Small data scattering for semi-relativistic equations with Hartree type nonlinearity
We prove that the initial value problem for the equation
is globally well-posed and the solution scatters to free waves asymptotically as if we start with initial data which is small in for , and if . Moreover, if the initial data is radially symmetric we can improve the above result to and , which is almost optimal, in the sense that is the critical space for the equation. The main ingredients in the proof are certain endpoint Strichartz estimates, bilinear estimates for free waves and an application of the and function spaces.
2010 Mathematics Subject Classification:35Q55
We consider the initial value problem (IVP) for the semi-relativistic equation with a cubic Hartree-type nonlinearity:
where is defined via its symbol in Fourier space, the constant is a physical mass parameter, the symbol denotes convolution in and is a potential, typically,
which is called a Coulomb potential if and a Yukawa potential if .
Equation (1.1) is used to describe the dynamics and gravitational collapse of relativistic boson stars and it is often referred to as the boson star equation; see [12, 14, 22, 24] and the references therein.
It is well-known that equation (1.1) exhibits the following conserved quantities of energy and -mass, which are given by
for fixed . This scaling symmetry leaves the -mass invariant, and so equation (1.1) is -critical.
There has been a considerable mathematical interest concerning the low regularity well-posedness (both local and global-in-time) and scattering theory of the initial value problem (1.1) in the past few years. A first well-posedness result was obtained by Lenzmann  for using energy methods. Moreover, he showed global well-posedness in for initial data sufficiently small in . There has been further well-posedness and scattering results for equation (1.1) with a more general potential, namely, ; see eg. [5, 6, 7, 8]. Recently, Pusateri  proved a modified scattering result in the case of the Coulomb potential in dimension , if .
Recently, Lenzmann and the first author  proved local well-posedness for initial data with (and if the data is radially symmetric). Moreover, these results are optimal up to end points, i.e., up to (and ), in the framework of perturbation methods. Strichartz estimates and space-time bilinear estimates were the main ingredients. This is in contrast with previous results where only energy methods and linear Strichartz estimates were used.
The aim of this paper is to prove global existence and scattering of solutions to the IVP (1.1). Our main result is the following.
Theorem 1.1 (Main Theorem).
Let in (1.2), i.e., is a Yukawa potential. Assume one of the following holds:
, and is radially symmetric,
Then, there exists such that for all satisfying
the IVP (1.1) has a global solution (spatially radial solution if is radial)
Moreover, the solution depends continuously on and scatters asymptotically as . Furthermore, it is unique in some smaller subspace of .
If , due to the modified scattering result in  (see also [5, Theorem 4.1]), it is known that the scattering result in Theorem 1.1 does not carry over to the case , i.e. if is the Coulomb potential.
Let us consider the IVP for the nonlinear Dirac equation with Hartree type nonlinearity:
where , is the Dirac spinor regarded as a column vector, , and and are the Dirac matrices. We refer the reader to  for a representation of the Dirac matrices and a recent result for a related problem with a cubic nonlinearity with null-structure.
Equation (1.3), with a Coulomb potential , was derived by Chadam and Glassey  by uncoupling the Maxwell-Dirac equations under the assumption of vanishing magnetic field. Then in two space dimensions they showed existence of a unique global solution for smooth initial data with compact support. They also conjectured [3, see pp. 507] equation (1.3) with a Yukawa potential can be derived by uncoupling the Dirac-Klein-Gordon equations, see also [4, 1] for certain global and scattering results in this context. Later, Dias and Figueira [10, 11] proved existence of weak solution for (1.3) with a Yukawa potential in the massless case (). We now comment on how to conclude a similar result as in Theorem 1.1b for the IVP (1.3) with a Yukawa potential.
Following , we define the projections
Then , where . Now, if we apply to (1.3), the IVP transforms to
where . Notice that these equations are of the form (1.1), and hence an easy modification111One has to be careful in the radial case since the Dirac operator does not preserve spherical symmetry in the classical sense (see e.g. [23, 25]). However, we do not consider this case here. of the proof of Theorem 1.1b will give the following result.
Recently, in [2, 1] small data scattering results for low regularity initial data have been proven for the nonlinear Dirac equation and a massive Dirac-Klein-Gordon system. These problems exhibit null-structure and a non-resonant behavior. Note that there is also null-structure in (1.4), so we expect that the regularity threshold in Corollary 1.3 can be lowered, but we do not pursue this here. We will use some ideas introduced in [2, 1], in particular localized Strichartz estimates in the non-radial case, but otherwise the analysis differs significantly.
The rest of the paper is organized as follows. In the next Section, we give some notation, define the and -spaces and collect their properties. In Section 3, we prove some linear and bilinear estimates for solutions of free Klein-Gordon equation. In Section 4, we state a key proposition and then give the proof of Theorem 1.1. In Section 5–6, we give the proof of the key proposition.
2. Notation, and -spaces and their properties
We denote the spatial Fourier transform by or and the space-time Fourier transform by . Frequencies will be denoted by Greek letters and , which we will assume to be dyadic, that is of the form for .
Consider an even function such that if , and define for ,
Thus, whereas for . We define
For fixed , we denote to be the linear propagator of the Boson star equation (1.1) defined by
2.2. and spaces
These function spaces were originally introduced in the unpublished work of Tataru on the wave map problem and then in Koch-Tataru  in the context of NLS. The spaces have since been used to obtain critical results in different problems related to dispersive equations (see eg. [15, 16, 27, 18]) and they serve as a useful replacement of -spaces in the limiting cases. For the convenience of the reader we list the definitions and some properties of these spaces.
Let be the collection of finite partitions of . If , we use the convention for all functions . We use to denote the sharp characteristic function of a set .
Let . A -atom is defined by a step function of the form
The atomic space is defined to be the collection of functions of the form
with the norm
define as the space of all functions for which the norm
Likewise, let denote the normed space of all functions such that and , endowed with the norm (2.2).
We let () denote the closed subspace of all right-continuous functions ( functions).
Let . Then we have the following:
is a Banach space.
The embeddings are continuous.
Every is right-continuous. Moreover, .
Let . Then we have the following:
The spaces , , and are Banach spaces.
The embedding is continuous.
The embeddings and are continuous.
The embedding is continuous.
 Let and . There exists such that for all , there exist and with
We now introduce -type spaces that are adapted to the linear propagator of equation (1.1):
We define (and , respectively) to be the spaces of all functions such that is in (resp. ), with the respective norms:
We use to denote the subspace of right-continuous functions in .
(Transfer principle) Let
be a multilinear operator and suppose that we have
for some . Then
3. Linear and bilinear estimates
In this section, we prove linear and bilinear estimates for free solutions of the Klein-Gordon equation both for radial and non-radial data, which are key to prove our main Theorem.
3.1. Estimates in the radial case
Let . Consider , where is radial. Then
Moreover, for all radial function , we have
The following Lemma extends the result of Foschi-Klainerman for [13, Lemma 4.4] to the massive case.
Let and consider the integral
The proof given here is a modification of the argument for from [13, Lemma 4.4]. For a smooth function , define the hypersurface
If for , then
For a nonnegative smooth function which does not vanish on , (3.4) also implies
Now using (3.5), we can write
where in the first line we multiplied the argument of the delta function on the left by .
Introduce polar coordinate , where . Then
If we also set , then
where . In the change of variables, we obtain , which in turn implies With these transformations our integral becomes
The delta function sets the value of to
which implies , and thus we are restricted to . This forces us to integrate over
Using these facts and (3.4) gives the desired estimate. ∎
Let . Consider and , where and are radial. Then for any , we have
Let . Suppose and are radial functions such that . Then for any , we have
Proof of Lemma 3.3.
First assume . By symmetry we may assume . Then by Hölder, (3.1) and the energy inequality, we obtain
Therefore, we assume from now on that . If and are radial, then their Fourier transforms will also be radial. Let and . Applying the space-time Fourier transform, we write
Then using Lemma 3.2, we obtain
where to get the third line we used the change of variable , for , and to obtain the fourth inequality we used Cauchy-Schwarz with respect to and the fact that .
We now use the change of variable , which implies
(where , since ) to write
We thus obtain
where to get the last equality we used the identities
3.2. Estimates in the non-radial case
The first part of the next Lemma is well-known.
Lemma 3.5 (KG-Strichartz).
Let and . Then
Moreover, for all , we have
Lemma 3.6 (KG-Localized Strichartz).
Let . Consider , where is supported in a cube of side length at a distance from the origin, with . Then for all , we have
Moreover, for all , we have
Without loss of generality, we take . By a standard argument, (3.9) is equivalent to the estimate
We then interpolate these estimates, for , to obtain
where we used . Hence
which implies (applying also Young’s inequality in and )
Choosing yields the desired estimate (3.11). ∎
Let , , be the collection of cubes (as in ) which induce a disjoint covering of . Then we have
4. Proof of the Main Theorem
We also define by the corresponding space where is replaced by with a norm
On the time interval , we define the restricted space by