The Nonlinear Stability of the
Maxwell-Born-Infeld System
(Arxiv reference: 1008.5018)
Jared Speck
jspeck@math.princeton.edu
Princeton University
September 1, 2010
The MBI system
The MBI system is a nonlinear theory of classical
electromagnetism. As shown (independently) by Boillat
and Plebański, it is the unique theory that is derivable from
an action principle, and that satisfies the following 5
postulates:
1
2
3
4
5
The electromagnetic energy associated to a
stationary point charge is finite.
The field equations transform covariantly under the
Poincaré group.
The field equations reduce to the linear
Maxwell-Maxwell equations in the weak field limit.
The field equations are covariant under a Weyl
(gauge) group.
The solutions to the field equations are not
birefringent.
References of relevance
Electromagnetic field theory without divergence
problems. I. The Born legacy (Kiessling, 2004)
Electromagnetic field theory without divergence
problems. II. A least invasively quantized theory
(Kiessling, 2004)
Asymptotic properties of linear field equations in
Minkowski space (Christodoulou & Klainerman, 1990)
The Action Principle and Partial Differential Equations
(Christodoulou, 2000)
Global existence for small initial data in the
Born-Infeld equations (Chae & Huh, 2003)
A possible error in the literature
Previous authors have used Lorenz gauge:
F = dA, ∇κ Aκ = 0.
The L2 energy estimate for ∇A appears to be
incorrect, and is not fixable in any obvious manner.
It is not clear whether or not the MBI equations are
hyperbolic in A in the Lorenz gauge.
This same problem appears to exist for typical
quasilinear perturbations of linear Maxwell-Maxwell
theory.
We resolve this difficulty by working directly with F.
Our method has other advantages: very geometric +
sharp decay estimates.
The MBI equations in 1 + 3 dimensional
Minkowski space
The unknown is the Faraday tensor F, which is a two-form.
The equations are
dF = 0,
} due to Maxwell
dM = 0
def
(⋆F µν + (2) Fµν )
Mµν = `−1
(MBI)
(Maxwell√tensor; relation due to Born and Infeld)
def
`(MBI) = 1 + (1) − 2(2)
def
(1) = 12 (g −1 )ζκ (g −1 )ηλ Fζη Fκλ = ∣E∣2 − ∣B∣2
def
(2) = 14 (g −1 )ζκ (g −1 )ηλ Fζη ⋆F κλ = Ei B i
The Lagrangian and hµνκλ
√
=
1
−
1 + (1) − 2(2) = 1 − `(MBI)
(MBI)
= − 21 (1) + quartic terms = ⋆L (linear theory ) + quartic(F)
⋆L
dM = 0 ⇐⇒ hµνκλ ∇µ Fκλ = 0,
(ν = 0, 1, 2, 3)
∂ 2⋆L
hµνκλ = − 12 (∂Fµν )(∂Fκλ )
= − 12 [(g −1 )µκ (g −1 )νλ + (g −1 )µλ (g −1 )νκ ] + quadratic(F)
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
linear theory
Heuristics
Moral reason for stability:
MBI = Maxwell-Maxwell + cubic(F, ∇F)
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
quasilinear
Moral reason for recovering the linear decay properties:
The nonlinearities depend on the (i) , which have a
special quadratic null form structure.
Important surfaces in Minkowski space
t ∈ R, x ∈ R3
def
Cs− = {(t ′ , x ′ ) ∣ ∣x ′ ∣ + t ′ = s} are the ingoing null cones
def
Cq+ = {(t ′ , x ′ ) ∣ ∣x ′ ∣ − t ′ = q} are the outgoing null cones
def
Σt = {(t ′ , x ′ ) ∣ t ′ = t} are the constant time slices
def
Sr ,t = {(t ′ , x ′ ) ∣ t ′ = t, ∣x ′ ∣ = r } are the Euclidean
spheres
Null frame and null coordinates
Null frame: {L, L, e1 , e2 }
def
L = ∂t − ∂r is tangent to the ingoing cones
def
L = ∂t + ∂r is tangent to the outgoing cones
e1 , e2 are orthonormal, & tangent to the spheres
Null coordinates (useful for expressing decay rates)
q = r − t (constant on outgoing cones)
s = r + t (constant on ingoing cones)
Null decomposition of F
With g/ µν = gµν + 12 (Lµ Lν + Lµ Lν ), µν = 12 µνκλ Lκ Lλ , we
define
def
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
The 6 components of F ⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
ν
def
ν
αµ = g/ µ Fνλ Lλ = BAD,
def
ν
αµ = g/ µ Fνλ Lλ = good,
def 1
ρ
= 2 Fκλ Lκ Lλ = good,
def 1 κλ
σ
= 2 Fκλ = good.
def
g/ µ projects g−orthogonally onto the Sr ,t
Null forms
(i) = ci Q(i) (F, F), each Q(i) (F, F) is a null form
def
Q(1) (F, G) = F κλ Gκλ
= −δ AB αA [F]αB [G] − δ AB αA [G]αB [F]
− 2ρ[F]ρ[G] + 2σ[F]σ[G]
Q(2) (F, G) = ⋆F κλ Gκλ
def
= AB αA [F]αB [G] + AB αA [G]αB [F]
− 2σ[F]ρ[G] − 2ρ[F]σ[G]
Conformal Killing fields
CKFs satisfy LZ gµν = φZ gµν , φZ is a function.
Lie algebra of Minkowski CKFs has 15 generators:
T(µ) = ∂µ , translations
Ω(µν) = xµ ∂ν − xν ∂µ , rotations and boosts
S = x κ ∂κ , scaling
K(µ) = −2xµ S + gκλ x κ x λ ∂µ , accelerations
def
Z = {T(µ) , Ω(µν) , S}0≤µ<ν≤3 = {Z 1 , ⋯, Z 11 }
If I = (ι1 , ⋯, ιk ), ιi ∈ {1, 2, ⋯, 11} for 1 ≤ i ≤ k , then
def
LIZ = LZ ι1 ○ ⋯ ○ LZ ιk
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
iterated Lie derivatives
Norms and seminorms (q = r − t, s = r + t)
∣F∣VW = ∑V ∈V,W ∈W ∣V κ W λ Fκλ ∣,
V, W ∈ {L, T , U},
def
def
def
L = {L},
T = {L, e1 , e2 },
U = {L, L, e1 , e2 }
∣Q(1) (F, G)∣ = ∣c1 F κλ Gκλ ∣
≲ ∣F∣LU ∣G∣ + ∣F∣∣G∣LU + ∣F∣T T ∣G∣T T
def
∤ F ∤2 = (1 + q 2 )∣α∣2 + (1 + s2 )∣α∣2 + (2 + q 2 + s2 )(ρ2 + σ 2 )
def
∤ F ∤2LZ ;N = ∑∣I∣≤N ∤ LIZ F ∤2
def
∦ F(t) ∦2LZ ;N
= ∫R3 ∤ F(t, x) ∤2LZ ;N d 3 x
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
control for global existence
Global Sobolev inequality of C & K
Let F be any two-form on R1+3 , and recall that
def
def
def
r = ∣x∣, q = r − t, s = r + t.
Lie derivative version: if ∣I∣ ≤ N − 2, then
∣LIZ F∣ ≲ (1 + s)−1 (1 + ∣q∣)−3/2 ∦ F ∦LZ ;N ,
∣LIZ F∣LU ≲ (1 + s)−2 (1 + ∣q∣)−1/2 ∦ F ∦LZ ;N ,
∣LIZ F∣T T ≲ (1 + s)−2 (1 + ∣q∣)−1/2 ∦ F ∦LZ ;N .
Covariant derivative version: if 0 ≤ k + l + m ≤ N − 2, then
∣∇kL ∇lL ∇
/ (m) α(t, x)∣ ≲ (1 + s)−1−l−m (1 + ∣q∣)−3/2−k
× ∦ F ∦LZ ;N ,
∣∇kL ∇lL ∇
/ (m) (α(t, x), ρ(t, x), σ(t, x))∣ ≲ (1 + s)−2−l−m (1 + ∣q∣)−1/2−k
× ∦ F ∦LZ ;N .
The main stability theorem
Theorem
Let N ≥ 3. If ∥(B̊, D̊)∥H N is sufficiently small, then these
1
data launch a unique classical solution F to the MBI
system existing on the spacetime slab (t, x) ∈ [0, ∞) × R3 .
Furthermore, there exists a C∗ > 0 such that
∦ F(t) ∦LZ ;N ≤ C∗ ∥(B̊, D̊)∥H N
1
holds for all t ≥ 0.
Finally, the solution decays according to the global
Sobolev inequality. This is the same rate of decay
possessed by solutions to the linear Maxwell-Maxwell
equations.
Blowup results
Yann Brenier (2002) and J. Speck (2008) gave sharp
blow-up criteria for plane-symmetric solutions to the MBI
system.
Basic idea of the stability proof
Proposition
If the solution blows up at time Tmax , then
limt↑Tmax ∦ F(t) ∦LZ ;3 = ∞.
We will rule out the blow-up scenario by studying
energies EN [F(t)] ≈∦ F(t) ∦LZ ;N .
EN [F(t)] will constructed out of the canonical stress
ν
Q̇ µν and a well-chosen timelike vectorfield K .
We will apply the divergence theorem to the current
ν
def
J̇ µ = −Q̇ µν K and use the special structure of J̇ 0 and
of ∇µ J̇ µ .
The result will be a differential inequality for EN [F(t)],
which will force it to remain small.
Equations of variation
The equations of variation are defined to be
∇λ Ḟµν + ∇µ Ḟνλ + ∇ν Ḟλµ = Jλµν (= 0 for us),
ν
hµνκλ ∇µ Ḟκλ = `−1
(MBI) I .
The latter are the Euler-Lagrange equations of a
linearized Lagrangian L˙ ∶
∂ 2⋆L
1
def 1
Ḟζη Ḟκλ = − hζηκλ (F)Ḟζη Ḟκλ .
L˙ =
2 (∂Fζη )(∂Fκλ )
4
The specific algebraic structure of the Iν in the case
Ḟ = LIZ F is extremely important.
Canonical stress
∂ L˙
Ḟνζ + δνµ L˙ }
Ḟµζ
1
= H µζκλ Ḟκλ Ḟνζ − δνµ H ζηκλ Ḟζη Ḟκλ ,
4
def
Q̇ µν = `(MBI) { − 2
where
def
H µζκλ = `(MBI) hµζκλ .
Properties of Q̇ µν
Bad properties of Q̇ µν
Q̇µν ≠ Q̇νµ
∇µ Q̇ µν ≠ 0
Good properties of Q̇F [Ḟ, Ḟ] ∶
Q̇ µν ξµ X ν ≥ 0 for well-chosen ξ, X
∇µ Q̇ µν does not depend on ∇Ḟ
MBI canonical stress
1
1
+ `−2
{ − Fµζ Ḟνζ F κλ Ḟκλ + gµν (F κλ Ḟκλ )2 }
2 (MBI)
4
1
1
⋆ ζ
⋆ κλ
⋆ ζη
2
+ (1 + 2(2) `−2
(MBI) ){ − F µ Ḟνζ F Ḟκλ + gµν ( F Ḟζη ) }
2
4
1
1
ζ
⋆ κλ
ζη
⋆ κλ
+ (2) `−2
(MBI) {Fµ Ḟνζ F Ḟκλ − gµν F Ḟζη F Ḟκλ }
2
4
1
1
⋆ ζ
κλ
ζη
⋆ κλ
+ (2) `−2
(MBI) { F µ Ḟνζ F Ḟκλ − gµν F Ḟζη F Ḟκλ },
2
4
1
(Maxwell) def
Q̇µν
= Ḟµζ Ḟνζ − gµν Ḟζη Ḟ ζη .
4
(Maxwell)
Q̇µν = Q̇µν
Energy current
Morawetz type conformal Killing field:
1
def
K = K(0) + T(0) = {(1 + s2 )L + (1 + q 2 )L}.
2
The energy current:
ν
J̇Fµ [Ḟ] = −Q̇ µν K ,
def
J̇F0 [Ḟ] = Q̇(T(0) , K ) = “density of energy"
=∤ Ḟ ∤2 , from linear theory
³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹·¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ
1
= {(1 + q 2 )∣α̇∣2 + (1 + s2 )∣α̇∣2 + (2 + q 2 + s2 )(ρ̇2 + σ̇ 2 )}
2
+ ∣α̇∣2 O((1 + s2 )∣F∣2LU ) + (1 + s2 )(∣α̇∣2 + ρ̇2 + σ̇ 2 )O(∣F∣2 ) .
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
Nonlinear junk
∇µ(J̇Fµ [Ḟ]) (for controlling the time derivative
of the energy)
Inhomogeneous term
³¹¹ ¹ ν¹ ¹ ¹ ¹ ¹ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹µ
ν
µ
∇µ (J̇F [Ḟ]) = −
K Ḟνη Iη
−(∇µ H µζκλ )Ḟκλ Ḟνζ K
1 ν
+ (K ∇ν H ζηκλ )Ḟζη Ḟκλ
4
1 −2
− {`(MBI) F κλ Ḟκλ (F νζ Ḟ µζ − F µζ Ḟ νζ )
4
⋆ κλ
⋆ νζ µ
⋆ µζ ν
+ (1 + 2(2) `−2
(MBI) ) F Ḟκλ ( F Ḟ ζ − F Ḟ ζ )
⋆ κλ
µζ ν
νζ µ
+ (2) `−2
(MBI) F Ḟκλ (F Ḟ ζ − F Ḟ ζ )
κλ
⋆ µζ ν
⋆ νζ µ
+ (2) `−2
(MBI) F Ḟκλ ( F Ḟ ζ − F Ḟ ζ )}∇µ K ν .
A current for local existence
Lemma
If 1 + (1) − 2(2) > 0, then there exists a C > 0 such that
Q̇(T(0) , Xlocal ) ≥ C(∣Ė∣2 + ∣Ḃ∣2 ),
where
µ
Xlocal
= (b−1 )µν T(0)ν = −(b−1 )µ0 ,
def
(b−1 )µν = (g −1 )µν − (1 + (1) [F])−1 F µκ F νκ .
Furthermore 1 + (1) − 2(2) > 0 ⇐⇒ (B, D) is finite.
(b−1 )µν is the reciprocal Born-Infeld metric.
Hyperbolicity (regular hyperbolicity)
Conclusion
The MBI system is hyperbolic in all regimes where it is
well-defined. Furthermore, the equations are well-posed
in weighted Sobolev spaces.
The energy
def
EN2 [F(t)] = ∑∣I∣≤N ∫R3 J̇F0 [LIZ F(t, x)] d 3 x
"static norm"
³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ·¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ def
∦ F ∦2LZ ;N = ∑∣I∣≤N ∫R3 ∤ LIZ F ∤2 d 3 x
Lemma
If ∦ F(t) ∦LZ ;N is sufficiently small, then
EN [F(t)] ≈∦ F(t) ∦LZ ;N .
Fundamental energy estimate
Lemma
If N ≥ 3 and ∦ F(t) ∦LZ ;N is sufficiently small, then
E 2 [F(t)]
d 2
(EN [F(t)]) = ∑ ∫ ∇µ (J̇Fµ [LIZ F(t, x)]) d 3 x ≲ N
.
3
dt
1 + t2
∣I∣≤N R
Conclusion: EN [F(t)] remains uniformly bounded if
∦ F(t) ∦LZ ;N is small at t = 0.
The 1+t1 2 factor comes from the global Sobolev
inequality and the null structure of the nonlinearities.
A sample term from ∣∇µ(J̇Fµ [LIZ F])∣
∇L K L = −4s,
∇L K L = 4q,
∇A K B = 2tδAB
Q(1) (F ,LIZ F )
³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ·¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ
Ô⇒ ∣ F κλ (LIZ F)κλ F νζ (LIZ F)µζ ∇µ K ν ∣
≲ s{∣F∣2LU ∣LIZ F∣2 + ∣F∣2 ∣LIZ F∣2LU + ∣F∣2T T ∣LIZ F∣2T T } (N.C.)
≲
≲
∦ F(t) ∦2LZ ;N
(1 + s)3
{∣LIZ F∣2 + (1 + s)2 (∣LIZ F∣2LU + ∣LIZ F∣2T T )} (G.S.)
1
∤ LIZ F ∤2
(1 + s)3
Future projects
Stability of the Einstein-Nonlinear electromagnetic
system à la Lindblad-Rodnianski (almost finished)
The complete geometry of the MBI system (w/ Willie
Wong)
Positive energy densities for linearized systems;
regular hyperbolicity (w/ Willie Wong)
Thank you