Mass concentration for the $L^2$-critical Nonlinear Schr\"odinger equations of higher orders

We consider the mass concentration phenomenon for the $L^2$-critical nonlinear Schr\"odinger equations of higher orders. We show that any solution $u$ to $iu_{t} + (-\Delta)^{\frac\alpha 2} u =\pm |u|^\frac{2\alpha}{d}u$, $u(0,\cdot)\in L^2$ for $\alpha>2$, which blows up in a finite time, satisfies a mass concentration phenomenon near the blow-up time. We verify that as $\alpha$ increases, the size of region capturing a mass concentration gets wider due to the stronger dispersive effect.


introduction
We consider the L 2 -critical Cauchy problem in Here (−∆) α 2 is the pseudo-differential operator defined by e ixξ |ξ| α f (ξ)dξ and f (ξ) = R d e −ixξ f (ξ)dξ. The equation (1.1) is L 2 -critical in the sense the that the equation is invariant under the rescaling transformation u → u λ , u λ (t, x) = λ d 2 u(λ α t, λx), which preserves L 2 norm. The system conserves the mass M(t) and the energy E(t) a priori ; The Fourth order Schrödinger equations were initially studied by Karpman [7] and Karpman and Shagalov [8]. They considered the fourth order Schrödinger equation to take into account the role of small fourth order dispersion terms in the propagation of intense laser beams in a bulk medium with a cubic nonlinearity (Kerr nonlinearity). The fourth order L 2 -critical case with nonlinearity |u| 8 d u in (1.1) was studied in [6,12]. When α = 2, it is unknown whether there exists a blow up solution of (1.1) except the numerical evidence of [6]; unlike α = 2 case a virial type inequality or a pseudo conformal type symmetry are not yet known to hold.
In this paper we are concerned with the mass concentration phenomena of blowup solutions to (1.1), especially when the initial datum u 0 ∈ L 2 and its mixed L q t L r xnorm blows up in a finite time. When α = 2 and d = 2, Bourgain in his seminal paper [3] showed that if the L 2 -wellposed solution in R 2 breaks down at a maximal time 0 < T * < ∞ with u for some M > 0. Later, this was extended to higher dimensions by Bégout and Vargas [1]. A generalization in mixed norm spaces L q t L r x was obtained in [4].
In order to get a sufficiently broad band solution, still around ξ 0 , we define u(t, x) = e ix·ξ 0 +it|ξ 0 | α φ(ǫ(x + α|ξ 0 | α−1 ξ 0 t)) for a smooth bounded function φ, then u can be shown to satisfy i∂ t u + (−∆) α 2 u = O φ (ǫ 2 ). This means that the profile of |u| moves roughly at velocity −α|ξ 0 | α−1 ξ 0 . Assuming high frequency initial data (|ξ 0 | ≫ 1), the propagation speed increases as α increases. In other words, the wave tends to spend shorter time in a fixed region. So, when α > 2 it is reasonable to expect that we need a larger set for concentration region than B(x, (T * − t) for a solution u which blows up at T * . If we impose further condition that ǫ depends only on u 0 , then among the power type sizes, (T * − t) β , we can see that the size (T * − t) 1 α is optimal by a simple scaling argument.
Before giving precise statement of our results, we briefly clarify the issue of wellposedness of (1.1). The local well-posedness in H s (R d ), s ≥ 0 relies on the space time estimate for the free propagator which is called Strichartz's estimate(see (2.2) in Lemma 2.1). We call that a pair (q, r) is α-admissible if , q, r ≥ 2 and r = ∞.
Then by the usual argument it is possible to show the inhomogeneous Strichartz's estimates (2.3) (Lemma 2.1) for α-admissible (q, r) and (q,r). By Duhamel principle the solution can be written as When the initial datum u 0 ∈ L 2 x (R d ), following the standard argument for local wellposedness, we see that there exists the unique solution u(t, x) on a small time interval [0, T ] such that The existence time interval [0, T ] is extended as long as u L q t L r x ([0,T ]×R d ) < ∞. If the solution blows up at T * , then Indeed, using (1.3), the inhomogeneous Strichartz's estimate (2.3) and Hölder's inequality, one get for any α-admissible (q, r) and ( q, r) . Hence the nonlinear map becomes a contraction map if there are α-admissible pairs (q, r) and (q,r) satisfying It is possible as long as the condition (1.4) is satisfied (see Figure 1). The following is our first result.    The results in [1,3,4] were obtained by the use of refinement of Strichartz's estimates for e it∆ f which come from bilinear restriction estimate for the paraboloid [11,10,14,16]. To deal with the case α > 2 we need similar estimates for e it(−∆) α 2 . It turns out that the related analysis is simpler than [1,3,4] due to a stronger dispersion effect so that we give a direct proof of refinement of Strichartz's estimates for e it(−∆) α 2 exploiting bilinear interaction of Schrödinger waves. In particular we have the refinement (Proposition 2.3) in terms of dyadic shells, instead of cubes as in the previous work [1,3,4] for which the Galilean invariance of the operator e it∆ f played a role, which is no longer available when α = 2.
Secondly, we consider the L 2 -critical Hartree equation, which is given by for One can easily check that the equation (1.7) is also L 2 -critical, that is, invariant under u → u λ . One may be interested in a mass concentration for the finite time  blow-up solutions for (1.7). The local wellposedness can be established by following the standard argument. In fact, using the Strichartz estimates (2.3) and triangle inequality By Hölder's inequality and Hardy-Littlwood-Sobolev inequality the last of the above is bounded by Let us take q 1 = q 2 and r 1 = r 2 . Then we find that the nonlinear map is a contraction if there is an α-admissible pair ( q, r) such that ).
An easy calculation shows that the line segment [A, B] in Figure 2 is parallel to the segment [e, f ] corresponding to the set and moreover |e−f | = 3|A−B|. So it is possible to find ( q, r) satisfying (1.9) as long as (q, r) is contained in [A, B], that is, For these (q, r) we also get a blowup alternative; If T * < ∞, then (1.5) should be satisfied. As it was shown in [4], the mass concentration phenomenon is mostly involved with the homogeneous part of the solution. The argument used in [1,3] works for (1.7) without much modifications if the nonlinear term can be controlled properly. This is actually equivalent to showing the local wellposedness of (1.7) under the condition (1.10).
The paper is organized as follows. In Section 2 we obtain some preliminary estimates which are to be used for the proofs of Theorems. In Section 3 we give the proofs of Theorems 1.1 and 1.2.

preliminary
In this section we show several lemmas which will be used later for the proofs of the theorems. For q, r ≥ 2, r = ∞ and 2 Let ρ be a smooth function supported in [1/2, 4] and satisfying ∞ −∞ ρ(x/2 k ) = 1 for all x > 0. Then we define a projection operator by The following lemma is a version of Strichartz estimates for Schrödinger equations of higher orders α with α > 2. It seems well known but for a convenience of the readers we include the proof. The arguments are based on rescaling and Littlewood-Paley theorem.
In particular, if (q, r) is α-admissible, then Also if (q, r) and (q,r) are α-admissible, then we have Proof. Once we get (2.1), then (2.2) follows from Plancherel's theorem. Also (2.3) can be shown by duality and the argument due to Christ and Kiselev ([5]). We now show (2.1). Since α > 2, by the stationary phase method (see p.344 in Hence, from the argument of Keel-Tao in [9], we have and r, q ≥ 2 (with exception r = ∞ when d = 2). Then by rescaling we observe that Since f = k P k f and q, r ≥ 2, from Littlewood-Paley theorem followed by Minkowski's inequality we have Putting (2.5) in the right hand side of the above, we get the desired.
This means that it is possible to obtain better bounds than the one trivially obtained by rescaling (Lemma 2.1) when the waves interact at different frequency levels. Such observation was first made by Bourgain [3].
Proof. By rescaling it is enough to show that Let us set L = M − N ≤ 0. Hence Fourier supports of P 0 f , P L f are contained in the sets {|ξ| ∼ 1}, {|ξ| ∼ 2 L }, respectively. For d r + 2 q ≤ d 2 , and r, q ≥ 2, by Hölder's inequality and (2.5) one can see If one interpolates this with x ≤ C2 L(d−1)/2 f 2 g 2 which will be proven later, one get the desired estimat (2.6). Indeed, note that the bound in the above is better than the trivial bounds follows from rescaling. That is, for some ǫ > 0 because α, d ≥ 2. Hence via interpolation we get the desired estimate with some ǫ > 0 as long as d/r + 2/q < d/2 and q > 2.
Proof of (2.7). We may assume that f is supported in the set {ξ : |ξ| ∼ 1}. When 2 L ∼ 1, the estimate (2.7) is trivial from (2.5) and Hölder's inequality. So we also may assume 2 L ≪ 1. By decomposing the Fourier support of f into finite number of sets, rotation and mild dilation, it is enough to show that is the open ball centered at x with radius r. We write Freezingη = (η 2 , . . . , η d ), we consider an operator We now make the change of variables Then by a direct computation one can see that on the supports of f and g. Hence making change of variables (ξ, η 1 ) → ζ, applying Plancherel's theorem and reversing the change variables (ζ → (ξ, η 1 )), we get Since by Minkowski's inequality we see This gives the desired bound by Schwartz's inequality, because of |η| ≤ 2 L . Here Proof of Proposition 2.3. In fact, for the proof it is sufficient to show that By dividing the support of P k f into three dyadic shells B k−1 , B k , and B k+1 , we get the desired. This actually can be shown by using (2.1) and the following two estimates: If (q, r) is an α-admissible with q > 2, then with some p < 2 < q. Interpolation among (2.1) and these two estimates gives as long as (1/p * , 1/q * ) is contained in the triangle Γ with vertices (1/2, 1/2), (1/2, 1/q) and (1/p, 1/2). Obviously one can find a point (1/p 0 , 1/q 0 ) contained in the interior of Γ so that it lies on the line segment joining (1/2, 1/2) and (1/p, 0) for some p < 2. Then by interpolation among the mixed norm spaces * ([2]) we see Therefore, using (2.10) which is valid with (q * , p * ) = (q 0 , p 0 ) together with the above and Plancherel's theorem we get the desired inequality. Now it remains to show (2.8) and (2.9). We first show (2.9) which is easier. Note that α > 2. By interpolation between (2.4) and the trivial L 1 → L ∞ bound, one can see that for each α-admissible (q, r), q > 2, there is a p < 2 such that Here we used the fact that α > 2. Then by rescaling we see that By using Littlewood-Paley theorem, Minkowski's inequality and the above we get In particular, when (q, r) is α-admissible we get (2.9).
Now we turn to (2.8). We start with the inequality (2.1) which reads as However in the right hand side the norm in k is ℓ 2 . We need to upgrade this slightly so that the norm in k is replaced by ℓq for somẽ q > 2. To do this it is enough to show that there is a pair (q, r) satisfying q, r ≥ 2, r = ∞ and 2 q + d r ≤ d 2 , such that for someq > 2. The interpolation between this and (2.1) gives the desired. In particular when (q, r) is α-admissible we get (2.8).

Then by triangle inequality
By symmetry it is enough to deal with the first one because the second can be handled similarly. Hence it is enough to show that First we handle the case j = 0, 1, 2. By Cauchy-Schwarz's inequality we have So, squaring both sides we get Hence it follows that Then by Lemma 2.2 we see Therefore by Schwarz's inequality and summation in l we get Now we turn to case j ≥ 3. Observe the Fourier supports of are boundedly overlapping. Hence by Plancherel's theorem, we see that x .
Using Lemma 2.2, the right hand side is bounded by Therefore, Schwarz's inequality gives us the desired bound (2.12). Proposition 2.3 can be combined with the following elementary lemma to find out the region where the given L 2 function is not severely concentrating but still containing a moderate amount of mass.
Lemma 2.4. Let ǫ > 0, f ∈ L 2 (R 2 ) and suppose that there is a measurable subset Q such that ǫ ≤ (|Q| Here all the implicit constants are independent of f , Q, ǫ and λ.

Proof of Theorems
As in α = 2 case ([1, 3, 4]) the following two lemmas play crucial roles in showing the mass concentration. The first one is concerned with decomposition of the initial datum into functions of which Fourier transforms are spreading rather than concentrating. In view of uncertainty principle the spreading part of the initial datum may concentrate on some spatial region. The second one enables us to find regions where the linear Schrödinger wave concentrates in the mixed norm space L q t L r x (here (q, r) is α admissible) when the Fourier transform of the initial data does not severely concentrate.
Lemma 3.1. Let (q, r) be an α-admissible pair satisfying q > 2 and α/q+d/r = d/2. Suppose f ∈ L 2 (R d ) and for some ǫ > 0. Then there exist a f k ∈ L 2 (R d ) and a dyadic shell B n k for k = 1, 2, · · · , N with N = N( f L 2 , d, ǫ) such that Here the constants C, µ, and ν depend only on d.
Lemma 3.2. Let (q, r) be an α-admissible pair satisfying 2 < q ≤ r. Suppose g ∈ L 2 (R d ) and supp g ⊂ B k and | g| < C 0 2 − kd 2 for C 0 > 0. Then for any ǫ > 0, there exist N 1 ∈ N, N 1 ≤ C(d, C 0 , ǫ), and sets (Q n ) 1≤n≤N 1 ⊂ R × R d which is given by where I n ⊂ R is an interval with |I n | = 2 −kα and C n is a cube with the side length l (C n ) = 2 −k such that Notation. Let E be a measurable set in R d+1 and f : Once we have the refinement of Strichartz estimates (Proposition 2.3) the proofs of Lemma 3.1 and 3.2 can be given by a modification of the argument in [1,3]. The proofs of lemmas are given in Appendix.
Proof of Theorem 1.1. The proof consists of following steps: · Controlling the inhomogeneous part, · Decomposition to the initial datum with non-concentration Fourier transforms, · Figuring out the concentrating region, · Determining the size of mass concentration region. The two lemmas (Lemmas 3.1 and 3.2) will be incorporated into the second and the third step respectively.
To prove Theorem 1.1 it is enough to consider the case q ≤ r in which q ≤ 2(d + α)/d. Let u be the maximal solution to (1.1) over the maximal forward existence time interval [0, T * ) so that (1.5) is satisfied for an α-admissible pair (q, r), 2 < q ≤ r and u L q Then for a fixed small η > 0 there is a strictly increasing sequence {t n } ∞ n=1 in [0, T * ) such that lim n→∞ t n = T * and for every n ∈ N By Duhamel's formula, we have for t ∈ (0, T * ) Applying Strichartz's estimate with (3.3), we have where (1.4) holds. Hence from (3.3), (3.4) and time translation invariance property we obtain for sufficiently small η. Fix n ∈ N and the time interval (t n , t n+1 ). We denote f = u(t n ) and then by the mass conservation we have where L = L( f L 2 , d, η). By Hölder's inequality with 2 r + r−2 r = 1, we have t n+1 tn By using Hölder's inequality with 2 r + r−2 r = 1 again, the last term of (3.7) is bounded by In order to estimate E and F , we apply (3.3), (3.4) and (3.6). Since (2α+d)r−4α d > r for r ≥ q > 2, we see that We may split (3.3) into two integrals such as From (3.8) and (3.9) we obtain that Since L = L( u 0 L 2 (R d ) , η), there exists an n 0 and an f 0 = f n 0 supported on a dyadic shell B k for some k such that where we denote by ǫ 0 = 1 2 η q L (r−2)q/r 0 . Then from (3.5) we have | f 0 | ≤ C ǫ −ν 2 − kd 2 . By Lemma 3.2, there is a L 1 = L 1 ( f 0 L 2 , η) and a set of regions {Q n } 1≤n≤L 1 defined by where C n is a cube of side length l(C n ) = 2 −k and I n is an interval of length |I n | = 2 −kα such that Then by Hölder's inequality with 2 r + r−2 r = 1 repeatedly, we have Thus from (3.10) it follows that This implies that there is a region Q 0 ∈ {Q n } L 1 n=1 such that where we set Since | f 0 | ≤ C 2 − kd 2 and f 0 is supported in a dyadic shell of measure 2 kd , we have where we use dq(r − 2)/r = 2α and |I 0 | = 2 −kα . Thus we have (3.12) and in view of (3.11) Thus we find the lower bound We divide the integral in the left hand side of (3.11) into two integrals such that By (3.12), similarly we can choose A small enough so that In view of this and (3.11), we obtain that The inequality (3.12) leads to us that Hence we obtain that Thus, for each t n there are t 0 ∈ (t n , t n+1 − A |I 0 |ǫ 2 ] and a cube Q t 0 0 such that Hence Q t 0 0 can be covered by a finite number (depending on η, d and u 0 2 ) of balls of radius r = (T * − t 0 ) 1 α . Therefore, there exists x 0 ∈ R d such that where ε is ε ( u 0 L 2 (R d ) , d, η) and independent of t n . This completes the proof.
Proof of Theorem 1.2. We proceed as in proof of Theorem 1.1. Let u be the maximal solution to (1.7) over the maximal forward existence time interval [0, T * ) so that (1.5) holds for some Strichartz admissible pairs (q, r) satisfying (1.9), and u L q Let η and sequence t 1 , . . . , t n , . . . be given as before such that t n ր T * and (3.3) is satisfied for every n ∈ N. By the Duhamel's formula we may write for t ∈ (0, T * ) That is to say, for the solution u of (1.7) there is a constant C > 0 such that for (q, r) satisfying (1.9) and for some 0 < θ < 1. We note that the inequality above is obtained by repeating the local wellposement argument. See the argument around (1.8) † in Section 1. After achieving this we only need to deal with the homogeneous part of the solution to show the mass concentration. Hence, the remaining parts are the same as those for Theorem 1.1. We omit the details.

Appendix
To prove Lemma 3.1 and Lemma 3.2 we modify Bourgain's arguments in [3] (also see [1]) for the Schrödinger operator of higher orders α with α > 2. The proof of Proof of Lemma 3.1. From (3.1) and Proposition 2.3, we see that there are 0 < θ < 1 and p < 2 such that So there exists a dyadic shell B n 1 for some n 1 such that Applying Lemma 2.4 to f and B n 1 , we have .
We now define f 1 by f 1 = f λ Bn 1 and insert where the first inequality follows from f 2 = f 1 2 + f −f 1 2 . The L 2 orthogonality holds as well, f − f 1 Recursively we can find f k supported on B n k in the frequency space for k = 1, 2, . . . , N such that This process will stop within a finite number of steps. The number of steps depends on ǫ and f L 2 because This completes the proof.
Proof of Lemma 3.2. We follow closely the argument for the proof Lemma 3.3 in [1]. Let g ′ ∈ L 2 (R d ) be the normalized function of g defined by g ′ (ξ ′ ) = 2 kd 2 g(2 k ξ ′ ).
We will keep track of the free evolution of g ′ . Let E ⊂ R × R d be the set Since α > 2, we now note that the α-admissible line is properly contained in the region of 2/q + d/r ≤ d/2. Hence, we can pick up a pair (q * , r * ) in the region d r * + 2 q * ≤ d 2 such that q * < q, r * < r and r * /q * = r/q. Such choice may not be possible for the end point 1 r = d−α 2d but it was excluded because we are assuming q > 2 and r = ∞ (see (1.4)). Then for α-admissible (q, r), (2.4) yields where the second inequality follows from the fact that supp g ′ ⊂ B 1 . Since r * < r, by choosing λ = λ(C 0 , ǫ) small enough, we have Due to the normalization, supp g ′ ⊂ B 1 and g ′ L ∞ ≤ C 0 . Hence the function (x, t) → e it(−∆) α 2 g ′ (x) is smooth with bounded derivatives. In particular, the map is Lipschitz. That is, where C = C(C 0 , d) ≥ 1. Hence, if (t ′ , x ′ ) ∈ E and |x ′ − x ′′ |, |t ′ − t ′′ | ≤ λ 2C < 1 2 , then (t ′′ , x ′′ ) is in E. In other words, for (t ′ , x ′ ) ∈ (R × R d )\Ẽ, there is a space-time cube P = J ×K centered at (t ′ , x ′ ) with |J| = λ C and l (K ) = λ C such that P ∈ (R×R d )\E.
Let us cover (R × R d )\Ẽ with the family of (P r ) r∈I such that Int(P r ) ∩ Int(P s ) = ∅ for r = s, and where Int(P r ) denotes the interior of the set P r . Note that the index set I is finite. We set N 1 = ♯I. It follows from (4.3) and the Strichartz's estimate that from which we deduce that N 1 ≤ C( g L 2 , d, C 0 , ǫ). Actually, since our hypothesis implies that g L 2 ≤ C 0 , we can also write N 1 ≤ C(d, C 0 , ǫ). For simplicity let {1, . . . , N 1 } denote the index set I. For any integer 1 ≤ n ≤ N 1 , let (t n , x n ) be the center of P n and let I n ⊂ R be the interval of center tn 2 kα with |I n | = 1 2 kα . Also set I ′ n = 2 kα I n . Let C n ∈ C of center 2 −k x n with ℓ(C n ) = 2 −k and let C ′ n = 2 k C n . Finally let Q n be defined by (3.2). Then from the choice of λ it follows that By (4.2) and reversing the change of variables (t ′ , x ′ ) → (t, x), we have < ǫ q since (q, r) is admissible. This concludes the proof of the lemma.