Improved almost Morawetz estimates for the cubic nonlinear Schrodinger equation

We prove global well-posedness for the cubic, defocusing, nonlinear Schr{\"o}dinger equation on $\mathbf{R}^{2}$ with data $u_{0} \in H^{s}(\mathbf{R}^{2})$, $s>1/4$. We accomplish this by improving the almost Morawetz estimates in [9].


Introduction
The cubic, defocusing, nonlinear Schrödinger equation on R 2 , iu t + ∆u = |u| 2 u, u(0, x) = u 0 (x) ∈ H s (R 2 ), (1.1) has been the subject of a great deal of research in recent years. It was proved in [4] that for any s > 0, ( (1. 2) The first progress to proving the existence of a global solution was proved in [3]. The reader will notice there is a gap between the regularity necessary to prove local well-posedness (s > 0), [4] and the regularity needed in Theorem 1.1 to prove a global solution, [3]. Many have undertaken to close this gap. The first progress was made in [2]. In this case the method of proof was the Fourier truncation method. Take φ(ξ) ∈ C ∞ 0 , φ(ξ) = 1, |ξ| ≤ 1; 0, |ξ| > 2.
Then split the initial data into low frequency and high frequency components.û Since u l H 1 N 1−s u 0 H s , the equation has a global solution with Also, if s > 3/5, the equation has a solution on [0, T ] of the form This approach was modified in [6] to produce the I-method. The I-operator, is the smooth, radial Fourier multiplier From this point on, we will understand that I refers to the I -operator I N .
was a conserved quantity then the existence of a global solution would follow for any s > 0. This is not the case, however. Instead, it was proved in [6] that (1.11) This implies global well-posedness for u 0 ∈ H s (R 2 ), s > 4/7. Subsequent papers (see [8], [5], [9]) have decreased the necessary regularity to This was proved by combining the I-method, a modified energy functional, and almost Morawetz estimates. The method will be described in more detail in the subsequent sections. In addition, the almost Morawetz estimates will be improved, thus improving Theorem 1.4 to In §2 the modified energy functional of [8] will be recalled, as well as a modified local well-posedness theorem. In §3, the Morawetz inequality for u(t, x) will be proved (originally proved in [5]), (1.12) In §4, the known almost-Morawetz estimate in [9] for Iu(t, x) will be improved. Finally, in §5, this improvement will be used to prove Theorem 1.5.
In [8], the authors proved the existence of a modified energy functional E(u(t)) satisfying the properties: 1.Ẽ(u(t)) has a slower variation than E(Iu(t)).

Proposition 2.2
There exists a modified energy functionalẼ satisfying the fixed time estimate, Proposition 2.3Ẽ(u(t)) has the energy increment for a time interval J, Proof: See §7 and §8 of [8].
The X 1,1/2+ norm will not be defined in this paper, because it will not be needed.

Morawetz inequalities
In this section we will recall the proof of the following Morawetz inequality from [5]. This recollection will be useful for the arguments given in the next section.
Proof: Suppose that v(t, z) solves the partial differential equation Then define the quantities These quantities obey the relation, Let v(t, z) be a tensor product of solutions to (1.1) on R 2 × R 2 , Define the Morawetz action, following the convention that repeated indices are summed.
Since M will be large, |∇a(z)| is uniformly bounded on R 2 × R 2 , and (3.14) The proof will be complete once we prove (3.11) and (3.12) are positive.
gives a positive definite matrix for all z ∈ R 2 × R 2 if a(z) = f (|x − y|). Proof: Take the inner product defined by this matrix.
This proves the lemma.

Almost Morawetz Inequalities
In this section, the almost Morawetz estimate in [5], [9] will be improved.
where J k is a partition of [0, T ].
Proof: Split the nonlinearity (4.4) After taking a tensor product of solutions v(t, z) = Iu(t, x)Iu(t, y), repeat the procedure from §3 to obtain Once again, the second term 8 Re(∂ j v(t, z)∂ k v(t, z))dz is strictly positive and can be discarded, as well as the parts of the third term with N g in place of F. Therefore it suffices to handle terms of the form as well as terms of the form Integrating by parts in x, (4.8) is a sum of terms of the form (4.7), along with terms of the form (4.7) will be tackled first.
is a sum of terms of the form This implies (4.10) (4.11) The quantity can be estimated by making a Littlewood-Paley partition of u(t, x). Define a quantity F (t, ξ) Supposeû(t, ξ i ) is supported on the frequency region |ξ i | ∼ N i , and without loss of generality suppose N 1 ≥ N 2 ≥ N 3 . Consider four regions separately.
In this case, make the trivial multiplier estimate, This uses the fact that m(ξ)ξ is monotone increasing for any s > 0 and m(N )N = N . Therefore, Finally, consider the region N N 3 ≤ N 2 ≤ N 1 : Doing the same analysis, This proves the proposition for terms of the form (4.7).
(4.14) 6 . This completes the proof of the proposition.

Proof of Theorem 1.5
Fix a time interval [0, T 0 ]. We wish to show that (1.1) has a solution on that time interval. If u(t, x) is a solution on [0, T ] then is a solution on [0, λ 2 T ]. Let u 0,λ denote the rescaled solution at t = 0, and let u λ (t) be the rescaled solution.