Global solutions of quasilinear wave equations

Abstract

We show that a general class of quasilinear wave equations have global solutions for small initial data as we conjectured in an earlier paper.

Full text

Page 1

arXiv:math/0511461v1 [math.AP] 18 Nov 2005
Global solutions of quasilinear wave equations
Hans Lindblad
University of California at San Diego
February 2, 2008
1Introduction
We show that the Cauchy problem in R1+3:
(1.1)
??g(φ)φ = 0,φ??t=0= φ0,∂tφ??t=0= φ1
has a global solution for all t ≥ 0 if initial data are sufficiently small. Here the curved wave operator is
??g= gαβ∂α∂β, where we used the convention that repeated upper and lower indices are summed over
of φ such that gαβ(0)= mαβ, where m00=−1, m11= m22= m33=1 and mαβ= 0, if α?=β. The result
holds for vector valued φ, in particular for the principal part of Einstein’s equations; φαβ= gαβ−mαβ.
This result was conjectured in [L2] where it was also shown in the spherically symmetric case for
α,β = 0,1,2,3, and ∂0= ∂/∂t, ∂i= ∂/∂xi, i = 1,2,3. We assume that gαβ(φ) are smooth functions
(1.2)−∂2
tφ + c(φ)2△xφ = 0,wherec(0) = 1.
In [L2] there was also a heuristic argument for why the conjecture should be true in general: Consider
(1.3)
?φ = aαβ∂αφ∂βφ + cubic terms,
where here we used multiindex notation and the sum is over 0≤|α|≤|β|≤2 and aαβare constants. If we
neglect derivatives tangential to the outgoing Minkowski light cones and cubic terms, that are known
to decay faster, we get the asymptotic equation for Φ = rφ, introduced by H¨ ormander [H1, H2, H3]:
(1.4) (∂t+∂r)(∂t−∂r)Φ ∼ r−1Amn(∂t−∂r)mΦ (∂t−∂r)nΦ,Amn=1
4
?
|α|=m|,|β|=n
aαβˆ ωαˆ ωβ,ˆ ω=(−1,ω).
Here we have introduced polar coordinates x = rω, ω ∈ S2. The classical null condition introduced by
Klainerman [K1] is that Anm≡ 0 under which Klainerman [K2] and Christodoulou [C] proved global
existence. In [L2] it was observed that the asymptotic equation corresponding to (1.1) has global
solution1, contrary to other cases like ?φ = φt△φ or ?φ = φ2
for all small data, see John [J1, J2]. However, unlike for the classical null condition, the solution of
(1.1) do not behave asymptotically like a solution of a free linear wave equation.
The method of proof of [L2] is integration along characteristics so it does not directly generalize to
the non-symmetric case. However, as observed in [L1], the method of integration along characteristics
t, where solutions are known to blow up
1In [L-R1] we in general say that (1.3) satisfy the weak null condition, if (1.4) has global solution with some decay.
1

Page 2

can still be used to obtain sharp decay estimates assuming weaker decay estimates that can be obtained
from energy estimates for vector fields applied to the solution. This then has to be combined with some
refined energy estimates that take into account that the characteristic surfaces curve asymptotically,
since the solution do not decay as much as a solution of a free linear wave equation.
Recently Alinhac [A2] generalized the result in [L2] to general data for the special case (1.2)2.
[A2] combines ideas from [L1, L2] of how to obtain decay estimates with ideas from [A1] for energy
estimates with weights. Because the asymptotic behavior is different from that of solutions to a free
linear wave equation, [A2] constructs vector fields adapted to the characteristic surfaces at infinity,
which in spirit is similar to the work of Christodoulou-Klainerman [C-K]. Since these depend on the
solution itself, commuting the vector fields with the wave operator leads to a loss of regularity so it has
to be combined with a smoothing procedure, which leads to long schematic commutator estimates.
There is however no need to construct vector fields adapted to the geometry at infinity. In fact we
just use the vector fields for the Minkowski space time. In [L-R3], for Einstein’s equations, we also
got away with just using the regular vector fields, but only because we got additional control from the
wave coordinate condition. The observations here will hopefully will lead to a proof that uses less of
the special structure and applies to a more general class of equations, which is useful in applications.
As mentioned above the proof involves obtaining sharp decay estimates for low derivatives just
assuming a weak decay estimate that later will be obtained from energy estimates for higher derivatives.
The sharp decay estimates uses integration along characteristics as in [L1, L2, L-R2, L-R3]. We adopt
the energy method with weights of [A2], depending on the solution of an approximate eikonal equation.
This is a much easier substitute for energies on characteristic surfaces as I originally planned to use.
The construction of vector fields adapted to the asymptotic behavior of the characteristic surfaces
of [A2] is avoided by considering a family of energy and decay estimates for the vector fields of flat
Minkowski space time, with different decays for different types of derivatives. We prove the following:
Theorem 1.1. Suppose that φ0and φ are smooth functions such that φ0(x) = φ1(x) = 0, when |x| ≥ 1,
and let N ≥ 14. Then there is a constant ε0> 0, such that if?
in Proposition 6.1 with ν=1−cε, for some c>0, and the energy estimates in Proposition 9.1.
|α|≤N
??∂∂αφ0?L2+?∂αφ1?L2?= ε ≤ ε0,
the Cauchy problem (1.1) has a global solution φ for all t ≥ 0. The solution satisfies the decay estimates
We remark that the result is true also for systems φ = (φ1,··· ,φM), in particular the principal
quasilinear part of Einstein’s equations. We also remark that the assumptions on compact support is
not needed and we can include decaying data using energy norms with weights as in [L-R2, L-R3].
Let us now give the strategy of the proof and the main ideas. The proof involves getting sharp decay
estimates for low derivatives assuming weak decay estimates, and energy estimates for high derivatives
assuming sharp decay estimates for low derivatives. The weak decay estimates can then be obtained
from energy estimates using a bootstrap or continuity argument that we describe below. Let
?
(1.5)EN(t) =
?
|I|≤N
|∂ZIφ(t,x)|2dx,
where ZIis a product of |I| of the vector fields, Ωαβ= xα∂β− xβ∂α, S = xα∂αthat span the tangent
space of the forward light cone and have good commutators with the wave operator, and ∂α. (Here
xi= xi, i ≥ 1, x0= −x0= −t.) In view of local existence results it suffices to give a bound for EN(t),
2As mentioned in [A3] the method of [A2] use the special structure of (1.2) and is unlikely to work in general.
2

Page 3

which will be obtained through a continuity argument, see section 11. Fix 0<δ<1. Assuming that
(1.6)EN(t) ≤ 16Nε2(1 + t)δ,
for 0 ≤ t ≤ T, which holds for T = 0, we will show that this bound, implies the same bound with 16
replaced by 8 if ε is sufficient small (independently of T). Using Klainerman-Sobolev inequality and
the assumption of compactly supported data, see sections 10, 11, this gives weak decay estimates:
(1.7)|ZIφ(t,x)| ≤ c0ε(1 + t)−ν, ν > 0,|I| ≤ N − 2.
These weak decay estimates imply the sharp decay estimates in Proposition 6.1, as well as the estimates
for the approximate radial characteristic surfaces in Proposition 5.1, Lemma 5.2 and Lemma 5.3. These
sharp decay estimates for low derivatives are sufficient for the energy estimate in Proposition 9.1 to
hold and we therefore get back a stronger energy estimate if ε > 0 is sufficient small:
(1.8)EN(t) ≤ 8Nε2(1 + t)C0,Nε.
We now give the main ideas for the sharp decay estimates. We will try to mimic the integration
along characteristic that was done in the radial case in [L2], by expressing the wave operator in spherical
coordinates and a null-frame, using the weak decay estimates to control the angular derivatives. The
discussion below will be a bit technical, but it is useful to get a feeling for how the different kind of
terms are dealt with since the structure of the argument is the same also for the energy estimates.
In section 2 we express the inverse of the metric in terms of a nullframe:
(1.9)gαβ= −1
2
?Lα
1Lβ+ LαLβ
1
?+ γαβ,
where
Lα
γαβ= δABAαBβ+ HLLLαLβ+ HALAαLβ+ HALLαAβ+ HABAαBβ.
1= Lα− HLLLα− 2HLLLα− 2HLAAα,(1.10)
(1.11)
Here HLLetc. are the components of Hαβ= gαβ− mαβin the Minkowski null frame
(1.12)L = (1,−ω),L = (1,ω),A,B ∈ S2,δαβAαBβ= δAB.
In section 3 we use (1.9) to decompose the wave operator:
(1.13)
???
?
2Lα
1∂α+ℓ
r
?
(r∂qφ) − r??gφ
??? ?
?
1≤|k|≤2
r|k|−1|¯∂kφ|,∂q=1
2(∂r−∂t),
where ℓ = δABHAB+ 4HLL− 2HLL, and ∂ ∈ {L,A,B} are derivatives tangential to the outgoing
Minkowski light cones, that can be estimated in terms of the vector fields:
(1.14)(1 + |t − r|)|∂φ| + (1 + t + r)|∂φ| ?
?
|I|=1
|ZIφ|.
3

Page 4

Note that when |t − r| > t/2 this together with (1.7) gives the sufficient εt−1decay for all derivatives
but when |t−r| is close to the light cone we are missing one derivative perpendicular to the light cone.
In section 4 we integrate (1.13) along the flow lines of the vector field 2Lα
also get an estimate for a derivative perpendicular to the outgoing light cones r∂qφ which yields
1∂α, from |t−r| = t/2, to
(1.15) (1 + t + r)|∂φ(t,x)| ≤ C sup
0≤τ≤t
?t
?
|I|≤1
?ZIφ(τ,·)?L∞
+ C
0
?
(1 + τ)???gφ(τ,·)?L∞(Dτ)+
?
|I|≤2
(1 + τ)−1?ZIφ(τ,·)?L∞(Dτ)
?
dτ,
If H=0, (1.13) is the decomposition in radial and spherical coordinates and (1.15) was used in [L1].
In section 6.1 we use the weak decay estimates (1.7) in (1.15) to get the sharp decay estimates
(1.16)|∂φ| ≤ c1ε(1 + t)−1,|φ| ? c1ε(1 + |t − r|)(1 + t)−1.
The last inequality follows by integrating the first from r=t+1 where φ vanishes. If (1.16) had been
true also for Zφ it would have been easy, but there is a small loss that requires a delicate analysis.
With the sharper decay (1.16) for H(φ) the decomposition of the wave operator (1.13) simplifies
to
???Lα
where p = r + t and
(1.17)
2∂α(p∂qφ) − r??gφ
??? ≤
C
1+ t
?
|I|≤2
|ZIφ|,
(1.18)Lα
2∂α=?Lα− HLLLα?∂α= (∂t+ ∂r) + HLL(∂r− ∂t).
In section 5 we study the integral curves of the vector field (1.18) since we will integrate (1.17).
Let q = r − t, p = r + t and ω = x/|x|, and introduce the radial characteristics q = q(s,ρ,ω) by
(1.19)dq/ds = 2HLL, when |t − r| ≤ t/2,q = ρ,when |t − r| ≥ t/2,p = 2s.
Equivalently let ρ be the solution of a radial eikonal equation:
(1.20)Lα
2∂αρ = 0,when|t − r| ≦ t/2,ρ = r − t,when|t − r| > t/2.
ρ behaves roughly like q:
(1.21)
?1 + t
1+|ρ|
?−c1ε≤∂ρ
∂q≤
?1 + t
1+|ρ|
?c1ε,
?1 + t
1+|ρ|
?−c1ε≤1 + |ρ|
1 + |q|≤
?1 + t
1+|ρ|
?c1ε.
This comes from differentiating (1.20): Lα
factor using the estimate (1.16) for H(φ), as was observed in the spherically symmetric case in [L2].
In section 6.2 we prove the following sharp decay estimates for second derivatives:
2∂α∂qρ+2∂qHLL∂qρ = 0, and multiplying by the integrating
(1.22)|∂φ| ≤
c1ε
1 + t
1
(1 + |ρ|)ν,|∂2φ| ≤
c2ε
1 + t
???∂ρ
∂q
???
1
(1 + |ρ|)1+ν,ν > 0.
4

Page 5

The first estimate follows from integrating (1.17) along the integral curves of L2from t = 2|ρ| using
(1.7). For the proof of the second we note that since [∂ρ,Lα
2∂α] = 0 it follows from (1.17) (c.f. [L2])
(1.23)
???Lα
2∂α(p∂ρ∂qφ) − r∂ρ??gφ
??? ≤
Cρ−1
1 + t + r
q
?
|I|≤2
|∂ZIφ|,where∂ρ= ρ−1
q∂q.
The second estimate in (1.22) follows from integrating (1.23) using (1.14) and (1.21).
For vector fields we are not quite as lucky and there is a loss in the strong decay estimate:
(1.24)|Zφ| ≤ c2ε(1 + t)−1+c2ε(1 + |q|)1−ν≤
c2ε
1 + t
?
(1 + |q|) + (1 + t)c2ε/ν?
.
In fact, by (1.22) the commutator is
(1.25)
?
|I|≤1
????gZIφ??≤ C
?
|I|≤1
|ZIH||∂2φ| ≤c′
2ερq
1 + t
?
|I|≤1
|ZIφ|
1 + |ρ|.
If we use (1.17) applied to ZIφ in place of φ and also commute ρ−1
term due to that Lα
q
through (1.17), we get an extra
2∂αρq= −2ρq∂qHLL:
(1.26)
?
|I|≤1
???Lα
2∂α(p∂ρZIφ)
??? ≤ c2ε
?
|I|≤1
?|ZIφ|
1 + |ρ|+ |∂ρZIφ|
?
+
Cρ−1
1 + t + r
q
?
|I|≤3
|ZIφ|.
Since φ = 0 when ρ ≤ −1 we can estimate |ZIφ|/(1 + |ρ|) by the derivative |∂ρZIφ| we get if we first
integrate (1.26) from t = 2|ρ| where we can use (1.7) and (1.14):
(1.27)M(t) ≤
?t
1
c2ε
1 + τM(τ)dτ + c0ε whereM(t) = (1 + t) sup
|ρ|≤t/2
(1 + |ρ|)ν?
|I|≤1
|∂ρZIφ|
which by a Gronwall type of argument implies that M(t) ≤ c0(1 + t)c2εfrom which (1.24) follows.
For more vector fields there is a problem with the most straightforward approach. We have
?
(1.28)|??gZIφ| ? |ZIφ||∂2φ| +
|J|≤1, |K|=|I|−1
|ZJφ||∂2ZKφ| +
?
|J|+|K|≤|I|, |J|<|I|, |K|<|I|−1
|ZJφ||∂2ZKφ|.
The first term can be handled as above and the terms in the last are lower order. However the problem
is the term with |K| = |I| − 1 and |J| = 1 which is highest order. Using (1.14) and (1.24)
(1.29)|Zφ|
?
|K|=|I|−1
|∂2ZKφ| ≤
c2ε
1 + t
?
|J|=|I|
|∂ZJφ| +
c2ε
(1 + t)1−c2ε/ν
?
|K|=|I|−1
|∂2ZKφ|.
The first sum can be handled as above, however the lack of decay in the last sum cause a problem. The
estimate (1.24) can not be improved to get the needed ε(1 + t)−1decay close to the light cone. One
could use modified vector fields that take into account the bending of the light cones at infinity. The
modified rotations are defined by (?Ωφ)(q,p,ω)=Ω(φ(q(ρ,s,ω),2s,ω)), where q(ρ,s,ω) is as in (1.19).
This however leads to the loss of regularity encountered by [A2] for the energy estimate. We will
5