Bootstrapped Morawetz Estimates And Resonant Decomposition For Low Regularity Global Solutions Of Cubic NLS On R^{2}

We prove global well-posedness for the L^{2}-critical cubic defocusing nonlinear Schr\"odinger equation on R^{2} with data u_{0} \in H^{s}(R^{2}) for s>{1/3}.


Introduction
We shall consider the L 2 -critical Schrödinger equation on R 2 . This problem is known to be locally well-posed [3] for any s ≥ 0. If s > 0 then local well-posedness means that for any data u 0 ∈ H s (R 2 ), there exists a time of local existence T l = T l ( u 0 H s (R 2 ) ) depending only on the norm of the initial data and a unique solution u lying in a Banach space X ⊂ C [0, T l ], H s (R 2 ) such that u(t) satisfies for t ∈ [0, T l ] the Duhamel formula (1.6) E (u(t)) := 1 2 R 2 |∇u(t, x)| 2 dx + 1 4 R 2 |u(t, x)| 4 dx.
In this paper we are interested in proving that H s -solutions to (1.1) with s ≥ 0 exist for all time T ≥ 0. If s > 0 then in view of the local well-posedness theory it suffices to find an a priori bound of the form with Q a function depending only on the norm of the initial data and time T . If s = 1 then the energy conservation law immediately yields the bound (1. 7). No blowup solutions are known for (1.1). It is conjectured that (1.1) is globally well-posed in H s (R 2 ) for 1 > s ≥ 0. The first breakthrough to establish global well-posedness below the energy threshold, by using what is now referred to as the Fourier truncation method, appears in [1]. He showed global well-posedness for data in H s (R 2 ) with s > 3 5 . A sequence of works ( [5,6,7,4]) has lowered the regularity requirements for global well-posedness for (1.1) down to s > 2 5 . Recently, the conjecture was proved in [8], in the case of spherically symmetric initial data. The main result of this paper is the following improvement: Theorem 1.1. The L 2 -critical Schrödinger equation on R 2 is globally well-posed in H s (R 2 ), 1 > s > 1 3 . Moreover there exists a constant C depending only on u 0 H s (R 2 ) such that for all times T .
Before sketching the main ideas underpinning this theorem, we set up some notation. Given A, B two nonnegative numbers, A B means that there exists a universal nonnegative constant K such that A ≤ KB. We say that K 0 is the constant determined by the relation A B if K 0 is the smallest K such that A ≤ KB is true. We write A ∼ B when A B and B A. A << B denotes A ≤ KB for some universal constant K < 1 100 . We also use the notations A+ = A + ǫ, A + + = A + 2ǫ, A− = A − ǫ and A − − = A − 2ǫ, etc. for some universal constant 0 < ǫ ≪ 1. We shall abuse the notation and write +, − for 0+, 0− respectively.
Let λ ∈ R and let u λ denote the following function (1.9) u λ (t, x) := 1 λ u t λ 2 , x λ . We recall that if u satisfies (1.1) with data u 0 then u λ also satisfies (1.1) but with data 1 λ u 0 x λ . If J := [a, b] is an interval then |J| is its size. A partition P µ (J) = (J i ) i∈{1,...,l} of a finite interval J is of size µ, µ > 0 if three conditions are satisfied If u if a solution of (1.1) on J then we can write u as the sum of its linear part and its nonlinear part; more precisely Let dt denote the standard Lebesgue measure and let dµ l 1 be the following measure (1.13) dµ l := δ(t − a)dt.
If u 1 = ... = u 2p = u then we abbreviate Λ 2p (σ; u) := Λ 2p (σ; u 1 , .., u 2p ) and Λ 2p,J (σ; u) := Λ 2p,J (σ; u 1 , .., u 2p ). Let Ω 1→2p k denote the set of unordered subsets of size k from the set {0, ..., 2p}. [1,..,4] , |ξ j | ≤ N 100 and (1.20) Γ := (ξ j ) j∈ [1,..,4] , |cos (ξ 12 , ξ 14 )| ≥ θ where 0 < θ/ll1 is a parameter to be determined. Here we use the convention ξ ab := ξ a + ξ b , ξ abc := ξ a + ξ b + ξ c , etc. We constantly use the Imethod [5] throughout this paper in order to find a pointwise-in-time upper bound of the H s -norm of the solution to (1.1) with data u(0) = u 0 ∈ H s (R 2 ). We recall it now. Let I be the following multiplier where m(ξ) := η ξ N , η is a smooth, radial, nonincreasing in |ξ| such that (1.22) η(ξ) := 1, |ξ| ≤ 1 1 |ξ| 1−s , |ξ| ≥ 2 1 µ l since this measure will be applied to be the linear part u l J of u By plugging the multiplier I into the energy conservation law (1.6) we define the so-called modified energy x)| 4 dx. The following proposition [5] shows that it suffices to estimate the modified energy at time T in order to find an upper bound of the pointwise-in-time H s -norm of the solution u to (1.1) with data u(0) = u 0 ∈ H s (R 2 ); more precisely Proposition 1.2 (H s (R 2 ) norm and modified energy are comparable [5]). For all time T ≥ 0 Since the symbol of I approaches one as N goes to infinity we expect the variation of the modified energy to be slower and slower as N increases. Therefore we estimate the modified energy by using the fundamental theorem of calculus and we use Proposition 1.2 to control u(T ) H s (R 2 ) .
The paper is organized as follows: In Section 2 we recall the main ideas of [6]. In particular, we explain their construction of a new almost conservation lawẼ(u(t)) which is close to the modified energy E(Iu(t)) at each time t and how they estimated the variation ofẼ(u(t)) on an interval of size, roughly speaking, equal to one. In Section 3 we recall the main results of [4] and, in particular, the Morawetz-type estimates. We would like to combine the ideas from [6] with those from [4]. However there are two non-trivial difficulties that appear.
The I-method is based upon an estimate of the variation of an almost conservation law on a small interval where we have a control of a large number of norms. Then the variation of the almost conservation law on an arbitrary large time interval [0, T ] is estimated by iteration on each subinterval of a partition of [0, T ] where this local control holds. This total variation must be controlled at the end of the process. Therefore, if we can establish a local control on a subinterval as large as possible then the number of iterations is reduced and we have a better control of the total variation, which implies global well-posedness for rougher data. Unfortunately we cannot use the result established in [6] (see Proposition 2.2) to estimate the variation ofẼ(u(t)) since the local control of the solution in X s,b spaces is only true for short time intervals (see Proposition 2.4). This is due to the nature of these spaces: they describe very well the solution locally in time but not on long time intervals. Proposition 3.1 shows that we have a local control on intervals J where the L 4 t L 4 x norm of Iu is small. The first idea would be to divide [0, T ] into subintervals J where the L 4 t L 4 x norm of Iu is small. Indeed their size is expected to be, roughly speaking, larger than one, because the Morawetz-type estimates provide good control of the L 4 t L 4 x norm of Iu on [0, T ]. In Section 5 we estimate the variation ofẼ on J. The proof has similarities with that of (2.10) but there are differences in the method. We write the variation ofẼ on J in the spacetime Fourier domain, we decompose u into the sum of its linear part u l J and its nonlinear part u nl J , and after some measure rearrangements performed via the use of Fubini's theorem, we use some refined bilinear estimates [1,6]. These estimates are key estimates to get a slow increase ofẼ. At the end of the process we can bound the variation ofẼ by some quantities that are estimated by the local control theory in turn.
Unfortunately, if we use the Morawetz-type estimate on the whole [0, T ] then an error term appears and, as time T goes to infinity, it grows at a faster rate than that generated by the variation of the modified energy on the same interval. The control of the error term is possible if and only if s > 2 5 (see [4]). We would like to use the Morawetz-type estimates in a better way. To this end we perform a double layer in time decomposition. First we divide [0, T ] into subintervals J of size, roughly speaking, equal to N 3− . This enables us to control the error term of the Morawetz estimate on J by its main term. Then we decompose each J into subintervals J k where the L 4 t L 4 x norm of Iu is small. By applying the local control theory and the almost conservation law in L q t L r x spaces (see Proposition 4.1) we can estimate the variation ofẼ on J k and then on J by iteration. The final step is to bootstrap the Morawetz estimates. More precisely, we use for every J the corresponding Morawetz estimate and we iterate to estimate the variation ofẼ on the whole interval [0, T ]. At the end of the proof we can control the modified energy on [0, T ], provided that s > 1 3 . The whole process is explained in Section 4.
Acknowledgements : We would like to thank T. Tao and N. Tzirakis for discussions related to this work.
2. Summary of [6] In this section we recall the main ideas and results of [6] since we will often refer to them throughout this paper.
The variation of the modified energy E(Iu(t)) is not equal to zero, because of the presence of the commutator Iu 3 − (Iu) 3 . The control of this variation is possible if the Sobolev exponent s is larger than a threshold s 0 . This control implies global existence for data in H s (R 2 ), s > s 0 . In [6] the authors aimed at designing a new almost conservation lawẼ(u(t)) that would satisfy two properties (1) Almost conservation law:Ẽ(u(t)) would have a slower variation than E(Iu(t)) (2) Proximity to E(Iu(t)) at each time t: this property would allow to control E( To this end they searched for a candidateẼ that would have the following form ) with σ 2 denoting the following multiplier and σ 4 to be determined. Notice that Λ 2 (σ 2 )(u(t)) is nothing else but the kinetic part of the modified energy, i.e Λ 2 (σ 2 )(u(t)) = 1 2 Iu(t) 2Ḣ 1 . The idea is to substitute the potential term V (t) := 1 4 R 2 |Iu(t, x)| 4 dx of the modified energy E(Iu) for a new quadrilinear term Λ 4 (σ 4 ; u(t)) and to search for some cancellations in the computation of the derivative ∂ tẼ (u(t)). If we compute the derivative of Λ 2 (σ 2 ; u(t)) and Λ 4 (σ 4 ; u(t)) then we find, by using (1.1) The authors tried to cancel the quadrilinear terms resulting from the derivative of E(u(t)) by letting σ 4 := − µ α4 . The problem is that the singularity α 4 = 0 appears. Therefore they had to truncate optimally σ 4 away from α 4 = 0 so that the truncation does not totally lose the effect of these cancellations. This requires a detailed study of the singularity. Recall that the corrective term Λ 4 (σ 4 ; u(t)) is a quadrilinear integral evaluated on the convolution surface ξ 1 + ... + ξ 4 = 0. They observed that α 4 = 2iξ 12 ξ 14 cos (ξ 12 , ξ 14 ) on this surface and that if |ξ i | ≪ N , i ∈ {1, .., 4} then the singularity disappears and α 4 = 1 4 2 . Therefore they truncated − µ α4 in the following way With this value for σ 4 , E(u(t)) is well-defined by (2.1). They showed thatẼ(u(t)) and E(Iu(t)) are closed to each other at each time t; more precisely Then, by using a delicate mutilinear analysis, they proved the following result The definition of the X 1, 1 2 + spaces can be found in [2] for example. The proof of Proposition 2.2 extensively relies upon two refined bilinear estimates Proposition 2.3 (Bilinear estimates [1], [6]). Let f , g be two Schwartz functions. Let N 1 , N 2 be two dyadic numbers such that N 1 < N 2 . Let θ be a parameter such that 0 < θ ≪ 1. If The same conclusions hold if −|ξ 1 | 2 − |ξ 2 | 2 is substituted for |ξ 1 | 2 + |ξ 2 | 2 in (2.11) and (2.12).
Proposition 2.2 shows that we can estimate the variation of E(u(t)) on an interval J provided that we can control the X 1, 1 2 + norm of Iu. The next proposition shows that such a control is possible as long as the size of J is, roughly speaking, bounded by one Proposition 2.4 (Modified Local Well-Posedness in X s,b spaces [6]). There exists 1 < ǫ 1 such that if sup t∈J E(Iu(t)) 1 and |J| ≤ ǫ then Finally, by choosing the optimal parameter θ = 1 N , they estimated the variation of the almost conservation lawẼ on an interval J of size one The variation is slower than that of the modified energy.
increase for the variation of the modified energy [5].
3. Summary of [4] In this section we recall two results from [4] that we use in the proof of Theorem 1.1. The first one shows that if we the L 4 t L 4 x norm of a solution to (1.1 ) is small then we control several norms. This result will be extensively used in establishing the almost conservation law: see Proposition 4.1.
Proposition 3.1 (Modified Local Well-Posedness [4] ). Let u be a solution to ( 1.1 ). Assume that (q, r) is admissible, i.e (q, r) ∈ (2, ∞] × [2, ∞) and 1 q + 1 r = 1 2 . Assume also that Then there exists 0 < ǫ ≪ 1 such that if This inequality results from the two following estimates In this section we prove the global existence of (1.1) in H s (R 2 ) × H s−1 (R 2 ), 1 > s > 1 3 . Our proof relies on an intermediate result that we prove in the next sections. More precisely we shall show the following  (1) Scaling. We recall (see [5])that there exists C 0 : with u λ defined in (1.9). (2) Bootstrap. Let F T denote the following set with λ defined in (4.2) and C mor the constant determined by in (3.5). We claim that F T is the whole set [0, T ] for N = N (T, u 0 H s (R 2 ) ) ≫ 1 to be chosen later. Indeed • F T = ∅ since 0 ∈ F T by (4.3) • F T is closed by the dominated convergence theorem.
• F T is open. Let T ′ ∈ F T . Then by continuity there exists δ > 0 such that for every .
Let C f ix be the constant determined by in (2.9). Let θ = 1 N . Then by Proposition 2.1, (4.3) and the triangle inequality we have Now we iterate again to cover [0, λ 2 T ′ ]. The number of intervals J is bounded by λ 2 T N 3− . Therefore by this observation, (4.7) and (4.9) we have By (4.10) and Proposition 2.1 we have Let C tot be the constant determined by in (4.11). Since s > 1 3 then for every T > 0 we can always choose N = N (T ) ≫ 1 such that C tot It remains to prove Iu λ 4 We get from Proposition 3.2, (4.8) and the elementary inequality Accounting. Following the Imethod described in [5] (4.14) sup t∈[0, T ] E (Iu(t)) λ 2 sup t∈[0, λ 2 T ] E(Iu λ (t)) λ 2 By Proposition 1.2 we have global well-posedness of the defocusing cubic Schrödinger equa- we have N ∼ T s 6s−2 + . Plugging this value of N into (4.14) and using (1.24) we obtain (1.8).
Then we need the following lemma Let us postpone the proof of this lemma to later and let us assume that it is true for the moment. Then we have (5.9) by the fractional Leibnitz rule and by Hölder inequality. Moreover This proves the quadrilinear estimate (5.2).
We have We perform a Paley-Littlewood decomposition to prove (5.17). Let X denote the left-hand side of (5.17) after decomposition. By Plancherel's theorem and we want to prove Since the L 2 -norm only depends on the magnitude of the Fourier transform we may assume that w(t j ) ≥ 0, j ∈ {1, ..., 4}. There are two cases: • Case 1: N 3 ≫ N 4 . Recall (see [6]) that There are two subcases -Case 1.a: θ N4
Observe also that (5.52) x by Plancherel and Bernstein inequalities. By (5.43), (5.44), (5.52), the claim and Haussdorf-Young we have D Iw j (t j ) L 2 x with q being the logarithmic weight introduced in (5.23).