Minimal mass non-scattering solutions of the focusing $L^2$-critical Hartree equations with radial data

We prove that for the Cauchy problem of focusing \begin{document}$L^2$\end{document} -critical Hartree equations with spherically symmetric \begin{document}$H^1$\end{document} data in dimensions \begin{document}$3$\end{document} and \begin{document}$4$\end{document} , the global non-scattering solution with ground state mass must be a solitary wave up to symmetries of the equation. The approach is a linearization analysis around the ground state combined with an in-out spherical wave decomposition technique.


1.
Introduction. Consider the Cauchy problem of focusing L 2 -critical nonlinear Schrödinger equation with Hartree type nonlinearity (Hartree equation) i∂ t u + ∆u = −(| · | −2 * |u| 2 )u, (t, x) ∈ R × R d , where d ≥ 3, u(t, x) is a complex-valued function, and * denotes convolution on R d . The nonlinear Schrödinger equation of Hartree type describes the dynamics of mean-field limits of many-body quantum systems, such that coherent states, condensates. In particular, it provides an effective models for quantum systems with long-range interactions, see, e.g., [4]. The equation in (1) is L 2 -critical since it is preserved by the scaling u λ (t, x) = λ d/2 u(λ 2 t, λx) which keeps the L 2 -norm of the initial datum invariant. The equation also admits the following symmetries: • Translation: u(t, x) → u(t + t 0 , x + x 0 ), t 0 ∈ R, x 0 ∈ R d , • Phase rotation: u(t, x) → e iθ u(t, x), θ ∈ R, • Time reversal: u(t, x) → u(−t, x). The phase rotation and the time translation symmetries respectively lead to mass and energy conservation laws: • Mass: • Energy: Throughout this paper, we always use P(u) to denote 1 4 R d (| · | −2 * |u| 2 )|u| 2 (x) dx. Let I be a time interval containing 0. By u(t, x) : I × R d → C a strong L 2 solution of (1), we mean u ∈ C 0 t L 2 x ∩ L 6 t L 6d 3d−2 x (K × R d ) for any compact K ⊂ I and u obeys the Duhamel formula u(t) = e it∆ u 0 + i t 0 e i(t−τ )∆ (| · | −2 * |u| 2 )u(τ ) dτ, ∀t ∈ I.
I is called the maximal life-span if it can not be extended to any strictly larger interval. If I = R, we say u is global.
The local theory for (1) with L 2 x data has been established in [27] (see also [1]). We summarize the results in the following.
Conversely, given u + ∈ L 2 x , there exists a unique solution u of (1) such that (4) holds. The analog statement holds for the negative time direction.
(v)(Small data global existence and scattering) If N (u 0 ) is sufficiently small, then u is global and u For the Cauchy problem (1), there is a solitary wave e it Q. Here Q is the unique positive radial smooth decreasing solution [17], called ground state, of the following elliptic equation The ground state Q is a minimizer of the functional

MINIMAL MASS NON-SCATTERING SOLUTIONS 1981
In [17], the uniqueness of the ground state was solved when d = 4. In this case, the nonlinearity is a Newtonian potential, and by the Newton's charge theorem [22], the uniqueness can be achieved using the ODE technique in [18]. If d = 3, the nonlinearity is non-Newtonian, and the ODE trick is no longer available. In this paper, we always assume the resolution of the uniqueness for d = 3. Assumption: The ground state of (5) is unique when d = 3.
In [29], Miao-Xu-Zhao established the variational characterization of the ground state when d = 4. The argument can be adapted to the case d = 3 on the condition that the ground state is unique. In particular, the following holds.
Proposition 1 ( [29]). Let d = 3, 4. For f ∈ H 1 (R d ), If N (f ) = N (Q) and E(f ) = 0. Then f (x) = e iθ λ d/2 Q(λx + b) for some θ ∈ R, The scattering problem for critical Hartree equations, especially energy-critical and mass-critical equations, has been well studied by Miao C., et al. Specifically, in [26], Miao-Xu-Zhao used the induction on energy method to prove for defocusing energy critical Hartree equation that spherically symmetric solutions are globally well-posed and scatter. The radial assumption was removed by these authors in [30]. The scattering theory for the focusing energy-critical Hartree equation was established in [19], using a concentration-compactness technique. For the mass critical equation, Miao-Xu-Zhao showed in [28] that the spherically symmetric solution is globally wellposed and scatters if its mass is below the ground state. For the scattering result of NLS, see e.g., [2,3,10,14,13].
Notice that the equation in (1) admits the pseudo-conformal invariance Applying this transformation to e it Q, one obtains a finite time blow-up solution with mass N (Q). It is believed that e it Q and P S(e it Q) are the only obstructions to scattering when the solution has ground state mass. The characterization of minimal mass blow-up solution for the mass critical dispersive equation was first studied by F. Merle who proved in [24] that if an H 1 x solution of mass-critical NLS with minimal mass blows up in finite time, then it equals to P S(e it Q) up to symmetries of the equation. In [12], Killip-Li-Visan-Zhang showed for NLS in d ≥ 4 that the minimal mass global non-scattering solutions with spherically symmetric H 1 x data must be coincide with solitary wave up to the symmetries of the equation. Definition 1.2 (Non-scattering). Let I be a time interval. We say a solution u of (1) with maximal life-span I is non-scattering forward in time if sup I < ∞ and can define non-scattering backward in time.
In lower dimensions, i.e., 1 ≤ d ≤ 3, the difficulty arises from the low dispersion of the solutions. In [21], Li-Zhang employed the linearized analysis around the ground state beyond the frequency localization technique, and generalized the results to d = 2, 3. The problem for d = 1 remains open.
For the Hartree equation, Miao-Xu-Zhao [29] used a slightly different approach from [24] and proved that if an H 1 solution with minimal mass blows up in finite time, then it equals to P S(e it Q) up to symmetries of the equation. There they used a simplified argument from [8]. In [20], Li-Zhang showed that when d ≥ 5 and u 0 ∈ H 1 rad (R d ), the only non-scattering solution of (1) with ground state mass is the solitary wave or the pseudo-conformal solitary wave up to the symmetries of the equation.
Inspired by [21], we consider in this paper (1) in dimensions 3 and 4. The main result sounds as follows.
. Assume that the corresponding solution u of (1) exists globally but does not scatter in at least one direction. Then there exist θ 0 ∈ R and λ 0 > 0 such that This theorem combined with the results in [20] gives a complete characterization of minimal mass non-scattering solutions for L 2 -critical Hartree equations with radial data in H 1 .
The main ingredient in proving Theorem 1.3 is the following localization property of kinetic energy.
Assume the corresponding solution u globally exists but does not scatter. Then for any η > 0, there exists C(η) > 0 such that Here φ >C(η) is a radial bump function (see Section 2.1).
Outline of proof for Theorem 1.3: Note that by Proposition 1, N (u 0 ) = N (Q) implies E(u 0 ) ≥ 0. Moreover, if E(u 0 ) = 0, then u 0 = Q up to rotation, scaling and translation. By uniqueness, u equals to the solitary wave up to the symmetries. Thus, we only need to exclude the case where E(u 0 ) > 0. This is done by using a localized virial argument. On one hand, the localized virial quantity is always bounded above. On the other hand, by virtue of Theorem 1.4, the second derivative of the localized virial quantity has positive lower bound. These yield a contradiction.
So the main task of the paper is to achieve Theorem 1.4. Following the idea of [21], we first employ the ideas of modulation stability theory from Merle-Raphaël [25] to show a regular decomposition for minimal mass solutions of Hartree equation. This allows us to capture weak localization of the kinetic energy of the solution. In particular, we shall show the following proposition.
Proposition 2 (Weak localization of kinetic energy). Let d = 3, 4, and let u 0 ∈ H 1 rad (R d ) satisfy N (u 0 ) = N (Q). Let u be the corresponding solution with maximal life-span I. Then for all t ∈ I, we have where φ is a bump function (see Section 2.1).
In this first step, we shall make use of the structure of the linearized operator L + , and L − around the ground state, From the discussion in [5], L + has exactly one negative eigenvalue and the continuous spectra of L − and L + are [1, ∞), see also [17]. Moreover, in [17], Krieger-Lenzmann-Raphaël proved for d = 4 the non-degeneracy of L + , and obtained The argument for the non-degeneracy property relies on the uniqueness of the ground state. In this paper, we assume the resolution of this null space property for d = 3 . With this and by an adaption of [31] (see also [23]), one may infer the following coercivity of L + , and L − .
Let ω be the eigenfunction corresponding to the negative eigenvalue of L + . There exists a constant σ > 0 such that the following holds.
The second step for achieving the main result is to invoke the compactness property of a non-scattering solution with minimal mass to upgrade Proposition 2 to Theorem 1.4. Indeed, it has been shown in [15,13,28] that the non-scattering solution with minimal mass is an almost periodic solution in the following sense: there exist functions N : I → R + , ξ : I → R d , x : I → R d and C : R + → R + such that for all t ∈ I and η > 0, An important consequence of the almost periodic solution is the following no waste Duhamel formula. Lemma 1.6. Let u be an almost periodic solution of (1) on [0, ∞). Then for all t ≥ 0, as a weak limit in L 2 x . We shall use (8) together with the in-out decomposition technique to derive both frequency and spatial decay estimates for the minimal mass solution and then prove Theorem 1.4. The nonlocal nonlinearity makes things a bit complicated. To deal with it, we shall do delicate analysis in different integration regions. This paper is structured as follows: In Section 2, we present some preliminaries. In Section 3, we give a regular decomposition for the solution with minimal mass, and as a consequence, we obtain Proposition 2. In Section 4, we prove a frequency and spatial decay for the global non-scattering solution. Theorems 1.3 and 1.4 shall be proved in Section 5.
2. Preliminaries. The notation X Y or Y X denotes X ≤ CY , where C > 0 is a constant that can depend on exponents (such as dimension), as well as the energy and the mass, but not parameters t or functions u. We use O(Y ) to denote any quantity X such that |X| Y . We denote by X± any quantity of the form X ± ε for any ε > 0. We use the Japanese bracket convection x := (1 + |x| 2 ) 1/2 . (·, ·) means the inner product in L 2 .
We use L q t L r x to denote the Banach space with norm with the usual modifications when q or r is equal to infinity, or the region R × R d is replaced by some I × R d . When q = r we abbreviate L q t L r x as L q t,x . For s ∈ R, we use H s rad (R d ) to denote the space of functions in H s (R d ) that are spherically symmetric.
2.1. Basic harmonic analysis. Let φ ∈ C ∞ (R d ) be a radial and non-negative bump function supported in the ball {x ∈ R d ; |x| ≤ 25 24 } and equal to 1 on the ball {x ∈ R d ; |x| ≤ 1}. For any constant c > 0, we denote φ ≤c (x) := φ( x c ) and φ >c (x) := 1 − φ ≤c (x). For each number N > 0, we define the multipliers and similarly P <N and P ≥N . We also define whenever M < N . We will always use these multipliers when M and N are dyadic numbers; in particular, all summations over N or M are understood to be over dyadic numbers. Note that P N is not truly a projection, P 2 N = P N . We shall use modified Littlewood-Paley operators: These obey P NPN =P N P N = P N . [12]). Let R > 0, and N, M > 0 be such that [7,9]). Let d ≥ 3. Let I be an interval containing 0, and x . Then the function u defined by .
We will also need the following weighted Strichartz estimate. [13,16]). Let I be an interval containing t 0 . Let P + and P − are regarded as projections onto outgoing and incoming spherical waves, respectively. For N > 0, let P ± N denote the product P ± P N where P N is the Littlewood-Paley operator.
with an N -independent constant.
Then there exist a subsequence of {ϕ n } (still denoted {ϕ n }), a family of sequence {x j n } in R d and a sequence {ψ j } of H 1 x functions such that (i) for every j = k, |x k n − x j n | → ∞ as n → ∞.
(ii) for every J ≥ 1 and every x ∈ R d , we have Moreover, we have for every J ≥ 1 that As a consequence of the above decomposition, we have the decoupling of the nonlinear energy.
3. Regular decomposition of the solution. In this section, we shall give a regular decomposition of the solution with ground state mass. We start by discussing where Q is the ground state of (5). E(Q) = 0.
Lemma 3.1. There exist θ 0 ∈ R and x 0 ∈ R d such that Proof. We argue by contradiction. Assume there exist an η 0 > 0 and a sequence and Applying the bubble decomposition (Lemma 2.7) to {v n }, we obtain Since Furthermore, by Corollary 1, we have Invoking the sharp Gargliardo-Nirenberg inequality (6) and (17), we get Note that from (16) N (ψ j0 ) ≤ N (Q). Thus, It follows that ψ j = 0 if j = j 0 and lim n→∞ N (r j0 Reviewing the argument from (18)- (19) and using (17), we also get ∇ψ j0 as n → ∞, which in turn gives by the Hardy-Littlewood-Sobolev inequality that Thus, On the other hand, since N (ψ j0 ) = N (Q), we have E(ψ j0 ) ≥ 0. Thus, By Proposition 1 and ∇ψ j0 which is in contradiction to our assumption (15).
Let e 0 be the negative eigenvalue of L + , and ω be the corresponding eigenfunction. Then there exist constants δ > 0, C > 1, satisfies the following The proof is achieved by applying the implicit function theorem and was essentially given in [25]. For sake of convenience, we present it.
With the above lemma, we shall get a decomposition for the solution with ground state mass. where The scaling parameter satisfies and If u is spherically symmetric, we may choose x(t) ≡ 0.
By our choice of λ(t), we see Finally, if u is spherically symmetric, then by Lemma 3.2,x(t) = 0 in (33). Hence, x(t) = 0. This completes the proof of the proposition.
Proof of Proposition 2. Let d = 3, 4, c 0 > 0 be a given number. Let u(t) ∈ H 1 (R d ) be a spherically symmetric solution of (1) with N (u) = N (Q) and maximal life-span I. Our aim is to show Proof. Let C 1 , C 2 > 0 be as in Proposition 3. For fixed t ∈ I, by Proposition 3, we may consider two cases: ∇u(t) 2 L 2 < C 1 E(u) and ∇u(t) 2 L 2 ≥ C 1 E(u). The former case is trivial. We consider the latter. By virtue of (29), (34), the support property of φ c0 and the decay of ground state Q, we obtain 4. Localization of the non-scattering solution. In this section, we prove a frequency decay estimate and a spatial decay estimate for the global non-scattering solution with minimal mass. 4.1. Frequency decay estimate. In this subsection, we prove the following proposition.
is a global solution of (1) that is nonscattering and satisfies Then there exists β = β(d) > 0 such that for any dyadic number N ≥ 1, we have for all t ≥ 0 that The proof of this proposition is completed by several lemmas. We first project u(t) onto the incoming and outgoing waves and use the Duhamel formula backward in time for the incoming wave and (8) forward in time for the outgoing wave. Namely, where the equality holds in the sense of weak L 2 limit. By the boundedness of φ >1 P − N e it∆ , we have We rewrite the second term of (37) and (38) as follows i(38) =φ >1 The analog holds for (42).

1.
We x are bounded. Thus, The claim shall be reformulated as Lemma 4.5 of nonlinear estimates and be proven in the end of this subsection.
To estimate I 2 , we shall use the equation to replace (|·| −2 * |u| 2 )u with −(i∂ τ +∆)u. And we need the following lemma, which is a consequence of integrating by parts.
By Lemma 4.2, we can write Now it suffices to estimate I 21 ∼ I 24 . We only estimate I 24 , since the estimates for the other three terms are in the same manner. By Lemma 2.6, the kernel satisfies where χ ≤2/3 is a characteristic function on the ball {x ∈ R d ; |x| ≤ 2/3}. These together with Minkowski's inequality, Schur's test lemma and Hölder's inequality give Thus, we get This ends the proof of Lemma 4.1.

The analog holds for (43).
Proof. We split the integral into three pieces: Estimate of (45): Noticing that φ >1 P − N is bounded on L 2 x (R d ), by Minkowski's inequality, mismatch estimate (Lemma 2.2), we have By Hölder's inequality, the nonlinear estimate (Lemma 4.5(iii)) and Sobolev's embedding theorem, we have Thus, Estimate of (46): We use weighted Strichartz estimate, Hölder's inequality to get where q = 2d ≥ 4. By Bernstein's inequality, the Leibniz rule and Lemma 4.5, it follows that This together with (48) yields

YANFANG GAO AND ZHIYONG WANG
Estimate of (47): By virtue of Lemma 4.2, Since the estimates for these terms are similar, we only give the estimate of (50). Note that the integral kernel obeys Using this, Minkowski's inequality and Schur's test lemma, we obtain Putting all these estimates together, we conclude Lemma 4.3.
To establish Proposition 4, we are now left to estimate (44). This is the following lemma.
Proof. We decompose the integral as follows Thus, to prove the lemma, it suffices to estimate (53)-(55). By Strichartz's inequality, mismatch estimate, nonlinear estimate (Lemma 4.5), we have for the k-th piece that We use weighted Strichartz estimate to estimate (54), and consider two cases according to the dimension. Indeed, we have by weighted Strichartz estimate, Hölder's inequality and Bernstein's inequality that where q = 4 if d = 3, and q = 6 if d = 4. Furthermore, by Hölder's inequality, Lemma 4.5, it follows that Hence, (54) k L 2 To estimate (55), using Lemma 4.2, we are reduced to estimating Invoking the kernel estimate and by Schur's test lemma, we obtain Thus, (55) L 2 Putting the estimate of (53)-(55) together, we achieve Lemma 4.4.
Combining the equations (37)-(44), and Lemmas 4.1, 4.3 and 4.4, we get the desired result in Proposition 4. We finish this subsection by proving the following lemma.
Proof. Using Hölder's inequality, we get This implies that (i) is a consequence of (iii) for p = d.
We first prove (iii). Since p > 2/d, we have |x| −2 So by the triangle inequality, Young's inequality, Hardy-Littlewood-Sobolev's inequality and Sobolev's embedding, we get x c0,d,p 1. To prove (ii), we also divide the proof into two cases: d = 4 and d = 3. In the case d = 4, we estimate by the triangle inequality, Minkowski's inequality and the Hardy-Littlewood-Sobolev inequality that x φ >c/8 u 2 H 1 + 1 1. In the case d = 3, by the triangle inequality, we have By the support property of φ, the Hardy-Littlewood-Sobolev inequality and Hölder's inequality, it follows that Using Young's inequality, we obtain Also, Thus, we establish (ii). This completes the proof of the lemma.

4.2.
Spatial decay estimate. In this subsection, we show a spatial decay estimate for the ground state mass solution.
is a global solution of (1) that is nonscattering and satisfies Let N 1 be a dyadic number. Then there exist γ = γ(d) > 0 and R 0 = R 0 (N 1 , u) such that for N > N 1 and R > R 0 , we have Proof. By Duhamel's formula and the in-out decomposition, where the last integral should be understood in weak L 2 sense. By the triangle inequality, It is easily seen that By Lemma 2.6, This together with Young's inequality implies that Next, we estimate the second and the third terms on the RHS of (58). We first split the integrals into 12 . We remark that the estimate for II 1 , II 3 , II 5 , namely those terms where the nonlinearity lies in small radii, can be done in the same way by substituting the nonlinearity with the linear part of the equation. II 2 , II 4 , II 6 in which the nonlinearity is confined on large radii will be treated by using weighted Strichartz estimate. We shall only give the estimate for II 5 , II 6 . Estimate of II 5 : Using Lemma 4.2, we turn to estimate Note that for |y| ≤R, |x| ≥ R, and τ ∈ [2 k , 2 k+1 ], by Young's inequality, Thus, the integral kernel has the estimate With this, using Young's inequality and Schur's test lemma, we get (60) L 2 where c is an absolute constant. Summing over k gives Estimate of II 6 : By weighted Strichartz estimate, Hölder's inequality, and Lemma 4.5, we obtain where q = 4 if d = 3, and q = 6 if d = 4. Combining (59), the estimate of II 5 , II 6 , and taking γ = 1 12 , we conclude the proposition.
Proof of Theorem 1.4. That is for any η > 0, there exists C(η) > 0 such that Let N 1 (η), N 2 (η) be dyadic numbers and C(η) a large constant to be determined momentarily. By the triangle inequality, we have By discarding the real space cutoff and Bernstein's inequality, we have for I 1 that For I 2 , it follows from the triangle inequality, Bernstein's estimate, mismatch estimate and Proposition 5 that To estimate I 3 , we first use the Leibniz rule to get
We are now in position to prove Theorem 1.3.