A Direct Proof of Global Existence for the
Dirac-Klein-Gordon Equations in
One Space Dimension
Yung-fu Fang1
Abstract. We establish local and global existence results for Dirac-Klein-Gordon
equations in one space dimension, employing a null form estimate and a fixed point
argument.
0. Introduction and Main Results.
In the present work, we want to consider the Cauchy problem for the DiracKlein-Gordon equations


 Dψ = φψ;
¤φ = ψψ;


ψ(0) = ψ0 ,
(t, x) ∈ R × R1 ,
(1)
φ(0) = φ0 ,
∂t φ(0) = φ1 ,
where ψ is a 2-spinor field takes value in C4 , φ is a scalar field takes value in R, D
is the Dirac operator, D = −iγ µ ∂µ , µ = 0, 1, and γ µ are the Dirac matrices, which
can be written as follows. First let us define the 2 × 2 matrix σ1 ,
µ
σ1 =
0 1
1 0
¶
.
(2)
The matrices γ µ are defined by
µ
0
γ =
I2
0
0
−I2
¶
µ
1
,
γ =
0
−σ1
σ1
0
¶
,
1991 Mathematics Subject Classification. 35L05.
Key words and phrases. null form estimate.
1 Partially supported by National Science Council at Taiwan, NSC90-2115-M006-025.
1
(3)
2
Y.F. FANG
where I2 is the 2 × 2 identity matrix, the wave operator ¤ = −∂tt + ∂xx , and
ψ = ψ † γ 0 , where † is the complex conjugate transpose.
The purpose of this work is to demonstrate a new proof of null form estimate, see
[1], by employing a solution representation of DKG equations, which apparently is
more simple. In a way, we give an interpretation of the null form structure depicted
within the nonlinear term ψψ, and it is different from that in [1]. This term has
been observed for possessing the null form structure, see[1] and [5, 6].
The energy for the DKG problem is not positive definite, therefore it is not
applicable. However we have the law of conservation of charge,
Z
|ψ(t)|2 dx = constant
(4)
which can be applied to derive the global existence result, once we establish the
local existence result. In this paper, the ”charge class” solutions mean solutions
with ψ(t, ·) ∈ L2 (R).
In ’73, Chadam showed that the Cauchy problem for the DKG equations has a
global unique solution if ψ0 ∈ H 1 , φ0 ∈ H 1 , φ1 ∈ L2 , see [2]. In ’93, Zheng proved
that there exists a global weak solution to the Cauchy problem of the modified DKG
equations, based on the technique of compensated compactness, with ψ0 ∈ L2 ,
φ0 ∈ H 1 , φ1 ∈ L2 , see [8]. In ’00, Bournaveas derived a new proof of a global
existence for the DKG equations, based on a null form estimate, if ψ0 ∈ L2 , φ0 ∈ H 1 ,
φ1 ∈ L2 , see [1].
The approach we adopt in this work is as follows: First, we will derive the
solution representation for the Dirac equation, while the solution representation for
the wave equation is well known. Next, we will use it to estimate the quadratic
term ψψ. In addition to this, the derivations of some necessary estimates become
DIRAC-KLEIN-GORDON EQUATIONS
3
straight forward. Finally we can prove the local and global existence results of DKG
equations, which are parallel to those in [1] and [8].
The main result in this work is as follows.
Theorem 1. (Global Existence) If the initial data of (1), ψ0 ∈ L2 , φ0 ∈ H 1 ,
φ1 ∈ L2 , then there is a unique global solution for (1).
1. Solution Representation.
Recall that, for the wave equation,
½
¤φ = F, (t, x) ∈ R1 × R1 ,
(5)
φ(0) = φ0 , φt (0) = φ1 ,
the solution representations is as follows:
Z tZ
h
i Z x+t
φ1 (y)dy +
2φ(t, x) = φ0 (x + t) + φ0 (x − t) +
x−t
Consider the Dirac equation,
½
Dψ = G;
ψ(0) = ψ0 .
0
x+t−s
F (s, y)dyds. (6)
x−t+s
(t, x) ∈ R1 × R1 ,
First we take D on the Dirac equation, then (7) becomes
½
¤ψ = DG;
(t, x) ∈ R1 × R1 ,
ψ(0) = ψ0 ,
d
ψ0 .
∂t ψ(0) = −γ 0 γ 1 dx
(7)
(8)
Applying the formula (6) and simplifying the expression, we finally have
2ψ(t, x) = (γ 0 + γ 1 )γ 0 ψ0 (x + t) + (γ 0 − γ 1 )γ 0 ψ0 (x − t)+
Z t
Z t
0
1
i
(γ + γ )G(s, x + t − s) ds + i
(γ 0 − γ 1 )G(s, x − t + s) ds. (9)
0
0
The formula (9) can also be derived in the approach via Fourier transform. First
we take the Fourier transform on the Dirac equation (9) over the space variable.
b ξ). Finally we take
Then we solve the resulting ODE to get a formula for ψ(t,
the inverse Fourier transform to obtain a solution formula for ψ(t, x). We will not
present the derivation of (9) via Fourier transform approach, due to the fact that it
is straight forward.
4
Y.F. FANG
2. Estimate.
Lemma 1. For the solution of the Dirac equation (7), we have
Z
³
kψ(t)kL2 ≤ C kψ0 kL2 +
T
0
´
kG(s)kL2 ds .
(13)
This can be shown directly.
Consider two Dirac equations,
½
Dψj
= Gj ,
ψj (0)
= ψ0j .
j = 1, 2,
(14)
Lemma 2. (Null Form Estimate)
Z
³
kψ 1 ψ2 kL2 ([0,T ),L2 ) ≤ C kψ01 kL2 +
T
0
Z
´³
kG1 (s)kL2 ds kψ02 kL2 +
T
0
´
kG2 (s)kL2 ds .
(15)
Proof. For simplicity, we demonstrate a special case when ψ1 = ψ2 , and the general
case will follow. Consider the Dirac equation (7) and write its solution as
2ψ(t, x) = U+ + U− + iV+ + iV− ,
(16)
U± (t, x) = (γ 0 ± γ 1 )γ 0 ψ0 (x ± t),
Z t
V± (t, x) =
(γ 0 ± γ 1 )G(s, x ± (t − s))ds.
(17)
where
(18)
0
Throughout calculations, we get
U + U+ = U − U− = V + V+ = V − V− = 0,
(19)
U + V+ = U − V− = V + U+ = V − U− = 0,
(20)
DIRAC-KLEIN-GORDON EQUATIONS
5
and
kU + U− kL2 ([0,T ),L2 ) ≤ Ckψ0 k2L2 ,
Z
kV + U− kL2 ([0,T ),L2 ) ≤ Ckψ0 kL2
T
Z
T
(21a)
0
kU + V− kL2 ([0,T ),L2 ) ≤ Ckψ0 kL2
ÃZ
kV + V− kL2 ([0,T ),L2 ) ≤ C
0
0
kG(s)kL2 ds,
(21b)
kG(s)kL2 ds,
(21c)
!2
T
kG(s)kL2 ds
.
(21d)
The rest cases are analogous. Among these cases, we only demonstrate the case of
V + V− . Since
Z tZ
t
V + V− = 2
0
G† (s, x + t − s)(γ 0 − γ 1 )G(r, x − t + r)drds,
(22)
0
hence we have
kV + V− kL2 ([0,T ),L2 )
ÃZ Z Z Z
! 21
T
³ t t
´2
≤C
|G† (s, x + t − s)(γ 0 − γ 1 )G(r, x − t + r)|drds dxdt
0
Z
0
T
Z
T
Z
T
ÃZ
≤C
0
Z
0
ÃZ
≤C
0
0
T
! 21
Z ¯
¯2 ¯
¯2
¯ †
¯ ¯
¯
drds
¯G (s, x + t − s)¯ ¯G(r, x − t + r)¯ dxdt
0
T
≤C
0
T
0
kG(s)kL2 kG(r)kL2 drds
!2
kG(s)kL2 ds
.
(23)
This completes the proof of the lemma.
¤
Lemma 3. For the wave equation (10), we have the energy estimate
Ã
!
Z T
kφ(t)kH 1 + kφt (t)kL2 ≤ C(T ) kφ0 kH 1 + kφ1 kL2 +
kF (s)kL2 ds .
0
This can be derived directly.
(25)
6
Y.F. FANG
3. Existence.
Let (ψ, φ) and (ψ 0 , φ0 ) be two charge class solutions of DKG equations. We define
the following quantities:
J(0) = kψ0 kL2 + kφ0 kH 1 + kφ1 kL2 ;
(26a)
J 0 (0) = kψ00 kL2 + kφ00 kH 1 + kφ01 kL2 ;
(26b)
J(T ) = sup (kψ(t)kL2 + kφ(t)kH 1 + kφ(t)kL2 ) ;
(26c)
J 0 (T ) = sup (kψ 0 (t)kL2 + kφ0 (t)kH 1 + kφ0 (t)kL2 ) ;
(26d)
[0,T ]
[0,T ]
∆(0) = kψ0 − ψ00 kL2 + kφ0 − φ00 kH 1 + kφ1 − φ01 kL2 ;
∆(T ) = sup (kψ(t) − ψ 0 (t)kL2 + kφ(t) − φ0 (t)kH 1 + kφt (t) − φ0t (t)kL2 ) .
[0,T ]
(26e)
(26f)
Lemma 4. For the solution of the DKG equations (1), we have
kφ(t)kL∞ (R) ≤ C(T, J(0)).
(27)
This can be obtained by using the solution representation for φ and the law of
conservation of charge, see [1].
Lemma 5. Let T > 0 and let (ψ, φ) be a charge class solution of (1). Then there
exists a constant C > 0, depending only on T and J(0), such that
J(T ) ≤ C(T, J(0)).
(28)
Proof. Since kψ(t)kL2 = kψ0 kL2 , this takes care the first term of J(T ). For the
DIRAC-KLEIN-GORDON EQUATIONS
7
terms involving φ, we first compute
Z T
Z T
kF (s)kL2 ds =
kψψ(s)kL2 ds
0
0
1
≤ T 2 kψψkL2 ([0,T ],L2 )
Z T
³
´2
1
≤ T 2 kψ0 kL2 +
kDψ(s)kL2 ds
0
Z
³
1
≤ T 2 J(0) +
T
0
´2
kφ(s)kL∞ kψ(s)kL2 ds
1
2
≤ C(T, J(0))T .
(29)
Now we apply the energy estimate to get
Ã
kφ(t)kH 1 + kφt (t)kL2 ≤ C(T ) J(0) +
Z
!
T
0
kF (s)kL2 ds
≤ C(T, J(0)).
This completes the proof of the lemma.
(30)
¤
Lemma 6. Let T > 0 and (ψ, φ), (ψ 0 , φ0 ) be two charge class solutions of (1). Then
there exists ² > 0, C > 0, depending only on J(0) and J 0 (0) such that if T < ², then
∆(T ) ≤ C∆(0).
(31)
Proof. Consider the difference of the two solutions, we have the following equations:
½
D(ψ − ψ 0 ) = (φ − φ0 )ψ + φ0 (ψ − ψ 0 ),
¤(φ − φ0 ) = (ψ − ψ 0 )ψ + ψ 0 (ψ − ψ 0 ).
(32)
For the first term of ∆(T ), first we compute
kD(ψ − ψ 0 )(s)kL2
≤kφ(s) − φ0 (s)kL∞ kψ(s)kL2 + kφ0 (s)kL∞ kψ(s) − ψ 0 (s)kL2
≤kφ(s) − φ0 (s)kH 1 J(0) + C(T, J 0 (0))kψ(s) − ψ 0 (s)kL2
¡
¢
≤C T, J(0), J 0 (0) ∆(T ).
(33)
8
Y.F. FANG
Invoke (13), we have
Z
³
T
´
kψ(t) − ψ (t)kL2 ≤C kψ0 −
+
kD(ψ − ψ 0 )(s)kL2 ds
0
³
´
¡
¢
≤C ∆(0) + C T, J(0), J 0 (0) T ∆(T ) .
0
ψ00 kL2
(34)
For the other two terms, first we calculate
Z T
k¤(φ − φ0 )(s)kL2 ds
Z
0
T
k(ψ − ψ 0 )ψ(s)kL2 + kψ 0 (ψ − ψ 0 )(s)kL2 ds
0
³
´
1
≤T 2 k(ψ − ψ 0 )ψkL2 + kψ 0 (ψ − ψ 0 )kL2 ,
≤
(35)
then we calculate, invoke (15),
k(ψ − ψ 0 )ψkL2
Z T
Z T
³
´³
´
0
0
≤C kψ0 − ψ0 kL2 +
kD(ψ − ψ )(s)kL2 ds kψ0 kL2 +
kDψ(s)kL2 ds
0
³
´³ 0
´
0
≤C T, J(0), J (0) ∆(0) + T ∆(T ) .
(36)
The calculation for kψ 0 (ψ − ψ 0 )kL2 is the same. Therefore, applying (25), we get
kφ(t) − φ0 (t)kH 1 + kφt (t) − φ0t (t)kL2
Ã
Z
0
0
≤C(T ) kφ0 − φ0 kH 1 + kφ1 − φ1 kL2 +
0
!
T
0
k¤(φ − φ )(s)kL2 ds
³
´³
´
1
≤C(T ) ∆(0) + T 2 C(T, J(0), J 0 (0)) ∆(0) + T ∆(T )
³
´³
´
≤C T, J(0), J 0 (0) ∆(0) + T ∆(T ) .
(37)
Now we assume T ≤ 1, thus
³
and
0
´
³
0
´
³
0
´
C T, J(0), J (0) ≤ C J(0), J (0) = C 1, J(0), J (0)
(38)
³
´³
´
∆(T ) ≤ C J(0), J 0 (0) ∆(0) + T ∆(T ) .
(39)
DIRAC-KLEIN-GORDON EQUATIONS
This concludes the proof.
9
¤
Combining (28), (31) and an iteration scheme, we have the following local existence result.
Theorem 2. (Local Existence) Let ψ0 ∈ L2 (R), φ0 ∈ H 1 (R), and φ1 ∈ L2 (R).
Then there exists a T > 0, depending only on J(0), and a unique charge class
solution of (1) defined on [0, T ) × R.
From Lemma 5, we have the inequality,
J(T ) ≤ C(T, J(0)),
(40)
which ensures that we can always extend the solution beyond T . Thus we have a
unique global solution for the Cauchy problem. The idea of this type of argument
is originated by Segal, see [7].
4. Remark.
Using the solution representation in Fourier transform, we can take full advantage
of the null form structure and derive the null form estimate,
Ã
!2
° G
°
° b °
kψψkL2 ≤ C kψ0 kL2 + ° 1 ° 2
,
c 2 −² L
W
(41)
¯
¯
c = ¯¯|τ | − |ξ|¯¯ + 1. Equipping with this estimate, together with other
where W
estimates, the result of global existence can be improved, i.e. if ψ0 ∈ L2 (R), φ0 ∈
1
1
H 2 (R), and φ1 ∈ H − 2 (R), then we have a unique global solution. The proof will
appear elsewhere.
Acknowledgement. The author wants to express his gratitude to M. Grillakis for his
encouragement and inspiring conversation, and also to Tai-ping Liu for his helps
when I visited Sinica in January 2002.
10
Y.F. FANG
References
1. N. Bournaveas, A New Proof of Global Existence for the Dirac Klein-Gordon Equations in
One Space Dimension, J. Funct. Anal. 173 (2000), 203-213.
2. J.M. Chadam, Global solutions of the Cauchy problem for the (classical) Maxwell-Dirac equations in one space dimension, J. Funct. Anal. 13 (1973), 173-184.
3. J.M. Chadam & R.T. Glassey, Properties of the solutions of the Cauchy problem for the
(classical) coupled Maxwell-Dirac equations in one space dimension, Proc. Amer. Math. Soc.
43 (1974), 373-378.
4.
, On Certain Global Solutions of the Cauchy Problem for the (Classical) Coupled
Klein-Gordon-Dirac Equations in One and Three Space Dimensions, Arch. Rational Mech.
Anal. 54 (1974), 223-237.
5. S. Klainerman & M. Machedon, Space-time estimates for the null form and the local existence
theorem, Comm. Pure Appl. Math. 46 (1993), 1221-1268.
6.
, On the Maxwell-Klein-Gordon equations with finite energy, Duke Math. J. 74 (1994),
19-44.
7. I.E. Segal, Non-linear semigroups, Ann. of Math. 78 (1963), 339-364.
8. Y. Zhang, Regularity of weak Solution to a Two-Dimensional Modified Dirac-Klein-Gordon
System of Equations, commun. Math. Phys. 151 (1993), 67-87.
Department of Mathematics, National Cheng Kung University, Tainan 701 Taiwan
E-mail address: [email protected]