On the Dirac-Klein-Gordon Equations in Three Space Dimensions

We establish a local existence result for Dirac-Klein-Gordon equations in three space dimensions, employing a null form estimate and a fixed point argument. 0. Introduction and Main Results. In the present work, we like to study the Cauchy problem for the Dirac-Klein-Gordon equations. The unknown quantities are a spinor field ψ : R × R 7→ C and a scalar field φ : R × R 7→ R. The evolution equations for the fields are given below, Dψ = φψ; (t, x) ∈ R× R (0.1a) ¤φ = ψψ; (0.1b) ψ(0, x) := ψ0(x), φ(0, x) = φ0(x), φ,t(0, x) = φ1(x), (0.1c) where D is the Dirac operator, D := −iγ∂μ, μ = 0, 1, 2, 3, and γ are the Dirac matrices, the wave operator ¤ = −∂tt + ∆, and ψ = ψ†γ0, and † is the complex conjugate transpose. The matrices can be written as follows. First let us define the 2× 2 matrices σ1, σ2, and σ3, σ1 := [ 0 1 1 0 ] , σ2 := [ 0 −i i 0 ] , σ3 := [ 1 0 0 −1 ] . (0.2a) The matrices γ are defined via γ = [ I2 0 0 −I2 ] , γ = [ 0 σj −σj 0 ] . (0.2b) 1 2 Y.F. FANG & M. GRILLAKIS 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. Such estimate can improve the existence result, see [Bo]. We will take advantage of the null form structure depicted in the nonlinear term ψψ, see [KM] and [Bo]. For the DKG system, there are many conserved quantities which are not positive definite, such as the energy, the momenta, and the angular momenta. However there is a known positive conserved quantity which is the law of conservation of charge, ∫ |ψ(t)|2 dx = constant, see [GS] In ’74, Chadam and Glassey showed that the Cauchy problem for the DKG equations has a unique local solution for ψ0 ∈ H, φ0 ∈ H, φ1 ∈ H, and global solution for a particular class of initial data, see [CG]. In ’81, Choquet-Bruhat proved the global existence result for the (massless) DKG equations by assuming small data, see [CB]. In ’88, Bachelot gave the global existence for DKG system with small data, see [Ba]. In ’99, Bournaveas derived a local existence for the DKG equations, based on a null form estimate, if ψ0 ∈ H 2 , φ0 ∈ H, φ1 ∈ L, see [Bo]. The outline of this paper is as follows. First we derive some solutions representations in Fourier transform. Next we prove some a priori estimates of solutions for Dirac equation and for wave equation. Then we show a local existence result for (0.1), employing the null form estimate together with some other estimates, and a fixed point argument. Finally we show the key estimate, namely the null form estimate. The main result in this work is as follows. Theorem 0.1. (Local Existence) Let 2 > 0. If the initial data of (0.1) ψ0 ∈ H 14+2(R3), φ0 ∈ H(R), φ1 ∈ L(R), then there is a unique local solution for (0.1). Remarks. 1. The DKG equations follow from the Lagrangian ∫ R3+1 { |∇φ|2 − |φt| − ψDψ − φψψ } dxdt. (0.3) DIRAC-KLEIN-GORDON EQUATIONS 1+3 D 3 2. The Dirac-Klein-Gordon system must be { Dψ = φψ; ¤φ + mφ = ψψ, (0.4) and the proof works for this system too. 3. Let I be the 4×4 identity matrix, D̂ = γτ +γξj , and ¤̂ = τ2−|ξ|2, thus we have D̂2 = ¤̂I. 4. ψψ = ψ†γ0ψ = |ψ1| + |ψ2| − |ψ3| − |ψ4|, where ψj are the component functions of the vector function ψ, which take values in C. 1. 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 of a given function. We will use μ = 1 4 + 2, α = 1 4 + δ, and ν = 2 − δ throughout the paper. We will also often skip the constant in the inequalities. For convenience, we denote the multipliers by Ê(τ, ξ) = |τ |+ |ξ|+ 1, (1.1a) Ŝ(τ, ξ) = ∣∣|τ | − |ξ| ∣∣ + 1, (1.1b) Ŵ (τ, ξ) = τ − |ξ|2, (1.1c) D̂(τ, ξ) = γτ + γξj , j = 1, 2, 3, (1.1d) M̂(ξ) = |ξ|+ 1. (1.1e) Notice that Ŵ and D̂ are the symbols of the wave and Dirac operators respectively. Also there is a summation over the upper and lower indeices. Consider the Dirac equation, { Dψ = G, (t, x) ∈ R × R, ψ(0) = ψ0. (1.2) First by taking the Fourier transform on (1.2) over the space variable and solving the resulting ODE, we can formally write down the solution 4 Y.F. FANG & M. GRILLAKIS as follows. ψ̃(t, ξ) = eit|ξ| 2|ξ| D̂(|ξ|, ξ)γ ψ̂0(ξ) + e−it|ξ| 2|ξ| D̂(|ξ|,−ξ)γ ψ̂0(ξ)+ ∫ t 0 ei(t−s)|ξ| 2|ξ| D̂(|ξ|, ξ)iG̃(s, ξ) ds + ∫ t 0 e−i(t−s)|ξ| 2|ξ| D̂(|ξ|,−ξ)iG̃(s, ξ) ds. (1.3) Rewriting the inhomogeneous terms in (1.3) gives ψ̃(t, ξ) = [eit|ξ| 2|ξ| D̂(|ξ|, ξ) + e−it|ξ| 2|ξ| D̂(|ξ|,−ξ) ] γψ̂0(ξ)+ ∫ [ e − eit|ξ| 2|ξ|(τ − |ξ|)D̂(|ξ|, ξ) + e − e−it|ξ| 2|ξ|(τ + |ξ|) D̂(|ξ|,−ξ) ] Ĝ(τ, ξ)dτ. (1.4) Now we split the function Ĝ into several parts in the following manner. Consider â(τ) a cut-off function equals 1 if |τ | ≤ 1 2 and equals 0 if |τ | ≥ 1, and denote by h(τ) the Heaviside function. For simplicity, let us write Ĝ±(τ, ξ) := h(±τ)â(τ ∓ |ξ|)Ĝ(τ, ξ), (1.5a) Ĝf (τ, ξ) := Ĝ(τ, ξ)− ( Ĝ+(τ, ξ) + Ĝ−(τ, ξ) ) , (1.5b) D̂± := D̂(|ξ|,±ξ). (1.5c) Notice that Ĝ± are supported in the regions {(τ, ξ) : ±τ > 0, |τ∓|ξ|| ≤ 1} respectively. Using the decomposition of the forcing term Ĝ = Ĝf +Ĝ+ + Ĝ−, the inhomogeneous term in (1.4) can be written as ∫ [ e − eit|ξ| 2|ξ|(τ − |ξ|)D̂(|ξ|, ξ) + e − e−it|ξ| 2|ξ|(τ + |ξ|) D̂(|ξ|,−ξ) ] Ĝf (τ, ξ)dτ = ∫ e D̂(τ, ξ) τ2 − |ξ|2 Ĝf dτ − e it|ξ| D̂+ 2|ξ| ∫ Ĝf τ − |ξ|dτ− e−it|ξ| D̂− 2|ξ| ∫ Ĝf τ + |ξ| (1.6a) ∫ e − eit|ξ| 2|ξ|(τ − |ξ|)++ + Ĝ−)dτ = eit|ξ| D̂+ 2|ξ| ∫ eit(τ−|ξ|) − 1 τ − |ξ| (Ĝ+ + â6(τ)Ĝ−)dτ+ ∫ e (1− â6(τ))D̂+Ĝ− 2|ξ|(τ − |ξ|) dτ − e it|ξ| D̂+ 2|ξ| ∫ (1− â6(τ))Ĝ− τ − |ξ| dτ, (1.6b) DIRAC-KLEIN-GORDON EQUATIONS 1+3 D 5 where â6(τ) = â( τ6 ) and â is the cut-off function defined previously. ∫ e − e−it|ξ| 2|ξ|(τ + |ξ|) D̂−(Ĝ+ + Ĝ−)dτ = e−it|ξ| D̂− 2|ξ| ∫ eit(τ+|ξ|) − 1 τ + |ξ| (â6(τ)Ĝ+ + Ĝ−)dτ+ ∫ e (1− â6(τ))D̂−Ĝ+ 2|ξ|(τ + |ξ|) dτ − e −it|ξ| D̂− 2|ξ| ∫ (1− â6(τ))Ĝ+ τ + |ξ| dτ. (1.6c) Recall the power expansion eit(τ±|ξ|) − 1 = ∞ ∑ k=1 1 k! (it)(τ ± |ξ|)k. (1.7) Combining (1.4)-(1.7), we can give a formula for ψ̂, namely