|ξ |

τ =0

|τ |=|ξ |

τ =0

|τ |=|ξ |

|τ |=|ξ |

And it suffices to prove the s = 1 case,

∥Acfj Acfj φ∥H 0,b−1+δ . ∥Acfj ∥X 1+α,b ∥Acfj ∥X 1+α,b ∥φ∥H 1,b τ =0

|τ |=|ξ |

τ =0

|τ |=|ξ |

This can be seen from cf ∞ ∥A ∥L∞ L∞ ∥φ∥ 2 2 ∥Acfj Acfj φ∥H 0,b−1+δ . ∥Acfj Acfj φ∥L2 L2x . ∥Acfj ∥L∞ j L Lx t x t Lx t

t

|τ |=|ξ |

. ∥

Acf j X 1+α,b τ =0

∥

∥

Acf j X 1+α,b τ =0

∥

∥φ∥H 1,b

|τ |=|ξ |

.

For the third inequality in the above, we use Theorem 2.6 and Sobolev embedding.

(3.39) 

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