Electronic Journal of Differential Equations, Vol. 2009(2009), No. 114, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
LOW REGULARITY SOLUTIONS OF THE
CHERN-SIMONS-HIGGS EQUATIONS IN THE LORENTZ
GAUGE
NIKOLAOS BOURNAVEAS
Abstract. We prove local well-posedness for the 2 + 1-dimensional ChernSimons-Higgs equations in the Lorentz gauge with initial data of low regularity.
Our result improves earlier results by Huh [10, 11].
1. Introduction
The Chern-Simon-Higgs model was proposed by Jackiw and Weinberg [12] and
Hong, Pac and Kim [9] in the context of their studies of vortex solutions in the
abelian Chern-Simons theory.
Local well-posedness of low regularity solutions was recently studied in Huh
[10, 11] using a null-form estimate for solutions of the linear wave equation due to
Foschi and Klainerman [8] as well as Strichartz estimates. Our aim in this paper
is to improve the results of [10, 11] in the Lorentz gauge. For this purpose we
use estimates in the restriction spaces X s,b introduced by Bourgain, Klainerman
and Machedon. A key ingredient in our proof is a modified version of a null-form
estimate of Zhou [19] and product rules in X s,b spaces due to D’Ancona, Foschi
and Selberg [6, 7] and Klainerman and Selberg [13]. The Higgs field has fractional
dimension (see below for details), a common feature of systems involving the Dirac
equation, see for example Bournaveas [1, 2], D’Ancona, Foschi and Selberg [6, 7],
Machihara [14, 15], Machihara, Nakamura, Nakanishi and Ozawa [16], Selberg and
Tesfahun [17], Tesfahun [18].
The Chern-Simon-Higgs equations are the Euler-Lagrange equations corresponding to the Lagrangian density
κ
L = µνρ Aµ Fνρ + Dµ φ Dµ φ − V |φ|2 .
4
Here Aµ is the gauge field, Fµν = ∂µ Aν − ∂ν Aµ is the curvature, Dµ = ∂µ − iAµ is
the covariant derivative, φ is the Higgs field, V is a given positive function and κ is
a positive coupling constant. Greek indices run through {0, 1, 2}, Latin indices run
through {1, 2} and repeated indices are summed. The Minkowski metric is defined
2000 Mathematics Subject Classification. 35L15, 35L70, 35Q40.
Key words and phrases. Chern-Simons-Higgs equations; Lorentz gauge; null-form estimates;
low regularity solutions.
c
2009
Texas State University - San Marcos.
Submitted February 22, 2009. Published September 12, 2009.
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N. BOURNAVEAS
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by (g µν ) = diag(1, −1, −1). We define µνρ = 0 if two of the indices coincide and
µνρ = ±1 according to whether (µ, ν, ρ) is an even or odd permutation of (0, 1, 2).
We define Klainerman’s null forms by
Qµν (u, v) = ∂µ u∂ν v − ∂ν u∂µ v,
(1.1a)
µν
Q0 (u, v) = g ∂µ u∂ν v.
(1.1b)
Let I µ = 2Im φDµ φ . Then the Euler-Lagrange equations are (we set κ = 2 for
simplicity)
1
Fµν = µνα I α ,
(1.2a)
2
µ
0
2
Dµ D φ = −φV |φ| .
(1.2b)
The system has the positive conserved energy given by
Z X
2
E=
|Dµ φ|2 + V (|φ|2 ) dx.
R2 µ=0
We are interested in the so-called ‘non-topological’ case in which |φ| → 0 as |x| →
+∞. For the sake of simplicity we follow [10, 11] and set V = 0. It will be clear from
our proof that for various classes of V ’s the term φV 0 (|φ|2 ) can easily be handled.
Under the Lorentz gauge condition ∂ µ Aµ = 0 the Euler-Lagrange equations (1.2)
become
∂0 Aj = ∂j A0 + 12 ij Ii ,
(1.3a)
∂1 A2 = ∂2 A1 + 12 I0 ,
(1.3b)
∂0 A0 = ∂1 A1 + ∂2 A2 ,
(1.3c)
µ
Dµ D φ = 0.
(1.3d)
Alternatively, they can be written as a system of two nonlinear wave equations:
1
1
Aα = αβγ Im(Dγ φDβ φ − Dβ φDγ φ) + αβγ (∂β Aγ − ∂γ Aβ )|φ|2 ,
(1.4a)
2
2
φ = 2iAα ∂α φ + Aα Aα φ.
(1.4b)
We prescribe initial data in the classical Sobolev spaces Aµ (0, x) = aµ0 (x) ∈ H a ,
∂t Aµ (0, x) = aµ1 (x) ∈ H a−1 , φ(0, x) = φ0 (x) ∈ H b , ∂t φ(0, x) = φ1 (x) ∈ H b−1 .
Dimensional analysis shows that the critical values of a and b are acr = 0 and
bcr = 21 . It is well known that in low space dimensions the Cauchy problem may
not be locally well posed for a and b close to the critical values due to lack of decay
at infinity. Observe also that φ has fractional dimension.
From the point of view of scaling it is natural to take b = a + 12 . With this
choice it was shown in Huh [10] that the Cauchy problem is locally well posed for
a = 34 + and b = 54 + . This was improved in Huh [11] to
3
9
+ , b = + (1.5)
4
8
(slightly violating b = a + 21 ). The proof relies on the null structure of the right
hand side of (1.4a). Indeed,
Dγ φDβ φ − Dβ φDγ φ = Qγβ (φ, φ) + i Aγ ∂β (|φ|2 ) − Aβ ∂γ (|φ|2 ) .
a=
On the other hand, since in (1.3) the Aµ satisfy first order equations and φ satisfies
a second order equation it is natural to investigate the case b = a + 1. It turns out
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CHERN-SIMONS-HIGGS EQUATIONS
3
that this choice allows us to improve on a at the expense of b. It is shown in Huh
[11] that we have local well posedness for
1
3
a= , b= .
(1.6)
2
2
To prove this result Huh uncovered the null structure in the right hand side of
equation (1.4b). Indeed, if we introduce Bµ by ∂µ B µ = 0 and ∂µ Bν − ∂ν Bµ =
µνλ Aλ , then the equations take the form:
B γ = − Im φ̄Dγ φ = − Im φ̄∂ γ φ + iµνγ ∂µ Bν |φ|2 ,
(1.7a)
φ = iαµν Qµα (Bν , φ) + Q0 (Bµ , B µ )φ + Qµν (B µ , B ν )φ .
(1.7b)
In this article we shall prove the Theorem stated below which corresponds to exponents a = 41 + and b = 45 + . This improves (1.6) by 14 − derivatives in both
a and b. Compared to (1.5), it improves a by 12 derivatives at the expense of 81
derivatives in b.
Theorem 1.1. Let n = 2 and 41 < s < 12 . Consider the Cauchy problem for the
system (1.7) with initial data in the following Sobolev spaces:
B γ (0) = bγ0 ∈ H s+1 (R2 ),
φ(0) = φ0 ∈ H
s+1
2
(R ),
∂t B γ (0) = bγ1 ∈ H s (R2 ),
s
2
∂t φ(0) = φ1 ∈ H (R ).
(1.8a)
(1.8b)
Then there exists a T > 0 and a solution (B, φ) of (1.7)-(1.8) in [0, T ] × R2 with
B, φ ∈ C 0 ([0, T ]; H s+1 (R2 )) ∩ C 1 ([0, T ]; H s (R2 )).
The solution is unique in a subspace of C 0 ([0, T ]; H s+1 (R2 )) ∩ C 1 ([0, T ]; H s (R2 )),
namely in Hs+1,θ , where 43 < θ < s + 21 (the definition of Hs+1,θ is given in the
next section).
Finally, we remark that the problem of global existence is much more difficult.
We refer the reader to Chae and Chae [4], Chae and Choe [5] and Huh [10, 11].
2. Bilinear Estimates
In this Section we collect the bilinear estimates we need for the proof of Theorem
1.1. We shall work with the spaces H s,θ and Hs,θ defined by
H s,θ = {u ∈ S 0 : Λs Λθ− u ∈ L2 (R2+1 )},
Hs,θ = {u ∈ H s,θ : ∂t u ∈ H s−1,θ }
where Λ and Λ− are defined by
s u(τ, ξ) = (1 + |ξ|2 )s/2 u
g
Λ
e(τ, ξ),
2
2 2 θ/2
θ u(τ, ξ) = 1 + (τ − |ξ| )
u
e(τ, ξ).
Λg
−
1 + τ 2 + |ξ|2
2
−|ξ|2 )2 θ/2
Notice that the weight 1 + (τ
is equivalent to the weight w− (τ, ξ)θ ,
1+τ 2 +|ξ|2
where we define
w± (τ, ξ) = 1 + ||τ | ± |ξ||.
We define the norms
kukH s,θ = khξis w− (τ, ξ)θ u
e(τ, ξ)kL2 (R2+1 ) ,
kukHs,θ = kukH s,θ + k∂t ukH s,θ .
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The last norm is equivalent to
khξis−1 w+ (τ, ξ)w− (τ, ξ)θ u
e(τ, ξ)kL2 (R2+1 ) .
We can now state the null form estimate we are going to use in the proof of Theorem
1.1.
Proposition 2.1. Let n = 2, 14 < s < 12 , 34 < θ < s + 21 . Let Q denote any of the
null forms defined by (1.1). Then for all sufficiently small positive δ we have
kQ(φ, ψ)kH s,θ−1+δ . kφkHs+1,θ kψkHs+1,θ .
(2.1)
If Q = Q0 there is a better estimate.
Proposition 2.2. Let n = 2, s > 0 and let θ and δ satisfy
1
1
< θ ≤ min{1, s + },
2
2
1
0 ≤ δ ≤ min{1 − θ, s + − θ}.
2
Then
kQ0 (φ, ψ)kH s,θ−1+δ . kφkHs+1,θ kψkHs+1,θ
For a proof of the above proposition, see [13, estimate (7.5)].
For Q = Qij , Q0j estimate (2.1) should be compared (if we set θ = s +
δ = 0) to the following estimate of Zhou [19]:
Ns,s− 12 (Qαβ (φ, ψ)) . Ns+1,s+ 12 (φ)Ns+1,s+ 12 (ψ),
where
1
4
<s<
1
2
(2.2)
1
2
and
(2.3)
and
Ns,θ (u) = kw+ (τ, ξ)s w− (τ, ξ)θ u
e(τ, ξ)kL2τ,ξ .
(2.4)
The spaces in estimate (2.1) are different, with φ and ψ slightly less regular in the
sense that kukHs,θ ≤ Ns,θ (u). Moreover we have to account for the extra hyperbolic
derivative of order δ on the left hand side.
Proof of Proposition 2.1. We only sketch the proof for Q = Q0j . The proof for
Q = Qij is similar. Let
θ
e ξ),
F (τ, ξ) = hξis w+ (τ, ξ)w−
(τ, ξ)φ(τ,
θ
e ξ).
G(τ, ξ) = hξis w+ (τ, ξ)w−
(τ, ξ)ψ(τ,
Let H(τ, ξ) be a test function. We may assume F, G, H ≥ 0. We need to show:
Z
θ−1+δ
hξ + ηis w−
(τ + λ, ξ + η)|τ ηj − λξj |
θ (τ, ξ)hηis w (λ, η)w θ (λ, η)
hξis w+ (τ, ξ)w−
+
−
(2.5)
× F (τ, ξ)G(λ, η)H(τ + λ, ξ + η)dτ dλ dξ dη
. kF kL2 kGkL2 kHkL2 .
Using
hξ + ηis ≤ hξis + hηis
we see that we need to estimate the following integral (and a symmetric one):
Z θ−1+δ
w−
(τ + λ, ξ + η)|τ ηj − λξj |F (τ, ξ)G(λ, η)H(τ + λ, ξ + η)
dτ dλ dξ dη (2.6)
θ (τ, ξ)hηis w (λ, η)w θ (λ, η)
w+ (τ, ξ)w−
+
−
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CHERN-SIMONS-HIGGS EQUATIONS
5
We restrict our attention to the region where τ ≥ 0, λ ≥ 0. The proof for all other
regions is similar. We use
τ η − λξ = (|ξ|η − |η|ξ) + (τ − |ξ|)η − (λ − |η|)ξ
= (|ξ|η − |η|ξ) + (|τ | − |ξ|)η − (|λ| − |η|)ξ
to see that, we need to estimate the following three integrals:
Z
||ξ|η − |η|ξ|F (τ, ξ)G(λ, η)H(τ + λ, ξ + η)dτ dλ dξ dη
,
R+ =
1−θ−δ
θ (τ, ξ)hηis w (λ, η)w θ (λ, η)
w−
(τ + λ, ξ + η)w+ (τ, ξ)w−
+
−
Z
||τ | − |ξ|||η|F (τ, ξ)G(λ, η)H(τ + λ, ξ + η)dτ dλ dξ dη
T+ =
,
1−θ−δ
θ (τ, ξ)hηis w (λ, η)w θ (λ, η)
w−
(τ + λ, ξ + η)w+ (τ, ξ)w−
+
−
Z
||λ| − |η|||ξ|F (τ, ξ)G(λ, η)H(τ + λ, ξ + η)dτ dλ dξ dη
+
L =
.
1−θ−δ
θ (τ, ξ)hηis w (λ, η)w θ (λ, η)
(τ + λ, ξ + η)w+ (τ, ξ)w−
w−
+
−
We start with R+ . We have
||η|ξ − |ξ|η|
1/2
. |ξ|1/2 |η|1/2 (|ξ| + |η|)
1/2
(||τ + λ| − |ξ + η|| + ||τ | − |ξ|| + ||λ| − |η||)
.
(2.7)
Indeed,
||η|ξ − |ξ|η|2 = 2|η||ξ| (|ξ||η| − ξ · η)
= |η||ξ| (|ξ| + |η| + |ξ + η|) (|ξ| + |η| − |ξ + η|) .
We have |ξ| + |η| + |ξ + η| ≤ 2 (|ξ| + |η|) and
|ξ| + |η| − |ξ + η| = (τ + λ − |ξ + η|) − (λ − |η|) − (τ − |ξ|)
≤ |τ + λ − |ξ + η|| + |λ − |η|| + |τ − |ξ||,
therefore (2.7) follows. Following Zhou [19] we use (2.7) to obtain
||η|ξ − |ξ|η| = ||η|ξ − |ξ|η|2s ||η|ξ − |ξ|η|1−2s
1/2−s
. ||η|ξ − |ξ|η|2s |ξ|1/2−s |η|1/2−s (|ξ| + |η|)
||τ + λ| − |ξ + η||1/2−s
1/2−s
||τ | − |ξ||1/2−s
1/2−s
||λ| − |η||1/2−s .
+ ||η|ξ − |ξ|η|2s |ξ|1/2−s |η|1/2−s (|ξ| + |η|)
+ ||η|ξ − |ξ|η|2s |ξ|1/2−s |η|1/2−s (|ξ| + |η|)
Therefore,
R+ . R1+ + R2+ + R3+ ,
where
R1+
1/2−s
1
||η|ξ − |ξ|η|2s |ξ|1/2−s |η|1/2−s (|ξ| + |η|)
||τ + λ| − |ξ + η|| 2 −s
=
1−θ−δ
θ (τ, ξ)hηis w (λ, η)w θ (λ, η)
w−
(τ + λ, ξ + η)w+ (τ, ξ)w−
+
−
× F (τ, ξ)G(λ, η)H(τ + λ, ξ + η) dτ dλ dξ dη
Z
1/2−s
||η|ξ − |ξ|η|2s (|ξ| + |η|)
≤
θ
θ
w− (τ, ξ)w− (λ, η)|ξ|s+1/2 |η|2s+1/2
× F (τ, ξ)G(λ, η)H(τ + λ, ξ + η) dτ dλ dξ dη
Z
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s+ 1 −θ−δ
(we have used the fact that w− 2
(τ + λ, ξ + η) ≥ 1. Indeed, s + 21 − θ − δ > 0
for small δ, because θ < s + 12 .)
Z
1/2−s
||η|ξ − |ξ|η|2s |ξ|1/2−s |η|1/2−s (|ξ| + |η|)
||τ | − |ξ||1/2−s
R2+ =
1−θ−δ
θ (τ, ξ)hηis w (λ, η)w θ (λ, η)
w−
(τ + λ, ξ + η)w+ (τ, ξ)w−
+
−
× F (τ, ξ)G(λ, η)H(τ + λ, ξ + η) dτ dλ dξ dη
Z
1/2−s
||η|ξ − |ξ|η|2s (|ξ| + |η|)
≤
θ+s− 12
θ (λ, η)|ξ|s+1/2 |η|2s+1/2
w−
(τ, ξ)w−
× F (τ, ξ)G(λ, η)H(τ + λ, ξ + η) dτ dλ dξ dη
1−θ−δ
(we have used the fact that w−
(τ + λ, ξ + η) ≥ 1. Indeed, 1 − θ − δ ≥ 0 for
1
small δ because θ < s + 2 < 1.)
Z
1/2−s
||η|ξ − |ξ|η|2s |ξ|1/2−s |η|1/2−s (|ξ| + |η|)
||λ| − |η||1/2−s
+
R3 =
1−θ−δ
θ (τ, ξ)hηis w (λ, η)w θ (λ, η)
w−
(τ + λ, ξ + η)w+ (τ, ξ)w−
+
−
× F (τ, ξ)G(λ, η)H(τ + λ, ξ + η) dτ dλ dξ dη
Z
1/2−s
||η|ξ − |ξ|η|2s (|ξ| + |η|)
≤
1
θ (τ, ξ)w θ+s− 2 (λ, η)|ξ|s+1/2 |η|2s+1/2
w−
−
× F (τ, ξ)G(λ, η)H(τ + λ, ξ + η) dτ dλ dξ dη
We present the proof for R2+ . The proofs for R1+ and R3+ are similar. We change
variables τ 7→ u := |τ | − |ξ| = τ − |ξ| and λ 7→ v := |λ| − |η| = λ − |η| and we use
the notation
fu (ξ) = F (u + |ξ|, ξ), gv (η) = G(v + |η|, η), Hu,v (τ 0 , ξ 0 ) = H(u + v + τ 0 , ξ 0 )
to get
R2+ =
hZ Z ||η|ξ − |ξ|η|2s (|ξ| + |η|)1/2−s
1
|ξ|s+1/2 |η|2s+1/2
(1 + |u|)θ+s− 2 (1 + |v|)θ
i
× fu (ξ)gv (η)Hu,v (|ξ| + |η|, ξ + η) dξ dη du dv.
ZZ
1
We have ||η|ξ − |ξ|η|2 = 2|ξ||η| (|ξ||η| − ξ · η) therefore
ZZ
s
1/2−s
(|ξ||η| − ξ · η) (|ξ| + |η|)
[· · · ] .
fu (ξ)gv (η)Hu,v (|ξ| + |η|, ξ + η)dξdη
|ξ|1/2 |η|s+1/2
ZZ
1/2
≤
fu (ξ)2 gv (η)2 dξdη
K 1/2
= kfu kL2 (R2 ) kgv kL2 (R2 ) K 1/2 ,
where
=
2s
1−2s
ZZ
(|ξ||η| − ξ · η) (|ξ| + |η|)
|ξ| |η|2s+1
ZZ
(|ξ 0 − η||η| − (ξ 0 − η) · η) (|ξ 0 − η| + |η|)
|ξ 0 − η| |η|2s+1
K=
2s
× Hu,v (|ξ 0 − η| + |η|, ξ 0 )2 dξ 0 dη.
Hu,v (|ξ| + |η|, ξ + η)2 dξ dη
1−2s
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CHERN-SIMONS-HIGGS EQUATIONS
7
We use polar coordinates η = ρω to get
ZZZ
2s
1−2s
(|ξ 0 − ρω| + ρ − ξ 0 · ω) (|ξ 0 − ρω| + ρ)
K.
|ξ 0 − ρω|
× Hu,v (|ξ 0 − ρω| + ρ, ξ 0 )2 dξ 0 dρ dω.
For fixed ξ 0 and ω, we change variables ρ 7→ τ 0 := |ξ 0 − ρω| + ρ to get
ZZ h
Z
i
1
K.
τ 01−2s
dω
H(τ 0 , ξ 0 )2 dξ 0 dτ 0 .
0
0
1−2s
S 1 (τ − ξ · ω)
From [19, estimate (3.22)] we know that
Z
1
τ 01−2s
dω . 1;
0
0
1−2s
S 1 (τ − ξ · ω)
therefore K . kHk2Ã . Putting everything together we get:
Z
Z kg k 2 2
kfu kL2 (R2 )
v L (R )
du
dv
kHk .
R2+ .
1
(1 + |v|)θ
(1 + |u|)θ+s− 2
Since 2θ + 2s − 1 > 2 · 34 + 2 · 41 − 1 = 1 and 2θ > 2 ·
Cauchy-Schwarz inequality to conclude:
3
4
> 1 we can use the
R2+ . kkfu kL2 (R2 ) kL2u kkgv kL2 (R2 ) kL2v kHk = kF kkGkà kHkà .
This completes the estimates for R2+ .
Next we estimate T + . We use ||τ | − |ξ|| ≤ w+ (τ, ξ)1−θ w− (τ, ξ)θ to get
Z
||τ | − |ξ|||η|F (τ, ξ)G(λ, η)H(τ + λ, ξ + η)dτ dλ dξ dη
T+ =
1−θ−δ
θ (τ, ξ)hηis w (λ, η)w θ (λ, η)
w−
(τ + λ, ξ + η)w+ (τ, ξ)w−
+
−
Z
F (τ, ξ)G(λ, η)H(τ + λ, ξ + η)
≤
dτ dλ dξ dη.
θ (λ, η)
hξiθ hηis w−
Changing variables τ 7→ u := |τ | − |ξ| = τ − |ξ| and λ 7→ v := |λ| − |η| = λ − |η| we
have
ZZ
1
T+ .
hξiθ hηis
i
h Z Z F (u + |ξ|, ξ)G(v + |η|, η)H(u + v + |ξ| + |η|, ξ + η)
du
dv
dξ dη.
(1 + |v|)θ
For fixed ξ and η we apply [19, Lemma A] in the (u, v)-variables to get
ZZ
1
T+ .
kF (u + |ξ|, ξ)kL2u kG(v + |η|, η)kL2v
hξiθ hηis
× kH(w + |ξ| + |η|, ξ + η)kL2w dξdη
ZZ
1
=
kF (·, ξ)kL2 (R) kG(·, η)kL2 (R) kH(·, ξ + η)kL2 (R) dξ dη.
θ
hξi hηis
Now we do the same in the (ξ, η)-variables to get
T + . kkF (·, ξ)kL2 (R) kL2ξ kkG(·, η)kL2 (R) kL2η kkH(·, ξ 0 )kL2 (R) kL20
ξ
= kF kà kGkà kHkà .
The proof for L+ is similar.
We are also going to need the following ‘product rules’ in H s,θ spaces.
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N. BOURNAVEAS
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Proposition 2.3. Let n = 2. Then
kuvkH −c,−γ . kukH a,α kvkH b,β ,
(2.8)
a+b+c>1
(2.9)
provided that
a + b ≥ 0,
b + c ≥ 0,
a+c≥0
(2.10)
α + β + γ > 1/2
(2.11)
α, β, γ ≥ 0.
(2.12)
Proof. If a, b, c ≥ 0, the result is contained in [13, Proposition A1]. If not, observe
that, due to (2.10), at most one of the a, b, c is negative. We deal with the case
c < 0, a, b ≥ 0. All other cases are similar. Observe that
hξi−c h|τ | − |ξ|i−γ |f
uv(τ, ξ)|
ZZ
. h|τ | − |ξ|i−γ
hξ − ηi−c |e
u(τ − λ, ξ − η)||e
v (λ, η)|dλdη
ZZ
+ h|τ | − |ξ|i−γ
|e
u(τ − λ, ξ − η)|hηi−c |e
v (λ, η)|dλdη,
therefore
kuvkH −c,−γ . kU v 0 kH 0,−γ + ku0 V kH 0,−γ ,
where
e (τ, ξ) = hξi−c |e
U
u(τ, ξ)|,
ue0 (τ, ξ) = |e
u(τ, ξ)|,
Ve (τ, ξ) = hξi−c |e
v (τ, ξ)|,
ve0 (τ, ξ) = |e
v (τ, ξ)|.
Since a + c ≥ 0, we have
kU v 0 kH 0,−γ . kU kH a+c,α kv 0 kH b,β . kukH a,α kvkH b,β .
Since b + c ≥ 0, we have
ku0 V kH 0,−γ . ku0 kH a,α kV kH b+c,β . kukH a,α kvkH b,β .
The result follows.
Proposition 2.4. Let n = 2. If s > 1 and
1
2
< θ ≤ s − 21 , then H s,θ is an algebra.
For the proof of the above proposition, see [13, Theorem 7.3].
Proposition 2.5. Let n = 2, s > 1,
−θ ≤ α ≤ 0
1
2
< θ ≤ s − 21 . Assume that
− s ≤ a < s + α.
(2.13)
Then
H a,α · H s,θ ,→ H a,α .
The proof can be found in [13, Theorem 7.2].
(2.14)
EJDE-2009/114
CHERN-SIMONS-HIGGS EQUATIONS
9
Proof of Theorem 1.1. Theorem 1.1 follows by well known methods from the
following a-priori estimates (together with the corresponding estimates for differences): For any space-time functions B, B 0 , φ, φ0 ∈ Hs+1,θ and any γ, µ ∈ {0, 1, 2}
we have:
kφ0 ∂ γ φkH s,θ−1+δ . kφ0 kHs+1,θ kφkHs+1,θ ,
0
µ
0
(2.15)
k(∂ B) φ φ kH s,θ−1+δ . kBkHs+1,θ kφkHs+1,θ kφ kHs+1,θ ,
(2.16)
kQµα (B, φ)kH s,θ−1+δ . kBkHs+1,θ kφkHs+1,θ ,
(2.17)
0
0
kQ0 (B, B )φkH s,θ−1+δ . kBkHs+1,θ kB kHs+1,θ kφkHs+1,θ ,
0
0
kQµν (B, B )φkH s,θ−1+δ . kBkHs+1,θ kB kHs+1,θ kφkHs+1,θ .
(2.18)
(2.19)
Here 14 < s < 12 , 34 < θ < s + 12 and δ is a sufficiently small positive number.
To prove (2.15) we use Proposition 2.3 to get:
kφ0 ∂ γ φkH s,θ−1+δ . kφ0 kH s+1,θ k∂ γ φkH s,θ . kφ0 kHs+1,θ kφkHs+1,θ .
Similarly, for (2.16) we have
k(∂ µ B) φ φ0 kH s,θ−1+δ . k∂ µ BkH s,θ kφ φ0 kH s+1,θ . kBkHs+1,θ kφ φ0 kH s+1,θ .
By Proposition 2.4 and our assumptions on s and θ it follows that the space H s+1,θ
is an algebra. Therefore,
kφ φ0 kH s+1,θ . kφkH s+1,θ kφ0 kH s+1,θ ,
and estimate (2.16) follows.
Estimate (2.17) follows from Proposition (2.1). Finally, we consider estimates
(2.18) and (2.19). We use the letter Q to denote any of the null forms Q0 , Qµν .
We have
(2.20)
kQ(B, B 0 ) φkH s,θ−1+δ . kQ(B, B 0 )kH s,θ−1+δ kφkH s+1,θ .
This follows from Proposition 2.5 with s replaced by s+1 and α replaced by θ−1+δ.
Next, by (2.1),
kQ(B, B 0 )kH s,θ−1+δ . kBkHs+1,θ kB 0 kHs+1,θ
(2.21)
therefore (2.18) and (2.19) follow.
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Nikolaos Bournaveas
University of Edinburgh, School of Mathematics, James Clerk Maxwell Building, King’s
Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK
E-mail address: n.bournaveas@ed.ac.uk