On Blowup solutions to the focusing mass-critical nonlinear fractional Schr\"odinger equation

In this paper we study dynamical properties of blowup solutions to the focusing mass-critical nonlinear fractional Schr\"odinger equation. We establish a profile decomposition and a compactness lemma related to the equation. As a result, we obtain the $L^2$-concentration and the limiting profile with minimal mass of blowup solutions.


Introduction
Consider the Cauchy problem for nonlinear fractional Schrödinger equations where u is a complex valued function defined on [0, +∞)× R d , s ∈ (0, 1) and α > 0. The parameter µ = 1 (resp.µ = −1) corresponds to the defocusing (resp.focusing) case.The operator (−∆) s is the fractional Laplacian which is the Fourier multiplier by |ξ| 2s .The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics, which was discovered by Laskin [29] as a result of extending the Feynmann path integral, from the Brownian-like to Lévylike quantum mechanical paths.The fractional Schrödinger equation also appears in the continuum limit of discrete models with long-range interactions (see e.g.[28]) and in the description of Boson stars as well as in water wave dynamics (see e.g.[16] or [25]).In the last decade, the fractional nonlinear Schrödinger equation has attracted a lot of interest in mathematics, numerics and physics (see e.g.[1,2,3,7,8,4,6,5,9,11,14,15,13,20,18,21,25,23,24,40,42,46,41] and references therein).The equation (1.1) enjoys the scaling invariance u λ (t, x) = λ 2s α u(λ 2s t, λx), λ > 0. A computation shows u λ (0) Ḣγ = λ γ+ 2s α − d 2 u 0 Ḣγ .We thus define the critical exponent 2) The equation (1.1) also enjoys the formal conservation laws for the mass and the energy: The local well-posedness for (1.1) in Sobolev spaces was studied in [21] (see also [7] for fractional Hartree equations).Note that the unitary group e −it(−∆) s enjoys several types of Strichartz estimates (see e.g.[6] or [11] for Strichartz estimates with non-radial data; and [19], [27] or [4] for Strichartz estimates with radially symmetric data; and [12] or [5] for weighted Strichartz estimates).For non-radial data, these Strichartz estimates have a loss of derivatives.This makes the study of local well-posedness more difficult and leads to a weak local theory comparing to the standard nonlinear Schrödinger equation (see e.g.[21] or [11]).One can remove the loss of derivatives in Strichartz estimates by considering radially symmetric initial data.However, these Strichartz estimates without loss of derivatives require an restriction on the validity of s, that is s ∈ d 2d−1 , 1 .We refer the reader to Section 2 for more details about Strichartz estimates and the local well-posedness in H s for (1.1).
• Mass-critical case, i.e. s c = 0 or α = 4s d : If E(u 0 ) < 0, then the solution u either blows up in finite time, i.e.T < +∞ or blows up infinite time, i.e.T = +∞ and with some C > 0 and t * > 0 that depend only on u 0 , s and d.
• Mass and energy intercritical case, i.e. 0 < s c < s or 4s d < α < 4s d−2s : If α < 4s and either E(u 0 ) < 0, or if E(u 0 ) ≥ 0, we assume that , where Q is the unique (modulo symmetries) positive radial solution to the elliptic equation then the solution blows up in finite time, i.e.T < +∞.
• Energy-critical case, i.e. s c = s or α = 4s d−2s : If α < 4s and either E(u 0 ) < 0, or if E(u 0 ) ≥ 0, we assume that where W is the unique (modulo symmetries) positive radial solution to the elliptic equation In this paper we are interested in dynamical properties of blow-up solutions in H s for the focusing mass-critical nonlinear fractional Schrödinger equation, i.e. s ∈ (0, 1)\{1/2}, α = 4s d and µ = −1 in (1.1).Before entering some details of our results, let us recall known results about blow-up solutions in H 1 for the focusing mass-critical nonlinear Schrödinger equation (mNLS) The existence of blow-up solutions in H 1 for (mNLS) was firstly proved by Glassey [17], where the author showed that for any negative energy initial data satisfying |x|v 0 ∈ L 2 , the corresponding solution blows up in finite time.Ogawa-Tsutsumi [38,39] showed the existence of blow-up solutions for negative energy radial data in dimensions d ≥ 2 and for any negative energy initial data (without radially symmetry) in the one dimensional case.The study of blow-up H 1 solution to (mNLS) is connected to the notion of ground state which is the unique (up to symmetries) positive radial solution to the elliptic equation By the variational characteristic of the ground state, Weinstein [45] showed the structure and formation of singularity of the minimal mass blow-up solution, i.e. v 0 L 2 = R L 2 .He proved that the blow-up solution remains close to the ground state R up to scaling and phase parameters, and also translation in the non-radial case.Merle-Tsutsumi [30], Tsutsumi [44] and Nava [37] proved the L 2 -concentration of blow-up solutions by using the variational characterization of ground state, that is, there exists x(t) ∈ R d such that for all r > 0, lim inf where T is the blow-up time.Merle [31,32] used the conformal invariance and compactness argument to characterize the finite time blow-up solutions with minimal mass.More precisely, he proved that up to symmetries of the equation, the only finite time blow-up solution with minimal mass is the pseudo-conformal transformation of the ground state.Hmidi-Keraani [22] gave a simplified proof of the characterization of blow-up solutions with minimal mass of Merle by means of the profile decomposition and a refined compactness lemma.Merle-Raphaël [33,34,35] established sharp blow-up rates, profiles of blow-up solutions by the help of spectral properties.
As for (mNLS), the study of blow-up solution to the focusing mass-critical nonlinear fractional Schrödinger equation is closely related to the notion of ground state which is the unique (modulo symmetries) positive radial solution of the elliptic equation (1. 3) The existence and uniqueness (up to symmetries) of ground state Q ∈ H s for (1.3) were recently shown in [14] and [15].In [2,15], the authors showed the sharp Gagliardo-Nirenberg inequality where Using this sharp Gagliardo-Nirenberg inequality together with the conservation of mass and energy, it is easy to see that if u 0 ∈ H s satisfies then the corresponding solution exists globally in time.This implies that Q L 2 is the critical mass for the formation of singularities.
To study blow-up dynamics for data in H s , we establish the profile decomposition for bounded sequences in H s in the same spirit of [22].With the help of this profile decomposition, we prove a compactness lemma related to the focusing mass-critical (NLFS).
Then there exists a sequence (x n ) n≥1 in R d such that up to a subsequence, where Q is the unique solution to the elliptic equation (1.3).
Note that the lower bound on the L 2 -norm of V is optimal.Indeed, if we take v n = Q, then we get the identity.
As a consequence of this compactness lemma, we show that the L 2 -norm of blow-up solutions must concentrate by an amount which is bounded from below by Q L 2 at the blow-up time.More precisely, we prove the following result.
Assume that the corresponding solution u to (1.1) blows up at finite time 0 < T < +∞.Let a(t) > 0 be such that as t ↑ T .Then there exists x(t) ∈ R d such that lim inf where Q is the unique solution to (1.3).
• The condition (1.6) comes from the local theory (see Table 1).• By the blow-up rate given in Corollary 2.7, we have we see that any function a(t) > 0 satisfying Finally, we show the limiting profile of blow-up solutions with minimal mass Q L 2 .More precisely, we show that up to symmetries of the equation, the ground state Q is the profile for blow-up solutions with minimal mass.Theorem 1.5 (Limiting profile with minimal mass).Let d, s, α and u 0 be as in (1.6).Assume that the corresponding solution u to (1.1) blows up at finite time 0 < T < +∞.
After submitting this manuscript, we are informed that a recent work of Feng [13] has considered the fractional nonlinear Schrödinger equation with combined power-types of nonlinearities.He studied blow-up dynamics in the case of a L 2 -critical nonlinear term perturbed by a L 2 -subcritical term.
The paper is oganized as follows.In Section 2, we recall Strichartz estimates for the fractional Schrödinger equation and the local well-posedness for (1.1) in non-radial and radial H s initial data.In Section 3, we show the profile decomposition for bounded sequences in H s and prove a compactness lemma related to the focusing mass-critical (1.1).The L 2 -concentration of blow-up solutions is proved in Section 4. Finally, we show the limiting profile of blow-up solutions with minimal mass in Section 5.

Preliminaries
with a usual modification when either p or q are infinity.We have three-types of Strichartz estimates for the fractional Schrödinger equation: • For general data (see e.g.[6] or [11]): the following estimates hold for d ≥ 1 and s ∈ (0, 1)\{1/2}, where (p, q) and (a, b) are Schrödinger admissible, i.e. and similarly for γ a ′ ,b ′ .Here (a, a ′ ) and (b, b ′ ) are conjugate pairs.It is worth noticing that for s ∈ (0, 1)\{1/2} the admissible condition 2 p + d q ≤ d 2 implies γ p,q > 0 for all admissible pairs (p, q) except (p, q) = (∞, 2).This means that the above Strichartz estimates have a loss of derivatives.In the local theory of the nonlinear fractional Schrödinger equation, this loss of derivatives makes the problem more difficult, and leads to a weak local well-posedness result comparing to the nonlinear Schrödinger equation (see Subsection 2.3).
• For radially symmetric data (see e.g.[27], [19] or [4]): the estimates (2.1) and (2.2) hold true for d ≥ 2, s ∈ (0, 1)\{1/2} and (p, q), (a, b) satisfy the radial Schödinger admissible condition: Note that the admissible condition allows us to choose (p, q) so that γ p,q = 0.More precisely, we have for d ≥ 2 and where ψ and f are radially symmetric and (p, q), (a, b) satisfy the fractional admissible condition, (2.5) These Strichartz estimates with no loss of derivatives allow us to give a similar local wellposedness result as for the nonlinear Schrödinger equation (see again Subsection 2.3).• Weighted Strichartz estimates (see e.g.[12] or [5]) Here ∇ ω = √ 1 − ∆ ω with ∆ ω is the Laplace-Beltrami operator on the unit sphere S d−1 .Here we use the notation These weighted estimates are important to show the well-posedness below L 2 at least for the fractional Hartree equation (see [7]).
2.2.Nonlinear estimates.We recall the following fractional chain rule which is needed in the local well-posedness for (1.1).
We refer the reader to [10, Proposition 3.1] for the proof of the above estimate when 1 < q 1 < ∞ and to [26] for the proof when q 1 = ∞.

2.3.
Local well-posedness in H s .In this section, we recall the local well-posedness in the energy space H s for (1.1).As mentioned in the introduction, we will separate two cases: non-radial initial data and radially symmetric initial data. 1 This condition follows by pluging γp,q = 0 to Non-radial H s initial data.We have the following result due to [21] (see also [11]).
The proof of this result is based on Strichartz estimates and the contraction mapping argument.The loss of derivatives in Strichartz estimates can be compensated for by using the Sobolev embedding.We refer the reader to [21] or [11] for more details.
Remark 2.3.It follows from (2.8) and s ∈ (0, 1)\{1/2} that the local well-posedness for non radial data in H s is available only for (2.9) In particular, in the mass-critical case α = 4s d , the (1.1) is locally well-posed in Proposition 2.4 (Non-radial global existence [11]).Let s, α and d be as in (2.9).Then for any u 0 ∈ H s , the solution to (1.1) given in Proposition 2.2 can be extended to the whole R if one of the following conditions is satisfied: Proof.The case µ = 1 follows easily from the blow-up alternative together with the conservation of mass and energy.The case µ = −1 and 0 < α < 4s d follows from the Gagliardo-Nirenberg inequality (see e.g.[43,Appendix]).Indeed, by Gagliardo-Nirenberg inequality and the mass conservation, The conservation of energy then implies If 0 < α < 4s d , then dα 2s ∈ (0, 2) and hence u(t) Ḣs 1.This combined with the conservation of mass yield the boundedness of u(t) H s for any t belongs to the existence time.The blowup alternative gives the global existence.The case µ = −1, α = 4s d and u 0 L 2 small is treated similarly.It remains to treat the case µ = −1 and u 0 Ḣs is small.Thanks to the Sobolev embedding with This shows in particular that E(u 0 ) is small if u 0 H s is small.Therefore, Since u 0 H s is small, the above estimate implies that u(t) H s is bounded from above and the proof is complete.
Radial H s initial data.Thanks to Strichartz estimates without loss of derivatives in the radial case, we have the following result.
Proof.It is easy to check that (p, q) satisfies the fractional admissible condition (2.5).We choose (m, n) so that

.11)
We see that (2.12) The later fact gives the Sobolev embedding Ẇ s,q ֒→ L n .Let us now consider where I = [0, ζ] and M, ζ > 0 to be chosen later.By Duhamel's formula, it suffices to prove that the functional is a contraction on (X, d).By radial Strichartz estimates (2.3) and (2.4), The fractional chain rule given in Lemma 2.1 and the Hölder inequality give This shows that for all u, v ∈ X, there exists C > 0 independent of T and u 0 ∈ H s such that If we set M = 2C u 0 Ḣs and choose ζ > 0 so that then Φ is a strict contraction on (X, d).This proves the existence of solution u ∈ C(I, H s ) ∩ L p (I, W s,q ).By radial Strichartz estimates, we see that u ∈ L a (I, W s,b ) for any fractional admissible pairs (a, b).The blow-up alternative follows easily since the existence time depends only on the Ḣs -norm of initial data.The proof is complete.
As in Proposition 2.4, we have the following criteria for global existence of radial solutions in H s . .Then for any u 0 ∈ H s radial, the solution to (1.1) given in Proposition 2.5 can be extended to the whole R if one of the following conditions is satisfied: Combining the local well-posedness for non-radial and radial initial data, we obtain the following summary.for all 0 < t < T .
Proof.We follow the argument of Merle-Raphael [36].Let 0 < t < T be fixed.We define with λ(t) to be chosen shortly.We see that v t is well-defined for Moreover, v t solves Since s > s c , we choose λ(t) so that v t (0) Ḣs = 1.Thanks to the local theory, there exists τ 0 > 0 such that v t is defined on [0, τ 0 ].This shows that .
The proof is complete.

Profile decomposition
In this subsection, we use the profile decomposition for bounded consequences in H s to show a compactness lemma related to the focusing mass-critical (1.1).Theorem 3.1 (Profile decomposition).Let d ≥ 1 and 0 < s < 1.Let (v n ) n≥1 be a bounded sequence in H s .Then there exist a subsequence of (v n ) n≥1 (still denoted (v n ) n≥1 ), a family (x j n ) j≥1 of sequences in R d and a sequence (V j ) j≥1 of H s functions such that • for every k = j, • for every l ≥ 1 and every for every q ∈ (2, 2 ⋆ ), where Ḣs + o n (1), (3.4) as n → ∞.
Proof.The proof is similar to the one given by Hmidi-Keraani [22,Proposition 3.1].For reader's convenience, we recall some details.Since H s is a Hilbert space, we denote Ω(v n ) the set of functions obtained as weak limits of sequences of the translated We shall prove that there exist a sequence (V j ) j≥1 of Ω(v n ) and a family ( as n → ∞, and up to a subsequence, the sequence (v n ) n≥1 can be written as for every l ≥ 1 and every with η(v l n ) → 0 as l → ∞.Moreover, the identities (3.3) and (3.4) hold as n → ∞.Indeed, if η(v n ) = 0, then we can take V j = 0 for all j ≥ 1. Otherwise we choose By the definition of Ω(v n ), there exists a sequence ( In fact, if it is not true, then up to a subsequence, ) converges weakly to 0, we see that V 2 = 0.This implies that η(v 1 n ) = 0 and it is a contradiction.An argument of iteration and orthogonal extraction allows us to construct the family (x j n ) j≥1 of sequences in R d and the sequence (V j ) j≥1 of H s functions satisfying the claim above.Furthermore, the convergence of the series To complete the proof of Theorem 3.1, it remains to show (3.2).To do so, we introduce θ : where • is the Fourier transform of χ.In particular, we have χR where * is the convolution operator.Let q ∈ (2, 2 ⋆ ) be fixed.By Sobolev embedding and the Plancherel formula, we have On the other hand, the Hölder interpolation inequality implies Thus, by the definition of Ω(v l n ), we infer that lim sup By the Plancherel formula, we have for some ǫ > 0 small enough, we see that Letting l → ∞ and using the fact that η(v l n ) → 0 as l → ∞ and the uniform boundedness in H s of (v l n ) l≥1 , we obtain lim sup The proof is complete.
We are now able to give the proof of the concentration compactness lemma given in Theorem 1.2.Proof of Theorem 1.2.According to Theorem 3.1, there exist a sequence (V j ) j≥1 of H s functions and a family (x j n ) j≥1 of sequences in R d such that up to a subsequence, the sequence (v n ) n≥1 can be written as , (3.4) hold.This implies that we have Using the pairwise orthogonality (3.1), the Hölder inequality implies that V j (• + x k n − x j n ) ⇀ 0 in H s as n → ∞ for any j = k.This leads to the mixed terms in the sum (3.5) vanish as n → ∞.We thus get We next use the sharp Gagliardo-Nirenberg inequality (1.4) to estimate By (3.4), we infer that Therefore, Since the series j≥1 V j 2 L 2 is convergent, the supremum above is attained.In particular, there exists j 0 such that By a change of variables, we write where ṽl n (x) := v l n (x + x j0 n ).The pairwise orthogonality of the family (x j n ) j≥1 implies V j (• + x j0 n − x j n ) ⇀ 0 weakly in H s , as n → ∞ for every j = j 0 .We thus get where ṽl is the weak limit of (ṽ l n ) n≥1 .On the other hand, ṽl By the uniqueness of the weak limit (3.7), we get ṽl = 0 for every l ≥ j 0 .Therefore, we obtain n ) n≥1 and the function V j0 now fulfill the conditions of Theorem 1.2.The proof is complete.

Blow-up concentration
In this section, we give the proof of the mass-concentration of finite time blow-up solutions given in Theorem 1.3.Proof of Theorem 1.3.Let (t n ) n≥1 be a sequence such that t n ↑ T .Set By the blow-up alternative, we see that λ n → 0 as n → ∞.Moreover, we have This implies in particular that The sequence (v n ) n≥1 satisfies the conditions of Theorem 1.2 with Therefore, there exists a sequence (x n ) n≥1 in R d such that up to a subsequence, This implies for every R > 0, lim inf Ḣs , the assumption (1.7) implies for some x(t) ∈ R d .This shows (1.8).The proof is complete.

Limiting profile with minimal mass
In this section, we give the proof of the limiting profile given in Theorem 1.5.Let us start with the following characterization of solution with minimal mass.
where C GN is the sharp constant in (1.4).By the characterization of the sharp constant to the Gagliardo-Nirenberg inequality (1.4) (see e.g.[15, Section 3]), we learn that u is of the form u(x) = aQ(λx + x 0 ) for some a ∈ C ⋆ , λ > 0 and x 0 ∈ R d .On the other hand, since u L 2 = Q L 2 , we have |a| = λ d 2 .This shows the result.
We are now able to prove the limiting profile of finite time blow-up solutions with minimal mass given in Theorem 1.5.Proof of Theorem 1.5.We will show that for any (t n ) n≥1 satisfying t n ↑ T , there exist a subsequence still denoted by (t n ) n≥1 , sequences of θ n ∈ R, λ n > 0 and x n ∈ R d such that e itθn λ d 2 n u(t n , λ n • +x n ) → Q strongly in H s as n → ∞.
(5.1) Let (t n ) n≥1 be a sequence such that t n ↑ T .Set By the blow-up alternative, we see that λ n → 0 as n → ∞.Moreover, we have and and E(v n ) = λ 2s n E(u(t n )) = λ 2s n E(u 0 ) → 0, as n → ∞.This yields in particular that Ḣs , as n → ∞. (5.4) The sequence (v n ) n≥1 satisfies the conditions of Theorem 1.2 with Therefore, there exists a sequence (x n ) n≥1 in R d such that up to a subsequence, as n → ∞ with V L 2 ≥ Q L 2 .Since v n (•+x n ) ⇀ V weakly in H s as n → ∞, the semi-continuity of weak convergence and (5.2) imply This together with the fact V L 2 ≥ Q L 2 show that (5.5) Therefore v n (• + x n ) → V strongly in L 2 as n → ∞.On the other hand, the Gagliardo-Nirenberg inequality (1.4) shows that v n (• + x n ) → V strongly in L 4s d +2 as n → ∞.Indeed, by (5.3), L 2 → 0, as n → ∞.Moreover, using (5.4) and (5.5), the sharp Gagliardo-Nirenberg inequality (1.4) yields or Q Ḣs ≤ V Ḣs .By the semi-continuity of weak convergence and (5.3), Therefore, (5.6) Combining (5.5), (5.6) and using the fact v n (• + x n ) ⇀ V weakly in H s , we conclude that v n (• + x n ) → V strongly in H s as n → ∞.
In particular, we have This shows that there exists V ∈ H s such that By Lemma 5.1, we have V (x) = e iθ λ d 2 Q(λx + x 0 ) for some θ ∈ R, λ > 0 and x 0 ∈ R d .Thus

Table 1 .
Local well-posedness (LWP) in H s for NLFS Corollary 2.7 (Blow-up rate).Let d, α and u 0 ∈ H s be as in Table1.Assume that the corresponding solution u to (1.1) given in Proposition 2.2 and Proposition 2.5 blows up at finite time 0 < T < +∞.Then there exists C > 0 such that