Asymptotic stability for the Dirac--Klein-Gordon system in two space dimensions

We study the Dirac--Klein-Gordon system in $1+2$ spacetime dimensions. We show global existence of the solutions, as well as sharp time decay and linear scattering. One key advance is that we provide the first asymptotic stability result for the Dirac--Klein-Gordon system in $1+2$ spacetime dimensions in the case of a massive Klein-Gordon field and a massless Dirac field. The nonlinearities are below-critical in two spatial dimensions, and so our method requires the identification of special structures within the system and novel weighted energy estimates. Another key advance, is that our proof allows us to weaken certain conditions on the nonlinear structures that have been assumed in the literature.

We define certain assumptions on the constant matrices F, H: Conditions H1a and H2a are natural in the sense that H1a guarantees the conservation of charge while H2a ensures that the nonlinear term in the Klein-Gordon equation is real valued.Conditions H1b and H2b are, respectively, special cases of H1a and H2a.We also note that there exist non-trivial examples of the matrix F satisfying H1a, for instance F = γ µ for µ = 0, 1, 2.

Main results
We first state our main theorems and then discuss their relation to previous results in the literature, followed by an outline of the novel ideas used in our proof.
Remark 1.3.In both Theorem 1.1 and Theorem 1.2, the pointwise decay of the solutions is sharp in time in the sense that the solutions enjoy the same decay rates in time as the linear equations.Thus, we prove asymptotic stability for the 2D DKG system (1.1) under the relevant assumptions stated in the theorems.Indeed, our result provides the first asymptotic stability result for the DKG system for the case M = 0, m = 1 for smooth, small, and compactly supported initial data.
Remark 1.4.Grünrock and Pecher [23] have shown global existence for the 2D DKG system (1.1) under the assumptions M, m ∈ R, H1b and H2b and with (large) low-regularity data Thus, our main contribution for the case M = 0, m = 1 is to show asymptotic stability and to weaken the structural assumptions on the nonlinearities considered in [23].In particular in Theorem 1.1 we can allow for H = γ 0 .It is not yet clear whether the most general case of H1a and H2a can be shown to admit small global solutions.

Previous work on the DKG system
The system (1.1) arises in particle physics as a model for Yukawa interactions between a scalar field and a Dirac spinor.It appears in the theory of pions and in the Higgs mechanism [2].We note that the nonlinearity ψ * γ 0 ψ is often writen as ψψ where ψ := ψ * γ 0 is the Dirac adjoint, and thus transforms as a scalar under Lorentz transformations.The Cauchy problem for the DKG system has been actively studied in various spacetime dimensions and for different cases of the Klein-Gordon and Dirac masses (i.e.m ≥ 0 and M ≥ 0).

Three spatial dimensions.
For high-regularity initial data, there are small-data results that show global existence for certain subcases of (1.1) with asymptotic decay rates [4,27].Similar results are also known for the closely related Dirac-Proca system [47,27].For low-regularity initial data, the problem is more difficult as the natural energy density associated to these DKG systems does not have a definite sign.The lack of positive definite conserved quantities makes it particularly difficult to prove global existence and scattering for low-regularity data.For results (note under conditions H1b and H2b), see for example [6,49] and references within, as well as for large-data results, see for example [13,9,12] and references cited within.
Two spatial dimensions.For high-regularity initial data, global existence and asymptotic stability to the DKG system (1.1) for the case M > 0, m > 0 was shown in the works [42,39] for smooth, small initial data.These results rely on transforming the DKG system (1.1) into two coupled Klein-Gordon equations.The asymptotic stability (even the stability) result for the other cases of M = 1, m = 0 or M = m = 0 remains open however.For low-regularity initial data, there are local existence results under the assumptions H1b and H2b [8,11].Global existence for low-regularity and possibly large data, again under the conditions H1b and H2b, is known by [23].
The study of two dimensional Dirac equations [28,5,36] is also relevant to our study.

Major difficulties and challenges
We first remind the reader of the important identity Thus we can think of (1.1) as encoding a coupled wave-like and Klein-Gordon system.Proving global existence and asymptotic decay results for coupled nonlinear wave and Klein-Gordon equations, such as in Theorems 1.1 and 1.2, is typically a challenging question in two spatial dimensions.This is because linear wave w and linear Klein-Gordon v equations have very slow pointwise decay rates in R 1+2 , namely (1.9) The identity (1.8) also indicates that a linear massless Dirac field should obey the same slow pointwise decay rates as |w| above.As a consequence of (1.9), when using Klainerman's vector field method [30] on quadratic nonlinearities, we might at best get an integral of t −1 .This leads to problems when closing the bootstrap argument and can possibly indicate finite-time blow-up.
Another obstacle when studying Klein-Gordon equations, in the framework of the vector field method, is that the scaling vector field L 0 = t∂ t + x 1 ∂ x 1 + x 2 ∂ x 2 does not commute with the Klein-Gordon operator − + 1.The scaling vector field can be avoided by using a spacetime foliation of surfaces H s of constant hyperboloidal time s = t 2 − |x| 2 .This idea originates in work of Klainerman [29,31] (see also Hörmander [24]) on Klein-Gordon equations, and was later reintroduced to treat coupled wave and Klein-Gordon equations by LeFloch and Ma in [33] under the name of the "hyperboloidal foliation method".This method can be regarded as Klainerman's vector field method on hyperboloids.We also remind the reader of the pioneering work by Tataru showing Strichartz estimates for wave equations in the hyperbolic space [46], and the work by Psarelli [40] on the Maxwell-Klein-Gordon equations.
Returning now to the DKG problem (1.1), we use the identity (1.8) to derive the following If we ignore the structure here (indeed, under H2a the term ψ * Hψ does not have any special structure), we roughly speaking have obtained a wave-Klein-Gordon system of the form The global existence of general small-data solutions to (1.11) is presently unknown in R 1+2 .Furthermore, if we assume that w and v obey the linear estimates (1.9), then the best we can expect from the nonlinearities (in the flat t = cst.slices) is Returning to the original PDE (1.1), for example under the assumptions H1b and H2a of Theorem 1.1, the best we can expect appears to be Thus one quantity is at the borderline of integrability and the other is strictly below the borderline of integrability.In previous work of the authors [18], such a situation was termed 'below-critical' in time decay, and indicates that if the classical vector field method is to be successful, then new ideas are required to close both the lower and higher order bootstraps.

Key ingredients and new ideas
To conquer the aforementioned difficulties in studying the DKG equations (1.1), we need several ingredients and novel observations that go beyond classical methods for Klein-Gordon equations such as in [29,31].The first ingredient is an energy functional, defined on hyperboloids, for solutions to the Dirac equation.This was first derived by the authors and LeFloch in [20].Using this Dirac-energy functional, we find that the best behaviour we can hope for is Here H s are constant s-surfaces defined in Section 2.1 and L 2 f (H s ) is defined in (2.1).Rough calculations, for instance under the assumptions H1b and H2a, lead us to the estimates (1.12) We see that one term is at, and the other is below, the borderline of integrability.We remark that the only other known work in the literature of coupled wave and Klein-Gordon equations studying such a situation is in our [18].
Our first new insight is to notice that a field can be thought of as 'Klein-Gordon type' if its L 2 f (H s )-norm is well-controlled by the natural energy functionals.We know that examples of 'Klein-Gordon type' fields include v, (s/t)∂ α v and we discover the further examples (s/t)ψ, ψ − (x a /t)γ 0 γ a ψ.
We then uncover a decomposition (see Lemma 3.2) in the following Dirac-Dirac interaction term The key observation is that terms on the RHS above always involve at least one 'Klein-Gordon type' factor.This observation is of vital importance in the proof of both Theorems 1.1 and 1.2.For example when H = γ 0 , (1.13) allows us to improve the initial estimate given in (1.12) to Interestingly, we find that several other Dirac-Dirac interactions, such as ψ * ψ, do not possess the same useful decomposition (see Remark 3.3).In addition, we find that the structure of the nonlinearity ψ * γ 0 ψ is preserved under commutation with the Lorentz boosts (see Lemma 3.4) and thus the decomposition (1.13) can be applied at higher orders.
The next ingredient comes from using nonlinear transformations to remove slowly-decaying nonlinearities (see Lemma 4.4 and a new transformation for the Dirac field given in Lemma 4.6) when estimating the low order energy.This comes at the expense of introducing cubic nonlinearities and quadratic null forms and we are able to close the bootstrap at lower-orders, provided we can control these null forms.
One more ingredient, needed to control the null forms introduced in the previous paragraph, is to obtain additional (t − r)-decay for the Dirac spinor.In the case of pure wave equations it is well-known that one can obtain extra (t − r)-decay with the aid of the full range of vector fields {∂ α , Ω ab , L a , L 0 } (defined in Section 2.1).For instance, for sufficiently regular functions φ we have the estimate [44] ∂∂φ If we cannot control certain vector fields acting on our solution, then it is usually more difficult to obtain extra (t − r)-control as in (1.14).We recall two examples of similar situations: 1) obtaining extra (t − r)-decay in the case of nonlinear elastic waves by Sideris [43] where Lorentz boosts L a = t∂ a + x a ∂ t are unavailable; 2) obtaining extra (t − r)-decay in the case of coupled wave-Klein-Gordon equations by LeFloch-Ma [33, §8.1, §8.2]where the scaling vector field L 0 is absent.
In the DKG model (1.1) we also cannot use L 0 , nor can we directly gain (t−r)-decay by studying the wave equation in (1.10).Our insight, inspired by the work [33], is to rewrite the Dirac operator in a frame adapted to the hyperboloidal foliation.The latter idea yields the following estimate This argument gives us the extra (t − r)-decay for ∂ψ (see Lemma 3.5 and Proposition 4.3) required to close the null form estimates.
The final ingredient, key to closing the highest order bootstrap for Theorem 1.1, is to derive weighted energy inequalities.We recall that we cannot rely on nonlinear transformations when estimating the highest order energy, and the nonlinearities are below-critical.Our idea is to derive and rely on a (t − r)-weighted Dirac energy functional (see Proposition 2.3).Such weighted estimates were introduced in [3], and have recently been adapted to the hyperboloidal setting, with applications to the Klein-Gordon-Zakharov system in [14].We utilise such weighted estimates here for the first time for Dirac equations.
Remark 1.6.We expect the ideas in the proof of Theorems 1.1 and 1.2 to have other applications.For instance, it can be used to show uniform energy bounds for the solution to the 3D Dirac-Klein-Gordon equations studied by Bachelot in [4] as well as the U (1)-Higgs model studied in [20].
Remark 1.7.Our decomposition approach in (1.13) in fact gives a reinterpretation of structure identified by Bournaveas [7,8].Suppose that there exists φ such that ψ = iγ ν ∂ ν φ.Using (1.8) one can show that − φ = iγ ν vF ∂ ν φ and also The two bracketed terms in (1.15) are semilinear null terms, which are known to obey better estimates (see for example Lemma 2.4).Such null structure played an essential role in the previous works, for example [23], mentioned in section 1.2 that rely on H2b.In the case of Theorem 1.1, however, our approach allows us to weaken the assumption of H2b to H2a.

Wave-Klein-Gordon Literature
To conclude the introduction, we remind the reader of some of the literature concerning global existence and decay for coupled wave-Klein-Gordon equations.In 3D these include wave-Klein-Gordon equations derived from mathematical physics, such as the Dirac-Klein-Gordon model, the Dirac-Proca and U (1)-electroweak model [20,27,47], the Einstein-Klein-Gordon equations [26,34,35,48], the Klein-Gordon-Zakharov equations [38], the Maxwell-Klein-Gordon equations [32,22] and certain geometric problems derived from wave maps [1].Very recently, there has been much research concerning global existence and decay for wave-Klein-Gordon equations in 2D.We mention for instance the works by Ma [37] and the present authors [18] for compactly supported initial data; see also the references therein.There have also been works [45,25,14] that have investigated wave and Klein-Gordon systems under certain null conditions without the restriction to compactly supported data.Other work has looked at the Klein-Gordon-Zakharov model in 1 + 2 dimensions [15,21,37,17] and the wave map model derived in [1] has been studied in the critical case of 1 + 2 dimensions in the recent works [19,21] (see also [50]).An analysis of general classes of cubic nonlinearities has also been given in [10].
Outline.We organise the rest of the paper as follows.In Section 2 we introduce some essential notation and the preliminaries of the hyperboloidal method.In Section 3 we present the essential hidden structure within the nonlinearities.Finally, Theorems 1.1 and 1.2 are proved in Section 4 and Appendix A, respectively, by using a classical bootstrap argument.

Basic notations
We denote a spacetime point in R 1+2 by (t, x) = (x 0 , x), and its spatial radius by r := (x 1 ) 2 + (x 2 ) 2 .Following Klainerman's vector field method [30], we introduce the following vector fields Such vector fields are referred to as translations, Lorentz boosts, rotations and scaling respectively.We also use the modified Lorentz boosts, first introduced by Bachelot [4], These are chosen to be compatible with the Dirac operator, in the sense that [ L a , iγ µ ∂ µ ] = 0 where we have used the standard notation for commutators [A, B] := AB − BA.
We restrict out study to functions supported within the spacetime region K := {(t, x) : t ≥ 2, t ≥ |x| + 1} which we foliate using hyperboloids.A hyperboloid H s with hyperboloidal time s ≥ s 0 = 2 is defined by H s := {(t, x) : t 2 = |x| 2 + s 2 }.We find that any point (t, x) ∈ K ∩ H s with s ≥ 2 obeys the following relations Without loss of generality we take s 0 = 2, and we use K [s 0 ,s 1 ] := s 0 ≤s≤s 1 H s K to denote the spacetime region between two hyperboloids H s 0 , H s 1 .We follow LeFloch and Ma [33] and introduce the semi-hyperboloidal frame The semi-hyperboloidal frame is adapted to the hyperboloidal foliation setting since the set ∂ a generate the tangent space to the hyperboloids.The usual partial derivatives, i.e. those in a Cartesian frame, can be expressed in terms of the semi-hyperboloidal frame as Standard notation.We use C to denote a universal constant, and A B to indicate the existence of a constant C > 0 such that A ≤ BC.For the ordered sets . Spacetime indices are represented by Greek letters while spatial indices are denoted by Roman letters.We adopt Einstein summation convention unless otherwise specified.We will often write |∂φ|, respectively |∂φ|, to denote an estimate on |∂ µ φ| for arbitrary µ, respectively |∂ a φ| for arbitrary a.

Energy estimates for wave and Klein-Gordon fields on hyperboloids
Given a function φ = φ(t, x) defined on a hyperboloid H s , we define its With this, the norm • L p f (Hs) for 1 ≤ p < +∞ can be defined.The subscript f comes from that the fact that the volume form in (2.1) comes from the standard flat metric in R 2 .
Following [24,33], we define the following L 2 -based energy of a function φ = φ(t, x), scalar-valued or vector-valued, on a hyperboloid Note in the above m ≥ 0 is a constant.From the last two equivalent expressions of the energy functional E m , we easily obtain We also adopt the abbreviation E(s, φ) = E 0 (s, φ).We have the following classical energy estimates for wave and Klein-Gordon equations.
Proposition 2.1.Let φ be a sufficiently regular function with defined in the region

Energy estimates for Dirac fields on hyperboloids
Let Ψ(t, x) : R 1+2 → C 2 be a complex-valued function defined in the region K [s 0 ,∞) .We introduce the energy functionals These were first introduced in [20], and the following useful identity was also derived From this identity we obtain the non-negativity of the functional E D (s, Ψ) and the inequality We have the following energy estimates (see [20,Prop. 2.3] for (2.4) and see [16] for an application of (2.5)).Proposition 2.2.Let Ψ(t, x) : R 1+2 → C 2 be a sufficiently regular function with support in the region

Weighted energy estimates
Following ideas of Alinhac [3], we next derive weighted energy estimates.These have been applied to coupled wave-Klein-Gordon systems in [15,Prop. 3.2], and here we pursue similar estimates for Dirac equations.We first define the (t − r)-weighted energy for a Dirac field The following useful identity holds: vanishing near ∂K [s 0 ,s] and satisfying Then for γ > 0 we have (2.8) Proof.As shown in [15], multiplying the Dirac equation by (t − r) −2γ ∂ t Ψ * γ 0 the proof follows from the differential identity and the fact that

Estimates for null forms and commutators
We next state a key estimate for null forms in terms of the hyperboloidal coordinates.The proof is standard and can be found in [33, §4].
Lemma 2.4.Let φ, ϕ be sufficiently regular functions with support in K and define We also have the useful property that for the Q 0 null form: Besides the well-known commutation relations valid for m ≥ 0, we also need the following lemma to control some other commutators.A proof can be found in [33, §3] and [34].

Weighted Sobolev inequalities on hyperboloids
We need certain weighted Sobolev inequalities to obtain pointwise decay estimates for the Dirac field and the Klein-Gordon field.
Proposition 2.6.Let φ = φ(t, x) be a sufficiently smooth function supported in the region K and γ ∈ R. Then for all s ≥ 2 we have (2.9) (2.11) We recall that such Sobolev inequalities involving hyperboloids were first introduced by Klainerman [29], and then later appeared in work of Hörmander [24].In the above Proposition we have used the version given by Hörmander in [24] where only the Lorentz boosts are required.The estimate (2.10) follows by combining (2.9) with the commutator estimates of Lemma 2.5 and is more convenient to use for wave components.
We also have the following modified Sobolev inequalities for spinors which make use of the modified Lorentz boosts L a .The proof follows from the fact that the difference between L a and L a is a constant matrix.
Corollary 2.7.Let Ψ = Ψ(t, x) be a sufficiently smooth C 2 -valued function supported in the region K. Then for all s ≥ 2 we have (2.12) as well as sup (2.13)

Linear scattering
To show linear scattering of the solution (v, ψ) in the energy space in Theorems 1.1 and 1.2, we need the following result, which gives a sufficient condition on linear scattering for Klein-Gordon and Dirac equations.
Lemma 2.8.Consider the Klein-Gordon equation If the source term satisfies then the solution u scatters linearly in the energy space.That is, there exists u + , such that in which u + is the solution to the free Klein-Gordon equation

Similarly, consider the Dirac equation
If the source term satisfies then the solution Ψ scatters linearly in the energy space, i.e., there exists Ψ + , such that in which Ψ + is the solution to the free Klein-Gordon equation ).The result in Lemma 2.8 is classical, and its proof can be found for instance in [17].We note that the scattering result is valid on constant t slices, while we work on constant s slices.
3 Hidden structure within the Dirac-Klein-Gordon equations

Transformations
In the present section we discuss three types of hidden structures which are present in the Dirac-Klein-Gordon equations.These are in the spirit of Shatah's normal form method [41].Identifying these structures plays an important role in our proof.
Type 1: Consider a Klein-Gordon equation of the type (− + 1)v = w 2 + F v , where w satisfies an unspecified semilinear wave equation.If we set v = v − w 2 , then we have In particular, we can remove the wave-wave interaction w 2 at the expenses of bringing in cubic and null terms.This strategy of treating wave-wave interactions in Klein-Gordon equations was first introduced by Tsutsumi [47] to study the Dirac-Proca equations in R 1+3 .
Type 2: Next we consider a wave equation with the form − w = wv + F w , where v satisfies an unspecified semilinear Klein-Gordon equation.If we set w = w + wv, then we have We can remove the interaction term wu at the expense of introducing null and cubic terms.
Thus we arrive at The nonlinear transformation has allowed us to cancel the Dirac-Klein-Gordon interaction vψ at the expense of introducing null and cubic terms.Such a transformation has, to the authors' knowledge, not been used before and is clearly inspired by the two prior transformations.

Hidden
Klein-Gordon structure in the Lorentz scalar ψ * γ 0 ψ We now consider the Dirac-Dirac interaction term ψ * γ 0 ψ and show that it can be decomposed into terms with Klein-Gordon type factors.Roughly speaking, we call a field φ of 'Klein-Gordon type' if its

If no confusion arises, we use the abbreviation
Lemma 3.2.Let Ψ, Φ be two C 2 -valued functions, then we have Proof.First we note 2Ψ = Ψ − + Ψ + and 2Φ = Φ − + Φ + .Thus we have We expand the last term above, noting (γ 0 γ a ) * = γ 0 γ a , and find Simple calculations give us and Thus we are led to Gathering together the above results finishes the proof.
Remark 3.3.The above Lemma gives the key improvement that the quadratic interaction term Ψ * γ 0 Φ can be written in terms of other quadratic interactions which always involve at least one 'Klein-Gordon type' field.It is also interesting to note that other Dirac-Dirac interactions terms do not possess the above useful decomposition.For example, replicating the argument for ψ * ψ in the proof of Lemma 3.2, we find (3.1)instead appears with a positive sign +(r/t) 2 Ψ * γ 0 Φ.This means that we cannot obtain a good factor of (s/t) 2 as in (3.2).Similar problems occur for ψ * γ 0 γ µ ψ.In this sense, general nonlinear terms ψ * Hψ under assumption H1a are more difficult to treat.
Since the Dirac-Dirac interaction term ψ * γ 0 ψ appears as a sourcing for the Klein-Gordon equation when H2b is assumed, we will need to act unmodified Lorentz boosts L on this term.The following Lemma surprisingly shows that when distributing these Lorentz boosts across the interaction term, they in fact turn into the modified boosts L.
Proof.Let Ψ, Φ be two C 2 -valued functions.We will only consider the case with Lorentz boosts acting on the nonlinearity.Since * denotes the conjugate transpose, and (γ 0 γ a ) * = (γ 0 γ a ), we have the identity and thus Similarly, Carrying on gives the general pattern.

Decay away from the light cone for differentiated Dirac components
The following lemma is inspired by a similar result in the context of wave equations obtained in [33, §8.Proof.We express the Dirac operator iγ µ ∂ µ in the semi-hyperboloidal frame to get Multiplying both sides by Simple calculations involving properties of the Dirac matrices imply This leads us to Finally we arrive at (3.3) by recalling the following relations, which hold within the cone K, 4 Proof of Theorem 1.1

Bootstrap assumptions and preliminary estimates
Fix N ∈ N a large integer (N ≥ 7 will end up working for our argument below).As shown by the local well-posedness theory in [33, §11], initial data posed on the hypersurface {t 0 = 2} and localised in the unit ball {x ∈ R 2 : |x| ≤ 1} can be developed as a solution of (1.1) up to the initial hyperboloid {s = s 0 } with the smallness (1.6) conserved.Thus there exists C 0 > 0 such that the following bounds hold for all |I| ≤ N : Next, we assume that the following bounds hold for s ∈ [s 0 , s 1 ): In the above, the constant C 1 ≫ 1 is to be determined, ǫ ≪ 1 measures the size of the initial data, and we let C 1 ǫ ≪ 1, and 0 < δ ≤ 1 10 .For the rest of section 4 we assume, without restating the fact, that (4.2) hold on a hyperboloidal time interval [s 0 , s 1 ) where With the bounds in (4.2), we obtain the following preliminary L 2 and L ∞ estimates.Proposition 4.1.For s ∈ [s 0 , s 1 ) we have Proof.The estimates for ψ follow from the definition of the energy functional E D (s, ψ), E D (s, ψ, 1), the decomposition (2.3), the commutator estimates in Lemma 2.5 and the fact that the difference between L a and L a is a constant matrix.The estimates for the Klein-Gordon field follow from the definition of the energy functional E 1 (s, v) and the commutator estimates in Lemma 2.5.
Next we derive the following pointwise estimates.
Proposition 4.2.For s ∈ [s 0 , s 1 ) we have Proof.To show the estimates for the Klein-Gordon components v and ∂v we combine the estimates from Proposition 4.1 with the Sobolev estimates from Proposition 2.6.To prove the estimates for Z I ψ, and thus Z I ψ, we combine Proposition 4.1 with the Dirac-type Sobolev estimates from Corollary 2.7.Finally to prove the estimates for (ψ) − and derivatives thereof, we note γ 0 γ 0 = I 2 in order to show the commutator identity We can control this error term since |x b /t| ≤ 1 in the cone.Using these calculations, we can compute Thus, using the first Sobolev estimate in Corollary 2.7, The estimates for ( Z I ψ) − follow in the same way and the proof is complete.
Proposition 4.3.The following weighted L 2 -estimates are valid for s ∈ [s 0 , s 1 ) and the following pointwise estimates also hold for s ∈ [s 0 , s 1 ) Proof.We first act Z I , with |I| ≤ N − 4, to the ψ equation in (1.1) to find Then by Lemma 3.5 we obtain in which we used the pointwise decay results of Proposition 4.2.The estimates ∂ t Z I ψ are a simple consequence of the above, while the case ∂ a Z I ψ (with a = 1, 2) can be seen from the relation Finally the L 2 -type estimates follow in a similar way, by combining Lemma 3.5 with Propositions 4.1 and 4.2.

Nonlinear transformations and corresponding estimates
Next, we introduce nonlinear transformations in the spirit of Shatah's normal form method [41].These are key to closing the low order bootstraps.
Proof.This nonlinear transformation was introduced in [47].The required result follows by using (1.1) to deduce Lemma 4.5.We have Proof.We estimate each of these three quantities in turn.
Step 1: Estimate of We first decompose the term into three pieces We now bound in which we used Lemma 2.5, Proposition 4.1, and Proposition 4.2.We continue to estimate in which we used Lemma 2.5, Proposition 4.1, and Proposition 4.2.We then get in which we used Propositions 4.1 and 4.2.
Thus we obtain Step 2: Estimate of We decompose the term into three pieces We first estimate in which we used Lemma 2.5, Proposition 4.1, and Proposition 4.2.We now bound in which we used Lemma 2.5, Proposition 4.2, and Proposition 4.3.We then obtain in which we used Proposition 4.1 and Proposition 4.2.
In conclusion, we get Step 3: Estimate of First, according to Lemma 2.5 and Lemma 2.4 we have We next estimate in which we used Proposition 4.3.We then bound in which we used Lemma 2.5, Proposition 4.1, and Proposition 4.3.Easily we can show To sum up, we get Next we introduce a nonlinear transformation of Type 3 as discussed in Section 3.1.
Lemma 4.7.We have Proof.We bound the terms one by one.
Step 1: We start by estimating Z I (ψ * Hψ)ψ L 2 f (Hτ ) for |I| ≤ N − 1.We find that Easily, we get in which we used Propositions 4.1 and 4.2.In the same way, we have and hence we arrive at Step 2: Next we estimate We note that For the term B 2a , we have in which we used Lemma 2.5, Proposition 4.2, and Proposition 4.3.For the term B 2b , we get in which we used Lemma 2.5, Proposition 4.1, and Proposition 4.2.For the third term B 2c , we obtain B 2c in which we used Lemma 2.5, Proposition 4.1, and Proposition 4.2.For the last term B 2d , we have in which we used Lemma 2.5, Proposition 4.1, and Proposition 4.2.
To conclude, we have Step 3: We now turn to Recalling Lemmas 2.5 and 2.4, we find that We have in which we used Propositions 4.1, 4.2, and 4.3.In succession, we get in which we used Lemma 2.5, Proposition 4.1, and Proposition 4.2.Similarly, we obtain in which we used Lemma 2.5, Proposition 4.1, Proposition 4.2, and Proposition 4.3.Easily, we get To conclude, we get Step 4: Finally, we estimate The estimate is very similar to Step 3 above, but we write it out for completeness.We first bound We have B 4a in which we used Propositions 4.1, 4.2, and 4.3.In succession, we get in which we used Lemma 2.5, Proposition 4.1, and Proposition 4.2.Similarly, we obtain in which we used Lemma 2.5, Proposition 4.1, Proposition 4.2, and Proposition 4.3.Easily, we get To conclude, we have

Improved estimates for low order energy
In order to improve the lower order energy bounds for Klein-Gordon and Dirac fields, we use nonlinear transformations (see Sections 3.1 and 4.2) to remove the slowly-decaying terms.This is at the expense of introducing null and cubic terms yet nevertheless allows us to obtain the desired energy bounds.Our strategy is to first estimate the new variables v, ψ in Lemmas 4.4 and 4.6, and then use these to estimate the original unknowns v, ψ.
Proof.Using the energy estimate in Proposition 2.1 for Klein-Gordon equations, together with the estimates in Lemma 4.5, we get for the v component that Proposition 4.9.We have Proof.We note that We find that B 1a in which we used Proposition 4.1 and Proposition 4.2.To proceed, we have in which we used Lemma 2.5, Proposition 4.1, and Proposition 4.2.We also get in which we used Proposition 4.1 and Proposition 4.2.Thus, we get In a similar way, we get Proof.According to the energy estimate (2.4) for Dirac equations, we have in which we used the estimates in Lemma 4.7.As a consequence, we obtain On the other hand, for |I| ≤ N − 2, we apply the energy estimate (2.5) for Dirac equations to get For the term D 3 , the estimates in Lemma 4.7 and (4.6) imply that Then we treat the term D 2 , and according to Lemma 3.2 we find that We proceed to have (recall 4 , in which we used the estimates in (4.6), Proposition 4.1, and Proposition 4.2.In turn we get, recall again in which we used again the estimates in (4.6), Proposition 4.1, and Proposition 4.2.Since the analysis for bounding the other two terms is very similar, we write directly the final estimates without further details To sum things up, we have shown and thus the proof is complete.
Proposition 4.11.We have Proof.We recall that so it suffices to show By the definition and decomposition of the energy functional E D in (2.6)-(2.7),we need to bound We only estimate for the case of |I| ≤ N − 2 as the case of |I| = N − 1 can be bounded in a very similar way.For |I| ≤ N − 2 we have We proceed to get in which we used the estimates in Lemma 2.5, Proposition 4.1, and Proposition 4.2.Next, we bound in which we used again the estimates in Lemma 2.5, Proposition 4.1, and Proposition 4.2.Thus we arrive at Analogously, we can show which concludes the proposition.

Improved estimates for the highest order energy
Our goal now is to close the highest order energy bootstrap.An essential difference compared with the lower order energy estimates is that nonlinear transformations are invalid due to issues with regularity.It seems impossible to close the highest order bootstrap at the first glance of the nonlinearities.Fortunately, the special structure of the DKG system, the Klein-Gordon decomposition within the nonlinearities and our (t − r) weighted energy estimate (see Proposition 2.3) will allow us to reach the desired goals.
Proposition 4.12.We have Proof.Recall the energy estimate for Klein-Gordon equations in Proposition 2.1, and for |I| = N we find that E 1 (s, Direct calculations show that in which we used the estimates in Propositions 4.1 and 4.2, and the fact 1.
Thus we arrive at Proposition 4.13.We have Proof.We apply a (t − r)-weighted energy estimate for the Dirac equation of Z I ψ (with |I| = N ) in Proposition 2.3 with γ = 1 to get We next estimate each of these four terms.
We start with the term A 1 , and we first decompose it into two parts In conjunction, we further get in which we used Propositions 4.1 and 4.2.We also find in which we used Propositions 4.1 and 4.2, as well as the fact that Thus we get Next, we bound the term A 2 as To proceed, we have in which we used Propositions 4.1 and 4.2, and in which we used Propositions 4.1 and 4.2, as well as the fact Thus we obtain In a very similar manner to estimating the term A 2 , we can show By gathering these estimates, we arrive at The proof is complete.

Proof of Theorem 1.1
Proof.Global existence and time decay.The results of Propositions 4.9, 4.11, 4.12, and 4.13 imply that for a fixed 0 < δ ≪ 1 and N ∋ N ≥ 7 there exists an ǫ 0 > 0 sufficiently small that for all 0 < ǫ ≤ ǫ 0 we have We can now conclude the bootstrap argument.By classical local existence results for nonlinear hyperbolic PDEs, the bounds (4.2) hold whenever the solution exists.Clearly s 1 > s 0 and, moreover, if s 1 < +∞ then one of the inequalities in (4.2) must be an equality.However we see from (4.8) that by choosing C 1 sufficiently large and ǫ 0 sufficiently small, the bounds (4.2) are in fact refined.This then implies that we must have s 1 = +∞.Finally the decay estimates (1.7) follow from (4.8) combined with the Sobolev estimates (2.9) and (2.13).
Scattering.We next show the scattering of the solution (v, ψ).We will only illustrate the proof for the Klein-Gordon field v, as the proof for the Dirac field ψ is analogous.Due to Lemma 2.8, it suffices to show that However, this does not seem possible.So we instead show the scattering for the variable ψ in Lemma 4.4.In any case, we need to first derive the bounds of Z I ψ L 2 (R 2 ) (i.e., on constant tslices) from the known ones Z I ψ L 2 f (Hs) (i.e., on constant s = √ t 2 − r 2 -slices).To do so, for any large T > t 0 + 2 the conservation of charge implies that In addition, for the Zψ equation we integrate the differential identity To proceed, we have Next, we use Lemma 2.4 to bound which is an integrable quantity.Thus we get Similarly, we can show Thus, there exists a free Klein-Gordon component v + , such that We note that for all t ≥ t 0 it holds Finally, we conclude that The proof is complete.

A Proof of Theorem 1.2
We note that the proof for Theorem 1.2 is similar to, and even easier than, the proof of Theorem 1.1.Given this, we omit some details for certain estimates in the proof.

A.1 Bootstrap assumptions and preliminary estimates
Fix N ∈ N a large integer (N ≥ 4 will work for our argument below).The local well-posedness theory guarantees that there exists C 0 > 0 such that the following bounds hold for all |I| ≤ N : Next we assume that the following bounds hold for s ∈ [s 0 , s 1 ): In the above, the constant C 1 ≫ 1 is to be determined, ǫ ≪ 1 measures the size the initial data, and we let C 1 ǫ ≪ 1, and 0 < δ ≤ 1 10 .For the rest of section A we assume, without restating the fact, that (A.2) hold on a hyperboloidal time interval [s 0 , s 1 ) where s 1 is defined as Proposition A.2.For s ∈ [s 0 , s 1 ) we have Proposition A.3.The following weighted L 2 -estimates are valid for s ∈ [s 0 , s 1 ) and the following pointwise estimates also hold for s ∈ [s 0 , s 1 )

A.2 Improved estimates for the Klein-Gordon field
In order to improve the energy bounds for the Klein-Gordon field, we apply two different arguments for the lower-order energy case and for the top-order energy case.For the lower-order case, we rely on a nonlinear transformation (of Type 1 in section 3.1) to remove the slowly-decaying term ψ * γ 0 ψ.This is at the expense of introducing null and cubic terms yet nevertheless allows us to obtain uniform energy bounds.
On the other hand, when deriving the refined bound for the top-order Klein-Gordon energy the nonlinear transformation is invalid due to issues with regularity.Thus in this case we need to utilise the hidden Klein-Gordon structure of the nonlinearities as shown in Lemma 3.2 and Lemma 3.4.Using this we can improve the energy bounds with the aid of the linear behavior of ψ in the lower-order case.
Proof.The proof is straightforward.
Lemma A.5.We have Proof.Acting Z I with |I| ≤ N − 1 on equation (A.3) produces The energy estimates of Proposition 2.1 then imply dτ.
The proof follows similar to Lemma 4.5, where we bound each of the terms to get the desired estimates.
The following lemma is the key to closing the top-order bootstraps for the Klein-Gordon field.
Lemma A.6.We have Proof.By Lemma 3.4 we find Next we apply Lemma 3.2 to reveal the hidden Klein-Gordon structure of the nonlinearity: We recall that Z I 1 ψ − can be regarded as a Klein-Gordon component in the sense that it enjoys the same L 2 -type and L ∞ estimates as Klein-Gordon components, while Z I 1 ψ + enjoys the same good bounds as Z I 1 ψ.We proceed to bound + (s/t) 2 Z I 1 ψ * γ 0 Z I 2 ψ L 2 f (Hs) .
We first show in which the assumption N ≥ 4 was used in the first inequality.Similarly, we also have Gathering the above estimates, we obtain Proposition A.7.We have Proof.We first show the improved energy estimates in the case of |I| ≤ N .We act the Klein-Gordon equation in (1.1) with Z I to get The energy estimates of Proposition 2.1 and the key result of Lemma A.6 imply We next turn to the uniform energy bounds for |I| ≤ N − 1. Due to the uniform estimates of Lemma A.5, we just need to study the difference between v and v.This is a quadratic term ψ * γ 0 ψ which, for |I| ≤ N − 1, is controlled using Lemma A.6 as In conclusion we find, for |I| ≤ N − 1,

A.3 Improved estimates for the Dirac field
In order to improve the energy bounds for the Dirac field, we also use two different arguments for the lower-order energy case and for the top-order energy case.For the lower-order case, our strategy is to introduce the new variable and derive a uniform energy bound for its lower-order energy.This is a nonlinear transformation of Type 3 in Section 3.1 and it allows us to remove the slowly-decaying nonlinearity vψ at the expense of introducing null and cubic terms.After obtaining lower-order uniform energy bounds for ψ we can then easily get improved bounds for the lower-order energy of ψ since the difference between ψ, ψ is a quadratic term.Similar to the strategy for the Klein-Gordon field, this transformation to ψ is not valid at toporder.Nevertheless with the linear behavior of the fields ψ, v in the bootstrap assumptions (A.2), we can also close the bootstrap for the top-order energy estimates.Proof.We begin with the estimate at top-order.For |I| ≤ N , and given N ≥ 4, we have As for the case of |I| ≤ N − 1, we can show (similar to the proof of Proposition 4.11), that

A.4 Proof of Theorem 1.2
Proof of Theorem 1.2.The results of Propositions A.7 and A.10 imply that for a fixed 0 < δ ≪ 1 and N ∋ N ≥ 4 there exists an ǫ 0 > 0 sufficiently small, such that for all 0 < ǫ ≤ ǫ 0 we have for all s ∈ [s 0 , s 1 ) E D (s, Z I ψ) 1/2 + E 1 (s, Similar to the argument in the proof of Theorem 1.1, we can deduce from the above that s 1 = +∞.As for the time decay and scattering, the proof is very similar to that of Theorem 1.1, and so we omit the details.

Lemma 3 . 4 .
For any multi-index |I| there exists a generic constant C = C(|I|) > 0 such that
1, §8.2].With the aid of Lemma 3.5, we will be able to prove better estimates for the ∂ψ component; see for instance Propositions 4.3 and A.3.