Electronic Journal of Differential Equations, Vol. 2004(2004), No. 102, pp. 1–19.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
LOW REGULARITY SOLUTIONS FOR DIRAC-KLEIN-GORDON
EQUATIONS IN ONE SPACE DIMENSION
YUNG-FU FANG
Abstract. We establish the existence of local and global solutions for DiracKlein-Gordon equations in one space dimension. This is done using a null form
estimate and a fixed point argument.
1. Introduction and Main Results
In this paper, we study the Cauchy problem for the Dirac-Klein-Gordon system.
The unknown quantities are a spinor field ψ : R × R1 7→ C4 and a scalar field
φ : R × R1 7→ R. The evolution equations for these fields are
Dψ = φψ,
(t, x) ∈ R × R1 ;
(1.1)
φ = ψψ;
ψ(0, x) := ψ0 (x),
φ(0, x) = φ0 (x),
φ,t (0, x) = φ1 (x),
where D is the Dirac operator, D := −iγ µ ∂µ , µ = 0, 1, and γ µ are the Dirac
matrices, the wave operator = −∂tt + ∂xx ,and ψ = ψ † γ 0 , and † is the complex
conjugate transpose.
The purpose of this work is to demonstrate a variant null form estimate, by
employing the solution representations in Fourier transform of the DKG equations.
We will take advantage of the null form structure depicted in the nonlinear term
ψψ, which has been observed by [11] and [3]. We interpret the null form in a way
that is different from that given in Bournaveas’ paper [3]. Equipping with this
estimate, we can lower the regularity of the spinor field.
For the DKG system, there are many conserved quantities which are not positive
definite, such as the energy. However the known positive conserved quantity is the
law of conservation of charge,
Z
|ψ(t)|2 dx = constant
(1.2)
which is applicable to lead to the global existence result, once the local existence
result is established, see [3] and [7].
In 1973, Chadam showed that the Cauchy problem for the DKG equations has
a global unique solution for ψ0 ∈ H 1 , φ0 ∈ H 1 , φ1 ∈ L2 , see [4]. In 1993, Zheng
2000 Mathematics Subject Classification. 35L70.
Key words and phrases. Dirac-Klein-Gordon; null form; low regularity; fixed point argument.
c
2004
Texas State University - San Marcos.
Submitted August 13, 2004. Published August 27, 2004.
1
2
YUNG-FU FANG
EJDE-2004/102
proved that there exists a global weak solution to the Cauchy problem of a modified
DKG equations, based on the technique of compensated compactness, with ψ0 ∈ L2 ,
φ0 ∈ H 1 , φ1 ∈ L2 , see [14]. In 2000, Bournaveas derived a new proof of a global
existence for the DKG equations, via a null form estimate, if ψ0 ∈ L2 , φ0 ∈ H 1 ,
φ1 ∈ L2 , see [1]. In 2002, Fang gave a direct proof for (1.1), based on a variant null
form estimate, which is straight forward, and the result is parallel to Bournaveas’,
see [7]
The outline of this paper is as follows. First we derive some solutions representations in Fourier transform, depending on various purposes. Next we prove some
a priori estimates of solutions for Dirac equation and for wave equation. Then we
show a local and global results for (1.1), employing the null form estimate together
with other estimates derived previously, and a fixed point argument. Finally we
show the null form estimate.
The main result in this work is as follows.
Theorem 1.1 (Local Existence). Let 0 < ≤ 41 and 0 < δ ≤ 2. If the initial
1
1
1
data of (1.1) ψ0 ∈ H − 4 + , φ0 ∈ H 2 +δ , φ1 ∈ H − 2 +δ , then there is a unique local
solution for (1.1).
Theorem 1.2 (Global Existence). Let δ > 0. If the initial data of (1.1) ψ0 ∈ L2 ,
1
1
φ0 ∈ H 2 +δ , φ1 ∈ H − 2 +δ , then there is a unique global solution for (1.1).
Remarks. 1. The DKG equations follow from the Lagrangian
Z
|∇φ|2 − |φt |2 − ψDψ − φψψ dx dt.
(1.3)
R1+1
2. The Dirac-Klein-Gordon system must be
Dψ = φψ; φ + m2 φ = ψψ,
(1.4)
b 2 = I,
b where I is the 4 × 4 identity matrix.
3. D
4. ψψ = ψ † γ 0 ψ = |ψ1 |2 +|ψ2 |2 −|ψ3 |2 −|ψ4 |2 , where ψj are the component functions
of the vector function ψ, which take values in C.
The case δ = 0 is critical in the following sense. Assuming that the initial data
1
(φ0 , φ1 ) are in H 2 × H −1/2 does not imply that φ(t, ·) is bounded. In fact, it is a
BMO function. One of the motivations for proving the existence of global solution
with low regularity, is based on an observation made by Grillakis, which is that the
1
1
initial data of (1.1): ψ0 ∈ L2 , φ0 ∈ H 2 , φ1 ∈ H − 2 , is a right space for the existence
of an invariant measure, see [1] and [12], resulted from the DKG equations.
2. Solution Representation
In what follows, we denote by (t, x) the time-space variables and by (τ, ξ) the
dual variables with respect to the Fourier transform. We will use α = 14 − in this
paper. We will also often skip the constant in the inequalities. For convenience, we
EJDE-2004/102
LOW REGULAIRTY FOR DIRAC-KLEIN-GORDON EQUATIONS
3
also denote the multipliers by
b ξ) = |τ | + |ξ| + 1
E(τ,
b ξ) = |τ | − |ξ| + 1
S(τ,
c (τ, ξ) = τ 2 − |ξ|2
W
b ξ) = γ 0 τ + γ 1 ξ
D(τ,
c(ξ) = |ξ| + 1 .
M
c and D
b as the symbols of the wave and Dirac operators respectively.
We use W
Consider the Dirac equation,
Dψ = G,
(t, x) ∈ R1 × R1 ,
ψ(0) = ψ0 .
(2.1)
First by taking the Fourier transform on (2.1) over the space variable and solving
the resulting ODE, we can formally write down the solution as follows.
it|ξ|
e−it|ξ| b
e ξ) = e
b
D(|ξ|,
ξ)γ 0 ψb0 (ξ) +
D(|ξ|, −ξ)γ 0 ψb0 (ξ)
ψ(t,
2|ξ|
2|ξ|
Z t i(t−s)|ξ|
Z t −i(t−s)|ξ|
e
e
b
e
b
e ξ) ds.
D(|ξ|, ξ)iG(s, ξ) ds +
D(|ξ|,
−ξ)iG(s,
+
2|ξ|
2|ξ|
0
0
(2.2)
Rewriting the inhomogeneous terms in (2.2) gives
it|ξ|
e−it|ξ| b
e ξ) = e
b
D(|ξ|,
ξ) +
D(|ξ|, −ξ) γ 0 ψb0 (ξ)
ψ(t,
2|ξ|
2|ξ|
Z
itτ
it|ξ|
e −e
eitτ − e−it|ξ| b
b ξ)dτ.
b
+
D(|ξ|,
ξ) +
D(|ξ|, −ξ) G(τ,
2|ξ|(τ − |ξ|)
2|ξ|(τ + |ξ|)
(2.3)
b into several parts in the following manner. Consider
Now we split the function G
b
a(τ ) a cut-off function equals 1 if |τ | ≤ 12 and equals 0 if |τ | ≥ 1, b
a6 (τ ) = b
a( τ6 ), and
denote by h(τ ) the Heaviside function. For simplicity, let us write
b ± (τ, ξ) := h(±τ )b
b ξ),
G
a(τ ± |ξ|)G(τ,
b f (τ, ξ) := G(τ,
b ξ) − G
b + (τ, ξ) + G
b − (τ, ξ) ,
G
b ± := D(|ξ|,
b
D
±ξ).
b ± are supported in the regions {(τ, ξ) : ±τ > 0, |τ ∓ |ξ|| ≤ 1} reNote that G
b=G
bf + G
b+ + G
b − , the
spectively. Using the decomposition of the forcing term G
inhomogeneous term in (2.3) can be written as
eitτ − eit|ξ|
eitτ − e−it|ξ| b
b
b f (τ, ξ)dτ
D(|ξ|,
ξ) +
D(|ξ|, −ξ) G
2|ξ|(τ − |ξ|)
2|ξ|(τ + |ξ|)
(2.4)
Z
b
b Z G
bf
b Z G
bf
itτ D(τ, ξ) b
it|ξ| D+
−it|ξ| D−
Gf dτ − e
dτ − e
dτ,
= e
τ 2 − |ξ|2
2|ξ|
τ − |ξ|
2|ξ|
τ + |ξ|
Z
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YUNG-FU FANG
EJDE-2004/102
eitτ − eit|ξ| b b
b − )dτ
D+ (G+ + G
2|ξ|(τ − |ξ|)
b Z eit(τ −|ξ|) − 1
it|ξ| D+
b+ + b
b − )dτ
=e
(G
a(τ )G
2|ξ|
τ − |ξ|
Z
b +G
b−
b−
b + Z (1 − b
(1 − b
a(τ ))D
D
a(τ ))G
+ eitτ
dτ − eit|ξ|
dτ,
2|ξ|(τ − |ξ|)
2|ξ|
τ − |ξ|
Z
eitτ − e−it|ξ| b b
b − )dτ
D− (G+ + G
2|ξ|(τ + |ξ|)
b − Z eit(τ +|ξ|) − 1
D
b+ + G
b − )dτ
= e−it|ξ|
(b
a(τ )G
2|ξ|
τ + |ξ|
Z
b −G
b+
b − Z (1 − b
b+
D
a(τ ))G
(1 − b
a(τ ))D
+ eitτ
dτ − e−it|ξ|
dτ,
2|ξ|(τ + |ξ|)
2|ξ|
τ + |ξ|
(2.5)
Z
b namely
Combining (2.3)-(2.6), we can give a formula for ψ,
∞
X
(k)
b ξ) =
b+,k (ξ) + δ (k) (τ, ξ)A
b−,k (ξ) + K(τ,
b ξ),
ψ(τ,
δ+ (τ, ξ)A
−
(2.6)
(2.7)
k=0
where δ± (τ, ξ) are the delta functions supported on {τ = ±|ξ|} respectively, δ (k)
mean derivatives of the delta function, and
b
b +G
b−
b −G
b+
(1 − b
a6 )D
a6 (τ ))D
b ξ) := D(τ, ξ) G
b f + (1 − b
K(τ,
+
,
(2.8)
c (τ, ξ)
2|ξ|(τ − |ξ|)
2|ξ|(τ + |ξ|)
W
Z b
b b∓ Gf + (1 − b
a6 (λ))G
b±,0 (ξ) := D± γ 0 ψb0 −
A
dλ ,
(2.9)
2|ξ|
λ ∓ |ξ|
b ± (−1)k Z
D
b
b± + b
b ∓ dλ.
A±,k (ξ) :=
(λ ∓ |ξ|)k−1 G
a6 (λ)G
(2.10)
2|ξ|k!
Now we split ψ in a different manner. Consider the cut-off function bb(τ ) equals
1 if |τ | < R, and equals 0 if |τ | > 2R. Let bb(τ ) + b
c(τ ) = 1. Applying (2.3), we can
b
give the following formula for ψ.
∞
X
(k)
b
b+,k (ξ) + δ (k) (τ, ξ)U
b−,k (ξ) + U
b (τ, ξ),
ψ(τ, ξ) =
δ+ (τ, ξ)U
(2.11)
−
k=0
where
Z
b b
c(λ ∓ |ξ|) b b±,0 (ξ) := D± γ 0 ψb0 −
U
Gdλ ,
2|ξ|
λ ∓ |ξ|
b ± (−1)k Z
D
b
b
U±,k (ξ) :=
(λ ∓ |ξ|)k−1bb(λ ∓ |ξ|)Gdλ,
2|ξ|k!
b −b
b c(τ − |ξ|) D
c(τ + |ξ|) b
b (τ, ξ) := D+ b
U
+
G(τ, ξ).
2|ξ|(τ − |ξ|)
2|ξ|(τ + |ξ|)
Consider the wave equation,
φ = F,
(t, x) ∈ R1 × R1 ,
φ(0) = φ0 ,
φt (0) = φ1 .
(2.12)
(2.13)
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LOW REGULAIRTY FOR DIRAC-KLEIN-GORDON EQUATIONS
Taking Fourier transform on (2.13) and solving the resulting ODE gives
Z t
sin (t − s)|ξ| e
e ξ) = cos t|ξ|φb0 (ξ) + sin t|ξ| φb1 (ξ) +
φ(t,
F (s, ξ)ds.
|ξ|
|ξ|
0
Thus we can rewrite it as follows.
it|ξ|
+ e−it|ξ| b
eit|ξ| − e−it|ξ| b
e ξ) = e
φ(t,
φ0 (ξ) +
φ1 (ξ)
2
i2|ξ|
Z itτ
Z itτ
−1
e − eit|ξ| b
1
e − e−it|ξ| b
+
F (τ, ξ)dτ +
F (τ, ξ)dτ.
2|ξ|
τ − |ξ|
2|ξ|
τ + |ξ|
5
(2.14)
(2.15)
For convenience, we define the following
Z
1b
1 b
1
b
c(λ ∓ |ξ|) b
φ0 (ξ) ±
φ1 (ξ) ±
F (λ, ξ)dλ,
2
i2|ξ|
2|ξ|
λ ∓ |ξ|
Z
(−1)k
vb±,k (ξ) := ∓
(λ ∓ |ξ|)k−1bb(λ ∓ |ξ|)Fb(λ, ξ)dλ,
2|ξ|k!
−1 b
c(τ − |ξ|) b
c(τ + |ξ|) b
−
F (τ, ξ).
vb(τ, ξ) :=
2|ξ| τ − |ξ|
τ + |ξ|
vb±,0 (ξ) :=
(2.16)
Combining (2.15) and (2.16), and invoking the cut-off function, we have
b ξ) =
φ(τ,
∞
X
∞
X
bk (τ, ξ) + N
b (τ, ξ),
Vb+,k (τ, ξ) + Vb−,k (τ, ξ) + Vb (τ, ξ) +
N
k=0
(2.17)
k=0
where
(k)
Vb±,k (τ, ξ) = 1 − b
a(ξ) δ± (τ, ξ)b
v±,k (ξ),
b
a(ξ) vb(τ, ξ),
V (τ, ξ) = 1 − b
(k)
(k)
bk (τ, ξ) = b
N
a(ξ) δ+ (τ, ξ)b
v+,k (ξ) + δ− (τ, ξ)b
v−,k (ξ) ,
b (τ, ξ) = b
N
a(ξ)b
v (τ, ξ).
Remark. We need to localize the solutions for Dirac equation and wave equation
due to the presence of the delta function.
3. Estimates
To localize the solution in time, let ϕ(t) be a cut-off function such that ϕ(t)
equals 1 if |t| ≤ 1/2, and equals 0 if |t| > 1, and ϕT (t) = ϕ(t/T ). Notice that, for
an arbitrary function f (t, x), we have
kϕ
bT ∗ fbkL2 = kϕT f kL2 ≤ kϕT kL∞ kf kL2 .
(3.1)
For the Dirac equation (2.1), using (2.11), we define
b T (τ, ξ) = ϕ
Ψ
bT ∗
∞
X
(k) b
(k) b
b
δ+ U
+,k + δ− U−,k (τ, ξ) + U (τ, ξ),
(3.2)
k=0
Lemma 3.1. Let > 0 and T R ∼ 1. If ψ0 ∈ H −α , then we have
1 −α c Ψ
b T 2 1 1 ≤ C kψ0 kH −α + T Sb 2 M
L (R ×R )
b
G
2 .
1
L
−
cα Sb 2
M
We will only outline the proof. For more details, please see [8].
(3.3)
6
YUNG-FU FANG
EJDE-2004/102
Proof. Applying formulae (2.11), we can derive the following bounds:
b
1 −α
G
c ϕ
b±,0 2 ≤ C kψ0 kH −α + T Sb 2 M
2 ,
bT ∗ δ± U
1
L
L
−
cα Sb 2
M
k
b
1 −α
G
(k) b
c ϕ
Sb 2 M
2 ≤ C (4RT ) T ,
bT ∗ δ± U
±,k
1
L
cα Sb 2 − L2
k!
M
b
1 −α G
c U
b 2 1 1 ≤ CT Sb 2 M
2,
1
L (R ×R )
L
−
α
c
b
M S2
Combining these estimates, we have (3.3).
Consider two Dirac equations,
Dψj = Gj ,
j = 1, 2,
(3.4)
ψj (0) = ψ0j .
For the solutions of this system, we have the following key estimate whose proof
will be presented in the last section.
Lemma 3.2 (Null Form Estimate). Let > 0, and ψ1 , ψ2 be the solutions for
(3.4). If ψ0j ∈ H −α , we have
(ϕ\
T ψ 1 ψ2 ) 2
L
α
2α
c
b
M S
b1 b G
2 kψ02 kH −α + G2 2 .
≤ C(T ) kψ01 kH −α + 1
L
α
cα Sb 14 L
c
b
M
M S4
(3.5)
For the wave equation (2.13), we define
∞
∞
X
X
bk (τ, ξ) + N
b (τ, ξ). (3.6)
b T (τ, ξ) = ϕ
Φ
bT ∗
(Vb+,k + Vb−,k )(τ, ξ) + Vb (τ, ξ) + ϕ
bT ∗
N
k=0
k=0
Thus we have the following estimate.
Lemma 3.3. Let > 0, δ ≥ 0, T R ∼ 1, and φ be the solution of (2.13). If
1
1
φ0 ∈ H 2 +δ and φ1 ∈ H − 2 +δ , then
1 1 +δ Fb
c2 Φ
b T 2 ≤ C kφ0 k 1 +δ + kφ1 k − 1 +δ + T 2 .
Sb 2 M
(3.7)
1
1
L
L
H2
H 2
−δ
−
b
c
M2 S2
Proof. Applying formula (2.17), we can derive the following bounds:
1 1 +δ
Fb
c2 ϕ
Sb 2 M
2 ,
bT ∗ Vb±,0 L2 ≤ C kφ0 k 12 +δ + kφ1 k − 21 +δ + T 1
1
L
H
H
−δ
−
c 2 Sb 2
M
1 1 +δ
Fb
c2 ϕ
b0 2 ≤ C kφ0 k 1 +δ + kφ1 k − 1 +δ + T Sb 2 M
,
bT ∗ N
1
1
L
H2
H 2
c 2 −δ Sb 2 − L2
M
1 1 +δ
(4RT )k Fb
c2 ϕ
Sb 2 M
2,
bT ∗ Vb±,k L2 ≤ C
T 1
1
L
−δ
−
c
b
k!
M2 S2
k
1 1 +δ
Fb
c2 ϕ
bk 2 ≤ C (4RT ) T Sb 2 M
2,
bT ∗ N
1
1
L
L
−δ
−
c 2 Sb 2
k!
M
1 1 +δ Fb
c 2 Vb 2 ≤ CT Sb 2 M
,
1
1
L
c 2 −δ Sb 2 − L2
M
1 1 +δ Fb
c2 N
b 2 ≤ CT Sb 2 M
2.
1
1
L
L
−δ
−
c
b
M2 S2
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LOW REGULAIRTY FOR DIRAC-KLEIN-GORDON EQUATIONS
Combining the above inequalitites, we complete the proof.
7
We will also need some technical lemmas.
Lemma 3.4 (Hardy-Littlewood-Polya). Let r = 2 − p1 − 1q . Then we have
Z
f (s)g(t)
dsdt ≤ Ckf kLp kgkLq .
r
R1 ×R1 |s − t|
(3.8)
Lemma 3.5. Let f (t, x) and g(t, x) be any functions such that f ∈ Lq (L2 (Rn ))
and Sbβ gb ∈ L2 (L2 (Rn )). Assume that > 0, 1q = 1 − , 1r = 12 − β, and 2 ≤ r < ∞.
Then we have
fb 1 2 2 n ≤ Ckf kLq (L2 (Rn )) ,
(3.9)
Sb 2 − L (L (R ))
kgkLr (L2 (Rn )) ≤ CkSbβ gbkL2 (L2 (Rn )) .
(3.10)
Proof. The proofs for (3.9) and (3.10) are analogous. Therefore, we will only prove
the case of (3.10). Taking the inverse Fourier transform in the time variable over
the identity
1 bβ
gb =
S gb
(3.11)
Sbβ
gives
Z ±i(t−s)|ξ|
e
ge(t, ξ) =
F −1 (Sbβ gb)(s, ξ) ds.
(3.12)
|t − s|1−β τ
Then we use duality and Hardy-Littlewood-Polya inequality to compute
Z Z Z ±i(t−s)|ξ|
e
g, ϕ = ge, ϕ
e =
Fτ−1 (Sbβ gb)(s, ξ)dsϕ(t,
e ξ)dtdξ 1−β
|t − s|
Z
−1 bβ
kFτ (S gb)(s)kL2 kϕ(t)k
e
L2
≤
dsdt
1−β
|t − s|
≤ CkF −1 (Sbβ gb)kL2 kϕk
e r0 2 = CkSbβ gbkL2 kϕk r0 2 .
τ
L (L )
(3.13)
L (L )
This completes the proof of (3.10).
4. Local and Global Existence
Now we are ready to prove the local existence for the (DKG) equations.
Proof of Theorem 1.1. Consider the DKG equations
Dψ = ϕT φψ,
(4.1)
φ = ϕT ψψ,
and the map T (ψ, φ) = (ΨT , ΦT ). We want to show that T is a contraction under
the norm
1 −α c ψb 2 + Sb 12 M
c 12 +δ φb 2 .
N (ψ, φ) = Sb 2 M
(4.2)
L
L
For convenience, we define
J(0) = kφ0 k
1
H 2 +δ
+ kφ1 k
1
H − 2 +δ
+ kψ0 k2H −α + 1.
(4.3)
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YUNG-FU FANG
EJDE-2004/102
First we apply (3.7) and (3.5) to compute
\ ϕ\
T ψψ
2 ≤ C J(0) + T ϕT φψ 2 2 .
1
1
L
−δ
−
c
b
cα Sb 14 L
M2 S2
M
(4.4)
To bound the term above, we first compute
1 1 +δ c2 Φ
b T 2 ≤ C J(0) + T Sb 2 M
L
−α
f 2 ∼ kGα ∗ (φψ)(t)kL2 ≤ kφ(t)kL∞ kGα ∗ ψ(t)kL2
c φψ(t)
M
L
≤ kφ(t)k
1
H 2 +δ
(4.5)
kψ(t)kH −α ,
where Gα (x) is an L1 -function such that
cα (ξ) ∼ (1 + |ξ|)−α ,
G
(4.6)
see [13]. Then we invoke (3.9), (3.10) and obtain
\ ϕ
^
T φψ 2 ≤ C ϕT φψ q 2
1
L
cα Sb 4
cα L (L )
M
M
≤ kφk 2q 12 +δ kψkL2q (H −α )
L (H
)
1 1 +δ 1 −α b
b
c
c ψb 2 .
2
2
≤ S M
φ L2 Sb 2 M
L
(4.7)
\ ϕ
T φψ 2 ≤ CN 2 (ψ, φ).
1
c
M α Sb 2 − L
(4.8)
Thus we get
b T . The estimate (3.3) implies
Next we want to bound the term involving Ψ
1 −α \ c Ψ
b T 2 1 1 ≤ C kψ0 kH −α + T ϕT φψ 2 .
Sb 2 M
L (R ×R )
cα Sb 14 L
M
(4.9)
Hence, using (4.3), (4.8), and (4.5), we have
N (T (ψ, φ)) ≤ C J(0) + T N 4 (ψ, φ) .
(4.10)
Choosing sufficiently large L, for suitable T , we have
N (ψ, φ) ≤ L =⇒ N T (ψ, φ) ≤ L,
(4.11)
provided that C(J(0) + T L4 ) ≤ L.
Now we consider the difference T (ψ, φ) − T (ψ 0 , φ0 ). Base on the observations
1
1
(ψ − ψ 0 )(ψ + ψ 0 ) + (ψ + ψ 0 )(ψ − ψ 0 ),
2
2
1
1
0 0
0
0
φψ − φ ψ = (φ − φ )(ψ + ψ ) + (φ + φ0 )(ψ − ψ 0 ),
2
2
ψψ − ψ 0 ψ 0 =
(4.12)
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9
Using (3.7), (3.5), (4.12), and (4.8), we first calculate
1 1 +δ
c 2 F(ΦT − Φ0 ) 2
Sb 2 M
T
L
F((ψ − ψ 0 )(ψ + ψ 0 ))
F((ψ + ψ 0 )(ψ − ψ 0 ))
kL2 + k
kL2
1
1
1
1
−δ
−
−δ
−
c
b
c
b
M2 S2
M2 S2
0
0 F((φ + φ0 )(ψ − ψ 0 )) F((φ − φ )(ψ + ψ )) 2
≤ CT
+
L2
L
cα Sb 14
cα Sb 14
M
M
F(φψ + φ0 ψ 0 ) 2
× J(0) + L
cα Sb 14
M
1
1
1
c 2 +δ φ\
c−α ψ\
≤ CT kSb 2 M
− φ0 kL2 + kSb 2 M
− ψ 0 kL2 L(J(0) + L2 )
1
1
c 12 +δ φ\
c−α ψ\
≤ CT L3 kSb 2 M
− φ0 kL2 + kSb 2 M
− ψ 0 kL2
≤ CT k
(4.13)
Analogously, we get
1 −α
1
1
c (ΨT\
c−α ψ\
c 12 +δ φ\
Sb 2 M
− Ψ0T )L2 ≤ CT L kSb 2 M
− ψ 0 kL2 + kSb 2 M
− φ0 kL2 .
(4.14)
Combining (4.13)and (4.14), we have
N T (ψ − ψ 0 , φ − φ0 ) ≤ CT L3 N (ψ − ψ 0 , φ − φ0 ).
(4.15)
Therefore for suitable T , we obtain
1
N T (ψ − ψ 0 , φ − φ0 ) ≤ N (ψ − ψ 0 , φ − φ0 ),
(4.16)
2
provided that CT L3 ≤ 21 . We can conclude that the map T is indeed a contraction
with respect to the norm N , thus it has a unique fixed point.
We now prove existence of a gobal solution.
Proof of Theorem 1.2. ¿From the law of conservation of charge, we have
sup kψ(t)kL2 = kψ0 kL2 .
(4.17)
[0,T ]
To bound φ we apply the formula (2.14),
Z x+t
Z tZ
2φ(t, x) = (φ0 (x+t)+φ0 (x−t))+
φ1 (y)dy +
x−t
0
x+t−s
ψψ(s, y)dyds. (4.18)
x−t+s
First we write φ = φL + φN , the homogeneous and inhomogeneous parts of the
solution, then we obtain
kφL (t)kL∞ ≤ kφL (t)k
1
H 2 +δ
and
Z tZ
≤ kφ0 k
≤ J(0),
(4.19)
ψψ(s, y)dy ds ≤ CT kψ0 k2 2 .
(4.20)
1
H 2 +δ
+ kφ1 k
1
H − 2 +δ
x+t−s
kφN (t)kL∞ ≤
L
0
x−t+s
Combining (4.19) and (4.20), we obtain
kφ(t)kL∞ ≤ C(T, J(0)).
Taking Fourier transform of the solution φ(t), we have
Z t
sin t|ξ| b
sin (t − s)|ξ| ^
e
b
φ(t, ξ) = cos t|ξ|φ0 (ξ) +
φ1 (ξ) +
ϕT ψψ(s, ξ)ds.
|ξ|
|ξ|
0
(4.21)
(4.22)
10
YUNG-FU FANG
EJDE-2004/102
Then we invoke (3.5), (3.9), (for = 41 ), and (4.20) to compute
Z
kφ(t)k
H
1 +δ
2
≤ kφ0 k
H
1 +δ
2
+ kφ1 k
1
H − 2 +δ
t
kϕT ψψ(s)k
+
0
1
H − 2 +δ
ds
\
1 ϕT ψψ ≤ J(0) + T 2 1 L2
c 2 −δ
M
1
≤ J(0) + T 2 kϕ\
ψψk 2
T
L
\
1 ϕ
T φψ 2 2
≤ J(0) + T 2 1
Sb 4 L
≤ J(0) + T ρ kϕT φψkL2
Z T
≤ J(0) + T ρ
kφ(t)k2L∞ kψ(t)k2L2 dt
0
≤ C T, J(0) ,
(4.23)
where ρ is some positive number. The calculation for kφt (t)k − 12 +δ is analogous.
H
Thus the above bounds ensure us to proceed the construction of solution beyond
T.
5. Null Form Estimate
In this section, we demonstrate the key estimate in Lemma 3.2: Let > 0 and ψ1 ,
1
ψ2 be the solutions for the Dirac equations (3.4). If the initial data ψ0j ∈ H − 4 + ,
j = 1, 2, then we have
b b ϕ\
T ψ 1 ψ2 2 ≤ C(T ) kψ01 kH −α + G1 2 kψ02 kH −α + G2 2 . (5.1)
1
cα Sb2α L
cα Sb 4 L
cα Sb 41 L
M
M
M
The proof of this estimate is based on the duality argument and it will be given
in a number of steps. Without loss of generality, we assume that ψ1 = ψ2 , and
prove: if ψ is a solution of the Dirac equation (2.1), then
ϕ\
b T ψψ 2 ≤ C(T ) kψ0 kH −α + G 2 2 .
cα Sb2α L
cα Sb 14 L
M
M
(5.2)
Recall the notation:
b ξ) := |τ | + |ξ| + 1,
E(τ,
c (τ, ξ) := τ 2 − |ξ|2 ,
W
b + := D(|ξ|,
b
D
+ξ),
b ξ) := |τ | − |ξ| + 1,
S(τ,
b ξ) := γ 0 τ + γ 1 ξ,
D(τ,
b − := D(|ξ|,
b
D
−ξ),
b as in (2.7), for the Dirac equation (2.1) is given by
The formula for ψ,
b ξ) =
ψ(τ,
∞
X
k=0
(k)
b+,k (ξ) + δ (k) (τ, ξ)A
b−,k (ξ) + K(τ,
b ξ),
δ+ (τ, ξ)A
−
(5.3)
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LOW REGULAIRTY FOR DIRAC-KLEIN-GORDON EQUATIONS
11
where δ± (τ, ξ) are the delta functions supported on {τ = ±|ξ|} respectively, δ (k)
mean derivatives of the delta function, and
b +G
b−
b −G
b+
b
a6 (τ ))D
(1 − b
a6 )D
b ξ) := D(τ, ξ) G
b f + (1 − b
+
,
K(τ,
c (τ, ξ)
2|ξ|(τ − |ξ|)
2|ξ|(τ + |ξ|)
W
Z b
b b∓ Gf + (1 − b
a6 (λ))G
b±,0 (ξ) := D± γ 0 ψb0 −
A
dλ ,
2|ξ|
λ ∓ |ξ|
b ± (−1)k Z
D
b
b± + b
b ∓ dλ.
A±,k (ξ) :=
(λ ∓ |ξ|)k−1 G
a6 (λ)G
2|ξ|k!
(5.4)
(5.5)
(5.6)
Moreover we write
b
b±,k (ξ) := D± fb±,k (ξ),
A
2|ξ|
(5.7)
b =K
b1 + K
b 2 , where
and set K
b
b 1 := D(τ, ξ) G
b ;
K
c (τ, ξ) f
W
b b
b b
b 2 := b1 D+ G− + b2 D− G+ ,
K
b Sb
E
(5.8)
and b1 , b2 are bounded functions. The Fourier transform of the quadratic expresc =ψ
b ∗ ψ,
b can be written as the sum of the following terms.
sion, ψψ
X (k) b (l) b
(5.9)
δ∓ A±,k ∗ δ± A
±,l ,
k,l
X
(k) b
(l) b
δ∓ A
±,k ∗ δ∓ A∓,l ,
(5.10)
k,l
X
(k) b
b
b X δ (k) A
b
b
b±,k , labele4.8c
δ∓ A
±,k ∗ K1 + K2 + K 1 + K 2 ∗
±
k
(5.11)
k
b ∗K
b ∗K
b ∗K
b ∗K
b1 + K
b2 + K
b1 + K
b2.
K
1
1
2
2
(5.12)
[
b† (−ξ),
A†±,k (ξ) = A
±,k
+
b†
fd
±,k (ξ) = f±,k (−ξ),
(5.13)
b
b † (−τ, −ξ)γ 0 ,
K(τ,
ξ) = K
(5.14)
Note that
b
[ (ξ) = fb† (−ξ) D± γ 0 ,
A
±,k
±,k
|ξ|
and
∞
b ξ) = X δ (k) (τ, ξ)A
b (ξ) + δ (k) (τ, ξ)A
b (ξ) + K(τ,
b
ψ(τ,
ξ),
+,k
−,k
−
+
(5.15)
k=0
Lemma 5.1. Let α < 1/4. Then the following estimate holds
b± 0
(l) b
(k) d
† D
∗ δ∓ D|ξ|∓ fb∓,l bT ∗ δ∓ f±,k
ϕ
|ξ| γ
2
L
cα Sb2α
M
1
≤ C(k + l + 1)T k+l− 2 kf±,k kH −α kf∓,l kH −α .
Proof. Let
b
b±,k ≡ δ (k) D± fb±,k = δ (k) A
b±,k .
Z
±
±
2|ξ|
(5.16)
12
YUNG-FU FANG
EJDE-2004/102
Using duality, we demonstrate the case (−, +), while the case (+, −) is being similar.
We first compute the fractional term
(
b
b
0,
if ξη > 0,
D(|ξ|,
−ξ)γ 0 D(|η|,
η)
=
0
1
|ξ||η|
2γ ± 2γ , if ξη < 0.
Throughout elementary analysis we have the bound:
(1 + |ξ|)α (1 + |η|)α
2α ≤ C,
(1 + |ξ + η|)α |ξ| + |η| − |ξ + η| + 1
(5.17)
for ξη < 0. Thus
ϕT Z −,k Z+,l , g Z
b
b
D(|ξ|,
−ξ)γ 0 D(|η|,
η) b
†
\
ϕT g(|ξ| + |η|, ξ + η)dξdη f+,l (η)tk+l
= fb−,k
(−ξ)
|ξ||η|
\
cα Sb2α tk+l
≤ Ckf−,k k −α kf+,l k −α kM
ϕT gkL2 ,
H
H
and through some computations, we have
1
\
cα Sb2α gbkL2 .
cα Sb2α tk+l
kM
ϕT gkL2 ≤ C(k + l + 1)T k+l− 2 kM
(5.18)
This completes the proof.
Lemma 5.2. Let α < 1/4. The following estimate holds
b± 0
(l) b
(k) d
† D
bT ∗ δ∓ f±,k
∗ δ± D|ξ|± fb±,l ϕ
|ξ| γ
2
L
cα Sb2α
M
1
≤ C(k + l + 1)T k+l− 2 kf±,k kH −α kf±,l kH −α .
Proof. Using duality, we demonstrate the case (+, +), while the case (−, −) is being
similar. We first compute the fractional term
(
b
b
0,
if ξη < 0,
D(|ξ|,
ξ)γ 0 D(|η|,
η)
=
0
1
|ξ||η|
2γ ∓ 2γ , if ξη > 0.
Throughout elementary analysis we have the bound:
(1 + |ξ|)α (1 + |η|)α
2α ≤ C
(1 + |ξ + η|)α − |ξ| + |η| − |ξ + η| + 1
(5.19)
for ξη > 0. Thus
ϕT Z +,k Z+,l , g Z
b
b
D(|ξ|,
ξ)γ 0 D(|η|,
η) b
†
\
= fb+,k
(−ξ)
f+,l (η)tk+l
ϕT g(−|ξ| + |η|, ξ + η)dξdη |ξ||η|
\
cα Sb2α tk+l
≤ Ckf+,k k −α kf+,l k −α kM
ϕT gkL2 .
H
H
This together with (5.18) complete the proof.
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13
Lemma 5.3. With the notation above, the following two estimates hold
kf±,0 kH −α ≤ C kψ0 kH −α + kf±,k kH −α ≤ C
b G
,
cα Sb 14 L2
M
b±
G
1
k
k 2.
cα Sb 14 L
k! M
(5.20)
(5.21)
The proof for the Lemma 5.3 is straight forward so that we skip it. Notice that,
b± .
in the (5.21), Sb ∼ 1 on the support of G
Lemma 5.4. With the notation above, the following estimate holds
b
b b ϕ
bT ∗ K 1 ∗ K1 2 ≤ C Gf 2 2 .
(5.22)
L
cα Sb2α
cα Sb 14 L
M
M
b := G
b f and K
b := K
b 1 . We use dyadic decomposiProof. For simplicity, we write G
tion to handle this case. Assume that
∞
X
b=
b ±,±,k ,
G
G
(5.23)
k=1
b ±,±,k (τ, ξ) is supported in one of the following four types of regions:
where G
Σ+,+ := {(τ, ξ) : τ > 0, +2k−1 < τ − |ξ| < +2k+1 },
Σ+,− := {(τ, ξ) : τ > 0, −2k+1 < τ − |ξ| < −2k−1 },
Σ−,+ := {(τ, ξ) : τ < 0, +2k−1 < τ + |ξ| < +2k+1 },
(5.24)
Σ−,− := {(τ, ξ) : τ < 0, −2k+1 < τ + |ξ| < −2k−1 }.
b induces a decomposition for K,
b namely
The decomposition of G
b
b ±,±,k = D G
b
K
.
c ±,±,k
W
To compute the convolution in (5.22),
Z
b
b
b
b
K
K
±,±,k ∗ K±,±,l (−τ, −ξ) =
±,±,k (−τ − σ, −ξ − η)K±,±,l (σ, η)dσdη
Z
0 b
b†
= K
±,±,k (τ + σ, ξ + η)γ K±,±,l (σ, η)dσdη,
(5.25)
(5.26)
we have 16 cases resulted from (5.24) and (5.26) as follows.
{(τ, σ, ξ, η) : τ + σ > 0, σ > 0, τ + σ − |ξ + η| ∼ ±2k , σ − |η| ∼ ±2l }
{(τ, σ, ξ, η) : τ + σ < 0, σ < 0, τ + σ + |ξ + η| ∼ ±2k , σ + |η| ∼ ±2l }
{(τ, σ, ξ, η) : τ + σ < 0, σ > 0, τ + σ + |ξ + η| ∼ ±2k , σ − |η| ∼ ±2l }
{(τ, σ, ξ, η) : τ + σ > 0, σ < 0, τ + σ − |ξ + η| ∼ ±2k , σ + |η| ∼ ±2l }
We label them as Σk,l [(±, ±); (±, ±)], and denote by Σk,l without specifying which
b k for abbreviation of K
b ±,±,k and G
b k for G
b ±,±,k .
one precisely. We also use K
Let g be an arbitrary function. We first compute
0
γ (τ +σ)−γ 1 (ξ+η) γ 0 γ 0 τ +γ 1 η = γ 0 (τ +σ)σ−(ξ+η)η +γ 1 (τ +σ)η−σ(ξ+η) .
14
YUNG-FU FANG
EJDE-2004/102
Thus, we have
E Z
0
1
0
1
Db
b l , gb = G
b † (τ + σ, ξ + η) γ (τ + σ) − γ (ξ + η) γ 0 γ σ + γ η G
b l (σ, η)
Kk ∗ K
k
(τ + σ)2 − (ξ + η)2
σ2 − η2
× gb(−τ, −ξ)dσ dη dτ dξ Z
1/2
bk
bl
2
G
G
≤ Ck
kL2 k
kL2
Ik,l (τ, ξ)gb(−τ, −ξ) dτ dξ
,
cα
cα
M
M
where
c2α (ξ + η)M
c2α (η)Q(τ, σ, ξ, η)
M
dσdη,
c 2 (τ + σ, ξ + η)W
c 2 (σ, η)
W
Dk,l
2 2
Q(τ, σ, ξ, η) := (τ + σ)σ − (ξ + η)η + (τ + σ)η − σ(ξ + η) ,
Z
Ik,l (τ, ξ) :=
(5.27)
(5.28)
and Dk,l (τ, ξ) is a slice of Σk,l for fixed (τ, ξ); i.e.,
Dk,l (τ, ξ) := {(σ, η) : (τ, σ, ξ, η) ∈ Σk,l }.
(5.29)
We need to sort the cases into two sets,
Σk,l [(±, ·); (±, ·)] and
Σk,l [(±, ·); (∓, ·)],
(5.30)
due to the fact that the computation for the 8 cases in each set is similar. For
simplicity, we will assume k ≥ l, while the other case is similar.
Cases H. We have the following estimate
b
b
b
G
b
K
+,·,k ∗ K+,·,l 2 ≤ Cl 1 1
+,·,k 2 G+,·,l 2 ,
L
L
L
(
−α)k
α
2α
α
α
c
b
c
c
2
22 2
M S
M
M
b
b
b
G
K
b
−,·,k 2 G−,·,l 2 ,
−,·,k ∗ K−,·,l 2 ≤ Cl 1 1
L
L
cα Sb2α
cα
cα L
2 2 2( 2 −α)k M
M
M
(5.31)
(5.32)
In these cases, we have (τ + σ)σ > 0. Throughout some algebraic manipulation,
the expression Q can be written as
2Q = (τ + σ − |ξ + η|)2 (σ + |η|)2 + (τ + σ + |ξ + η|)2 (σ − |η|)2
+ 8(τ + σ)σ |ξ + η||η| − (ξ + η)η .
(5.33)
Take the case of
b
b
K
+,+,k ∗ K+,+,l ,
(5.34)
as an example and in which Dk,l = {(η, σ) : τ + σ − |ξ + η| ∼ 2k , σ − |η| ∼
2l , (τ, σ, ξ, η) ∈ Σk,l [(+, +); (+, +)]}. In this case τ + σ > 0 and σ > 0. In the
ησ-plane, this is the region of the intersection of two forward cones. One has the
thickness of 2k and the translation of (−ξ, −τ ), while the other has thickness of 2l .
It is bounded mostly, except for the extreme case which is when one cone moves
along the other cone such that the intersection region is unbounded.
EJDE-2004/102
LOW REGULAIRTY FOR DIRAC-KLEIN-GORDON EQUATIONS
15
For the first part, we distinguish three cases: |ξ + η| ≤ |η|, |ξ + η| ≥ |η|, and the
extreme case. For the first two cases, we have
1
Ik,l
(τ, ξ) :=
Z
Dk,l
Z
=
Dk,l
∼
≤
≤
1
2l
Z
1
2l
Z
c2α (ξ + η)M
c2α (η)(τ + σ − |ξ + η|)2 (σ + |η|)2
M
dσ dη
c 2 (τ + σ, ξ + η)W
c 2 (σ, η)
W
c2α (ξ + η)M
c2α (η)
M
dσdη
(τ + σ + |ξ + η|)2 (σ − |η|)2
e k,l
D
e k,l
D
c2α (ξ + η)M
c2α (η)
M
dη
k
(2 + |ξ + η|)2
(5.35)
1
c2α (ξ)
dη M
(2k + |ξ + η|)2−2α
C
c2α (ξ)Sb4α (τ, ξ).
M
2(1−2α)k+l
For the extreme case, we obtain
1
Ik,l
(τ, ξ)
1
∼ l
2
Z
1
≤ l
2
Z
≤
e k,l
D
e k,l
D
c2α (ξ + η)M
c2α (η)
M
dη
(2k + |ξ + η|)2
(2k
C
2(1−2α)k+l
1
dη
+ |ξ + η|)2−4α
(5.36)
c2α (ξ)Sb4α (τ, ξ).
M
For the second part, again we distinguish three cases: |ξ + η| ≤ |η|, |ξ + η| ≥ |η|,
and the extreme case. For the first two cases, we get
2
Ik,l
(τ, ξ) :=
Z
Dk,l
c2α (ξ + η)M
c2α (η)(τ + σ + |ξ + η|)2 (σ − |η|)2
M
dσ dη
c 2 (τ + σ, ξ + η)W
c 2 (σ, η)
W
c2α (ξ + η)M
c2α (η)
M
dσdη
2
2
Dk,l (τ + σ − |ξ + η|) (σ + |η|)
Z
c2α (ξ + η)M
c2α (η)
1
M
∼ 2k−l
dη
l
2
(2 + |η|)2
e k,l
D
Z
1
1
c2α (ξ)
≤ 2k−l
dη M
l + |η|)2−2α
2
(2
e
Dk,l
Z
=
≤
(5.37)
C
c2α (ξ)Sb4α (τ, ξ).
M
2k+(1−2α)l
For the extreme case, we have
2
Ik,l
(τ, ξ)
∼
≤
≤
1
22k−l
1
22k−l
Z
e k,l
D
c2α (ξ + η)M
c2α (η)
M
dη
(2l + |η|)2
Z
e k,l
D
(2l
1
c2α (ξ)
dη M
+ |η|)2−4α
C
c2α (ξ)Sb4α (τ, ξ).
M
2k+(1−2α)l
(5.38)
16
YUNG-FU FANG
EJDE-2004/102
For the third part, we get
c2α (ξ + η)M
c2α (η)(τ + σ)σ |ξ + η||η| − (ξ + η)η
M
dσdη
c 2 (τ + σ, ξ + η)W
c 2 (σ, η)
W
Dk,l
Z
c2α (ξ + η)M
c2α (η)(τ + σ)σ|ξ + η||η|
C
M
dσdη
≤ 2k+2l
2
(τ + σ + |ξ + η|)2 (σ + |η|)2
Dk,l
(5.39)
Z
C
c2α (ξ + η)M
c2α (η)dσdη
≤ 2k+2l
M
2
Dk,l
3
Ik,l
(τ, ξ) :=
≤
Z
C
2(1−2α)k+l
c2α (ξ)Sb4α (τ, ξ).
M
The extreme case will not cause trouble since ξ + η and
η are of the same sign
except on a bounded region, i.e. |ξ + η||η| − (ξ + η)η = 0 except on a bounded
region. Let us denote the small region by R.
3
Ik,l
(τ, ξ) ≤
C
22k+2l
C
Z
ZR
c2α (ξ + η)M
c2α (η)(τ + σ)σ|ξ + η||η|
M
dσdη
(τ + σ + |ξ + η|)2 (σ + |η|)2
c2α (ξ + η)M
c2α (η)dσdη
M
22k+2l R
Z
C
c2α (ξ)
≤ 2k+2l
dσdη22αk M
2
R
C
c2α (ξ)Sb4α (τ, ξ).
≤ (1−2α)k+l M
2
≤
(5.40)
Cases E We have the following estimate
b
b
b
K
G
b
+,·,k ∗ K−,·,l 2 ≤ Cl 1 1
+,·,k 2 G−,·,l 2 ,
L
L
(
−α)k
cα Sb2α
cα
cα L
22 2 2
M
M
M
b
b
b
G
K
b
−,·,k 2 G+,·,l 2 ,
−,·,k ∗ K+,·,l 2 ≤ Cl 1 1
L
L
L
(
−α)k
α
2α
α
α
c
b
c
c
22 2 2
M S
M
M
(5.41)
(5.42)
In these cases, we have (τ + σ)σ < 0. Throughout some algebraic manipulation,
the expression Q can be written as
2Q = (τ + σ + |ξ + η|)2 (σ + |η|)2 + (τ + σ − |ξ + η|)2 (σ − |η|)2
− 8(τ + σ)σ |ξ + η||η| + (ξ + η)η .
Take the case of
b
b
K
−,+,k ∗ K+,+,l ,
(5.43)
as an example and in which Dk,l = {(η, σ) : τ + σ + |ξ + η| ∼ 2k , σ − |η| ∼
2l , (τ, σ, ξ, η) ∈ Σk,l [(−, +); (+, +)]}, In this case τ + σ < 0 and σ > 0. In ησ-plane,
this is the region of the intersection of a forward cone with a truncated backward
cone. One has the thickness of 2k and the translation of (−ξ, −τ ), while the other
has thickness of 2l . It is bounded for all cases. We still have the extreme case which
is when one cone moves along the other cone, though the region of intersection can
be as large as possible, nevertheless it is bounded.
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LOW REGULAIRTY FOR DIRAC-KLEIN-GORDON EQUATIONS
17
Again for the first part, we can estimate
c2α (ξ + η)M
c2α (η)(τ + σ + |ξ + η|)2 (σ + |η|)2
M
dσdη
c 2 (τ + σ, ξ + η)W
c 2 (σ, η)
W
Z
1
Ik,l
(τ, ξ) :=
Dk,l
c2α (ξ + η)M
c2α (η)
M
dσdη
(τ + σ − |ξ + η|)2 (σ − |η|)2
Z
=
Dk,l
1
≤ 2l
2
Z
Dk,l
(5.44)
(|ξ + η| + 1)2α (|η| + 1)2α
dσdη.
(τ + σ − |ξ + η|)2
To estimate the above integral, we separate the cases for |ξ + η| ≥ |η|, |ξ + η| ≤ |η|,
and the extreme case. Throughout some calculations, in each case, we have
1
Ik,l
(τ, ξ) ≤
1
1
c2α Sb4α .
M
l
(1−2α)k
2 2
(5.45)
For the second part, we derive
c2α (ξ + η)M
c2α (η)(τ + σ − |ξ + η|)2 (σ − |η|)2
M
dσdη
c 2 (τ + σ, ξ + η)W
c 2 (σ, η)
W
Z
2
Ik,l
(τ, ξ) :=
Dk,l
c2α (ξ + η)M
c2α (η)
M
dσdη
(τ + σ + |ξ + η|)2 (σ + |η|)2
Z
=
Dk,l
C
≤ 2k
2
Z
C2l
≤ 2k
2
Z
Dk,l
e k,l
D
(|ξ + η| + 1)2α (|η| + 1)2α
dσdη
(σ + |η|)2
(5.46)
(|ξ + η| + 1)2α
dη.
(2l + |η|)2+2α
To estimate the above integral, we separate the cases for |ξ + η| ≥ |η|, |ξ + η| ≤ |η|,
and the extreme case. Throughout some calculations, in each case, we have
2
Ik,l
(τ, ξ) ≤
1
1
c2α Sb4α .
M
2l 2(1−2α)k
(5.47)
For the third part, we have
3
Ik,l
(τ, ξ)
c2α (ξ + η)M
c2α (η)|τ + σ|σ |ξ + η||η| + (ξ + η)η
M
:=
dσdη
c 2 (τ + σ, ξ + η)W
c 2 (σ, η)
W
Dk,l
Z
c2α (ξ + η)M
c2α (η)|τ + σ|σ|ξ + η||η|
C
M
(5.48)
≤ 2k+2l
dσdη
2
(τ + σ − |ξ + η|)2 (σ + |η|)2
Dk,l
Z
C
≤ 2k+2l
(|ξ + η| + 1)2α (|η| + 1)2α dσdη.
2
Dk,l
Z
To estimate the above integral, we separate the cases for |ξ + η| ≥ |η|, |ξ + η| ≤ |η|,
and the extreme case. Notice that for the extreme case, we have |ξ + η||η| + (ξ +
η)η = 0 except on a small part of the region of the intersection. Throughout some
calculations, in each case, we have
3
Ik,l
(τ, ξ) ≤
1
1
c2α Sb4α .
M
2l 2(1−2α)k
(5.49)
18
YUNG-FU FANG
EJDE-2004/102
Now we return to the proof of (5.22). Combine the above, we get
Z
b b 2
1/2
K k Kl , g ≤C Gk 2 Gl 2
Ik,l (τ, ξ)gb(−τ, −ξ) dτ dξ
L
L
cα
cα
M
M
b bk G
G
1
C
cα Sb2α gbkL2
2 l 2 kM
≤ l ( 1 −α)k L
L
α
α
c
c
22 2 2
M
M
bk G
bl C 1 G
cα Sb2α gbkL2 .
2
kM
≤ 1 l k 1
cα Sb 4 L M
cα Sb 14 L2
24 2
M
Finally, we have
b
b ∗K
X K
b b K
2≤
k ∗ Kl 2
L
cα Sb2α L
cα Sb2α
M
M
k,l
≤
X
k,l
(5.50)
(5.51)
C
bk G
bl G
b 2
2 ≤ C G 2 2 .
1
1
1
L
L
k+
l
cα Sb 4
cα Sb 4
cα Sb 14 L
4
2
M
M
M
This completes the proof.
The estimates for the remaining cases are given in the following Lemma.
Lemma 5.5. For j = 1, 2 and k = 0, 1, 2, · · ·. The following estimates hold
b± 0
(k) d
† D
bj ∗ K
bT ∗ δ∓ f±,k
b ϕ
|ξ| γ
2 ≤ C(k + 1)T k− 12 kf±,k kH −α G 2 ,
L
α
2α
c
b
cα Sb 14 L
M S
M
b
b ∗ δ (k) D± fb
bT ∗ K
ϕ
b j
± |ξ| ±,k 2 ≤ C(k + 1)T k− 12 kf±,k kH −α G 2 ,
L
cα Sb2α
cα Sb 41 L
M
M
b
b b ϕ
bT ∗ K 1 ∗ K2 2 ≤ C G 2 2 ,
L
cα Sb2α
cα Sb 41 L
M
M
b
b b ϕ
bT ∗ K 2 ∗ Kj 2 ≤ C G 2 2 ,
L
α
2α
c
b
cα Sb 14 L
M S
M
The proof of this lemma is a repetition of the arguments presented in Lemmas
5.1, 5.2, and 5.4, so that we omit it.
Acknowledgement. The author want to express his gratitude to Manossos Grillakis and Chang-shou Lin for their encouragement and help.
References
[1] J. Bourgain, Invariant Measures for NLS in Infinite Volume, Commun. Math. Phys , Vol.
210 (2000), 605-620.
[2] A. Bachelot, Global existence of large amplitude solutions for Dirac-Klein-Gordon systems
in Minkowski space, Lecture Notes in Math. , Vol. 1402 (1989), 99-113. Springer, Berlin
[3] N. Bournaveas, A new proof of global existence for the Dirac-Klein-Gordon equations in one
space dimension, J. Funct. Anal. , Vol. 173 (2000), 203-213.
[4] J. Chadam, Global Solutions of the Cauchy Problem for the (Classical) Coupled MaxwellDirac Equations in one Space Dimension, J. Funct. Anal. , Vol. 13 (1973), 173-184.
[5] J. Chadam & R. Glassey, 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. , Vol. 54 (1974), 223-237.
[6] Yung-fu Fang, Local Existence for Semilinear Wave Equations and Applications to YangMills Equations, Ph.D dissertation , Vol. (1996),. University of Maryland
EJDE-2004/102
LOW REGULAIRTY FOR DIRAC-KLEIN-GORDON EQUATIONS
19
[7] Yung-fu Fang, A Direct Proof of Global Existence for the Dirac-Klein-Gordon Equations in
One Space Dimension, (2002), preprint.
[8] Yung-fu Fang & Manoussos Grillakis, Existence and Uniqueness for Boussinesq type Equations on a Circle, Comm. PDE, Vol. 21 (1996), 1253-1277.
[9] V. Georgiev, Small amplitude solutions of the Maxwell-Dirac equations, Indiana Univ. Math.
J., Vol. 40, (1991), 845-883.
[10] R. Glassey & W. Strauss, Conservation laws for the classical Maxwell-Dirac and KleinGordon-Dirac equations, J. Math. Phys. , Vol. 20 (1979), 454-458.
[11] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence
theorem, Comm. Pure Appl. Math. , Vol. XLVI (1993), 1221-1268.
[12] Sergej Kuksin, Infinite-Dimensional Symplectic Capacities and a Squeezing Theorem for
Hamiltonian PDE’s, Commun. Math. Phys , Vol. 167 (1995), 531-552.
[13] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
[14] Y. Zheng, Regularity of weak solutions to a two-dimensional modified Dirac-Klein-Gordon
system of equations, Commun. Math. Phys., Vol. 151 (1993), 67-87.
Department of Mathematics, National Cheng Kung University, Tainan 701 Taiwan
E-mail address: fang@math.ncku.edu.tw