Table Of ContentGLOBAL WELL-POSEDNESS OF THE CHERN-SIMONS-HIGGS
EQUATIONS WITH FINITE ENERGY
2 SIGMUNDSELBERGANDACHENEFTESFAHUN
1
0
2 Abstract. We prove that the Cauchy problem for the Chern-Simons-Higgs
equations on the (2+1)-dimensional Minkowski space-time is globally well
n
posed for initial data with finite energy. This improves a result of Chae and
a
Choe, whoprovedglobal well-posedness formoreregulardata. Moreover, we
J
provelocalwell-posednessevenbelowtheenergyregularity,usingthethenull
4 structureofthesysteminLorenzgauge andbilinearspace-timeestimates for
wave-Sobolev norms.
]
P
A
.
h 1. Introduction
t
a
The (2+1)-dimensional abelian Chern-Simons-Higgs model was proposed by
m
Hong, Kim and Pac [5] and Jackiw and Weinberg [8] in the study of vortex so-
[
lutions in the abelian Chern-Simons theory. The Lagrangianfor the model is
1 κ
L= ǫµνρA F +D φDµφ−V |φ|2 ,
v µ νρ µ
4
5 (cid:16) (cid:17)
7 on the Minkowski space-time R1+2 = Rt×R2x with metric gµν = diag(1,−1,−1).
9 HereD =∂ −iA isthecovariantderivativeassociatedtothegaugefieldA ∈R,
µ µ µ µ
0 F = ∂ A −∂ A is the curvature, φ ∈ C is the Higgs field, V(|φ|2) ∈ R is a
. µν µ ν ν µ
1 Higgs potential, κ >0 is a Chern-Simons coupling constant, and ǫµνρ is the skew-
0 symmetric tensor with ǫ012 = 1. Greek indices range from 0 to 2, Latin indices
2
from 1 to 2, and repeated upper/lower indices are implicitly summed.
1
The corresponding Euler-Lagrangeequations are
:
v
1
i (1.1) F = ǫ Jρ, D Dµφ=−φV′ |φ|2 ,
X µν µνρ µ
κ
r where (cid:16) (cid:17)
a
Jρ =2Im φDρφ .
There is a conserved energy,
(cid:0) (cid:1)
2
E(t)= |D φ(t,x)|2+V |φ(t,x)|2 dx,
µ
R2 !
Z µX=0 (cid:16) (cid:17)
and the equations are invariant under the gauge transformations
(1.2) A →A′ =A +∂ χ, φ→φ′ =eiχφ, D →D′ =∂ −iA′,
µ µ µ µ µ µ µ µ
hence we may impose an additional gauge condition. In this paper we rely on the
Lorenz condition ∂µA =0.
µ
2000 Mathematics Subject Classification. 35Q40;35L70.
Key words and phrases. Chern-Simons-Higgs;well-posedness;nullstructure;Lorenzgauge.
1
2 S.SELBERGANDA.TESFAHUN
AtypicalpotentialisV(r)=κ−2r(1−r)2 (see[5,8]),inwhichcasetherearetwo
possible boundaryconditions to makethe energy finite: Either |φ|→1 as |x|→∞
(the topological case) or |φ|→0 as |x|→∞ (the non-topological case).
We are interested in the Cauchy problem for the non-topological case, which
received considerable attention recently. Local well-posedness for low-regularity
data was studied in [6, 1, 11, 7], but the energy regularity was not quite reached;
Huh[7]camearbitrarilyclosetoenergyusingtheCoulombgauge. Inthispaperwe
closetheremaininggap,usingtheLorenzgauge. Infact,weprovethattheproblem
is locally well posed not only at the energy regularity but even a little below it.
From the local finite-energy well-posedness we get the corresponding global result
by exploiting the conservation of energy and the residual gauge freedom within
Lorenz gauge. In particular, we improve the earlier result of Chae and Choe [2],
whoprovedglobalwell-posednessformoreregulardata,namelywithonederivative
extra in L2 compared with energy.
In order to pose the Cauchy problem one should know the observables F , Jρ
µν
and E at time t = 0, so it suffices to specify φ(0) and D φ(0). Since we are
µ
interested in the non-topological case we assume V(0) = 0. Moreover we assume
that V′(r) has polynomial growth, hence E(0) is absolutely convergentif
(1.3) D φ(0)∈L2,
µ
(1.4) φ(0)∈Lp for all 2≤p<∞,
which imply
(1.5) Jρ(0)=2Im φ(0)Dρφ(0) ∈H˙−1/2,
since by the Hardy-Littlewood-Sobo(cid:0)lev inequality(cid:1) on R2,
(1.6) 2Im(fg) ≤C fg ≤kfk kgk .
H˙−1/2 L4/3 L4 L2
Here H˙s = H˙s(cid:13)(R2), |s| (cid:13)< 1, is the(cid:13)com(cid:13)pletion of S(R2) with respect to the
(cid:13) (cid:13) (cid:13) (cid:13)
normkfk = |ξ|sf(ξ) , where f(ξ) is the Fourier transformof f(x). A direct
H˙s L2
characterizationis
(cid:13) (cid:13)
(cid:13) bH˙(cid:13)s =F−1 L2b(|ξ|2sdξ) (|s|<1).
Here s>−1 ensures that S ⊂L2(cid:0)(|ξ|2sdξ) (d(cid:1)ensely), and s <1 ensures that func-
tions in L2(|ξ|2sdξ) are tempered, so the inverseFourier transformcan be applied.
We also need the inhomogeneous space Hs = Hs(R2), which is the completion of
S(R2) with respect to kfk = hξisf(ξ) , where hξi=(1+|ξ|2)1/2.
Hs L2
Recall the Hardy-Littlewood-Sobolev inequalities
(cid:13) (cid:13)
(cid:13) b (cid:13)
1 1 s
(1.7) kfk ≤Ck|∇|sfk 1<p<q <∞, − = ,
Lq Lp p q 2
(cid:18) (cid:19)
2
(1.8) kfk ≤Ckh∇isfk p≥1, s> ,
L∞ Lp p
(cid:18) (cid:19)
where |∇| = (−∆)1/2 and h∇i = (1 − ∆)1/2. In particular, H1 ⊂ Lp for all
2≤p<∞, and
(1.9) kfgk ≤kfk kgk .kfk kgk .
L2 L4 L4 H˙1/2 H1
The notation a.b stands for a≤Cb.
CHERN-SIMONS-HIGGS WITH FINITE ENERGY 3
2. Main results
Since the value of the positive constant κ is irrelevant for our analysis, we shall
set κ=1. Augmented with the Lorenz gauge condition ∂µA =0, (1.1) reads
µ
(2.1a) ∂ A −∂ A =ǫ Jk,
t j j 0 jk
(2.1b) ∂ A −∂ A =J ,
1 2 2 1 0
(2.1c) ∂ A +∂ A =∂ A ,
1 1 2 2 t 0
(2.1d) D Dµφ=−φV′ |φ|2 ,
µ
(cid:16) (cid:17)
where Jρ =2Im φDρφ and ǫ is the skew-symmetric tensor with ǫ =1.
jk 12
We posetheCauchyproblemintermsofdataforA and(φ,∂ φ). Thequestion
µ t
(cid:0) (cid:1)
then arises: What are the natural data spaces, given that (1.3)–(1.5) should hold?
Toanswerthis,firstnotethattheLorenzconditionleavessomegaugefreedom,since
it is preservedby (1.2) if (cid:3)χ=0, where (cid:3)=∂µ∂ =∂2−∆ is the d’Alembertian.
µ t
So formally, at least, we may impose the initial constraints
(2.2) A (0)=0, ∂jA (0)=0,
0 j
forifthesearenotalreadysatisfied,they willbeafter agaugetransformation(1.2)
with gauge function χ satisfying
(2.3) (cid:3)χ=0, ∆χ(0)=∂jA (0), ∂ χ(0)=−A (0).
j t 0
But from (2.2) and (2.1b) we get
(2.4) ∆A (0)=ǫ ∂kJ (0),
j jk 0
soA (0)shouldbe inH˙1/2,recalling(1.5). Thenfrom(1.3)and(1.4)weinferthat
j
(φ,∂ φ)(0) ∈ H1×L2, since ∂ φ(0) = D φ(0) and ∂ φ(0) = D φ(0)+iA (0)φ(0),
t t 0 j j j
and the last term is in L2 by (1.9).
So now we know what the correct data spaces for A and (φ,∂ φ) are. Note,
µ t
however, that (2.1b) imposes an initial constraint. The following lemma shows
that given any data for (φ,∂ φ) in H1×L2, there exists an initial potential A (0)
t µ
satisfying this constraint as well as the finite energy requirements (1.3) and (1.4).
Lemma 2.1. Given data
(φ,∂ φ)(0)∈H1×L2,
t
there exists an initial potential
A (0)∈H˙1/2
µ
satisfying (2.2), and (2.1b) at t=0. Moreover, (1.3) and (1.4) are satisfied.
Proof. First note that (1.4) holds by the embedding H1 ⊂ Lp, 2 ≤ p < ∞. Set
A (0)=0 and
0
A (0)=−(−∆)−1/2ǫ RkJ (0),
j jk 0
where R =(−∆)−1/2∂ is the Riesz transform. By (1.6),
k k
kA (0)k .kJ (0)k .kφ(0)k k∂ φ(0)k .kφ(0)k k∂ φ(0)k ,
j H˙1/2 0 H˙−1/2 L4 t L2 H1 t L2
and D φ(0) ∈ L2 follows from (1.9). By (2.4), ∆(∂ A (0)−∂ A (0)−J (0)) = 0
µ 1 2 2 1 0
and∆(∂ A (0)+∂ A (0))=0,andingeneral,∆f =0implies f =0iff ∈H˙−1/2,
1 1 2 2
since f is a tempered function. Thus (2.2) holds, as does (2.1b) at t=0. (cid:3)
b
4 S.SELBERGANDA.TESFAHUN
More generally, we shall prove local well-posedness for any data
(2.5) A (0)∈H˙1/2, (φ,∂ φ)(0)∈H1×L2,
µ t
satisfying (2.1b) initially:
(2.6) ∂ A (0)−∂ A (0)=J (0)=2Im φ(0)D φ(0) .
1 2 2 1 0 0
(cid:16) (cid:17)
Then D φ(0)=∂ φ(0)−iA (0)φ(0) is in L2, with norm bounded in terms of the
µ µ µ
norm of (2.5), in view of (1.9).
Theorem2.1. TheChern-Simons-Higgs-LorenzCauchyproblem (2.1),(2.5),(2.6)
is locally well posed, for any potential V ∈ C∞(R ;R) such that V(0) = 0 and all
+
derivatives of V have polynomial growth. More precisely, there exists a time T >0,
which is a decreasing and continuous function of the data norm
2
kA (0)k +kφ(0)k +k∂ φ(0)k ,
µ H˙1/2 H1 t L2
µ=0
X
and a solution (A,φ) of (2.1) on (−T,T)×R2 with the regularity
A ∈C([−T,T];H˙1/2),
µ
(2.7)
φ∈C([−T,T];H1), ∂ φ∈C([−T,T];L2).
t
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 par-
ticular, if the data are smooth, then so is the solution.
The proof is given in Section 4.
Ourplanis nowto showthat (i)the time T infactonly depends onI(0), where
2
I(t)=kφ(t)k + kD φ(t)k ,
L2 µ L2
µ=0
X
and(ii) I(t) is aprioricontrolledfor alltime interms of E(0)andkφ(0)k . Then
L2
it will of course follow that the solutions extend globally in time.
To prove (i) we apply the gauge transformation (1.2) with χ satisfying (2.3).
Lemma 2.2. Given data A (0)∈H˙1/2, there exists χ(t,x) with the regularity
µ
χ∈C(R1+2), ∂ χ∈C(R;H˙1/2),
µ
and satisfying (2.3).
Proof. The solution of (2.3) is
χ(t)=cos(t|∇|)f +sin(t|∇|)|∇|−1g,
where g =−A (0)∈H˙1/2 and f should satisfy
0
(2.8) ∆f =∂jA (0).
j
First, if the Fourier transform of A (0) is supported in {ξ ∈R2: |ξ|≥ 1}, then
µ
g ∈H1/2,and(2.8)hasauniquesolutionf ∈H3/2,soχ∈C(R;H3/2)⊂C(R1+2),
and ∂ χ∈C(R;H1/2).
µ
Now assume that A (0) has Fourier support in {ξ ∈ R2: |ξ| < 1}. Then A (0)
µ µ
is smooth, but it is not obvious that (2.8) has a solution (what is clear is that
CHERN-SIMONS-HIGGS WITH FINITE ENERGY 5
the solution, if it exists, will be smooth). Formally, f should be given by, with
R =(−∆)−1/2∂ the Riesz transform,
j j
f =−(−∆)−1/2RjA (0),
j
but it is not clear that this is meaningful. However, if we take the gradient we get
something well-defined:
∂ f =F ≡−R RjA (0)∈H˙1/2∩C∞.
k k k j
But (F ,F ) is a smooth vector field on R2 with zero curl:
1 2
∂ F −∂ F =0,
1 2 2 1
hence (F ,F ) is the gradient of a smooth function, which we denote f. Then
1 2
it follows that (2.8) is satisfied. So now f,g ∈ C∞(R2), hence χ ∈ C∞(R1+2).
Moreover, ∂ f,g ∈H˙1/2, so ∂ χ∈C(R;H˙1/2). (cid:3)
j µ
We also need the covariant Sobolev inequality, proved in [4],
1−2/p
2
(2.9) kφ(0)k ≤Ckφ(0)k2/p kD φ(0)k (2<p<∞),
Lp L2 j L2
j=1
X
which holds for all φ(0) ∈ H1 such that D φ(0) ∈ L2 (the regularity of the real-
j
valued functions A (0) is irrelevant here).
j
Theorem 2.2. The solution (A,φ) from Theorem 2.1 exists up to a time T > 0
which is a continuous and decreasing function of
2
I(0)=kφ(0)k + kD φ(0)k .
L2 µ L2
µ=0
X
Proof. Givendata(2.5)satisfying (2.6),apply the gaugetransformation(1.2) with
χ as in Lemma 2.2. Then (1.2) preserves the regularity (2.7), as does its inverse,
obtained by replacing χ by −χ. In the new gauge,
A′(0)=0, ∂jA′(0)=0,
0 j
and by the latter combined with (2.6) (which is gauge invariant),
∆A′(0)=ǫ ∂kJ (0).
j jk 0
Since we know that A′(0) belongs to H˙1/2, and since in general ∆f = 0 implies
j
f =0 if f ∈H˙1/2 (then f is a tempered function), we conclude that
A′(0)=−(−∆)−1/2ǫ RkJ (0),
b j jk 0
where R =(−∆)−1/2∂ is the Riesz transform. Thus, by (1.6),
k k
(2.10) A′(0) .kJ (0)k .kφ(0)k kD φ(0)k .I(0)2,
j H˙1/2 0 H˙−1/2 L4 0 L2
where we app(cid:13)lied (2.(cid:13)9) in the last step. Moreover,
(cid:13) (cid:13)
φ′(0)=eiχ(0)φ(0),
∂ φ′(0)=D′φ′(0)+iA′ (0)φ′(0)=eiχ(0)D φ(0)+ieiχ(0)A′ (0)φ(0),
µ µ µ µ µ
6 S.SELBERGANDA.TESFAHUN
hence
2 2
kφ′(0)k + k∂ φ′(0)k ≤I(0)+ A′(0) kφ(0)k .I(0)+I(0)3,
L2 µ L2 µ L4 L4
µ=0 µ=0
X X(cid:13) (cid:13)
(cid:13) (cid:13)
where we used (2.9) and (2.10).
Thus, applying Theorem 2.1 we get the solution (A′,φ′) up to a time T > 0
which is a continuous and decreasing function of I(0). Finally, reverse the gauge
transformation to get the solution (A,φ). (cid:3)
Finally, we show that the solutions extend globally in time.
Theorem 2.3. In addition to the hypotheses in Theorem 2.1, assume that
V(r)≥−α2r
for all r ≥ 0 and some α > 0. Then the solution (A,φ) from Theorem 2.1 exists
globally in time and has the regularity (2.7) for all T >0.
In view of Theorem 2.2 it suffices to show that
2
I(t)=kφ(t)k + kD φ(t)k
L2 µ L2
µ=0
X
is a priori bounded on every finite time interval. For this, we rely of course on
the conservation of energy (which is satisfied since our local solutions are limits of
smooth solutions with compact spatial support). First we note, using E(t)=E(0)
and the assumption V(r)≥−α2r, that
(2.11) kD φ(t)k2 =E(0)− V |φ(t,x)|2 dx≤|E(0)|+α2kφ(t)k2 .
µ L2 L2
Z (cid:16) (cid:17)
Then
d
kφ(t)k2 = 2Re φ(t,x)D φ(t,x) dx
dt L2 0
(cid:16) (cid:17) Z (cid:16) (cid:17)
≤2kφ(t)k kD φ(t)k
L2 0 L2
1/2
≤2kφ(t)k |E(0)|+α2kφ(t)k2
L2 L2
≤α−1|E(0)|(cid:16)+2αkφ(t)k2 , (cid:17)
L2
hence by Gr¨onwall’s lemma,
(2.12) kφ(t)k2 ≤e2α|t| kφ(0)k2 +|t|α−1|E(0)| .
L2 L2
(cid:16) (cid:17)
By (2.11) and (2.12) we control I(t), and Theorem 2.3 is proved.
It remains to prove Theorem 2.1. Note that in Lorenz gauge, ∂νF = −(cid:3)A ,
µν µ
so (2.1) implies
(2.13) (cid:3)A =−ǫ ∂νJρ, ∂µA =0, D Dµφ=−φV′ |φ|2 ,
µ µνρ µ µ
(cid:16) (cid:17)
and this is the system we actually solve.
Then we have to check that, conversely,(2.13) implies (2.1a) and (2.1b), assum-
ing that the latter two are satisfied at t=0. But then
v =∂ A −∂ A −ǫ Jk, w =∂ A −∂ A −J ,
j t j j 0 jk 1 2 2 1 0
CHERN-SIMONS-HIGGS WITH FINITE ENERGY 7
vanish at time t = 0. Moreover, using (2.1a), (cid:3)A = ǫ (−∂ Jk + ∂kJ ), and
j jk t 0
∂µJ =0 (which follows from the last equation in (2.13)), one finds
µ
∂ v =ǫ ∂kw, ∂ w =∂ v −∂ v ,
t j jk t 1 2 2 1
and these vanish at t=0 since v and w do. Taking another time derivative gives
j
(cid:3)v =∂ (∂kv ) and (cid:3)w =0. But (cid:3)A =−∂ J +∂ J implies ∂kv =0, hence
j j k 0 1 2 2 1 k
(cid:3)v =0, (cid:3)w=0.
j
Since the data vanish, we conclude that v =w =0, so (2.1a) and (2.1b) hold.
j
Before proving Theorem 2.1, we consider in the following section the problem
of local well-posedness with minimal regularity, and it turns out that we can get
below the energy regularity. Here we take data for A in inhomogeneous Sobolev
µ
spaces.
3. Low regularity local well-posedness
The system (2.13) expands to
(3.1a) ((cid:3)+1)A =−ǫ ImQνρ(∂φ,∂φ)+2ǫ ∂ν Aρ|φ|2 +A ,
µ µνρ µνρ µ
(3.1b) ∂µA =0, (cid:16) (cid:17)
µ
(3.1c) ((cid:3)+1)φ=2iA ∂µφ+A Aµφ−φV′ |φ|2 +φ,
µ µ
(cid:16) (cid:17)
where Q (∂u,∂v)= ∂ u∂ v−∂ u∂ v is the standard null form. Here we added
αβ α β β α
A and φ to eachside of (3.1a) and (3.1c), respectively, to get the operator (cid:3)+1;
µ
this is done to avoid a singularity in (3.7) below. We specify data
(3.2) A (0)∈Hs, (φ,∂ φ)(0)∈Hs+1/2×Hs−1/2.
µ t
The data for ∂ A are given by the constraints
t µ
(3.3) ∂ A (0)=∂ A (0)+∂ A (0)∈Hs−1,
t 0 1 1 2 2
(3.4) ∂ A (0)=∂ A (0)+ǫ Jk(0)∈Hs−1,
t j j 0 jk
whereJ =2Im φD φ =2Im φ∂ φ +2A |φ|2, hence Jk(0)∈Hs−1 with norm
k k k k
bounded in terms of the norm of (3.2), as follows from:
(cid:0) (cid:1) (cid:0) (cid:1)
Lemma 3.1. If s>0, the following estimates hold:
(3.5) kfgk .kfk kgk ,
Hs−1 Hs+1/2 Hs−1/2
(3.6) kfghk .kfk kgk khk .
Hs−1 Hs Hs+1/2 Hs+1/2
Proof. This follows from the Hs product law in two dimensions (see, e.g., [3]),
which states that, for s ,s ,s ∈R, the estimate
0 1 2
kfgk .kfk kgk
H−s0 Hs1 Hs2
holds if and only if (i) s +s +s ≥1, (ii) s +s +s ≥max(s ,s ,s ) and (iii)
0 1 2 0 1 2 0 1 2
at most one of (i) and (ii) is an equality. In particular, for s>0 this implies (3.5),
as well as
kfgk .kfk kgk
Hs−1/2 Hs Hs−1/2
and the latter combined with (3.5) gives (3.6). (cid:3)
8 S.SELBERGANDA.TESFAHUN
Theorem 3.1. The Chern-Simons-Higgs-Lorenz Cauchy problem (3.1)–(3.4) is lo-
cally well posed if s > 3/8, assuming that the potential V(r) is a polynomial of
degree n, where if s<1/2we assume n<1+2/(1−2s),whereas if s≥1/2there is
no restriction on n. To be precise, there exists a time T >0, which is a decreasing
and continuous function of the initial data norm
kA(0)k +kφ(0)k +k∂ φ(0)k ,
Hs Hs+1/2 t Hs−1/2
and a solution (A,φ) of (3.1) on (−T,T)×R2 with the regularity
A ∈C([−T,T];Hs), ∂ A ∈C([−T,T];Hs−1),
µ t µ
φ∈C([−T,T];Hs+1/2), ∂ φ∈C([−T,T];Hs−1/2).
t
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 par-
ticular, if the data are smooth, then so is the solution.
To prove this we iterate in Xs,b-spaces, so by standard methods we reduce to
provingestimatesfortherighthandsidesin(3.1). Themostdifficulttermsarethe
two bilinear ones, for which null structure is needed. The first term on the right
hand side of (3.1a) is already a null form, whereas the first term on the right hand
side of (3.1c) appears also in the Maxwell-Klein-Gordon system in Lorenz gauge,
and we showed in [10] that it has a null structure. To reveal this structure we
transform the variables:
A =A +A , φ=φ +φ ,
µ µ,+ µ,− + −
(3.7) 1 1
A = A ±i−1h∇i−1∂ A , φ = φ±i−1h∇i−1∂ φ .
µ,± µ t µ ± t
2 2
Then (3.1) transform(cid:0)s to (cid:1) (cid:0) (cid:1)
(3.8a) (i∂ ±h∇i)A =±2−1h∇i−1(R.H.S. (3.1a)),
t µ,±
(3.8b) ∂µA =0,
µ
(3.8c) (i∂ ±h∇i)φ =±2−1h∇i−1(R.H.S. (3.1c)).
t ±
We split the spatial part A=(A ,A ) of the potential into divergence-free and
1 2
curl-free parts and a smoother part:
(3.9) A=Adf+Acf+(1−∆)−1A,
(3.10) Adf =(R R A −R R A ,R R A −R R A ),
1 2 2 2 2 1 1 2 1 1 1 2
(3.11) Acf =(−R R A −R R A ,−R R A −R R A ),
1 2 2 1 1 1 1 2 1 2 2 2
where
R =(1−∆)−1/2∂
j j
is bounded on Lp, 1<p<∞. Now write
(3.12) A ∂µφ= A ∂ φ−Acf·∇φ −Adf·∇φ−h∇i−2A·∇φ≡B −B −B ,
µ 0 t 1 2 3
where B2 =Adf(cid:0)·∇φ was shown in(cid:1)[9] to be a null form:
(3.13) B =R ψ∂ φ−R ψ∂ φ, where ψ =R A −R A .
2 2 1 1 2 1 2 2 1
In [10] we found that B also has a null structure: By the Lorenz condition (3.1b)
1
we have R A +R A =h∇i−1∂ A , hence
1 1 2 2 t 0
(3.14) Acf =−R (R A +R A )=−iR (A −A ),
j j 1 1 2 2 j 0,+ 0,−
CHERN-SIMONS-HIGGS WITH FINITE ENERGY 9
wherewealsoused∂ A =ih∇i(A −A ). Thus,B =A ∂ φ+Acf∂jφbecomes
t 0 0,+ 0,− 1 0 t j
B =(A +A )ih∇i(φ −φ )−iR (A −A )∂j(φ +φ )
1 0,+ 0,− + − j 0,+ 0,− + −
(3.15) =i A h∇i(± φ )−R (± A )∂jφ ,
0,±1 2 ±2 j 1 0,±1 ±2
±X1,±2(cid:0) (cid:1)
where we used ∂ φ=ih∇i(φ −φ ).
t + −
Taking into account (3.13) and (3.15), we rewrite (3.8) as
(3.16a) (i∂ ±h∇i)A =±2−1h∇i−1M (A ,A ,φ ,φ ),
t µ,± µ + − + −
(3.16b) ∂µA =0,
µ
(3.16c) (i∂ ±h∇i)φ =±2−1h∇i−1N(A ,A ,φ ,φ ),
t ± + − + −
where
M (A ,A ,φ ,φ )=−ǫ ImQνρ(∂φ,∂φ)+2ǫ ∂ν Aρ|φ|2 +A ,
µ + − + − µνρ µνρ µ
(cid:16) (cid:17)
N(A ,A ,φ ,φ )=2i(B −B −B )+A Aµφ−φV′ |φ|2 +φ,
+ − + − 1 2 3 µ
with B and B given by (3.15) and (3.13), and B = h∇i−2(cid:16)A· φ(cid:17). Here it is
1 2 3
understoodthat A =A +A , φ=φ +φ , ∂ A =ih∇i(A −A ), and
µ µ,+ µ,− + − t µ µ,+ µ,−
∂ φ=ih∇i(φ −φ ).
t + −
The initial data are
1
A (0)= A (0)±i−1h∇i−1∂ A (0) ∈Hs,
µ,± µ t µ
2
(3.17)
1 (cid:0) (cid:1)
φ (0)= φ(0)±i−1h∇i−1∂ φ(0) ∈Hs+1/2.
± t
2
The systems (3.16) and (3(cid:0).1) are equivalent via th(cid:1)e transformation (3.7), so it
sufficestosolve(3.16). TheLorenzcondition(3.16b)reducestoaninitialconstraint,
since if (A ,A ,φ ,φ ) is a solution of (3.16), then setting A = A +A and
+ − + − + −
φ=φ +φ wehave((cid:3)+1)A =M ,so(3.1a)issatisfied,i.e.,(cid:3)A =−ǫ ∂νJρ.
+ − µ µ µ µνρ
Thus, u=∂µA satisfies (cid:3)u=0, and u(0)=∂ u(0)=0 by (3.3) and (3.4).
µ t
We prove local well-posedness of (3.16) by iterating in the Xs,b-spaces adapted
totheoperatorsi∂ ±h∇i,sobystandardarguments(see,e.g.,[10]formoredetails)
t
the proof of Theorem 3.1 reduces to proving, for some b,b′ ∈ (1/2,1), m ≥ 2, and
ε>0, the estimates
(3.18) kM(A ,A ,φ ,φ )k .B+Bm,
+ − + − Xs−1,b−1+ε
±
(3.19) kN(A ,A ,φ ,φ )k .B+Bm,
+ − + − Xs−1/2,b′−1+ε
±
where
2
B =kφ k +kφ k + kA k +kA k
+ X+s+1/2,b′ − X−s+1/2,b′ µ,+ X+s,b µ,− X−s,b
µX=0(cid:16) (cid:17)
and
kuk = hξish−τ ±|ξ|ibu(τ,ξ) .
Xs,b L2
± τ,ξ
Here u(τ,ξ) is the space-time Fo(cid:13)urier transform of u(t,(cid:13)x). Note that h−τ ±|ξ|i is
(cid:13) b (cid:13)
comparable to h−τ ±hξii.
Webalso need the wave-Sobolev norms
kuk = hξish|τ|−|ξ|ibu(τ,ξ) .
Hs,b L2
τ,ξ
(cid:13) (cid:13)
(cid:13) b (cid:13)
10 S.SELBERGANDA.TESFAHUN
Frequentusewillbemadeofthefactthatkuk ≤kuk ifα≤0,andthatthe
Xa,α Ha,α
±
reverseinequalityholdsifα≥0. Inparticular,itsufficestoprove(3.18)and(3.19)
with the X-norms on the left hand sides replaced by the corresponding H-norms.
3.1. Proof of (3.18) for M =−ǫ ImQνρ(∂φ,∂φ). We shall prove that
µ,1 µνρ
(3.20) Qνρ(∂φ,∂φ) .kφ k2 +kφ k2
Hs−1,b−1+ε + Xs+1/2,b′ − Xs+1/2,b′
+ −
holds if (cid:13) (cid:13)
(cid:13) (cid:13)
1 1 1 1 b b
(3.21) <b,b′ <1, s>max b− , , + , ,
2 2 4 6 3 2
(cid:18) (cid:19)
and ε>0 is sufficiently small.
Observe that
Q (∂φ,∂φ)= ∂ φ ∂ φ −∂ φ ∂ φ ,
jk j ±1 k ±2 k ±1 j ±2
±X1,±2(cid:0) (cid:1)
Q (∂φ,∂φ)= −ih∇i ± φ ∂ φ −∂ φ ih∇i(± φ ) ,
0j 1 ±1 j ±2 j ±1 2 ±2
±X1,±2(cid:0) (cid:0) (cid:1) (cid:1)
where we used ∂ φ=ih∇i(φ −φ ). Since kuk =kuk , it suffices to show
t + − Xs,b Xs,b
± ∓
(3.22) k∂ u∂ v−∂ u∂ vk .kuk kvk ,
j k k j Hs−1,b−1+ε Xs+1/2,b′ Xs+1/2,b′
±1 ±2
(3.23) k∂ (± u)h∇iv−h∇iu∂ (± v)k .kuk kvk .
j 1 j 2 Hs−1,b−1+ε Xs+1/2,b′ Xs+1/2,b′
±1 ±2
The left hand sides are bounded by kI(τ,ξ)k , where
L2
τ,ξ
σ(± η,± (ξ−η))
1 2
(3.24) I(τ,ξ)= |u(λ,η)||v(τ −λ,ξ−η)| dλdη
R1+2 hξi1−sh|τ|−|ξ|i1−b−ε
Z
and σ is either
b b
σ(η,ζ)=|η×ζ|≤|η||ζ|θ(η,ζ)
or
|ζ| |η|
σ(η,ζ)=|hηiζ −η hζi|≤|η||ζ|θ(η,ζ)+ + .
j j
hηi hζi
Here θ(η,ζ) denotes the angle between nonzero vectors η,ζ ∈R2.
We now use the following estimate from [10]:
Lemma 3.2. For all signs (± ,± ), all λ,µ∈R, and all nonzero η,ζ ∈R2,
1 2
1/2
h|λ+µ|−|η+ζ|i+h−λ± |η|i+h−µ± |ζ|i
θ(± η,± ζ). 1 2 .
1 2
min(hηi,hζi)
(cid:18) (cid:19)
Thus we reduce (3.22) and (3.23) to (recalling kuk ≤kuk , α≥0)
Ha,α Xa,α
±
kuvk .kuk kvk ,
Hs−1,b−1/2+ε Hs,b′ Hs−1/2,b′
kuvk .kuk kvk ,
Hs−1,b−1+ε Hs,b′−1/2 Hs−1/2,b′
kuvk .kuk kvk ,
Hs−1,b−1+ε Hs,b′ Hs−1/2,b′−1/2
kuvk .kuk kvk .
Hs−1,0 Hs+3/2,b′ Hs−1/2,b′
Assuming (3.21), all these estimates hold by the following product law.
