Scattering results for Dirac Hartree-type equations with small initial data

We consider the Dirac equation with cubic Hartree-type nonlinearity derived by uncoupling the Dirac-Klein-Gordon systems. We prove small data scattering result in full subcritical range. Main ingredients of the proof are the localized Strichartz estimates, improved bilinear estimates thanks to null-structure hidden in Dirac operator and $Up,Vp$ function spaces. We apply the projection operator and get a system which of linear part is the Klein-Gordon type. It enables us to exploit the null-structures in equation. This result is shown to be almost optimal by showing that iteration method based on Duhamel's formula over supercritical range fails.

In this paper we investigate the global behaviour of solution to (1.1), especially scattering problem when the initial data is sufficiently small. The scaling argument for the massless Dirac equation with Coulomb potential gives if ψ is a solution of (1.1), then so is ψ a (t, x) = a 3 2 ψ(at, ax) hence the scale invariant data space is ψ 0 ∈ L 2 (R 3 ). So our goal is to show the scattering result in L 2 (R 3 ) space.
There has been a considerable interest in low regularity well-posedness and scattering result in the past few years. First well-posedness result about Yukawa type was obtained in [7]. They showed existence of weak solution of (1.1) when mass is zero(m=0). Recently, Herr and Tesfahun [11] established the small data scattering result for ψ 0 ∈ H s for s > 1 2 (and s > 0 if the data is radially symmetric). In that paper they suggested that the regularity threshold can be lowered if the null structure of Dirac equation is taken accout into, which initiate our interest about this research. Remark 1.1. A few days ago, the author found that small data scattering result for Hartree Dirac equation with Yukawa type potential with initial data in H s for s > 0 was proved independently by Tesfahun [18]. In our paper, we consider more generalized Hartree type Dirac equation, establish same result with lower regularity condition on initial data(in L 2 ) and also provide the ill-posedness result. One of main ingredient in both paper is to exploit the null-structure in Dirac operator, but approach in application to estimates is quite different.
Before moving on to the result about Coulomb potential, let us write the solution of (1.1) using Duhamel's formula by where the linear propagator is defined on L 2 (R 3 ; C 4 ) by One can see that the essential parts of linear propagator U m (t) is e ±it(m 2 −∆) 1 2 . So one can check that some of wellposedness and scattering results for semirelativistic equation with hartree coulomb type nonlinearity are also valid for (1.1) with j = 0. The local well-posedness result was shown in [10] for s > 1 4 (and s > 0 if the initial data is radially symmetric). One can verify that the proof of [10] also apply for (1.1) with j = 1. Concerning the scattering, there is a negative result [4]. Our first main theorem says that the same behaviour is observed for Dirac equation. Theorem 1.2 (Non-existence of Scattering for Coulomb potential). Let µ = 0 and j = 0. Suppose that ψ is a smooth global solution in C(R + ; L 2 (R 3 )) ∩ C(R + ; H −1 (R 3 )) to (1.1) and there exist a smooth function Then ψ = U m (t)ψ + = 0. For j = 1, the same result holds if we further assume the third and fourth coordinate of ψ + is identically zero.
Recently, Pusateri [16] proved modified scattering result for Boson star eqaution. We think the method in that paper also might apply for (1.1) and give similar result. But we do not pursue it here.
We will prove the scattering result mainly depending on Littlewood-Paley dyadic decomposition and most of calculation will be done in Fourier side. Actually, major difference between two potentials in Fourier side occurs when we treat the low frequency part. Yukawa potential is bounded near zero in frequency side, on the other hand the coulomb potential is unbounded and behaves like |ξ| −2 as it approaches to zero. So in this paper we consider a generalized potential as follows: Definition 1.3. Let 0 ≤ a < 2. The potential V is C ∞ (R 3 /0) function whose Fourier transform satisfies the following: for some 0 < c < C.
The case a = 0 corresponds to Yukawa potential. And the case a = 2 corresponds to Couloumb potential, so we exclude it from our consideration because of Theorem 1.2. Now, we introduce our first main theorem. Theorem 1.4 (Scattering result). For 0 ≤ a ≤ 1, there exist δ > 0 such that for all ψ 0 ∈ L 2 (R 3 ) satisfying ψ 0 L 2 ≤ δ, the Dirac equation (1.1) has a global solution in C(R, L 2 (R 3 )), and furthermore the solution scatters in L 2 to a free solution as t → ±∞.
We lower the threshold of regularity than semirelativisitc equation by effectively exploiting the nullstructure in Dirac equation which is well arranged in [6] and [11]. When we estimate the hartree type nonlinearty term, the most difficult part is to bound the high-high-low interaction. Making use of null structure in this part enables us to prove the scattering result in scaling critical space. We also use localized Strichartz estimates and adapt the function space based on bounded quadratic variation spaces. But we cannot obtain the result in the full range for a, especially 1 < a < 2, where the singularity is so bad. The reason is that in the low frequency part, we cannot use null-structure anymore and also the Strichartz estimates localized to this region is not so satisfactory to get over the singularity. We think that we might overcome this difficulty and obtain the scattering result for 1 < a < 2 if we choose the initial data in some weighted spaces as [16].
In section 5 we will provide the ill-posedness result for initial data in H s (R 3 ) with s < 0, which implies our result in L 2 (R 3 ) is optimal. For the precise statement, See Theorem 5.1.
We also generalize the potential in different way similarly to semirelativistic equation from a mathematical point of view: By scaling argument for m = 0 in (1.11), it is easily verified that critical sobolev index is s = γ−1 2 . In [15] they showed the scattering result for 3 2 < γ < 3 if the initial data given in H s for s > γ−1 2 + 5 6 is sufficiently small. We improve the previous result for 2 < γ < 3 in Section 6 by proving the small data scattering in critical spaces H c (−2, 2) be even and satisfy ρ(s) = 1 for |s| ≤ 1. For ϕ(ξ) := ρ(|ξ|) − ρ(2|ξ|) define ϕ k = ϕ(2 −k ξ). Then, k∈Z ϕ k = 1 on R 3 \ {0} at it is locally finite. We define the (spatial) Fourier localization operator Now, we furthur make a decomposition involving the angular variable as described in [17,ChapterIX,Section4 ]. For each l ∈ N we consider a equally spaced set of points with grid length 2 −l on the unit sphere S 2 , that is we fix a collection Ω l := {ξ ν l } ν of unit vectors that satisfy |ξ ν l − ξ ν ′ l | ≥ 2 −l if ν = ν ′ and for each ξ ∈ S 2 there exists a ξ µ l such that |ξ − ξ ν l | < 2 −l . Let K ν l denote the corresponding cone in the ξ-space whose Then κ l is a smooth partition of unity subordinate to the covering of R 3 {0} with the cone K ν l such that each κ ν l is supported in 2K ν l and is homogeneous of degree 0 and satisfies (1.12) Let κ ν l with similar properties but slightly bigger support such that κ ν l κ ν l = 1. We define K ν l f :

Null structures
Following [6], [1] we introduce the projection operators Π m ± (D) with symbol We then define ψ ± := Π m ± (D)ψ and split ψ = ψ + + ψ − . By applying the operators Π m ± (D) to the equation (1.1), and using the identity we obtain the following system of equations Remark 2.1. From now on we fix m = 1 for simplicity. But it is clear that all arguments below carry over to the case m > 0 with modified constants depending on m. Also we simply denote D 1 and Π 1 ± (D) by D and Π ± (D) respectively.
We decompose ψ, β j ψ as A series of following Lemma analyzes the symbols of bilinear operator above. For more explanation about role of null structure in bilinear from see [1]. We first introduce the relation from [1, Lemma2.1].
From this relation we obtain the upperbound.
We can change the order of β and Π(D) as follows: Then the claim follows from (2.3) and Lemma 2.2.
If we restrict the region of symbol to some cubes we can also get lower bound from Lemma 2.2. More precisely, let us define a cube Proof. We write by (2.5) from which we obtain The same calculation as above with ξ + ζ instead of η implies (2.10) because in this case it holds that
Next we introduce the localized strichartz estimate. For detailed explanation and proof, See [2, Lemma3.1] and reference therein. We obey the notation in [2]. For the convenience of reader we explicitly organize here. For k ∈ Z let us consider the lattice point L k = 2 k Z 3 . Let η : R → [0, 1] be an even smooth function supported in the interval [− 2 3 , 2 3 ] with the property that n∈Z η(ξ − n) = 1 for ξ ∈ R. Let γ : ξ 2 , ξ 3 ). For k ∈ Z and n ∈ L k let γ k,n (ξ) = γ( (ξ−n) 2 k ). Clearly, n∈L k γ k,n ≡ 1 on R 3 . We define the projection operator Γ k,n by Γ k,n f = F −1 (γ k,n F f ). Then I = n∈L k Γ k,n . Finally define Γ k,n as before so that Γ k,n Γ k,n = Γ k,n Γ k,n = 1. Now we are ready to give a statement.
Lemma 3.2 (Localized Strichartz estimates). Let 1 p + 1 q = 1 2 with 2 < p < ∞. Then Proof. We follow the main stream of the proof in [2, Lemma3.1]. By orthogonality it suffices to show that uniformly in n ∈ L k ′ . Let T t be the operator on L 2 (R 3 ) into L p t L q x (R 1+3 ) defined by T t = Γ k ′ ,n P k e ±it D . The T t T * t is a space-time convolution operator with the kernel We claim that the kernel K k ′ ,k;n satisfies the following estimates uniformly in x and n ∈ L k . Suppose for a moment (3.5) holds. By the standard T T * argument to prove (3.3) is equivalent to showing where p ′ is hölder conjugate of p. By Young's inequality and Plancherel's theorem we have x . And by interpolation of these two estimates we obtain for q ≥ 2 Then we estimate Then Hardy-Littlewood-Sobolev inequality with 0 < 2 q = 1 p ′ − 1 p < 1 implies (3.6). Finally we prove (3.5). Rescaling yields K k ′ ,k;n (t, x) = 2 3k K k ′ −k,1,2 −k n (2 k t, 2 k x) Then it suffices to show that for |a| ∼ 1 where for ξ k = (|ξ| 2 + 2 −2k ) 1 2 , (3.9) K k ′ −k,1;a (s, y) = R 3 e ±is ξ k +iy·ξ ρ 2 1 (ξ)γ 2 k ′ −k,a (ξ)dξ, .
By rotation, we may assume that y = (0, 0, |y|). We change of variables using spherical coordinates: where ζ k ′ ,k is smooth cutoff function supported in a thickened spherical cap of size 2 k ′ −k located near unit sphere. The derivative of phase function with respect to r is |y| cos θ + s r r k . Thus the worst case occurs when 0 < θ ≪ 1 and |y| ∼ 2 k 2 k −1 |s|, otherwise since the derivative of phase function has a lower bound we can perform an integration by parts arbitrarily many times and get sufficient decay. So we only discuss the first case, where ζ k ′ ,k (r) and ζ k ′ ,k (θ) is supported in an interval of length 2 k ′ −k in ( 1 4 , 4) and [0, π) respectively and |∂ θ ζ k ′ ,k | 2 k−k ′ . We integrate by parts with respect to θ: The the support properties of ζ k ′ ,k imply (3.10) |K k ′ −k,1,a (s, y)| 2 k ′ −k |y| −1 , which implies (3.8).
3.2. Transference Principle. Let 1 ≤ p < ∞. We call a finite set {t 0 , . . . , t K } a partition if −∞ < t 0 < t 1 < . . . < t K ≤ ∞, and denote the set of all partitions by T . A corresponding step-function a : and U p ± D is the atomic space. Further, let V p ± D be the space of all right-continuous v : with the convention e ±itK D v(t K ) = 0 if t K = ∞. For the theory of U p ± D and V p ± D , see e.g. [8,9,12].
Proof. By the atomic structure of U p ± D , estimates in L 2 for free solutions transfer to U p ± D functions, hence to V 2 ± D . Thus from (3.3) we have Actually we can check (3.12) also holds with operator changed into K ν l P k from which we induce Since projection with respect to ν and n is almost disjoint, the definition of V 2 ± D implies Next we introduce the bilinear estimates which follows from Strichartz estimate (3.1) and (3.3).
Proposition 3.4. For all k i ∈ Z and ψ i ∈ V 2 ± D satisfying P ki ψ i = ψ i , i = 1, 2 the following bilinear estimates hold true for j = 0, 1 : Proof. By Bernstein inequality we estimate x . Then first inequality follows from (3.1) and the inclusion V 2 ± D ֒→ U p ± D for p > 2. On the other hand, we have by Hölder inequality x ψ 2 L 4 t L 4 x . And then apply (3.3) with k ′ = k and p = q = 4.
where in the last inequality null structure is exploited by Lemma 2.3 since Π θi (D)ψ i = ψ i and 2 ki > 1.
Applying the Cauchy-Schwarz inequality with n, n ′ and ν, ν ′ and then using (3.11) we finally get

Resolution space and Nonlinear estimates.
Our resolution space X s ± corresponding to the Sobolev spaces regularity s is the space of functions in C(R, H s (R 3 ; C 4 )) such that Briefly denote X 0 ± by X ± . We define the new potential And from now on, we denote P ki ψ i simply by ψ i,ki .
We estimate I 11 using (3.16) We estimate I 12 using (3.14) where the assumption a ≤ 1 is necessary. Estimate for I 2 . In this case, we use (3.15)  [13]. And since our equation also have the hartree cubic nonlinear term we can apply the same method used in that paper. Lastly, positiveness of inner product is obvious for β 0 , and also works for the case j = 1 thanks to further assumption.

4.2.
Proof of Theorem1.4. It suffices to consider positive times. We will construct a solution whenever the initial data satisfy ψ 0 L 2 ≤ δ. Let T (ψ + , ψ − ) denote the operator defined by the right side of above formula. For all ψ ∈ H s (R 3 ) we immediately have Next we study the nonlinear part. For all ψ i = Π θi (D)ψ i for i = 1, 2, 3 we have Indeed, the left side norm can be represented as a integral formula sup ψ∈X±: ψ X ± ≤1 Π ± (D) V * ψ 1 , ψ 2 ψ 3 ψ(t)dtdx by duality, see [8]. Further, we obtain by Corollary 3.6, which implies (4.3). We conclude from (4.2) and (4.3) T (ψ + , ψ − ) X ≤ δ + (ψ + , ψ − ) 3 X , and similar estimates for differences. Therefore Theorem 1.4 now follows from the standard approach via the contraction mapping principle. In particular, the scattering claim follows from the fact that functions in V 2 I have a limit at ∞. We omit the details.

ill-posedness
In this section we consider the supercritical range where the initial data is given in H s (R 3 ) for s < 0. We provide the ill-posed result which show the nonlinear term estimates (4.3) essential to occur the contraction fails for any resolution space X s ± . We adpat the argument in [14], where detailed explanation is well arranged .
We expect our resolution spaces X s ± to be continuously embedded in C([0, T ] : H s (R 3 )). And the free solution in X s ± is required to satisfy (4.2). So if we assume (4.3) holds true for some X s ± , we have for all ψ i = Π θi (D)ψ for i = 1, 2, 3 But the following theorem states that above inequality can not be true if s < 0.