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Global existence for a nonlinear Schroedinger–Chern–Simons system on a surface
L’existence d’une solution globale régulière pour un système non-linéaire d’équations de Schroedinger–Chern–Simons
sur une surface
Sophia Demoulini
Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OWB, UK Received 24 March 2005; accepted 25 January 2006
Available online 3 October 2006
Abstract
Global existence of regular solutions for a nonlinear Schroedinger–Chern–Simons system of equations on a two-dimensional compact Riemannian manifold is proved.
©2006 Elsevier Masson SAS. All rights reserved.
Résumé
L’existence d’une solution globale régulière est démontrée pour un système non-linéaire d’équations de Schroedinger–Chern–
Simons sur une variété Riemannienne compacte de deux dimensions.
©2006 Elsevier Masson SAS. All rights reserved.
Keywords:Nonlinear Schroedinger; Chern–Simons; Global existence; Regularity
1. Introduction and statement of global existence
The Ginzburg–Landau energy functional on an oriented two-dimensional compact surface Σ without boundary with a fixed Riemannian metricgis given by the integral
Σ
Vλ(A, Φ)dμg
E-mail address:s.demoulini@dpmms.cam.ac.uk (S. Demoulini).
0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2006.01.004
where dμgis the associated area form and Vλ(A, Φ)=1
2
|B|2+gij
(∇A)iΦ, (∇A)jΦ +λ
4
|Φ|2−12
. (1)
Here∇Ais anS1connection on a complex line bundleL→Σ of whichΦ is a section and the 2-formBdμg is the curvature associated to∇A. If∇ais a fixed connection with curvaturebdμg then there exists a real 1-formAsuch that∇A= ∇a−iAandBdμg=bdμg+dA. Various time dependent models associated to this functional have been considered:
(i) The gradient flow, which is essentially parabolic (once gauge invariance is properly handled). This was studied in [6].
In addition to the gradient flow there are two associated conservative dynamical models:
(ii) the Abelian Higgs model which forms (modulo gauge invariance) a hyperbolic system of semi-linear wave equa- tions. Vortex dynamics for this model were studied in [12,13];
(iii) the Schroedinger–Chern–Simons equations (SCS) introduced in [10], to be studied here (see also [7]). These form (modulo gauge invariance) a system of coupled nonlinear Schroedinger equations for(A, Φ)together with a constraint.
1.1. The (SCS) system and statement of the main result
In the time-dependent caseLextends to a line bundleL=R×LoverR×Σ. Explicitly the dependent variables consist of a time-dependent 1-formA=(A0, A)≡(A0, A1, A2)onR×Σ whereA=A1dx1+A2dx2is a 1-form onΣ,A0(t, x)∈Rand a time dependent sectionΦ of L. Thus at each timet we have a sectionΦ(t )of L and a connection∇A(t )onLas well as the real valued functionA0(t ). The equations are
2μ(∂tA− ∇A0)+ ∇B= −iΦ,∇AΦ, iγ (∂t−iA0)Φ= −1
2AΦ−λ 4
1− |Φ|2
Φ (2)
together with a thirdconstraintequation 2μB=γ
1− |Φ|2 .
Here:T∗Σ→T∗Σ is the complex structure (Hodge dual operation),a, b denotes a real inner product onL, Φ, Φ = |Φ|2andγ , μ, λare positive constants. Note that the constraint equation is preserved by the evolution (since
∂tB=2μ1 iΦ, AΦ = 2μγ ∂t(1− |Φ|2)), however, it is slightly more general to consider the following additional equation forh∈H2(Σ ):
h(x)=B(t, x)− γ 2μ
1− |Φ|2
(t, x)=B(0, x)− γ 2μ
1− |Φ|2
(0, x). (3)
FromA=(A0, A)we form the operatorDAby DAΦ=(∂t−iA0)Φdt+ ∇AΦ
which is a connection onL→R×Σ and, writingEj =∂tAj−∂jA0, the 2-form −iEjdt∧dxj −iBdμg is the associated curvature. Thus by the first equation in (2) above
2μE+ ∇B= −iΦ,∇AΦ.
In conformal co-ordinatesg=e2ρ((dx1)2+(dx2)2)and the area form is then dμg=e2ρdx1∧dx2. For the connec- tion∇Adefined above we letA=e2ρ∇A·∇A. See the first appendix for further explanation of notational conventions and interpretation.
For the caseΣ=R2this system was proposed by Manton [10], who derived it as the Euler–Lagrange equations from a Lagrangian extending the Ginzburg–Landau functional by a Chern–Simons term and the Schroedinger term iΦ, (∂t−iA0)Φ. In [2] and [3] the authors prove local existence and blow-up in H2 (and global existence under
conditions on the initial data inH1) onΣ=R2for a closely related system having negative energy density (an “attrac- tive” nonlinearity, corresponding toλ <0). In this paper we study the case of a positive energy density (a “repulsive”
nonlinearity, corresponding toλ >0) withΣ a two-dimensional surface as described above. We prove a local exis- tence theorem for (2), (3), and then global existence for Φ inH2. In [5] the authors show a global existence for a related system forΣ=R2in whichΦ solves a wave equation. A study of vortex dynamics of (2), (3) is currently undertaken along the lines of [12,13] for the Abelian Higgs model.
Two useful ways to think of (2), (3) are:
(i) In Coulomb gauge divA=0 it is possible to reformulate this system as anonlocalSchroedinger equation forΦ, as (3) and the divergence of the first equation in (2) then determineA0andAin terms ofΦas nonlocal functionals (by solving elliptic equations) inxat each timet. This is the approach adopted in [2]. Here we make a different gauge choice, the parabolic gaugeA0=divAin whichA0andA(through the same equations) are determined byΦnonlocally in(t, x)(by solving a heat equation).
(ii) With appropriate choice of symplectic structure the equation is a constrained Hamiltonian system with Vλ as Hamiltonian and (3) is a constraint (i.e. its time-derivative vanishes identically as a consequence of (2)). A useful consequence of this second formulation is the conservation ofVλwhich will be used later.
In view of interpretation (i) it is to be expected that control ofΦ in a sufficiently strong norm for all time ensures the existence of a global solution and this idea is used to prove the following in the parabolic gauge (the spaces used are explained below):
Theorem 1.1 (Global existence). Consider the Cauchy problem for (2), (3) with initial data Φ(0)∈H2(Σ ) and A(0)=(A0(0), A(0))∈H3(Σ )satisfyingA0(0)=0=divA(0). In the parabolic gauge withA0=divA, there exists a unique global solution(A, Φ)∈C([0,∞);H1(Σ )×H2(Σ ))∩C1([0,∞);L2(Σ )×L2(Σ ))such thatΦsatisfies the estimate
Φ(t )
H2(Σ )ceαeβt (4)
for some positive constantsc, α, βdepending only on(Σ, g), the constantsγ , μ, λand the initial data.
Brezis and Gallouet in [4] prove an analogous result for the nonlinear Schroedinger equation i∂tu−u+ |u|2u=0.
The crucial point there was the use of the inequality (valid, e.g., foru∈H2(R2))
|u|L∞C
1+
ln(1+ |u|H2)
(5) withC=C(|u|H1), in conjunction with standardL2 estimates for the differentiated equation to provide control of theH2norm of the solution at each time. For this to work the cubic structure of the nonlinearity and the square root in (5) turn out to be important. The main point of the present article is that for (2)the same balancing of nonlinear effects occurs even in the presence of the additional nonlinearity provided by the presence of A in the equation:
the constraint equations ensure that the overall strength of these can be estimated in the same manner as the cubic nonlinearity. This is achieved by careful use of the constraint equations to estimate the various commutator terms which appear on differentiation of the equation, together with a covariant form of the Brezis–Gallouet inequality (5).
Before stating the inequality we discuss the spaces in which we work (more details can be found in the appendix).
The space ofHk connections is the space of operators∇Aof the form∇A= ∇a−iAwithA∈Hk(Ω1(Σ )), which usually will be written Hk or Hk(Σ )suppressingΩ1. For any given measurable connection 1-formA=(A1, A2) with measurablek-order derivatives, we define the space ofHAk sectionsΦ as
HAk(Σ )=
Φ: Σ→C:
|α|k
|∇AαΦ| ∈L2
with the usual norm. For x ∈Σ,|Φ(x)|2= Φ, Φh(x)and |∇AΦ|2= |∇AΦ|2g×h=gij∇AiΦ,∇AjΦh whereg is the metric onΣ andh is the inner product on L (cf. Appendix A); we suppressh in notation. Also when the
background connectiona is implied we often write only ∇ in place of∇a and the norm|Φ|Hk rather than |Φ|Hak. Certain Sobolev imbedding theorems are valid, see Lemma 1.2 and Appendix A. IfAis time dependent thenHA(t )1 is a time-dependent norm. Supposing (as will be the case below) thatA(t,·)varies continuously witht inH1(Σ )then the correspondingHA(t )1 norms forΦ∈H1(Σ )are equivalent and continuously varying int, both of which can be seen from
|∇A(t )Φ|L2|∇A(τ )Φ|L2+ A(t )−A(τ ) Φ
L2
|∇A(τ )Φ|L2+c A(t )−A(τ )
H1|Φ|H1
by the Kato and Sobolev inequalities (see below).
In the remainder of this section in Lemmas 1.2, 1.3 and 1.4 we show covariant versions of known inequalities.
These are derived for a complex sectionΦ of a line bundleLwith connectionA(the time variable is fixed andA, Φ are time-independent) and obtained in two stages from their corresponding statements on Euclidean space: once derived for the two-dimensional Riemannian manifoldΣ, the covariant version onLis then derived from that. The first stage is easily achieved in the usual way with a partition of unity, see Appendix A.
Lemma 1.2(Covariant version of the Sobolev and Gagliardo–Nirenberg inequalities). For(Σ, g)as above and for (A, Φ)∈(H1×HA2)(Σ )then∇AΦ∈L4(Σ )and
|∇AΦ|L4c|∇AΦ|H1
A (6)
and also for all1p <∞,HA2→WA1,p→L∞continuously onΣ. Also
|∇AΦ|L4c|∇AΦ|1/2L2
|∇AΦ|1/2L2 + |∇A∇AΦ|1/2L2
(7)
wherecdepends only on(Σ, g).
Proof. For real valuedu∈H1(Σ )we have the Sobolev and Gagliardo–Nirenberg inequalities, respectively,
|u|L4c|u|H1 and |u|L4c|u|1/2L2|u|1/2H1. (8) (Both of these follow the corresponding standard forms of the Sobolev and Gagliardo–Nirenberg inequalities onR2 by a partition of unity cf. Appendix A.) We recall the Kato inequality,
∇|Φ|
Lp|∇AΦ|Lp
and letu= |∇AΦ| ∈H1(Σ ). By (8) we have
|∇AΦ|2L4c|∇AΦ|L2
|∇AΦ|L2+ ∇|∇AΦ|
L2
and by the Kato inequality
c|∇AΦ|L2
|∇AΦ|L2+ |∇A∇AΦ|L2
which proves (7). The Sobolev inequality and imbeddings follow in the same way. 2
Lemma 1.3(Covariant version of the Garding inequality). ForΨ =(A, Φ)such that the norms onΣ appearing below are finite we have
|∇A∇AΦ|L2|AΦ|L2+c|B|1/2L∞|∇AΦ|L2+c|Φ|1/2L∞|∇AΦ|1/2L2 |∇B|1/2L2 (9) wherecis a number depending only on(Σ, g).
Proof. Recall that ·,· is the inner product onL. Using a local co-ordinate system{xj}2j=1 onΣ and using the upper/lower index notation we can define two real 1-forms by
αj≡
(∇A)jΦ, AΦ
, βk≡
(∇A)jΦ, (∇A)k(∇A)jΦ . Stokes theorem implies
∇jαjdμg=
∇kβkdμg=0 since∂Σ= ∅; but expanding out these divergences as
∇jαj− ∇kβk= AΦ, AΦ −
(∇A)j(∇A)kΦ, (∇A)k(∇A)jΦ
+gj lgkm
(∇A)mΦ,
(∇A)j, (∇A)k
(∇A)lΦ +gj lgkm
(∇A)jΦ, (∇A)k
(∇A)m, (∇A)l Φ and integrating then implies (9) by use of (A.5) and (A.6). 2
Lemma 1.4(Covariant version of the Brezis–Gallouet inequality). IfA∈H1(Σ )andΦ∈HA2(Σ )then
|Φ|L∞(Σ )c
1+ |Φ|H1
A
ln
1+ |Φ|H2
A (10)
wherecdepends only on(Σ, g).
Proof. The form of this inequality for realu∈H2(Σ )is
|u|L∞(Σ )c
1+ |u|H1(Σ )
ln
1+ |u|H2(Σ ) (11)
where, throughout in this proof, cis a generic constant independent of udepending only on (Σ, g). This follows from the inequality foru∈H2∩H1(R2)in [4] using a partition of unity (see Appendix A). Now apply (11) with u(x)= |Φ(x)|2forx∈Σ; by the Kato inequality (and using the unitarity property ofA, cf. Appendix A)
|∇u|L2c|Φ|L∞|∇AΦ|L2 and
|∇∇u|L2c
|∇AΦ|2L4+ |Φ|L∞|∇A∇AΦ|L2
c
|∇AΦ|2H1
A+ |Φ|L∞|∇A∇AΦ|L2
c|Φ|2H2 A
using Lemma 7. Altogether with (11) this leads to the inequality
|Φ|2L∞c
1+ |Φ|L∞|Φ|H1
A
ln
1+ |Φ|H2
A .
Takec >1 without loss of generality, then this leads to (10). 2 2. Statement of local existence
The system comprising (2), (3) is gauge invariant: for smooth real valued functionsgonR×Σthe triple(A0, A, Φ) is a smooth solution if and only if
eig·(A0, A, Φ)≡
A0+∂tg, A+dg,eigΦ
(12) is also a smooth solution. (Clearly this action can be extended to more general weak solutions.) To circumvent this degeneracy we consider theparabolic gaugein which
divA=A0.
This choice of gauge fixes the positive direction in time, so from now on we solve fort0. (The choice divA= −A0 fixes the opposite direction; the existence result obtained here is then similarly valid fort0.)
As mentioned in Section 1, we will prove local and global existence for the augmented system of equations, coupling the equations of (2) with a constraint equation
B− γ 2μ
1− |Φ|2
=h
for generalh∈H2(Σ )and determined by the initial data. (The existence for the original system follows as the special case ofh=0.) Local existence is established by the following theorem which is proved in Section 4.
Theorem 2.1(Local existence). For initial dataΦ(0)∈H2(Σ ), A(0)∈H3(Σ )together withA0(0)=divA(0)=0, there is a positive timeTlocwhich depends continuously on the above norms of the initial data, and there is a solution (A0, A, Φ)of(2),(3)satisfying the gauge conditionA0=divA, and of regularity
A∈C
[0, Tloc], H1(Σ )
∩C1
[0, Tloc], L2(Σ ) , Φ∈C
[0, Tloc], H2(Σ )
∩C1
[0, Tloc], L2(Σ ) .
Furthermore, the solution is unique in these spaces and satisfies the conservation laws Vλ
A(t ), Φ(t )
=Vλ
A(0), Φ(0)
, (13)
Φ(t )
L2= Φ(0)
L2. (14)
Remark.The regularity ofΦ implies by the constraint equation (3) that
∗dA∈C
[0, Tloc], H2(Σ )
∩C1
[0, Tloc], L2(Σ )
(however, divAis only proved (Lemma 4.1) to be continuous intoL2 in this gauge, thus overallAis continuous intoH1).
3. Proof of the global existence theorem
Assuming Theorem 2.1, we have a local solution(A, Φ)of (2)–(3) defined on an interval[0, Tloc]. In this section it is shown that this solution can be extended (in the same spaces) to a solution for infinite time. From the construction in the proof of Theorem 2.1 (in Section 4), the timeTlocdepends continuously on the norms of the initial data, and we haveA(0)∈H3andΦ(0)∈H2. By standard local existence theory there is a maximal timeTmaxTlocsuch that fort∈ [0, Tmax)a solution(A, Φ)of the system exists in the same gauge and spaces of Theorem 2.1 and for the same initial data. The bounds in the norm defined in Theorem 2.1 are not yet proved valid up to timeTmax; however, a priori bounds derived below from the energy and the equations do hold and will be used to showTmax= +∞.
Denote byca generic constant which depends on(Σ, g), the Sobolev norms of the initial data,h, the constants γ , μ, λ, and the energy. Unless stated otherwise the norms below are taken overΣat fixed timet and the dependence ontis omitted where no confusion is possible.
Differentiate in time equation (2) forΦlettingV =4γλ (1− |Φ|2)
∂t−iA0− i 2γA
(∂t−iA0)Φ=
− i
2γA, ∂t−iA0
Φ+iV (∂t−iA0)Φ+i(∂tV )Φ
=2 i
2γE· ∇AΦ+ i
2γ(divE)Φ+iV (∂t−iA0)Φ+i(∂tV )Φn (15) (withE· ∇AΦ =gijEi∇AjΦ) and Lemma B.5 applies to give an estimate for |(∂t −iA0)Φ(t )|L2 as in (B.11).
Algebraically from (2) this implies an estimate for|AΦ(t )|L2 and by the Garding inequality (9) this in turn gives an estimate for|∇A∇AΦ(t )|L2. Altogether we have fort∈ [0, Tmax),
|∇A∇AΦ|L2c
1+ A(0)Φ(0)
L2+ V (0)Φ(0)
L2+ |V Φ|L2+ |B|1/2L∞|∇AΦ|L2+ |Φ|1/2L∞|∇AΦ|1/2L2 |∇B|1/2L2
+ t
0
|E· ∇AΦ|L2+ |divEΦ|L2+ (∂tV )Φ
L2
ds
. (16)
We recall from Theorem 2.1 that the local solutions have constant energy, Vλ
A(t ), Φ(t )
=Vλ
A(0), Φ(0) . Observing that for arbitrary >0
4 λV
1− |Φ|22
1− 1 2
+(1−2)|Φ|4
we infer that|Φ|L∞(L4)and|∇AΦ|L∞(L2)are bounded uniformly int and hence sup
t >0
|Φ|H1
A(t )+ |Φ|Lp(t )
c=c
p,V(0), Σ, g, λ
(17)
for all 1p <∞. Now observe from Eqs. (2), (3) that E,∇E,∇B are “schematically” given by E= ∇B+ Φ,∇AΦ,∇B= Φ,∇AΦ,∇E= Φ,∇A∇AΦ + |∇AΦ|2), so that the Sobolev (6) and Holder inequalities im- ply bounds in terms of the H2 andL∞ norms ofΦ (where any choice made is with the view that (10) will be eventually used on|Φ|L∞)
|V Φ|L2c
|Φ|L2+ |Φ|3L6
c
|Φ|L2+ |Φ|3H1 A
,
|B|L∞c1
1+ |Φ|2L∞
,
|E|L2c
1+ |Φ|L∞|∇AΦ|L2 ,
|∇E|L2c
1+ |Φ|L∞|∇A∇AΦ|L2+ |∇AΦ|L2|∇AΦ|H1
A
.
(18)
Based on these, and using the Sobolev (6) and the interpolation (7) inequalities (where the choice between the two is essential toavoid superlinearterms in|∇A∇AΦ|L2 which would cause the following argument to fail), we obtain
|E· ∇AΦ|L2c|E|L4|∇AΦ|L4, (by (7)) c|E|1/2L2|E|1/2H1
A
|∇AΦ|1/2L2 |∇AΦ|1/2H1 A
,
(by (18)) c
1+ |Φ|1/2L∞|∇AΦ|1/2L2
1+ |Φ|1/2L∞|∇AΦ|1/2L2 + |Φ|1/2L∞|∇A∇AΦ|1/2L2
+ |∇AΦ|1/2L2
|∇AΦ|1/2L2 + |∇A∇AΦ|L2
|∇AΦ|1/2L2
|∇AΦ|1/2L2 + |∇A∇AΦ|1/2L2
which by (17) can be estimated as
c
1+ |Φ|L∞|∇A∇AΦ|L2
. (19)
(Here all norms bounded by the energy are absorbed in the constantc.) Similarly,
|divEΦ|L2 c|∇E|L2|Φ|L∞
c
1+ |Φ|2L∞|∇A∇AΦ|L2
. (20)
The final term under the integral is estimated using (15) (∂tV )Φ
L2c Φ, (∂t−iA0)ΦΦ
L2
c|Φ|2L∞|AΦ|L2
c|Φ|2L∞|∇A∇AΦ|L2 (21)
using thatVt=2Φ,iAΦ. Altogether we obtain from (16)–(21)
|∇A∇AΦ|L2c
1+ |Φ|L∞+ t
0
1+ |Φ|L∞|∇A∇AΦ|L2+ |Φ|2L∞|∇A∇AΦ|L2
ds
c
1+ |Φ|L∞+ t
0
1+ |Φ|2L∞
|∇A∇AΦ|L2ds
. (22)
To this now apply the inequality (10):
Φ(t )
HA(t)2 c
1+ ln
1+ Φ(t )
HA(t)2
+ t
0
1+ Φ(s)
HA2 + Φ(s)
HA2
ln
1+ Φ(s)
HA2 ds
.
Now since√
ln(1+x)/x→0 asx→ +∞so there existsL(c)such that for|Φ(t )|H2
A(t)> L(c)
ln
1+ |Φ|H2
A
1
2c|Φ|H2
A
and hence there exists (another) constantcsuch that Φ(t )
HA(t)2 c
1+ t
0
1+ Φ(s)
HA2+ Φ(s)
HA2
ln
1+ Φ(s)
HA2 ds
≡G(t ).
As the functions in the integrand are increasing we have G(t )c
1+G(t ) 1+ln
1+G(t )
or ln(1+ln(1+G(t )))ct+k(forka constant). Hence|Φ(t )|H2
A(t)G(t )kekect as claimed in (4) for all time t < Tmaxwhich by a standard continuation argument implies that the solution(A, Φ)exists on[0,∞), and this proves the theorem. 2
4. Proof of local existence
The proof of Theorem 2.1 follows a fixed point argument for an iteration as in the procedure layed out for conserva- tion laws in [9]. Here the proof is based on the following two Lemmas 4.1, 4.2. The first lemma shows that a uniform time exists in which all the iteratesΨn=(An, Φn)are bounded in terms of the initial data only in the (high) norm Ψn∈C([0, Tloc], H1×H2(Σ )). The second lemma shows that the iteration (24), (25) is a contraction in the (low) normC([0, Tloc], L2×H1(Σ )), to be precise,Ψnis proved to be Cauchy in this space.
To start, we define and solve the iteration scheme.
Smoothing of the initial data: consider a smooth sequenceΨ0n=(An(0), Φn(0)which approximates the initial data Ψ (0)=(A(0), Φ(0))in the sense that:
Ψ0n−Ψ (0)
H1×H2(Σ )02−n (23)
(implying also thatΨ0nare bounded, uniformly inn, in the same norm in terms of the initial data). Smoothing of the initial data ensures that the iterates below are smooth and well-defined.
Definition of the iteration scheme: given(An−1, Φn−1), letΨn=(An, Φn)be the solution of the approximating system
∂tAn− ∇divAn=F
Φn−1,∇An−1Φn−1
≡Fn−1, (24)
∂t−iAn0− i 2γAn
Φn= λi 4γ
1− Φn−1 2
Φn (25)
with initial dataΨ0nas above and where Fn≡ − 1
2μ∇
h+ γ 2μ
1− Φn 2
− 1 2μ
iΦn,∇AnΦn
(26) where as aboveis the antisymmetric 2×2 tensor. Differentiating (24) and (26) will allow estimation of norms of higher order derivatives: first, takingdof (24) (lettingBn=b+∗dAn) we have∂t∗dAn=2μ1iΦn−1, An−1Φn−1 =
γ
2μ∂t(1− |Φn−1|2)and hence Bn= γ
2μ
1− Φn−1 2
+h(x). (27)
Taking divergence of (24) we obtain an inhomogeneous heat equation
∂tdivAn−divAn= ∇ ·F
Φn−1,∇An−1Φn−1
= ∇ ·Fn−1 (28)
where the right-hand side depends on(Φn−1,∇An−1Φn−1, An−1Φn−1, Bn−1).
Finally we will use the following equations obtained by differentiation of (25). LetEn=∂tAn− ∇An0andVn=
λ
4γ(1− |Φn|2). Differentiation in time gives
∂t−iAn0− i
2γAn
∂t−iAn0 Φn
=
− i
2γAn, ∂t−iAn0
Φn+iVn−1
∂t−iAn0 Φn+i
∂tVn−1 Φn
=2 i
2γEn· ∇AnΦn+ i 2γ
divEn
Φn+iVn−1
∂t−iAn0 Φn+i
∂tVn−1
Φn. (29)
(HereEn· ∇AnΦn=gijEi∇AnjΦn.) Similarly differentiation inxgives
∂t−iAn0− i
2γAn
∇AnΦn
=
∂t−iAn0,∇An
Φn− i
2γ[An,∇An]Φn+i
∇Vn−1
Φn+iVn−1∇AnΦn
=
−iEn+ i
2γ[An,∇An] +i∇Vn−1
Φn+iVn−1∇An
Φn−Φn−1 . (30) The unique solution of the iteration equations: we solve (28), (24), (25) with the understandingAn0=divAn, with smooth initial data (23) to obtain by standard linear theory smooth solutions
Ψn=
An,∇AnΦn
∈C∞
[0,∞)×Σ
forn=1,2, . . . . (31)
To see this, first solve the heat equation (28) which yields aC∞([0,∞)×Σ )solutiondivAn(the regularity att=0 follows as ∂Σ = ∅); following the o.d.e. (24) implies An is C∞ as well on[0,∞)×Σ. To solve the remaining equation (25) we apply Theorems 4.1 and 5.1 in [8] to the operator−iAn(t )−if (t,·):Hs(Σ )→L2(Σ )for smooth real f and heref =i(An0+4γλVn−1)Φn; then the evolution operator at each time 0t <∞preservesHs(Σ )in L2(Σ )for eachs2). This implies that the solution is a smooth sectionΦn∈C∞([0,∞)×Σ )(recall that a fixed smooth background connection is implied for all derivatives ofΦ). The solution for each of these equations is unique and the regularity of the solutions justifies the manipulations in the following lemma.
Lemma 4.1(Uniform bounds for the iterates). There exists a timeT1>0and a constantM >0, both depending only on the initial data and the functionh(andΣ, g, γ , μ, λ), such that for eachnthe solution of (24),(25)given in(31) Ψn=(An, Φn), with smooth dataΨ0ndefined in(23), satisfies
Yn∈C [0, T1]
with Yn
C([0,T1])M (32)
whereYn(t )is the norm, Yn(t )≡ sup
s∈[0,t]
An
L2+ divAn
L2+ ∗dAn
L∞+ Φn
L∞+ ∇AnΦn
L2+ ∇An∇AnΦn
L2
(s).
Proof. We assume inductively thatT1exists for which (32) is valid forn=1, . . . , N−1; we denote byca generic constant depending only onΣ, g, h, γ , μ, λand the initial valuesYn(0)forn=0, . . . , N−1 in (i)–(iv) below. We show that for any 0t < T1, and
YN(t )c+tp
YN(t ), M
(33) wherepis a positive coefficient polynomial which depends (first argument) on the normYNof theNth iterateΨN, and (second argument) on the same norm of the previous 1, . . . , N−1 iterates, inductively assumed to be less thanM for all time 0tT1;pis taken as the sum of the polynomials in (i)–(iv) below. We also assume thatMis chosen so thatM > cand setT1=2p(M,M)M−c . If maxt∈[0,T1]YN(t )Mthen by continuity oft→YN(t )there is a first time 0< tNT1for whichYN(tN)=Mand hence
sup
ttN
YN(t )c+tNp(M, M)c+T1p(M, M) < M (34)
which implies thattN can be taken to beT1otherwise we contradict the choice ofM, T1. Below we also writep(M) in place ofp(M, M).
We now show Eq. (33) is valid to complete the induction argument. For the rest of this proof, all norms shown are implied to bespatially taken overΣpointwise in time fortfort∈ [0, T1]unless indicated by the notation, e.g.C(L2)
indicating the norm forC([0, t], L2(Σ )). The following estimates (35)–(37) will be used in (i)–(iv) below (and applied withn=N). The right-hand side of (24) is given in (26) and is seen to satisfy
Fn
L2c ∇h
L2+ Φn
L∞ ∇AnΦn
L2
(35)
and similarly inLp. Differentiating (26) once, ∇Fn
L2c ∇2h
L2+ Φn,∇An∇AnΦn
L2+ ∇AnΦn 2
L4
c ∇2h
L2+ Φn
L∞ ∇An∇AnΦn
L2+ Φn 2
HAn2
(36)
using the unitarity of the connection and the Sobolev inequality (6) applied to the last term. (Note here that either (6) or (7) can be used but for the corresponding calculation in the proof of global existence the quadratic rate for theH2 norm is not suitable and therefore (7) must be used.)
In the parabolic gaugeEn≡∂tAn− ∇An0=Fn−1from (24). Thus by (35) we can estimate (at eacht) En
L2 Fn−1
L2c
|∇h|L2+ Φn−1
L∞ ∇AnΦn−1
L2
(37)
and by (36) we have a similar bound for|∇En|L2. Also by the Sobolev inequality En
L4 Fn−1
L4c
|∇h|L4+ Φn−1,∇AnΦn−1
L4
(38)
c
|∇h|L4+ Φn−1
L∞ Φn−1
HAn−12
.
In Appendix B a priori estimates are shown for the heat, ordinary differential and Schroedinger equations which are now applied in turn to (28), (24), (25) and (29)–(30) leading to the following four estimates:
(i) Estimate for divAN, AN. Apply the a priori estimate (B.1) for the heat equation in Appendix B to (28) for 2p <∞to obtain
sup
st
divAN 2
L2(s)+ t
0
∇divAN 2
L2(s)ds divAN(0) 2
L2+ct
|∇h|2L2+ ΦN−1 2
L∞(L∞) ∇AN−1ΦN−1 2
L∞(L2)
(39)
and the right-hand side is of the form (33). Moreover, divAn∈H1and by (24) and (26) and with the o.d.e. esti- mate (B.2),
sup
τt
AN
L2(τ )c+tp YN
C([0,t]), M
which is of the form (33). (Alternatively, note that by (39) and (27) we deduce fromLp estimates for the div–curl system ((A.10)) that
AN 2
L2+ ∇AN 2
L2
(τ )c+tp
ΨN, M
(40) withc, p, Mas defined above, which is again (33).)
(ii) Estimate for∇ANΦN. From (25), and the Schroedinger equation estimate (B.5) in Lemma B.4 withVN−1=
λi
4γ(1− |ΦN−1|2), and by (37), sup
st
∇ANΦN 2
L2(s) ∇ANΦN 2
L2(0)+tp(M)sup
st
ΦN(s)
L∞ ∇ANΦN(s)
L2
, (41)
which is of the general form of (33). Also note that as in (B.4) theL2norm is conserved, ΦN
L2(t )= ΦN
L2(0). (42)
(iii) Estimate for(∂t−iAN0)ΦN. We consider (29) (forn=N) ∂t−iAN0 −iAN
∂t−iAN0 ΦN
=2iEN∇ANΦN+i divENΦN+i
∂tVN−1
ΦN+iVN−1
∂t−iAN0 ΦN. To this we apply the estimate (B.11) (the last term does not contribute) to obtain,
