Low regularity well-posedness for the 3D Klein-Gordon-Schr\"odinger system

The Klein-Gordon-Schr\"odinger system in 3D is shown to be locally well-posed for Schr\"odinger data in H^s and wave data in H^{\sigma} \times H^{\sigma -1}, if s>- 1/4, \sigma>- 1/2, \sigma -2s>3/2 and \sigma -2<s<\sigma +1 . This result is optimal up to the endpoints in the sense that the local flow map is not C^2 otherwise. It is also shown that (unconditional) uniqueness holds for s=\sigma=0 in the natural solution space C^0([0,T],L^2) \times C^0([0,T],L^2) \times C^0([0,T],H^{-1/2}) . This solution exists even globally by Colliander, Holmer and Tzirakis. The proofs are based on new well-posedness results for the Zakharov system by Bejenaru, Herr, Holmer and Tataru, and Bejenaru and Herr.


Introduction and main results
We consider the (3+1)-dimensional Cauchy problem for the Klein -Gordon -Schrödinger system with Yukawa coupling i∂ t u + ∆u = nu (1) with initial data u(0) = u 0 , n(0) = n 0 , ∂ t n(0) = n 1 , where u is a complex-valued and n a real-valued function defined for (x, t) ∈ R 3 × [0, T ] . This is a classical model which describes a system of scalar nucleons interacting with neutral scalar mesons. The nucleons are described by the complex scalar field u and the mesons by the real scalar field n. The mass of the meson is normalized to be 1.
Our results do not use the energy conservation law but only charge conservation u(t) L 2 (R 3 ) ≡ const (for the global existence result), so they are equally true if one replaces nu and |u| 2 by −nu and/or −|u| 2 , respectively.
Thus three questions arise: (a) Can we show local well-posedness for even rougher data ? (b) Is it possible to show the sharpness of this local well-posedness result ? (c) Under which assumptions on the data can we show unconditional uniqueness, i.e. uniqueness in the natural solution space ?
Concerning (a) we prove that local well-posedness holds in Bourgain type spaces which are subsets of the natural spaces, provided the data fulfill Especially the choice s = − 1 4 + and σ = − 1 2 + is possible, thus we relax the regularity assumptions for the Schrödinger and wave parameters by almost 1 4 and 1 2 order of derivatives, respectively. Concerning (b) the estimates for the nonlinearities which lead to the local well-posedness result fail if s < − 1 4 or σ < − 1 2 or σ − 2s > 3 2 . Using ideas of Holmer [8] and Bejenaru, Herr, Holmer and Tataru [3] we can even show that the solution map (u 0 , n 0 , n 1 ) → (u(t), n(t), ∂ t n(t)) is not C 2 in these cases, i.e. some type of ill-posedness holds.
Concerning (c) we show that for data u 0 ∈ L 2 (R 3 ) , n 0 ∈ L 2 (R 3 ) , n 1 ∈ H −1 (R 3 ) unconditional uniqueness holds in the space (4). Using the global existence result of [6] we get unconditional global well-posedness in this case.
The question of unconditional uniqueness was considered among others by Yi Zhou for the KdV equation [16] and nonlinear wave equations [17], by N. Masmoudi and K. Nakanishi for the Maxwell-Dirac, the Maxwell-Klein-Gordon equations [9], the Klein-Gordon-Zakharov system and the Zakharov system [10], and by F. Planchon [14] for semilinear wave equations.
The results in this paper are based on the (3+1)-dimensional estimates by Bejenaru and Herr [4] which they recently used to show a sharp well-posedness result for the Zakharov system. We also use the corresponding sharp (2+1)-dimensional local well-posedness results for the Zakharov system by Bejenaru, Herr, Holmer and Tataru [3], especially their counterexamples.
We use the standard Bourgain spaces X m,b for the Schrödinger equation, which are defined as the completion of S(R 3 × R) with respect to ± (I) . We often skip I from the notation. In the following we mean by a solution of a system of differential equation always a solution of the corresponding system of integral equations.
Our local well-posedness result reads as follow: Theorem 1.1. The Cauchy problem for the Klein -Gordon -Schrödinger system (1), (2),(3) is locally well-posed for data under the assumptions This solution has the property These conditions are sharp up to the endpoints, more precisely we get . Then the flow map (u 0 , n 0 , n 1 ) → (u(t), n(t), ∂ t n(t)) , t ∈ [0, T ] , does not belong to C 2 for any The necessary estimates for the nonlinearities required in the local existence results are false if the assumptions regarding the smoothness of the data are violated. This is proven in section 4, Prop. 4.1 and Prop. 4.2.
The unconditional uniqueness result is the following: We use the following notation. The Fourier transform is denoted by or F and its inverse byˇor F −1 , where it should be clear from the context, whether it is taken with respect to the space and time variables simultaneously or only with respect to the space variables. For real numbers a we denote by a+ and a− a number sufficiently close to a, but larger and smaller than a, respectively. Acknowlegment: The author thanks the referee for many valuable suggestions which improved the paper.

(Conditional) local well-posedness
. Remark: A possible choice of the parameters is: Because we are going to use dyadic decompositions of u and v we take the notation from [4] and start by choosing a function ψ ∈ C ∞ 0 ((−2, 2)) , which is even and nonnegative with ψ(r) = 1 for |r| ≤ 1. Defining ψ N (r) = ψ( r N ) − ψ( 2r N ) for dyadic numbers N = 2 n ≥ 2 and in the Schrödinger case and the wave case.

Proof of Theorem 2.1.
Defining We use dyadic decompositions Defining 2 ) . : Dyadic summation over L, L 1 , L 2 ≥ 1 and N, : as in case a. Dyadic summation gives the claimed estimate.
a. In the case L 2 ≪ N 2 2 we get by [4, formula (3.26) and (3.28)] : max(L, L 1 , L 2 ) N 2 2 and (10). Dyadic summation gives the desired bound as in case 1. b. In the case L 2 N 2 2 we consider 3 subcases using the proof of A again using (11),(10) and b 2 ≥ 1 2 . Case 3: Interchanging the roles of N 1 and N 2 as well as L 1 and L 2 we consider different cases.
a. L 1 ≪ N 2 1 . We use [4, formula (3.26) and (3.28)] and get max(L, L 1 , L 2 ) N 2 1 and If s ≤ 1 we get the bound AN A using (12). Dyadic summation gives the claimed estimate. b. In the case L 1 N 2 1 we consider 3 subcases using the proof of [4, Prop.

Proof. We have to show
Using dyadic decompositions as in the proof of Theorem 2.1 we consider different cases. Case 1: N 1 ∼ N 2 N ≫ 1 a. In the case L, L 1 , L 2 ≤ N 2 1 we get by [4, formula (3.24)] If σ ≤ 1 we get the bound BN , so that the same argument applies. Case 2: N 1 ≪ N 2 (=⇒ N ∼ N 2 ) (or N 2 ≪ N 1 , which is the same problem). a. In the case L 2 N 2 2 we get by [4, formula (3.27)] : In this case we can estimate this by B , because σ − 2s < 3 2 , whereas in the case s > 1 2 we estimate by BN σ−2−s+ B , because σ − 2 < s . Dyadic summation gives the desired bound. a2. max(L, L 1 ) > N 2 1 We estimate in this case by the same bound as in a1. b. In the case L 2 ≪ N 2 2 we get by [4, formula (3.26) and (3.28)] : max(L, L 1 , L 2 ) N 2 2 and |I(f L,N , g L1,N1 1 , g L2,N2 2 )| the same bound as in a1.
Assuming without loss of generality L 1 ≤ L 2 and using the bilinear Strichartz type estimate [4,Prop. 4.3] we get 2 L 2 . Furthermore we get by [4, formula (4.22)] 2 L 2 , so that by interpolation we arrive at 4 . Dyadic summation in all cases completes the proof of Theorem 2.2.
Proof of Theorem 1.1. It is by now standard to use (the remark to) Theorem 2.1 and Theorem 2.2 to show the local well-posedness result (Theorem 1.1) for the system (5),(6), (7) as an application of the contraction mapping principle. For details of the method we refer to [7]. This solution then immediately leads to a solution of the Klein-Gordon-Schrödinger system (1),(2),(3) with the required properties as explained before Theorem 1.1.

Unconditional uniqueness
In this section we show that solutions of the KGS system are unique in its natural solution spaces in the important case, where the Cauchy data for the Schrödinger part belong to L 2 and the data for the Klein-Gordon-part belong to respectively. This is of particular interest, because we know that in this case the solution exists globally in time by the result of [6].
First we show and T > 0 be given. Any solution of the system (5),(6), (7) belongs to Proof of Theorem 1.3. We again remark that any solution of the Klein-Gordon-Schrödinger system (1),(2),(3) leads to a corresponding solution of the system (5),(6),(7) with Thus combining Prop. 3.1 with the local well-posedness result Theorem 1.1 and the global existence result of [6] we immediately get Theorem 1.3.

Sharpness of the well-posedness result
In this section we show that the local well-posedness result is sharp up to the endpoints. First we construct counterexamples which show that the threshold on the parameters s and σ in Theorem 2.1 and Theorem 2.2 is essentially necessary. We follow the arguments of [8] and [3].
is false.
The following Propositions 4.3, 4.4 and 4.5 show that the flow map of our Cauchy problem is not C 2 so that the problem is ill-posed in this sense. This strategy of proof goes back to Bourgain [3], Tzvetkov [15] and Molinet-Saut-Tzvetkov [11], [12] and is taken up by Holmer [8]. Thus the proof of Theorem 1.2 immediately follows from these propositions by the arguments of Holmer [8].