Global well-posedness for the $L^2$ critical Hartree equation on $\bbr^n$, $n\ge 3$

We consider the initial value problem for the L^2-critical defocusing Hartree equation in R^n, n \ge 3. We show that the problem is globally well posed in H^s(R^n) when 1>s>\frac{2(n-2)}{3n-4}$. We use the"I-method"combined with a local in time Morawetz estimate for the smoothed out solution.


Introduction
In this paper we study the initial value problem of the L 2 -critical defocusing Hartree equation,    i∂ t φ + 1 2 ∆φ = (|x| −2 * |φ| 2 )φ, x ∈ R n , t > 0, φ(x, 0) = φ 0 (x) ∈ H s (R n ). (1.1) Here H s (R n ) denotes the usual inhomogeneous Sobolev space. (1.1) is meaningful in dimension n ≥ 3, where the Hartree potential is locally integrable. The Hartree type equations arise in atomic and nuclear physics and is related to the mean-field theory with respect to wave functions describing boson systems. ( [14], [27]) The local well-posedness results for s ≥ 0 is shown by the Strichartz estimates similarly as polynomial type NLS. For s > 0 (1.1) is locally well-posed in the subcritical sense. More precisely, for any φ 0 ∈ H s (R n ), the lifetime span of the solution depends on the norm of the initial data, φ 0 H s . Whereas, for s = 0 the lifetime span depends on the profile of the initial data as well. The classical solutions to (1.1) enjoy the mass conservation law, φ(·, t) L 2 (R n ) = φ 0 (·) L 2 (R n ) , and the energy conservation law, When s ≥ 1, the energy conservation law (1.2) together with the subcritical local theory immediately yields the global well-posedness. But when 0 ≤ s < 1, where the energy could be infinite, the mass conservation law cannot imply the global well-posedness, since in the local theory for L 2 initial data, the lifetime T = T (φ 0 ) could go to zero for a fixed L 2 norm.
The purpose of this paper is to extend the global well-posedness result below the energy norm. Our main theorem is as follows: Theorem 1.1. Let n ≥ 3. The initial value problem of (1.1) is globally well-posed for initial data φ 0 ∈ H s (R n ) when 2(n−2) 3n−4 < s < 1. We use the I-method and the interaction Morawetz inequality, which were used in several literatures of the same type of results, [5,7,9,12,13,25]. The idea of I-method, which introduced by Colliander et.al. [8], is to use a smoothing operator I which regularizes a rough solution up to the regularity level of a conservation law by damping high frequency part. In our example, when φ ∈ H s for s < 1, E(φ) may not be finite, but for a smoothed function Iφ, E(Iφ) is finite. Here, one doesn't expect that E(Iφ) is conserved, since Iφ is not a solution to (1.1). But if I operator is close to the identity operator in some sense, Iφ is close to a solution and E(Iφ) is almost conserved. In fact, we control the growth of E(Iφ)(t) in time.
In addition to I-method, we use the interaction Morawetz inequality. Colliander et al. introduced in [9] a new Morawetz interaction potential for the nonlinear Schrödinger equation in three dimension.
This is a generalization of the classical Morawetz potential, which has been studied in many literatures especially regarding on the dispersive property of the Schrödinger equations [1,16,22]. The above functional (1.3) generates a new space-time L 4 t,x estimate for the nonlinear Schrödinger equation with the relatively general defocusing power nonlinearity. Incorporating this with the almost conservation law, they proved the scattering of the equation and relaxed the low regularity assumption given in the previous work [8]. In [5] the authors showed the almost conservation law and Morawetz interaction potential approach worked as well with the Hartree equation in dimension 3. More precisely, when the defocusing Hartree nonlinearity is mass supercritical and energy subcritical case, which is (|x| −γ * |φ| 2 )φ, 2 < γ < 3, the equation is globally well posed in H s (R 3 ), 1 > s > max( 1 2 , 4(γ−2) 3γ−4 ) and has scattering as well. In the H 1 (R 3 ) case, the same result was shown in [18] and later the scattering part was simplified in [26].
The interaction Morawetz inequality is extended to other dimensions [29,12,7]. But in the mass critical case, where the admissible norm is critical, the space-time norm grows in time. We follow the similar way to [7,12]. Due to local in time Morawetz inequality we are able to control for an admissible pair ( 4(n−1) n , 2(n−1) n−2 ). The same machinery in [12] with the above inequality (1.4) would yield the result that the global well-posedness of (1.1) holds when 1 > s > max 1 2 , 2(n−2) 3n−4 . Since we allow the admissible space-time norm grows in time, we do not know whether scattering holds true. Note that the number 2(n−2) 3n−4 is lower than 1 2 in dimension 3. The restriction s > 1 2 is inevitable if relying on the inequality (1.4). In order to remove this restriction, we use the the inequality (1.4) for the smoothed out solution Iφ. This idea was first introduced in [7,13]. They showed it still holds true with negligible error. In our case we have (For detail see Lemma 4.2) Since Iφ is in H 1 (in particular inḢ 1 2 ), s may go below 1 2 . We show that on the time interval where the local well-posedness the error therm is very small. At the time we prepare this paper we are informed that Miao et.al. [25] use the same idea to remove the restriction s > 1 2 in the result ofḢ 1 2 -subcritical Hartree equation as an improvement of [5]. On the other hand, Miao et. al. [23,24] studied the focusing or defocusing L 2 critical Hartree equations as well. They established the global well-posedness and scattering for L 2 radial initial data and the blow up criterion to the focusing L 2 critical Hartree equation in R 3 .
Before we close the introduction, we would like to add some remark on the L 2 -critical focusing case, (1.5) Note that the local well-posedness proof in Section 2 equally works for the focusing case. The equation is known to have a ground state solution Q, which solves The existence of Q is proven in [24] with the decisive property of being the sharp constant of the Gagliardo-Nirenberg inequality such as The uniqueness is open except n = 4, which was settled in [20] adapting E. Lieb's uniqueness proof in [21]. The paper is organized as follows. In Section 2, we review the local well-posedness theorem using the Strichartz estimate. In Section 3 we give the definition of I operator, show the modified local well-posedness of Iφ, and obtain the upper bound of time increment of the modified energy. In Section 4 we recall the almost interaction Morawetz inequality for Iφ and show the error bound. In Section 5 we conclude the proof of global well-posedness in Theorem 1.1.
Notations. Given A, B, we write A B to mean that for some universal constant K > 2, A ≤ K · B. We write A ∼ B when both A B and B A. The notation A ≪ B denotes B > 3 · A. We write A ≡ (1 + A 2 ) 1 2 , and ∇ for the operator with Fourier multiplier (1 + |ξ| 2 ) 1 2 . The symbol ∇ denote the spatial gradient. We will often use the notation 1 2 + ≡ 1 2 + ǫ for some universal 0 < ε ≪ 1. Similarly, we write 1 2 − ≡ 1 2 − ε. We use the function space L q t L r x and H s,p given norms by where F is a fourier transform, 1 ≤ p, q, r ≤ ∞.

The local well-posedness
We refer (q, r) the admissible pair when 2 ≤ q < ∞, 2 ≤ r ≤ 2n n−2 and 2 q + n r = n 2 and state the Strichartz inequality in dimension n.
Proposition 2.1. Suppose that (q, r), (λ, η) are any two admissible pairs. Suppose that u(x, t) is a solution of the problem where λ ′ and η ′ are the Hölder conjugates of λ and η, respectively. Then u belongs to L q t H s,r x ([0, T ]×R n )∩C t H s,r x ([0, T ]×R 3 ) and we have the estimate For the pure power nonlinearity λ|u| α u, the local well-posedness of i∂ t u + 1 2 ∆u = λ|u| α u with the rough data u(0) ∈ H s , 0 < s < 1 was proven in [2] (See also [3,28]).
We define the Strichartz norm of functions φ : [0, T ] × R n → C by In particular S 0 Then the Strihartz estimates may be written as is the conjugate of an admissible pair (q, r).
The local existence theorem of (1.1) is as follows.
Proof. Let S L (t) be the flow map e it∆ corresponding to the the linear Schrödinger equation.
Then the Duhamel formulation of (1.1) is We will show that the map Applying the linear and the dual Strichartz estimates, we have for any admissible (λ, η). We recall the Leibnitz rule for fractional Sobolev spaces [6,30]: Let us choose (λ ′ , η ′ ) = ( 4 3+s , 2n n−s+1 ). The fractional Leibnitz rule, Hardy-Sobolev inequality and Hölder's inequality lead to (2.8) By the Sobolev embedding we have Combining this with (2.7) we find

The local well-posedness time T is chosen as
Similarly, one can show that A is a contraction. And uniqueness assertion and continuous dependence on data follow in the same manner.

Almost conservation law of the modified energy
In this section, we define the smoothing operator I N , which sends an H s function to an H 1 function. We find a bound of the growth of E(I N φ)(t) in time. The operator I N is defined as in [9]. Let N ≫ 1 be a parameter to be chosen later. Define where the multiplier m(ξ) is smooth, radially symmetric, nonincreasing in |ξ| and satisfies We note that m(ξ) satisfies the Hörmander multiplier condition. As intended, the definition of m(ξ) gives the following relations between I N φ H 1 and φ H s for 0 < s < 1; where P k φ is defined by P k φ(ξ) = ϕ(ξ/2 k ) φ(ξ) for a nonnegative smooth function ϕ with supp φ = {ξ|2 −1 ≤ |ξ| ≤ 2} and k∈Z ϕ(2 −k ξ) = 1. What it follows we write I for I N suppressing N . Let us define the iteration space Z I (t) as

Modified local theory.
First of all, we prove a local well-posedness result for the modofied solution Iφ. This theorem is essentially similar to the local well-posedness proof at the critical regularity in [2]. But here we assume critical Strichartz norm of Iφ is small, instead of φ. Similar proofs are found in [7], [13].
Lemma 3.1. For given initial data φ 0 ∈ H s (R n ) for 0 < s, there are time T * > 0 and a universal constant δ > 0 satisfying the following: Proof. The first part is from the local well-posedness theorem, Theorem 2.1. The second part is also done by the Strichartz estimate (2.1) in the Duhamel formula with ∇ I operator: where (γ, ρ) is admissible. In the previous step we have used Leibniz's rule for ∇ I. Note that in the high frequency (|ξ| > N ), I is a negative derivative, but ∇ I is a positive fractional derivative. A simple modification of the proof of the fractional Leibniz rule works for it. Let us choose (γ, ρ) = (4, 2n n−1 ). In fact we can use any admissible pair satisfying γ ≥ 2(n−1) n−2 . We first estimate (|x| −2 * |φ| 2 )∇Iφ . By using Hölder's, fractional Sobolev's inequalities, we obtain . In a similar way, the other term is also estimated as follows: Now we estimate φ 2 L 4(n−1) n t L 2(n−1) n−2 x . We decompose φ into its frequency localized pieces, where N j = 2 k j and and k j 's are consecutive integers starting from [log N ] indexed by j = 1, 2, 3 · · · . By triangle inequality we get From the definition of I operator we have the followings: Putting these together into (3.15), we obtain Ignoring N s−1 ≤ 1 and using the fact that P N f L p f L p , one can sum up over j, if s > ǫ. Thus, we have Hence, from (3.12) we conclude Choosing sufficiently small δ and T * , we conclude the proof.

3.2.
Almost conservation law. We show the almost conservation law of the modified energy.
The usual energy (1.2) is shown to be conserved by differentiating in time .
Then we have 16) The following proposition shows that E(Iφ) is an almost conserved quantity. Assume in addition that ∇ Iφ 0 1. Then we conclude that for all t ∈ [0, T ],

Proof of Proposition 3.1.
We compute in the frequency space. Applying the Parseval formula to E T in (3.16), we obtain Now if we use equation (1.1) to substitute for ∂ t Iφ in (3.17), then it is split into two terms as follows: In both cases, we break down φ into Littlewood-Paley pieces φ j , each localized in 2 k j in frequency, ξ j ∼ 2 k j = N j , k j = 0, 1, 2, · · · , and then use a version of Coifman-Meyer estimate for a class of multiplier operators.
Proposition 3.2 (Proposition 6.1 in [5]). Let σ(ξ) be infinitely differentiable so that for all α ∈ N nk and all ξ = (ξ 1 , . . . , ξ k ) ∈ R nk . Then there is a constant c(α) with Let the multi-linear operator Λ be given We first estimate a pointwise bound on the symbol Factoring B(N 1 , N 2 , N 3 ) out of the integral in E T , it leaves a symbol σ 1 , which satisfies the condition of Proposition 3.2, as the following: .
We shall show that For this aim, we claim that From Proposition 3.2, we have where 1 p = 2 n − 1 + 1 p 1 + 1 p 2 + 1 p 3 . For the first term E a , we use (3.21) and Hölder inequality to get 1 4 , and using Bernstein inequlity, we obtain We reduce to show By symmetry we may assume N 2 ≥ N 3 ≥ N 4 . Then it suffices to consider the following three cases. Case 1: N ≫ N 2 . We have m(ξ i ) = 1 since i ξ i = 0. So, the symbol By the mean value theorem, Thus, Summing up with N 4 , N 3 , N 2 , we have (3.23). Case 3: N 2 ≥ N 3 N . In this case we need to consider two subcases N 1 ∼ N 2 and N 2 ≫ N 1 since by i ξ i = 0 the case N 1 ≫ N 2 cannot happen. For the first case, since xm(x) ≥ 1 for x ≥ 1. We can sum up N 4 , N 3 directly. But when summing up N 2 , we use the Cauchy-Schwartz inequality with φ i = P N i Iφ as follows: In the second case, N 2 ≫ N 1 , again by i ξ i = 0, we have N 2 ∼ N 3 .
For our purpose, we want to show If N 1 ≤ N , then m(ξ 1 ) = 1 and this is true. If N 1 N , then This conclude the proof of (3.19). Now we turn to the estimate of E b . The above analysis is applied to E b , once we show the following lemma.
In the case that all φ's are φ lo we simply estimate When all φ's are φ hi , we use Bernstein inequality, Sobolev embedding and the Leibniz rule as following: where we used 1 n − 1 3n−4 ≥ − 1 p + 3n−1 6n . The remaining lo − hi cases are controlled in a similar manner to the hi − hi case. We omit the detail here.
Hence, we have shown (3.19), (3.20) and so conclude the proof. 4. Almost interaction Morawetz estimate in R n , n ≥ 3 In this section, we show the almost interaction Morawetz inequality. Let us start by recalling the higher dimensional interaction Morawetz inequality for a general nonlinearity. The interaction Morawetz inequality was developed in [9] in R 3 and this higher dimensional extension was derived in [29]. We first recall higher dimensional interaction Morawetz inequality for a general nonlinearity.
First, we apply (4.25) to the solution to (1.1), where N = (|x| −2 * |φ| 2 )φ. A computation shows that the second term is positive. the second term of (4.25 By the same analysis as in [29], we obtain several estimates of space-time L q t L p x -norms.

From Lemma 5.6 in [29]
. (4.30) Interpolation between (4.30) and the trivial estimate and using the Hölder's inequality in time we have . (4.31) For the initial data belowḢ Then using (4.25) we obtain  By the same computation as above, one can see the second term of (4.32) is positive. We wish the third term involving N bad to be small. Similarly to (4.31) we have where Error is defined in the lemma below.
In particular, if we assume ∇ Iφ 0 L 2 1 and Iφ Proof. We rewrite the error term via N bad = I((|x| −2 * φφ)φ) − (|x| −2 * IφIφ)Iφ: (4.35) By Plancerel theorem in space, we have where we ignored complex conjugates since they don't make any differences. As we did in Section 3, we decompose φ into a sum of dyadic pieces. It is reduced to show We use Proposition 3.2. Note that the exponent numerology 3 · 3n−4 6n = 1 2 + 1 − 2 n and that (3, 6n 3n−4 ) is admissible. Thus, once we show that then we have The proof of (4.36) is very similar to the proof of Proposition 3.1. So, a sketch is enough. We assume N 2 ≥ N 3 ≥ N 4 by symmetry and consider the following cases. Case 1: N ≫ N 2 . The symbol is identically zero. Case 2: N 2 ≥ N ≫ N 3 ≥ N 4 . Since i ξ i = 0, we have N 1 ∼ N 2 . By the mean value theorem, we estimate Thus, Case 3: N 2 ≥ N 3 N . In this case we need to consider two subcases N 1 ∼ N 2 and N 2 ≫ N 1 due to i ξ i = 0.
For the first case, N 1 ∼ N 2 , we estimate where used xm(x) ≥ 1 for x ≥ 1. In the second case, For our purpose we want to show If N 1 ≤ N , then m(ξ 1 ) = 1 and N N 2

Proof of Main Theorem
We combine the interaction Morawetz estimate and Propotion 4.1 with a scaling argument to prove the following statement giving a uniform bound in terms of the H s -norm of the initial data.
Then for a given large T we have as long as 2(n−2) (3n−4) < s < 1. The positive number α(s, n) depends on s and n.
Remark 5.1. Since T is arbitrarily large, the a priori bound on the H s norm gives the global well-posedness in the range of 2(n−2) (3n−4) < s < 1.

Proof. The equation (1.1) is invariant over scaling of
we choose λ as λ ≈ N 1−s s . (5.38) in order to normalize ∇Iφ λ O(1). The second term of the modified energy E(Iφ λ 0 ) is treated as follows, by Sobolev embedding.
Hence we have E(Iφ λ 0 ) 1. The remaining proof is similar to the proof of Theorem 5.1 in [7] with necessary modification on exponents. As we have already seen, Hartree type nonlinearity behaves smoother than the polynomial type |φ| We now split the interval [0, T ] into consecutive subintervals J k , k = 1, · · · , L so that  Error dt.
We know that J k Error dt N −1+ on each J k . Hence summing up all the J k ′ s, we find