11institutetext: Tecnológico de Monterrey, School of Engineering and Sciences,
Mexico. 22institutetext: Universidad de las Americas Puebla, Department of
Chemical, Food and Environmental Engineering, Puebla, 72810, Mexico.
33institutetext: Universitat Politècnica de Catalunya. EEBE, Eduard Maristany
16, 08019 Barcelona. Catalonia, Spain.

# Comparing Machine Learning based Segmentation Models on Jet Fire Radiation
Zones††thanks: Currently under review for the Mexican Conference on AI (MICAI
2021)

Carmina Pérez-Guerrero 11 Adriana Palacios 22 Gilberto Ochoa-Ruiz 11 Christian
Mata 33 Miguel Gonzalez-Mendoza 11 Luis Eduardo Falcón-Morales 11

###### Abstract

Risk assessment is relevant in any workplace, however there is a degree of
unpredictability when dealing with flammable or hazardous materials so that
detection of fire accidents by itself may not be enough. An example of this is
the impingement of jet fires, where the heat fluxes of the flame could reach
nearby equipment and dramatically increase the probability of a domino effect
with catastrophic results. Because of this, the characterization of such fire
accidents is important from a risk management point of view. One such
characterization would be the segmentation of different radiation zones within
the flame, so this paper presents an exploratory research regarding several
traditional computer vision and Deep Learning segmentation approaches to solve
this specific problem. A data set of propane jet fires is used to train and
evaluate the different approaches and given the difference in the distribution
of the zones and background of the images, different loss functions, that seek
to alleviate data imbalance, are also explored. Additionally, different
metrics are correlated to a manual ranking performed by experts to make an
evaluation that closely resembles the expert’s criteria. The Hausdorff
Distance and Adjusted Random Index were the metrics with the highest
correlation and the best results were obtained from the UNet architecture with
a Weighted Cross-Entropy Loss. These results can be used in future research to
extract more geometric information from the segmentation masks or could even
be implemented on other types of fire accidents.

###### Keywords:

semantic segmentation deep learning computer vision jet fires.

##  1 Introduction

In the industrial environment there are certain activities such as the storage
of fuels or the transportation of hazardous material, that can be involved in
severe accidents that affect the industrial plant or activity border, as well
as external factors like human health, environmental damage, property damage,
and others. An overall knowledge of the features and characteristics of major
accidents is required to prevent and manage them or, in the worst case
scenario, take action to reduce and control their severity and aftermath.
There are some accidents that are relatively well known and researched,
however, there are other accidents that have not been broadly explored.

Sometimes the detection of fire, in an industrial setting, is not enough to
make the correct decisions when managing certain fire accidents. An example of
this, is the impingement of jet flames on nearby pipes, depending on the heat
fluxes of the flame, the pipe wall could heat up quickly and reach dangerous
temperatures that lead to severe domino effect sequences. These heat fluxes
can be defined as three main areas of interest within the flame [6], and the
definition of their geometric characteristics and localization becomes
valuable information in risk management.

Semantic segmentation could be an useful approach for this kind of flame
characterization, and the same procedure could even be employed in other fire
related accidents. To explore this proposal, different segmentation
methodologies are evaluated on a set of images from real jet fires of propane
to accurately segment the radiation zones within the flames. This different
zones are illustrated in Figure 1 and are defined across this paper as the
Central Zone, the Middle Zone and the Outer Zone.

The rest of this paper is organized as follows. Section 2 describes the
previous work present in the literature related to the problem and different
semantic segmentation methods. Section 3 describes the approaches used to
perform the exploration research. Section 4 explains the data set used for the
experiments and the pre-processing methods applied. Section 5 describes the
different evaluation metrics and loss functions explored during the
experiments. Section 6 contains the training protocol used for the experiments
with Deep Learning architectures. Section 7 explains the testing procedure
applied to the segmentation methods. Section 8 presents the results of the
exploration research. This section also offers a discussion of the future work
that can stem from the knowledge obtained. Finally, Section 9 summarizes the
major findings of the presented work.

(a)

(b)

Figure 1: Image (a) is an infrared visualization of an horizontal propane jet
flame. Image (b) is the corresponding ground truth segmentation, with the
segment names indicated. Modified from [6].

##  2 State of the Art

There has been previous research work regarding the use of Computer Vision and
image processing for the detection and monitoring of flares in the context of
industrial security. For example, Rodrigues and Yan [13] used imaging sensors
combined with digital image processing to determine the size, shape and region
of the flare, successfully characterizing the dynamism of the fire.

Another example is the work done in Janssen and Sepasian [7], where, by
separating the flare form the background using temperature thresholding, and
adding false colors to represent different temperature regions, a system was
created, capable of tracking the flare size for automated event signaling.

###  2.1 Deep Learning Architectures

Deep learning algorithms, such as Convolutional Neural Networks, have shown
outstanding performance in many complex tasks, such as image recognition,
object detection, and semantic and instance segmentation [9]. Advancements on
these methods have been increasing rapidly, with constant research being
published regarding better and more robust algorithms. Some important
architectures present in the literature are summarized in Table 1 with their
advantages and disadvantages described.

Table 1: Summary of the selected architectures. Source | Architecture | Pros | Cons
---|---|---|---
[4] | DeepLabv3 |  | Recovers detailed structures
---
lost due to spatial invariance.
Efficient approximations via
probabilistic inference.
Wider receptive fields.
| Low accuracy on small
---
scaled objects.
Has trouble capturing
delicate boundaries of objects.
[2] | SegNet |  | Efficient inference in terms of memory
---
and computational time.
Small number of trainable parameters.
Improved boundary delineation.
| Input image must be fixed.
---
Performance drops with a
large number of classes and
is sensitive to class imbalance.
[14] | UNet |  | Can handle inputs of arbitrary sizes.
---
Smaller model weight size.
Precise localization of regions.
| Size of network comparable
---
to the size of features.
Significant amount of time
to train.
High GPU memory footprint
in larger images.
[10] |  | Attention
---
UNet
| Avoids the use of multiple similar
---
feature maps.
Focuses on the most informative features
without additional supervision.
Enhances the results of the UNet architecture.
| Adds more weight parameters
---
The time for training increases,
especially for long sequences

In general, these architectures were selected due to their efficiency during
segmentation and their capacity to accurately portray the shape of such
dynamic figures obtained from the flames.

###  2.2 Traditional Computer Vision Methods

Different traditional segmentation methods are included in the analysis as a
baseline for the results given by the Deep Learning architectures, these
methods are: Gaussian Mixture Model (GMM), K-means clustering, Thresholding,
and Chan-Vese segmentation.

  * •

GMM is one of the main methods applied to fire and smoke image segmentation.
It offers a clear definition of their dynamic shape, but sometimes missed
pixels from the inner parts of the fire and smoke [1].

  * •

K-means clustering provides shape-based image segmentation [18] and has been
previously used for fire segmentation [15] [1]. However, this method does not
guarantee continuous areas [18]

  * •

Thresholding is the simplest method of image segmentation and has a fast
operation speed. It is most commonly used in region-based segmentation, but it
is sensitive to noise and gray scale unevenness [17].

  * •

The Chan-Vese segmentation [3] belongs to the group of active contour models,
which present some advantages for infrared image segmentation, since the edges
obtained are smooth and are represented with closed curves. However, this kind
of segmentation is very sensitive to noise and depends heavily on the location
of the initial contour, so it needs to be manually placed near the image of
interest [19].

##  3 Proposed Approach

To explore the semantic segmentation of radiation zones within the flames as
characterization of fire incidents, a group of 4 traditional segmentation
methods and 4 deep learning architectures are explored. These segmentation
methods are trained and tested using a data set of jet fire images obtained
from videos of an experiment performed in an open field.

To properly evaluate and compare the different segmentation approaches,
several metrics with different evaluation methods are correlated to manual
rankings performed by two experts in the field, this is to make sure that the
evaluation is the most representative of a fire engineer’s perception of good
segmentation.

The best model found from this exploratory research will then be used in
future work, not included in this paper, to extract other geometric
characteristics from the resulting segmentation masks.

##  4 Data set

Investigators from Universitat Politècnica de Catalunya performed an
experiment to produce horizontal jet fires at subsonic and sonic gas exit
rates. The experiment was filmed using an infrared thermographic camera, more
specifically, an AGEMA 570 by Flir Systems. The video was saved in four frames
per second, resulting in a total of 201 images with a resolution of 640 x 480
pixels. After obtaining the infrared visualizations of the flames, a
segmentation was performed of the three radiation zones within the fire, the
results were validated by experts in the field and the result was a ground-
truth for each of the infrared images. The images were saved as Matlab files
that contain a temperature matrix corresponding to the temperature values
detected by the camera for the infrared images, and the a label matrix
corresponding to the segments for the ground-truth segmentations. These files
were then exported as PNG files to be used by the different segmentation
algorithms described in Section 2.1.

###  4.1 Image Processing

To enhance the characteristics of the jet fires represented in the infrared
images, and to reduce their variance, a process of image normalization was
employed, which also helps in the convergence of the Deep Learning methods.
The ground truth images were transformed into labeled images, where a label id
was used instead of the original RGB values. Given the small number of samples
of the data set, and to increase the variability of the input during the
training of the models, data augmentation techniques were applied. These
processing of the images can increase the performance of the models, can avoid
the over-fitting to the training samples, and can help the models to also
perform well for instances that may not be present in the original data set.
Horizontal flipping, random cropping, and random scaling were applied in
parallel of the training workflow, so for each iteration of training, a
different augmented image was inputted. The probability of horizontal flipping
was set to 50%, and random scaling had values that ranged between 0.7 and 2.0.

##  5 Metrics and Loss Functions

###  5.1 Metrics

An analysis was performed to select evaluation metrics that were more
representative of an expert’s evaluation of the segmentation. The metrics
analysed are separated into groups that describe their evaluation method. This
diversity is important because the consideration of particular properties
could prevent the discovery of other particular errors or lead to over or
underestimating them. To compare the metrics enlisted in Table 2, a group of
images were evaluated with the metrics and two manual rankings performed by
two experts in the field. A Pearson pairwise correlation is used to perform
this comparison at segmentation level.

Table 2:  Summary of the metrics analysed in this paper. The ”Group” column describes the method group that the metric belongs to. Based on [16]. Metric | Group
---|---
Jaccard Index | Spatial Overlap Based
F-measure | Spatial Overlap Based
Adjusted Rand Index | Pair Counting Based
Mutual Information | Information Theoretic Based
Cohen’s Kappa | Probabilistic Based
Hausdorff Distance | Spatial Distance Based
Mean Absolute Error | Performance Based
Mean Square Error | Performance Based
Peak Signal to Noise Ratio | Performance Based

###  5.2 Loss Functions

The proportion of the different radiation zones within each flame is
different. The Outer Zone tends to be the largest segment and the central zone
is usually smaller than the Middle Zone, this differences could affect the
overall segmentation obtained during training, which is why the loss functions
employed in this research work were focused on dealing with this class
imbalance. The implemented loss functions are summarized in Table 3.

Table 3:  Summary of the loss functions implemented in this paper. Source | Loss Function | Description
---|---|---
[12] |  | Weighted Cross-Entropy
---
Loss
| Combines Log Softmax and Negative Log Likelihood.
---
Useful for multiple classes that are unbalanced.
[8] | Focal Loss |  | Addresses foreground-background class imbalance.
---
Training is focused on a sparse set of hard examples.
[5] |  | Generalized Wasserstein
---
Dice Loss (GWDL)
| Semantically-informed generalization of the Dice score.
---
Based on the Wasserstein distance on the probabilistic
label space.

##  6 Training

The PyTorch framework [12] was used for the implementation of the Deep
Learning architectures. To maintain the weights of the Convolutional Neural
Networks as small as possible, a weight decay strategy was used with L2
regularization. The learning rate had an initial value of 0.0001 and used an
ADAM optimizer during training. The class weights used by the Weighted Cross-
Entropy and Focal losses were computed according to the ENet custom class
weighing scheme [11] and are defined as the following: 1.59 for background,
10.61 for Outer zone, 17.13 for Middle zone, and 22.25 for Central zone. The
class distances used for the loss function of GWDL is defined as 1 between the
background and the radiation zones, and as 0.5 between the zones themselves.

The training was performed using an Nvidia DGX workstation that has 8 Nvidia
GPUs and allows a batch size of 4 as maximum, which is the batch size used for
the models. The data was split in 80% for training and 20% for testing and
validation, resulting in a total of 161 images for training, 20 images for
testing, and 20 images for validation. The models were trained for up to 5000
epochs with an Early Stopping strategy to avoid over fitting. The resulting
loss values for the best models can be visualized in Fig. 2.

(a) Loss values for the DeepLabv3 model. Early Stopping took place at epoch
701.

(b) Loss values for the SegNet model. Early Stopping took place at epoch 1274.

(c) Loss values for the UNet model. Early Stopping took place at epoch 1460.

(d) Loss values for the Attention UNet model. Early Stopping took place at
epoch 1560.

Figure 2: The Weighted Cross-Entropy loss values for the best models obtained
for each architecture explored.

##  7 Testing

Testing and Validation was performed on 20% of the data set, which represent a
total of 40 images. The results are compared using the metrics with the
highest correlation to the manual ranking of experts. The results for each
Deep Learning architecture are first compared across all 3 different loss
functions mentioned in Section 5.2. The best performing combination, for each
architecture, is then compared to the results obtained from the other 4
traditional segmentation models. The time each method takes to segment the
whole data set of 201 images is also taken into account. The goal of the
comparison is to find the best overall model for the segmentation of the
radiation zones within the flames.

##  8 Results and Discussion

###  8.1 Selected Metrics

The results of the correlation between the metrics, mentioned in Section 5,
and the manual rankings performed by experts can be observed in Fig. 3.

Figure 3: Heatmap representing the Pearson Correlation Coefficient for the
metrics in Section 5 and the manual rankings Rank1 and Rank2 done by experts
in the field of the problem.

The highest Pearson correlation value was given by the Hausdorff Distance,
with a value of -0.32 to the first manual ranking, this negative relationship
is because a smaller Hausdorff Distance is preferable and the manual ranking
assigned higher values as the segmentation improved. The second highest
correlations were given by the Adjusted Rand Index in both manual rankings,
with a value of 0.29 for the first one and 0.27 for the second one, this
positive relationship is because the values of the Adjusted Rand Index go from
0 to 1, with 1 being the best result, and showing a similar behaviour to the
manual ranking that assigns higher values to better segmentations.

###  8.2 Best Loss Function

Each combination of Deep Learning architecture and Loss Function was evaluated
using the Hausdorff Distance as main metric. The best results across all
architectures were obtained when using the Weighted Cross-Entropy Loss, this
can be observed in Fig 4. In general the Focal Loss also showed good results,
being close to the Weighted Cross-Entropy results and most of the times
surpassing the GWDL results, with the only exception being the Attention UNet
model, where the mean Hausdorff Distance with Focal Loss is larger than with
GWDL, but has a much closer distribution. The most dramatic difference
happened with the SegNet architecture, which showed very poor results with
GWDL.

(a) Hausdorff Distance for all the DeepLabv3 models.

(b) Hausdorff Distance for all the SegNet models.

(c) Hausdorff Distance for all the UNet models.

(d) Hausdorff Distance for all the Attention UNet models.

Figure 4: The Hausdorff Distance for all the model and loss function
combinations across the validation set. For each model the order of loss
functions is Weighted Cross-Entropy, Focal and GWDL from left to right.

###  8.3 Traditional and Deep Learning Segmentation

A comparison was done between the traditional Computer Vision methods
mentioned in Section 2.2 and the Deep Learning models mentioned in Section
2.1,using the Weighted Cross-Entropy loss. The Hausdorff Distance and the
Adjusted Rand Index are used to evaluate the models across the testing set and
the time each method takes to perform the segmentation on all the images from
the data set is also taken into account. These results are summarized in Table
4 and the distribution of the results can be visualized in Fig 5. Overall the
best performing models are observed to be the UNet and Attention UNet models.
Even if Attention UNet achieved a slightly better Adjusted Rand Index score,
UNet obtained a better Hausdorff distance and segmentation time, therefore we
can say that the best model is the one that uses the UNet architecture.

Table 4: Mean Hausdorff Distance and Adjusted Rand Index values for all the segmentation models across the testing set, as well as the time in seconds that each method takes to segment the whole data set. The best results are in bold. Method | Hausdorff Distance | Adjusted Rand Index | Time (s)
---|---|---|---
GMM | 1288.10 | 0.9156 | 2723.8
K-means | 1000.63 | 0.8855 | 3035.1
Thresholding | 1029.08 | 0.9152 | 30.7
Chan-Vese | 1031.90 | 0.8568 | 18177.5
DeepLabv3 | 784.86 | 0.9514 | 17.1
SegNet | 692.73 | 0.9381 | 16.4
UNet | 586.46 | 0.9504 | 15.7
Attention UNet | 601.05 | 0.9592 | 17.7

(a) Hausdorff distance for all segmentation models.

(b) Adjusted Rand Index for all segmentation models.

Figure 5: The Hausdorff Distance and Adjusted Rand Index values for all the
segmentation models across the testing set.

The difference in the segmentation of each method can also be visualized in
Fig 6. It can be observed that the shape of the Outer and Middle zones are
generally well represented in most of the segmentation models, however the
most important differences are found in the Central zone, where the Deep
Learning architectures defined more clearly it’s shape. The similar results of
the UNet and Attention UNet architectures are also observed in this sample
segmentation masks, with a really close resemblance to the ground truth.

(a) GMM.

(b) K-means.

(c) Thresholding.

(d) Chan-Vese.

(e) DeepLabv3.

(f) SegNet.

(g) UNet.

(h) Attention UNet.

Figure 6: Sample segmentations of all the models.

###  8.4 Discussion

Overall, the Deep Leaning algorithms greatly outperformed the traditional
Computer Vision methods and the best proposed model would be a UNet
architecture with a Weighted Cross-Entropy loss function. The segmentation
masks obtained from this model could be used in the future to further extract
other geometric characteristics of the flame, such as length, area and lift-
off distance. This additional information would improve greatly the decision
making process involved in the risk assessment and management of fire related
accidents that can take place in an industrial setting. Furthermore, the
metrics of Hausdorff Distance and Adjusted Rand Index can be used to evaluate
other segmentation approaches that may try to solve similar problems in the
future, having the certainty that the evaluation would be a close
representation of an expert’s opinion.

##  9 Conclusions

The semantic segmentation of radiation zones within the flames can be used to
characterize fire accidents, such as jet fires, and the information obtained
from the segmentation can prove to be critical when dealing with risk
management in industrial settings. The exploratory research presented in this
paper continued to show that Deep Learning architectures greatly outperform
other traditional Computer Vision approaches. It was also found that for this
specific problem, the best loss function to train a Deep Learning model is a
Weighted Cross-Entropy Loss, the best architecture to be used is UNet, and the
best evaluation metrics are both the Hausdorff Distance and the Adjusted Rand
Index. All this knowledge can be later used in future research focused on
extracting even more geometric information from the segmentation masks. This
could bring about a more complete characterization analysis of jet fires and
the methods applied to this types of fire accidents could then be used on
other fire scenarios.

#### Acknowledgements

This research is supported in part by the Mexican National Council of Science
and Technology (CONACYT).

This research is part of the project 7817-2019 funded by the Jalisco State
Council of Science and Technology (COECYTJAL).

## References

  * [1] Ajith, M., Martínez-Ramón, M.: Unsupervised segmentation of fire and smoke from infra-red videos. IEEE Access 7, 182381–182394 (2019). https://doi.org/10.1109/ACCESS.2019.2960209
  * [2] Badrinarayanan, V., Kendall, A., Cipolla, R.: Segnet: A deep convolutional encoder-decoder architecture for image segmentation (2016)
  * [3] Chan, T., Vese, L.: Active contours without edges. IEEE Transactions on Image Processing 10(2), 266–277 (2001). https://doi.org/10.1109/83.902291
  * [4] Chen, L.C., Papandreou, G., Schroff, F., Adam, H.: Rethinking atrous convolution for semantic image segmentation (2017)
  * [5] Fidon, L., Li, W., Garcia-Peraza-Herrera, L.C., Ekanayake, J., Kitchen, N., Ourselin, S., Vercauteren, T.: Generalised Wasserstein dice score for imbalanced multi-class segmentation using holistic convolutional networks. In: International MICCAI Brain Lesion Workshop. pp. 64–76. Springer (2017). https://doi.org/10.1007/978-3-319-75238-9_6
  * [6] Foroughi, V., Palacios, A., Barraza, C., Àgueda, A., Mata, C., Pastor, E., Casal, J.: Thermal effects of a sonic jet fire impingement on a pipe. Journal of Loss Prevention in the Process Industries 71, 104449 (2021). https://doi.org/10.1016/j.jlp.2021.104449
  * [7] Janssen, R., Sepasian, N.: Automatic Flare-Stack Monitoring. SPE Production & Operations 34(01), 18–23 (09 2018). https://doi.org/10.2118/187257-PA
  * [8] Lin, T., Goyal, P., Girshick, R., He, K., Dollar, P.: Focal loss for dense object detection. IEEE Transactions on Pattern Analysis & Machine Intelligence 42(02), 318–327 (2020). https://doi.org/10.1109/TPAMI.2018.2858826
  * [9] Litjens, G., Kooi, T., Bejnordi, B.E., Setio, A.A.A., Ciompi, F., Ghafoorian, M., van der Laak, J.A., van Ginneken, B., Sánchez, C.I.: A survey on deep learning in medical image analysis. Medical Image Analysis 42, 60–88 (Dec 2017). https://doi.org/10.1016/j.media.2017.07.005
  * [10] Oktay, O., Schlemper, J., Folgoc, L.L., Lee, M., Heinrich, M., Misawa, K., Mori, K., McDonagh, S., Hammerla, N.Y., Kainz, B., Glocker, B., Rueckert, D.: Attention u-net: Learning where to look for the pancreas (2018)
  * [11] Paszke, A., Chaurasia, A., Kim, S., Culurciello, E.: Enet: A deep neural network architecture for real-time semantic segmentation. CoRR 1606.02147 (2016)
  * [12] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Kopf, A., Yang, E., DeVito, Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., Chintala, S.: Pytorch: An imperative style, high-performance deep learning library. In: Wallach, H., Larochelle, H., Beygelzimer, A., d'Alché-Buc, F., Fox, E., Garnett, R. (eds.) Advances in Neural Information Processing Systems 32, pp. 8024–8035. Curran Associates, Inc. (2019), http://papers.neurips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf
  * [13] Rodrigues, S.J., Yan, Y.: Application of digital imaging techniques to flare monitoring. Journal of Physics: Conference Series 307, 012048 (aug 2011). https://doi.org/10.1088/1742-6596/307/1/012048
  * [14] Ronneberger, O., Fischer, P., Brox, T.: U-net: Convolutional networks for biomedical image segmentation (2015)
  * [15] Rudz, S., Chetehouna, K., Hafiane, A., Laurent, H., Séro-Guillaume, O.: Investigation of a novel image segmentation method dedicated to forest fire applications*. Measurement Science and Technology 24, 075403 (06 2013). https://doi.org/10.1088/0957-0233/24/7/075403
  * [16] Taha, A.A., Hanbury, A.: Metrics for evaluating 3d medical image segmentation: Analysis, selection, and tool. BMC Medical Imaging 15(29) (08 2015). https://doi.org/10.1186/s12880-015-0068-x
  * [17] Yuheng, S., Hao, Y.: Image segmentation algorithms overview (2017)
  * [18] Zaitoun, N., Aqel, M.: Survey on image segmentation techniques. Procedia Computer Science 65, 797–806 (12 2015). https://doi.org/10.1016/j.procs.2015.09.027
  * [19] Zhang, R., Zhu, S., Zhou, Q.: A novel gradient vector flow snake model based on convex function for infrared image segmentation. Sensors 16, 1756 (10 2016). https://doi.org/10.3390/s16101756


# Long time solutions for 1D cubic dispersive equations, Part II: the focusing
case

Mihaela Ifrim  Department of Mathematics, University of Wisconsin, Madison
<EMAIL_ADDRESS>and  Daniel Tataru  Department of Mathematics, University of
California at Berkeley<EMAIL_ADDRESS>
###### Abstract.

This article is concerned with one dimensional dispersive flows with cubic
nonlinearities on the real line. In a very recent work, the authors have
introduced a broad conjecture for such flows, asserting that in the defocusing
case, small initial data yields global, scattering solutions. Then this
conjecture was proved in the case of a Schrödinger dispersion relation. In
terms of scattering, our global solutions were proved to satisfy both global
$L^{6}$ Strichartz estimates and bilinear $L^{2}$ bounds. Notably, no
localization assumption is made on the initial data.

In this article we consider the focusing scenario. There potentially one may
have small solitons, so one cannot hope to have global scattering solutions in
general. Instead, we look for long time solutions, and ask what is the time-
scale on which the solutions exist and satisfy good dispersive estimates. Our
main result, which also applies in the case of the Schrödinger dispersion
relation, asserts that for initial data of size $\epsilon$, the solutions
exist on the time-scale $\epsilon^{-8}$, and satisfy the desired $L^{6}$
Strichartz estimates and bilinear $L^{2}$ bounds on the time-scale
$\epsilon^{-6}$. To the best of our knowledge, this is the first result to
reach such a threshold.

###### Contents

  1. 1 Introduction
  2. 2 Density-flux and interaction Morawetz identities
  3. 3 Strichartz and bilinear $L^{2}$ bounds
  4. 4 Long time energy estimates
  5. 5 Optimality remarks

##  1\. Introduction

The question of obtaining long time solutions for one dimensional dispersive
flows with quadratic/cubic nonlinearities has attracted a lot of attention in
recent years. One can distinguish two different but closely related types of
results that have emerged, as well as several successful approaches.

On one hand, _normal form methods_ have been developed in order to extend the
lifespan of solutions, beginning with [28] in the late ’80’s. Somewhat later,
around 2000, the _I-method_ , introduced in [5] brought forth the idea of
constructing better almost conserved quantities. These two ideas serve well in
the study of semilinear flows, where it was later understood that they are
connected [3].

Neither of these techniques can be directly applied to quasilinear problems.
Addressing this problem, it was discovered in the work of the authors and
collaborators [16], [18] that one can adapt the normal form method to
quasilinear problems by constructing energies which simultaneously capture
both the quasilinear and the normal form structures. This idea was called the
_modified energy method_ , and can also be seen in some way as a quasilinear
adaptation of the I-method. An alternate approach, also in the quasilinear
setting, is provided by the _flow method_ of [15], where a better normal form
transformation is constructed using a well chosen auxiliary flow.

On the other hand, the further goal of obtaining scattering, global in time
solutions for one dimensional dispersive flows with quadratic/cubic
nonlinearities has also been extensively studied in the last two decades for a
number of models, under the assumption that the initial data is both _small_
and _localized_ ; without being exhaustive, see for instance [13, 14, 25, 23,
17]. The nonlinearities in these models are primarily cubic, though the
analysis has also been extended via normal form and modified energy methods to
problems which also have nonresonant quadratic interactions; several such
examples are [1, 18, 10, 19, 24], see also further references therein, as well
as the authors’ expository paper [22].

If instead one considers initial data which is just _small_ , without any
localization assumption, then the problem becomes much more difficult, because
this allows for far stronger nonlinear interactions over long time-scales. One
also needs to distinguish between the focusing and the defocusing problems. In
a recent paper [21], the authors have introduced a broad global well-posedness
(GWP) conjecture, which applies to both semilinear and quasilinear problems:

######  Conjecture 1 (Non-localized data defocusing GWP conjecture).

One dimensional dispersive flows on the real line with cubic defocusing
nonlinearities and small initial data have global in time, scattering
solutions.

The main result of [21] asserts that this conjecture is true under suitable
assumptions, most notably that the dispersion relation is the Schrödinger
dispersion relation. That was the first global in time well-posedness result
of this type. Notably, scattering here is interpreted in a weak sense, to mean
that the solution satisfies global $L^{6}$ Strichartz estimates and bilinear
$L^{2}$ bounds. This is because of the strong nonlinear effects, which
preclude any kind of classical scattering. The precise result is stated later
in Theorem 3.

Our interest in this article is instead in the focusing case of the same
problem. Since 1D focusing dispersive problems typically admit small solitons,
a global result as stated in the above conjecture simply cannot hold. Even if
global solutions exist (as it is the case for instance for the cubic focusing
NLS problem) the presence of solitons will defeat any kind of global decay
estimates. For this reason, in the focusing case we will rethink the problem
as a question about the lifespan of solutions with small initial data.
Precisely, if the initial data has size $\epsilon$ when measured in a suitable
$H^{s}$ Sobolev norm, what can be said about the lifespan of the solutions as
a function of $\epsilon$ ?

Following the lead of our earlier paper, we begin by formulating the focusing
counterpart of the previous conjecture. Then we will prove that the conjecture
is true under suitable assumptions. Our main conjecture is as follows:

######  Conjecture 2 (Non-localized data focusing conjecture).

One dimensional dispersive flows on the real line with cubic nonlinearities
and small initial data of size $\epsilon$ have solutions which remain of
comparable size at least on an $\epsilon^{-8}$ time-scale.

The main result of this paper, see Theorem 1 below, asserts that this
conjecture is valid under the additional assumption that the dispersion
relation is of Schrödinger type. As part of this result, we also prove that
our long time solutions satisfy both $L^{6}$ Strichartz estimates and bilinear
$L^{2}$ bounds on suitable time-scales. This is akin to our earlier work on
the defocusing case, but with the difference that in the defocusing case such
estimates were proved globally in time.

For reference purposes, we note some intermediate lifespan thresholds which
can be reached with methods which were developed earlier:

  * •

A cubic lifespan $\epsilon^{-2}$ can be reached using direct energy estimates,
using only the fact that the nonlinearity is cubic.

  * •

A quintic lifespan $\epsilon^{-4}$ can be reached by more accurate energy
estimates using a quartic energy correction. This requires the cubic
nonlinearity to be conservative, as defined later in the introduction.

  * •

In the case of a perturbative nonlinearity, a quintic lifespan can also be
obtained by directly using Strichartz estimates.

Heuristically, if it were possible to directly combine the last two ideas
above, that would lead to the $\epsilon^{-8}$ threshold in the present paper.
However, the price to achieve that would be very steep, as one would need to
assume both a perturbative nonlinearity and high regularity. By comparison,
our new result provides a much more robust approach, which is both of a
nonperturbative nature and far more efficient in terms of the regularity
requirements. We show here how dispersive and normal form tools can be
combined very efficiently in order to make substantial gains.

In the present paper we aim for a reasonably simple setting, where our model
problem is borderline semilinear, and where we prove results we expect to be
optimal. This should also serve as a baseline for further developments. In
particular, we believe that our methods can be also applied in non-
perturbative, quasilinear settings.

###  1.1. Cubic NLS problems in one space dimension

The fundamental model for one-dimensional dispersive flows with cubic
nonlinearity in one space dimension is the cubic nonlinear Schrödinger (NLS)
flow,

(1.1) $\left\\{\begin{aligned} &iu_{t}+u_{xx}=\pm 2u|u|^{2}\\\
&u(0)=u_{0},\end{aligned}\right.$

with $u:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{C}$. This comes in a
defocusing (+) and a focusing (-) version.

The above cubic NLS flows are globally well-posed in $L^{2}$ both in the
focusing and in the defocusing case, though the global behavior differs in the
two cases. Both of these model problems are completely integrable, and one may
study their global behavior using inverse scattering tools [9], [2].

In the defocusing case, the inverse scattering approach allows one to treat
the case of localized data, and show that global solutions scatter at
infinity, see for instance [9]. This can also be proved in a more robust way,
without using inverse scattering, under the assumption that the initial data
is small and localized, see [17] and references therein. Much less is known in
terms of scattering for nonlocalized $L^{2}$ data. However, if more regularity
is assumed for the data, then we have the following estimate due to Planchon-
Vega [26], see also the work of Colliander-Grillakis-Tzirakis [4]:

(1.2)
$\|u\|_{L^{6}_{t,x}}^{6}+\|\partial_{x}|u|^{2}\|_{L^{2}_{t,x}}^{2}\lesssim\|u_{0}\|_{L^{2}_{x}}^{3}\|u_{0}\|_{H^{1}_{x}}.$

This allows one to estimate the $L^{6}$ Strichartz norm of the solution, i.e.
to prove some type of scattering or dispersive decay. This estimate was
improved and extended to $L^{2}$ solutions as a corollary of the results in
our previous paper [21]. Precisely, we have

(1.3)
$\|u\|_{L^{6}_{t,x}}^{2}+\|\partial_{x}|u|^{2}\|_{L^{2}_{t}(\dot{H}^{-\frac{1}{2}}_{x}+cL^{2}_{x})}\lesssim\|u_{0}\|_{L^{2}_{x}}^{2},\qquad
c=\|u_{0}\|_{L^{2}_{x}}.$

On the other hand, the focusing problem admits small solitons, so the
solutions cannot in general scatter at infinity. If in addition the initial
data is localized, then one expects the solution to resolve into a
superposition of (finitely many) solitons, and a dispersive part; this is
called _the soliton resolution conjecture_ , and is known to hold in a
restrictive setting, via the method of inverse scattering, see e.g. [2].

Our interest here is in focusing problems, but without any integrability
assumptions, and even without assuming any conservation laws. The model we
consider is similar to the one in [21], namely

(1.4) $\left\\{\begin{aligned} &iu_{t}+u_{xx}=C(u,\bar{u},u)\\\
&u(0)=u_{0},\end{aligned}\right.$

where $u$ is a complex valued function,
$u:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{C}$. Here $C$ is a trilinear
translation invariant form, whose symbol $c(\xi_{1},\xi_{2},\xi_{3})$ can
always be assumed to be symmetric in $\xi_{1},\xi_{3}$; see [21] for an
expanded discussion of multilinear forms. The arguments $u,\bar{u}$ and $u$ of
$C$ are chosen so that our equation (1.4) has the phase rotation symmetry,
$u\to ue^{i\theta}$, as it is the case in many examples of interest. The
symbol $c(\xi_{1},\xi_{2},\xi_{3})$ will be required to satisfy the following
set of assumptions, which are similar to [21]:

  1. (H1)

Bounded and regular:

(1.5) $|\partial_{\xi}^{\alpha}c(\xi_{1},\xi_{2},\xi_{3})|\leq
c_{\alpha},\qquad\xi_{1},\xi_{2},\xi_{3}\in\mathbb{R},\,\mbox{ for every
multi-index $\alpha$}.$

  2. (H2)

Conservative:

(1.6) $\Im c(\xi,\xi,\eta)=0,\qquad\xi,\eta\in\mathbb{R},\mbox{ where }\Im
z=\mbox{imaginary part of }z\in\mathbb{C}.$

In addition to these two conditions, in [21] we have also employed a
defocusing assumption, namely

  1. (H3)

Defocusing:

(1.7) $c(\xi,\xi,\xi)\geq c_{0}>0,\qquad\xi\in\mathbb{R}\mbox{ and
}c_{0}\in\mathbb{R^{+}}.$

Here one might think that we should require the opposite, namely

  1. (H4)

Focusing:

(1.8) $-c(\xi,\xi,\xi)\geq c_{0}>0,\qquad\xi\in\mathbb{R}\mbox{ and
}c_{0}\in\mathbb{R^{+}}.$

But as it turns out, no such assumption is needed here, as the result of this
paper applies equally regardless of any sign condition; so we will simply drop
it.

Using the same assumptions (H1), (H2) as in [21] is convenient here because it
will allow us to reuse a good part of the analysis there, up to the point
where the defocusing assumption is needed.

Repeating a similar comment in [21], one should view both our choices of the
Schrödinger dispersion relation and the uniform bounds in (H1) not as
fundamental, but rather as a balance between the generality of the result on
one hand, and a streamlined exposition on the other hand. This choice places
our model in the semilinear class, but just barely so.

The simplest example of such a trilinear form $C$ is of course $C=\pm 1$,
which corresponds to the classical one-dimensional cubic NLS problem. But this
problem has too much structure, in particular it is completely integrable, and
also globally well-posed in $L^{2}$.

At the other end, both our use of the linear Schrödinger operator and the
boundedness condition (H1) are non-optimal, and we hope to relax both of these
restrictions in subsequent work.

###  1.2.  The main result

Our main result asserts that long time well-posedness holds for our problem
for small $L^{2}$ data. In addition, our solutions not only satisfy uniform
$L^{2}$, but also space-time $L^{6}$ Strichartz estimates, as well as bilinear
$L^{2}$ bounds, on appropriate time-scales:

######  Theorem 1.

Consider the problem (1.4) where the cubic nonlinearity $C$ satisfies the
assumptions (H1) and (H2). Assume that the initial data $u_{0}$ is small,

$\|u_{0}\|_{L^{2}_{x}}\leq\epsilon\ll 1,$

Then the solution $u$ exists on on a time interval
$I_{\epsilon}=[0,c\epsilon^{-8}]$ and has the following properties for every
interval $I\subset I_{\epsilon}$ of size $|I|\leq\epsilon^{-6}$:

  1. (i)

Uniform $L^{2}$ bound:

(1.9) $\|u\|_{L^{\infty}_{t}(I_{\epsilon};L^{2}_{x})}\lesssim\epsilon.$

  2. (ii)

Strichartz bound:

(1.10) $\|u\|_{L^{6}_{t,x}(I\times\mathbb{R})}\lesssim\epsilon^{\frac{2}{3}}.$

  3. (iii)

Bilinear Strichartz bound:

(1.11)
$\|\partial_{x}(u\bar{u}(\cdot+x_{0}))\|_{L^{2}_{t}(I;H_{x}^{-\frac{1}{2}})}\lesssim\epsilon^{2},\qquad
x_{0}\in\mathbb{R}.$

The local well-posedness in $L^{2}$ for the problem (1.4) was already proved
in [21], so the emphasis here and later in the proof is on the lifespan bound
and the long time estimates in the theorem.

We remark that the intermediate time-scale $\epsilon^{-6}$ does not have an
intrinsic meaning from a scaling perspective, but is instead connected to the
unit frequency-scale which is implicit in (H1), and which motivates the
frequency decomposition on the unit frequency-scale which is used in the proof
of both the present result and the earlier result in [21]. One could also use
a smaller frequency-scale for this decomposition, which in turn corresponds to
a smaller size for $|I|$. This is however not needed in the proof of the
$\epsilon^{-8}$ result, so, in order to avoid cluttering the theorem, we omit
the details. But the interested reader should see Remark 1.1 below.

A natural question to ask is whether this result is optimal. On one hand, the
bounds (1.10) and (1.11) are sharp for the cubic NLS, and likely for any
focusing flow (i.e. which satisfies (H4) at least in some region); this is
discussed in Section 5. But the $\epsilon^{-8}$ lifespan bound is not optimal
for any flow satisfying our hypotheses. Indeed, if for instance the $L^{2}$
norm is conserved (as is the case for the focusing NLS) then global well-
posedness follows. However, we conjecture that

######  Conjecture 3.

The result in Theorem 1 is sharp for generic focusing flows satisfying our
hypotheses.

It it also interesting to see how our theorem applies to the focusing cubic
NLS problem. There we can also consider large data simply by scaling. Global
well-posedness in $L^{2}$ is relatively straightforward there, but some of the
estimates we prove are new:

######  Theorem 2.

Consider the focusing 1-d cubic NLS problem (1.1)(-) with $L^{2}$ initial data
$u_{0}$. Then the global solution $u$ satisfies the following bounds in all
intervals $I$ with size $|I|>\|u_{0}\|_{L^{2}_{x}}^{-4}$:

  1. (i)

Uniform $L^{2}$ bound:

(1.12) $\|u\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim\|u_{0}\|_{L^{2}_{x}}.$

  2. (ii)

Strichartz bound:

(1.13)
$\|u\|_{L^{6}_{t,x}(I\times\mathbb{R})}\lesssim\|u_{0}\|_{L^{2}_{x}}(I\|u_{0}\|_{L^{2}_{x}}^{4})^{\frac{1}{6}}.$

  3. (iii)

Bilinear Strichartz bound:

(1.14)
$\|\partial_{x}|u|^{2}\|_{L^{2}_{t}(I;\dot{H}_{x}^{-\frac{1}{2}}+cL^{2}_{x})}\lesssim\|u_{0}\|_{L^{2}_{x}}^{2},\qquad
c^{2}=\|u_{0}\|_{L^{2}_{x}}^{2}(|I|\|u_{0}\|_{L^{2}_{x}}^{4}).$

Here the Strichartz estimates are fairly easy to obtain directly, but the
bilinear Strichartz bounds are new. Returning to our discussion after Theorem
1, the numerology in this application helps clarify the earlier comment about
the choice of the time-scales in Theorem 1:

######  Remark 1.1.

In the context of Theorem 1, a scaling argument shows that the bounds (1.13)
and (1.14) hold for all intervals $I$ so that
$\epsilon^{-4}\leq|I|\leq\epsilon^{-6}$.

One may gain further insights into our result for focusing problems by
comparing it with our earlier result in [21].

######  Theorem 3 ([21]).

Under the above assumptions (H1), (H2) and (H3) on the symbol of the cubic
form $C$, small initial data

$\|u_{0}\|_{L^{2}_{x}}\leq\epsilon\ll 1,$

yields a unique global solution $u$ for (1.4), which satisfies the following
bounds:

  1. (i)

Uniform $L^{2}$ bound:

(1.15) $\|u\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim\epsilon.$

  2. (ii)

Strichartz bound:

(1.16) $\|u\|_{L^{6}_{t,x}}\lesssim\epsilon^{\frac{2}{3}}.$

  3. (iii)

Bilinear Strichartz bound:

(1.17)
$\|\partial_{x}(u\bar{u}(\cdot+x_{0}))\|_{L^{2}_{t}H_{x}^{-\frac{1}{2}}}\lesssim\epsilon^{2},\qquad
x_{0}\in\mathbb{R}.$

One may observe here that the estimates are similar in the two cases, and the
only difference is the time-scale on which the estimates hold: in the
defocusing case this is global, while in the focusing case it is finite and
depends on the solution size. For this reason, the proofs of Theorem 3 and (1)
are closely related, and we will take advantage of this within the proof.

For the convenience of the reader, we also recall the main ideas in the proof
of the last theorem in [21], which are equally employed here:

_1\. Energy estimates via density flux identities._ This is a classical idea
in pde’s, and particularly in the study of conservation laws. The novelty in
[21] is that this analysis is carried out in a nonlocal setting, where both
the densities and the fluxes involve translation invariant multilinear forms.
The densities and the fluxes are not uniquely determined here, so careful
choices need to be made.

_2\. The use of energy corrections._ This is an idea originally developed in
the context of the so called I-method [5] or more precisely the second
generation I-method [8], whose aim was to construct more accurate almost
conserved quantities. In [21] this idea is instead implemented at the level of
density-flux identities.

_3\. Interaction Morawetz bounds._ These were originally developed in the
context of the three-dimensional NLS problems by Colliander-Keel-Stafillani-
Takaoka-Tao in [6], and have played a fundamental role in the study of many
nonlinear Schrödinger flows, see e,g. [7, 27], and also for one-dimensional
quintic flows in the work of Dodson [11, 12]. Our take on this is somewhat
closer to the one-dimensional approach of Planchon-Vega [26], though recast in
the setting and language of nonlocal multilinear forms.

_4\. Tao’s frequency envelope method._ This is used as a way to accurately
track the evolution of the energy distribution across frequencies. Unlike the
classical implementation relative to dyadic Littlewood-Paley decompositions,
in [21] we adapt and refine this notion for lattice decompositions instead.
This is also very convenient as a bootstrap tool, see e.g. Tao [29], [30] but
with the added twist of also bootstrapping bilinear Strichartz bounds, as in
the authors’ paper [20].

###  1.3.  An outline of the paper

To a large extent, the proof of our main result mirrors the proof of the
global result in the defocusing case in [21]. The primary difference is in how
the $L^{6}$ Strichartz norms are handled, both globally and in a frequency
localized setting.

Section 2 reviews two of the main ideas in [21], namely the construction of
modified density-flux identities for the mass/momentum in a frequency
localized setting, as well as the interaction Morawetz identities associated
to those density-flux relations.

The proof of the main $L^{6}$ Strichartz bounds and the bilinear $L^{2}$
estimates is done in a frequency localized setting, using a bootstrap argument
based on a frequency decomposition on the unit scale, where the components are
measured using a maximal frequency envelope, another notion introduced in
[21]. This is described in Section 3, and leads to dispersive bounds on the
$\epsilon^{-6}$ time-scale. It is within this argument where the $L^{6}$ norms
are treated differently from the defocusing case.

In order to advance from the $\epsilon^{-6}$ to the $\epsilon^{-8}$ time-scale
it suffices to propagate the $L^{2}$ bound (i.e. the mass) on the larger time-
scale. However the mass is not a conserved quantity, so instead it is better
to propagate the bounds for the modified mass. This analysis is carried out in
Section 4.

Finally, in the last section of the paper we discuss the optimality of our
result, or rather the optimality of the $L^{6}$ and the bilinear $L^{2}$
estimates on the $\epsilon^{-6}$ time-scale. This is done by considering the
obvious enemies, namely the solitons, in the focusing NLS context.

###  1.4. Acknowledgements

The first author was supported by the Sloan Foundation, and by an NSF CAREER
grant DMS-1845037. The second author was supported by the NSF grant
DMS-2054975 as well as by a Simons Investigator grant from the Simons
Foundation.

##  2\. Density-flux and interaction Morawetz identities

A key role in the proof of the results in both [21] and in the present paper
is played by the approximate conservation laws for the mass and the momentum.
Rather than considering them directly, we instead consider several
improvements:

  * •

the conservation laws are written in density-flux form, rather than as
integral identities, where both the densities and the fluxes are multilinear
forms.

  * •

we improve the accuracy of these conservation laws by using well chosen
quartic corrections for both the densities and the fluxes, with $6$-linear
errors.

  * •

we use these densities and fluxes not only globally in frequency, but also in
a frequency localized setting.

The aim of this section is to provide an overview of these density-flux
identities, following the set-up of [21]. We conclude the section with an
overview of the interaction Morawetz identities obtained in [21] from the
above density flux identities.

###  2.1. Resonances and multilinear forms

A key role in our analysis is played by four wave resonances. Given three
input frequencies $(\xi_{1},\xi_{2},\xi_{3})$ in the cubic nonlinearity $C$,
the output is at frequency

$\xi_{4}=\xi_{1}-\xi_{2}+\xi_{3}.$

The three wave interaction is resonant if

$\xi_{4}^{2}=\xi_{1}^{2}-\xi_{2}^{2}+\xi_{3}^{2}.$

To rewrite these relations in a symmetric fashion we use the notations

$\Delta^{4}\xi=\xi_{1}-\xi_{2}+\xi_{3}-\xi_{4},\qquad\Delta^{4}\xi^{2}=\xi_{1}^{2}-\xi_{2}^{2}+\xi_{3}^{2}-\xi_{4}^{2},$

The first expression is Galilean invariant but not the second, which is why we
also use the adjusted, Galilean invariant expression

${\tilde{\Delta}}^{4}\xi^{2}=\Delta^{4}\xi^{2}-2\xi_{avg}\Delta^{4}\xi,$

where $\xi_{avg}$ represents the average of the four frequencies.

With these notations, the resonant set is described as

$\mathcal{R}=\\{[\xi]=(\xi_{1},\xi_{2},\xi_{3},\xi_{4});\
\Delta^{4}\xi=0,\Delta^{4}\xi^{2}=0\\}$

which can be explicitly characterized as

$\mathcal{R}=\\{[\xi];(\xi_{1},\xi_{3})=(\xi_{2},\xi_{4})\\}$

Quadruples in the resonant set can be described by two parameters, namely

(2.1) $\displaystyle{\delta\xi^{\text{hi}}}=$ $\displaystyle\
\max\\{|\xi_{1}-\xi_{2}|+|\xi_{3}-\xi_{4}|,|\xi_{1}-\xi_{4}|+|\xi_{3}-\xi_{2}|\\},$
$\displaystyle{\delta\xi^{\text{med}}}=$ $\displaystyle\
\min\\{|\xi_{1}-\xi_{2}|+|\xi_{3}-\xi_{4}|,|\xi_{1}-\xi_{4}|+|\xi_{3}-\xi_{2}|\\},$

These distance parameters are carefully defined so that they can also be used
outside the resonant set to characterize frequency quadruples. This is very
useful in the density-flux relations later on.

###  2.2. The conservation of mass

The starting point of the analysis in [21] is to consider energy estimates for
our flow from a density-flux perspective. In the simplest case, we start with
the mass density

$M(u,\bar{u})=|u|^{2},$

whose linear flux is given by the momentum

$P(u,\bar{u})=2i\Im(u\partial_{x}\bar{u}).$

These can be viewed as translation invariant bilinear forms with symbols

$m(\xi,\eta)=1,\qquad p(\xi,\eta)=\xi+\eta.$

Integrating the densities we obtain the familiar mass and momentum,

$\mathbf{M}(u)=\int_{\mathbb{R}}M(u,\bar{u})\,dx,\quad\mathbf{P}(u)=\int_{\mathbb{R}}M(u,\bar{u})\,dx.$

At the nonlinear level, we have the density flux relation

$\partial_{t}M(u,\bar{u})=\partial_{x}P(u,\bar{u})+C^{4}_{m}(u,\bar{u},u,\bar{u}),$

where $C^{4}_{m}$ is a symmetric translation invariant real multilinear form
which depends on our cubic nonlinearity $C$. A key observation in [21] is
that, under the conservative assumption (H2) on the nonlinearity, the mass
density admits a quartic correction which is accurate to sixth order.
Precisely, the correction has the form

(2.2) $M^{\sharp}(u)=M(u)+B^{4}_{m}(u,\bar{u},u,\bar{u}),$

and the associated density flux relation has the form

(2.3)
$\partial_{t}M^{\sharp}(u)=\partial_{x}(P(u)+R^{4}_{m}(u,\bar{u},u,\bar{u}))+R^{6}_{m}(u,\bar{u},u,\bar{u},u,\bar{u}).$

with suitable translation invariant multilinear forms $R^{4}_{m}$ and
$R^{6}_{m}$. The corresponding integral corrected mass is

$\mathbf{M}^{\sharp}(u)=\int_{\mathbb{R}}M^{\sharp}(u)\,dx.$

The choice of the symbols $b^{4}_{m}$ and $r^{4}_{m}$ above depends on the
behavior of $c^{4}_{m}$ near the resonant set $\mathcal{R}$, precisely they
have to solve the division problem

(2.4) $c^{4}_{m}+i\Delta^{4}\xi^{2}\,b^{4}_{m}=i\Delta^{4}\xi\,r^{4}_{m}.$

This is possible due to our condition (H2), which implies that $c^{4}_{m}=0$
on $\mathcal{R}$. But the choice is not uniquely determined, so it is
important to make a good one, i.e. which insures good symbol bounds. To
achieve this, in [21] we decompose the phase space for frequency quadruples
into three overlapping regions which can be separated using cutoff functions
which are smooth on the unit scale:

  1. i)

The full division region,

$\Omega_{1}=\\{{\delta\xi^{\text{med}}}\lesssim 1\\},$

which represents a full unit size neighbourhood of the resonant set
$\mathcal{R}$.

  2. ii)

The region

$\Omega_{2}=\\{1+|\Delta^{4}\xi|\ll{\delta\xi^{\text{med}}}\\},$

where ${\tilde{\Delta}}^{4}\xi^{2}$ must be elliptic,
${\tilde{\Delta}}^{4}\xi^{2}\approx{\delta\xi^{\text{hi}}}{\delta\xi^{\text{med}}}$,
and thus we will favor division by the symbol ${\tilde{\Delta}}^{4}\xi^{2}$.

  3. iii)

The region

$\Omega_{3}=\\{1\ll{\delta\xi^{\text{med}}}\lesssim|\Delta^{4}\xi|\\},$

we will instead divide by $\Delta^{4}\xi$; this is compensated by the
relatively small size of this region.

Since we will also need this in the present paper, we state the result in the
following

######  Proposition 2.1 ([21]).

Assume that the nonlinearity $C$ satisfies the conditions (H1), (H2). Then
there exist multilinear forms $B^{4}_{m}$ $R^{4}_{m}$ and $R^{6}_{m}$ so that
the relation (2.3) holds for solutions $u$ to (1.4), and so that the symbols
$c^{4}_{m}$ and $r^{4}_{m}$ satisfy the bounds

  1. i)

Size

(2.5) $\displaystyle|\partial^{\alpha}r^{4}_{m}|\lesssim$ $\displaystyle\
\frac{1}{\langle{\delta\xi^{\text{med}}}\rangle},$
$\displaystyle|\partial^{\alpha}b^{4}_{m}|\lesssim$ $\displaystyle\
\frac{1}{\langle{\delta\xi^{\text{hi}}}\rangle\langle{\delta\xi^{\text{med}}}\rangle}.$

  2. ii)

Support: $b^{4}$ is supported in $\Omega_{1}\cup\Omega_{2}$ and
$\tilde{r}^{4}$ is supported in $\Omega_{1}\cup\Omega_{3}$.

In addition, we have the fixed time bound

(2.6)
$\left|\int_{\mathbb{R}}B^{4}_{m}(u,u,u,u)\,dx\right|\lesssim\|u\|_{L^{2}_{x}}^{4}.$

We make several remarks concerning this result:

  * •

No bound for the symbol $r^{6}_{m}$ is provided in the proposition. This is
because $r^{6}_{m}$ is obtained directly as the contribution of $C$ to the
time derivative of $b^{4}_{m}$.

  * •

This proposition is a consequence of Lemma 4.1 and Lemma 7.1 in [21].

  * •

The defocusing hypothesis (H3), which is used for the final result in [21],
plays no role here.

  * •

The estimate (2.6) shows that the mass correction is perturbative for as long
as the solution $u$ remains small in $L^{2}$.

  * •

A similar analysis applies for the momentum conservation law. But the
counterpart of the above Proposition for the momentum is less useful directly,
and instead it is used in [21] only in a frequency localized context.

###  2.3. Frequency localized density-flux identities

Instead of relying on the more standard Littlewood-Paley decomposition, the
analysis in [21] uses a frequency decomposition on the unit scale in
frequency. Given any integer $j$, we will use localized versions of the mass
in a unit size region around $j$. More generally, for an interval
$A\subset{\mathbb{Z}}$, we use a symbol $a_{0}$ which is frequency localized
in a unit neighbourhood of $A$. At the level of bilinear forms, we will use
the symbol

$a(\xi,\eta)=a_{0}(\xi)a_{0}(\eta).$

Corresponding to such $a$ we define quadratic localized mass, momentum and
energy densities by the symbols

$m_{a}(\xi,\eta)=a(\xi,\eta),\qquad
p_{a}(\xi,\eta)=(\xi+\eta)a(\xi,\eta),\qquad
e_{a}(\xi,\eta)=(\xi+\eta)^{2}a(\xi,\eta).$

The associated bilinear forms are denoted by $M_{a}$, $P_{a}$, respectively
$E_{a}$. If $A=\\{j\\}$ then we simply replace the subscript $a$ with $j$.

It is shown in [21], again under the assumptions (H1) and (H2), that one may
find quartic corrections $M^{\sharp}_{a}$ and $P^{\sharp}_{a}$ of the form

(2.7) $M^{\sharp}_{a}(u)=M_{a}(u)+B^{4}_{m,a}(u,\bar{u},u,\bar{u}),$ (2.8)
$P^{\sharp}_{a}(u)=P_{a}(u)+B^{4}_{p,a}(u,\bar{u},u,\bar{u}),$

for which we obtain density-flux identities akin to (2.3), namely

(2.9)
$\partial_{t}M^{\sharp}_{a}(u)=\partial_{x}(P_{a}(u)+R^{4}_{m,a}(u))+R^{6}_{m,a}(u),$

and

(2.10)
$\partial_{t}P^{\sharp}_{a}(u)=\partial_{x}(E_{a}(u)+R^{4}_{p,a}(u))+R^{6}_{p,a}(u).$

We will consider these relations together with their Galilean shifts obtaining
relations of the form

(2.11)
$(\partial_{t}-2\xi_{0}\partial_{x})M^{\sharp}_{a}(u)=\partial_{x}(P_{a,\xi_{0}}(u)+R^{4}_{m,a,\xi_{0}}(u))+R^{6}_{m,a,\xi_{0}}(u),$

respectively

(2.12)
$(\partial_{t}-2\xi_{0}\partial_{x})P^{\sharp}_{a,\xi_{0}}(u)=\partial_{x}(E_{a,\xi_{0}}(u)+R^{4}_{p,a,\xi_{0}}(u))+R^{6}_{p,a,\xi_{0}}(u).$

These correspond to the algebraic division relations

(2.13) $c^{4}_{m,a}+i\Delta^{4}(\xi-\xi_{0})^{2}b^{4}_{m,a}=i\Delta^{4}\xi
r^{4}_{m,a,\xi_{0}},$

respectively

(2.14)
$c^{4}_{p,a,\xi_{0}}+i\Delta^{4}(\xi-\xi_{0})^{2}b^{4}_{p,a,\xi_{0}}=i\Delta^{4}\xi
r^{4}_{p,a,\xi_{0}},$

where $c^{4}_{m,a}$ and $c^{4}_{p,a,\xi_{0}}$ are the density-flux sources
corresponding to uncorrected mass, respectively momentum.

The symbols above are connected in the obvious way. Precisely, we have

(2.15) $r^{4}_{m,a,\xi_{0}}=r^{4}_{m,a}-2\xi_{0}b^{4}_{m,a},\qquad$

and

(2.16) $P^{\sharp}_{a,\xi_{0}}=P^{\sharp}_{a}-2\xi_{0}M^{\sharp}_{a},\qquad
b^{4}_{p,a,\xi_{0}}=b^{4}_{p,a}-2\xi_{0}b^{4}_{m,a},$

and finally

(2.17)
$r^{4}_{p,a,\xi_{0}}=r^{4}_{p,a}-2\xi_{0}b^{4}_{p,a}-2\xi_{0}r^{4}_{m,a,\xi_{0}}.$

To use these density flux relations we need to have appropriate bounds for our
symbols:

######  Proposition 2.2.

Let $J\subset\mathbb{R}$ be an interval of length $r$, and
$d(\xi_{0},J)\lesssim r$. Assume that $a$ is supported in $J\times J$, with
bounded and uniformly smooth symbol. Then the relations (2.11) and (2.12) hold
with symbols $b^{4}_{m,a}$, $b^{4}_{p,a,\xi_{0}}$, $r^{4}_{m,a,\xi_{0}}$ and
$r^{4}_{p,a,\xi_{0}}$ which can be chosen to have the following properties:

  1. i)

Support: they are all supported in the region where at least one of the
frequencies is in $J$.

  2. ii)

Size:

(2.18)
$|b^{4}_{m,a}|\lesssim\frac{1}{\langle{\delta\xi^{\text{hi}}}\rangle\langle{\delta\xi^{\text{med}}}\rangle},\qquad|b^{4}_{p,a,\xi_{0}}|\lesssim\frac{r}{\langle{\delta\xi^{\text{hi}}}\rangle\langle{\delta\xi^{\text{med}}}\rangle},$
(2.19) $\displaystyle|r^{4}_{m,a,\xi_{0}}|\lesssim$ $\displaystyle\
\frac{1}{\langle{\delta\xi^{\text{med}}}\rangle}1_{\Omega_{1}\cup\Omega_{2}}+\frac{r}{\langle{\delta\xi^{\text{hi}}}\rangle\langle{\delta\xi^{\text{med}}}\rangle}1_{\Omega_{1}\cup\Omega_{3}},$
$\displaystyle|R^{4}_{p,a,\xi_{0}}|\lesssim$ $\displaystyle\
\frac{r}{\langle{\delta\xi^{\text{med}}}\rangle}1_{\Omega_{1}\cup\Omega_{2}}+\frac{r^{2}}{\langle{\delta\xi^{\text{hi}}}\rangle\langle{\delta\xi^{\text{med}}}\rangle}1_{\Omega_{1}\cup\Omega_{3}}.$

  3. iii)

Regularity: similar bounds hold for all derivatives.

This is Proposition 4.3 in [21].

###  2.4.  Interaction Morawetz identities

One way the desity flux relations above are used is to obtain more accurate
bounds for the mass propagation. However, another way to use them is via
interaction Morawetz identities, which yield bilinear estimates for the
interaction of different frequency portions of the solutions, or even the
self-interaction of unit frequency portions of solutions.

Given two frequency intervals $A$ and $B$ and corresponding mass/momentum
modified density associated to these intervals for two solutions $u,v$ to
(1.4), in [21] we define the associated _interaction Morawetz functional_ by

(2.20)
$\mathbf{I}_{AB}=\iint_{x>y}M^{\sharp}_{a}(u)(x)P^{\sharp}_{b,\xi_{0}}(v)(y)-P^{\sharp}_{a,\xi_{0}}(u)(x)M^{\sharp}_{b}(v)(y)\,dxdy,$

where we add several remarks:

  * •

In applications, for the second solution $v$ we will simply choose a spatial
translation of the solution $u$.

  * •

A velocity parameter $\xi_{0}$ is introduced on the right, but the functional
does not depend on $\xi_{0}$. This parameter plays a role, however, in
estimating the time derivative of $\mathbf{I}_{AB}$, and will be chosen to be
close to the sets $A$ and $B$.

  * •

Such interaction functionals will be used in two settings:

    1. (i)

the separated case $1\ll|A|\approx|B|\approx dist(A,B)$, and

    2. (ii)

the self-interaction case $|A|+|B|+dist(A,B)\lesssim 1$.

The time derivative of the interaction Morawetz functional is computed using
the frequency localized mass density-flux (2.11) and the corresponding
momentum density-flux (2.12). This yields a localized _interaction Morawetz
identity_ ,

(2.21)
$\frac{d}{dt}\mathbf{I}_{AB}=\mathbf{J}^{4}_{AB}+\mathbf{J}^{6}_{AB}+\mathbf{J}^{8}_{AB}+\mathbf{K}^{8}_{AB},$

where the terms on the right are described as follows:

  1. a)

the quartic contribution $\mathbf{J}^{4}_{a}$ is

$\mathbf{J}^{4}_{AB}=\int_{\mathbb{R}}M_{a}(u)(x)E_{b,\xi_{0}}(v)(x)+M_{b}(v)(x)E_{a,\xi_{0}}(u)(x)-2P_{a,\xi_{0}}(u)(x)P_{b,\xi_{0}}(v)(x)\,dx,$

and is used to capture bilinear $L^{2}$ bounds.

  2. b)

The sixth order term $\mathbf{J}^{6}_{a}$ has the form

$\mathbf{J}^{6}_{AB}=\int_{\mathbb{R}}-(P_{a,\xi_{0}}B^{4}_{p,b,\xi_{0}}+P_{b,\xi_{0}}R^{4}_{m,a,\xi_{0}})+(M_{a}R^{4}_{p,b,\xi_{0}}+E_{b,\xi_{0}}B^{4}_{m,a,\xi_{0}})-\text{symmetric}\,dx,$

where in the symmetric part we interchange both the indices $a,b$ and the
functions $u,v$. In the defocusing case its symbol has a favorable sign on the
diagonal, and is used to capture the $L^{6}$ bound in the self-interaction
case. Here, it is estimated perturbatively.

  3. c)

The eight-linear term

$\mathbf{J}^{8}_{AB}=\int_{\mathbb{R}}-R^{4}_{m,a,\xi_{0}}B^{4}_{p,b,\xi_{0}}+B^{4}_{m,a,\xi_{0}}R^{4}_{p,b,\xi_{0}}-\text{symmetric}\,dx.$

plays a perturbative role.

  4. d)

The $8^{+}$-linear term $\mathbf{K}^{8}_{AB}$ has the form

$\mathbf{K}^{8}_{AB}=\iint_{x>y}M^{\sharp}_{a}(x)R^{6}_{p,a,\xi_{0}}+P^{\sharp}_{a,\xi_{0}}R^{6}_{m,a,\xi_{0}}-\text{symmetric}\,dxdy.$

This is a double integral, which also includes a $10$-linear term. It is also
estimated perturbatively.

##  3\. Strichartz and bilinear $L^{2}$ bounds

To obtain estimates for $L^{2}$ solutions $u$ to (1.4), a unit scale frequency
decomposition is needed,

$u=\sum_{k\in{\mathbb{Z}}}u_{k},\qquad u_{k}:=P_{k}u,$

where $P_{k}$ are multipliers with smooth symbols localized in a unit
neighbourhood of the integer frequency $k$. To measure the components $u_{k}$
we use a frequency envelope $\\{c_{k}\\}\in\ell^{2}$ in order to transfer
bounds from the initial data to the solutions.

These frequency envelopes are chosen to satisfy an adapted version of the
slowly varying property, originally introduced by Tao [29] in the context of
dyadic decompositions. Such a property is needed in order to account for the
nonlinear leakage of energy between nearby frequencies.

We recall the frequency envelope set-up in [21], associated to lattice
decompositions:

######  Definition 3.1.

A lattice frequency envelope $\\{c_{k}\\}$ is said to have the maximal
property if

(3.1) $Mc\leq Cc,$

where $Mc$ represents the maximal function of $c$. Here $C$ is a universal
constant.

Frequency envelopes that have this property will be called _admissible_. The
proof of the $L^{6}$ Strichartz and the bilinear $L^{2}$ bounds will be
phrased as a bootstrap argument relative to an admissible frequency envelope
for the initial data.

######  Theorem 4.

Let $u\in C[0,T;L^{2}]$ be a solution for the equation (1.4) with initial data
$u_{0}$ which has $L^{2}$ size at most $\epsilon$. Let $\\{c_{k}\\}$ be a
maximal frequency envelope for the initial data in $L^{2}$, also of size
$\epsilon$,

$\|u_{0k}\|_{L^{2}}\lesssim\epsilon c_{k}.$

Assume that

(3.2) $T\ll\epsilon^{-6}.$

Then the solution $u$ satisfies the following bounds in $[0,T]$:

  1. (i)

Uniform frequency envelope bound:

(3.3) $\|u_{k}\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim\epsilon c_{k},$

  2. (ii)

Localized Strichartz bound:

(3.4) $\|u_{k}\|_{L_{t,x}^{6}}\lesssim(\epsilon c_{k})^{\frac{2}{3}},$

  3. (iii)

Localized Interaction Morawetz:

(3.5)
$\|\partial_{x}|u_{k}|^{2}\|_{L^{2}_{t,x}}\lesssim\epsilon^{2}c_{k}^{2},$

  4. (iv)

Transversal bilinear $L^{2}$ bound:

(3.6)
$\|\partial_{x}(u_{A}{\bar{u}}_{B}(\cdot+x_{0}))\|_{L^{2}_{t,x}}\lesssim\epsilon^{2}c_{A}c_{B}\,\langle
dist(A,B)\rangle^{\frac{1}{2}},$

for all $x_{0}\in\mathbb{R}$ whenever $|A|+|B|\lesssim\langle
dist(A,B)\rangle$.

This proposition mirrors a similar result in Section 7 of [21], with two key
differences. On one hand we drop the defocusing assumption (H3), and on the
other hand we limit the size of the time interval in (3.2).

The $L^{6}$ Strichartz estimates (1.10) and the bilinear $L^{2}$ bounds in
(1.11) follow from the estimates in the above proposition, by the same
arguments as those in Section 8 of [21].

To prove this theorem, we make a bootstrap assumption where we assume the same
bounds but with a worse constant $C$, as follows:

  1. (i)

Uniform frequency envelope bound,

(3.7) $\|u_{k}\|_{L^{\infty}_{t}L^{2}_{x}}\lesssim C\epsilon c_{k},$

  2. (ii)

Localized Strichartz bound,

(3.8) $\|u_{k}\|_{L^{6}_{t,x}}\lesssim C(\epsilon c_{k})^{\frac{2}{3}},$

  3. (iii)

Localized Interaction Morawetz,

(3.9) $\|\partial_{x}|u_{k}|^{2}\|_{L^{2}_{t,x}}\lesssim
C\epsilon^{2}c_{k}^{2},$

  4. (iv)

Transversal Interaction Morawetz,

(3.10)
$\|\partial_{x}(u_{k_{1}}\bar{u}_{k_{2}}(\cdot+x_{0}))\|_{L^{2}_{t,x}}\lesssim
C\epsilon^{2}c_{k_{1}}c_{k_{2}}\langle k_{1}-k_{2}\rangle^{\frac{1}{2}}$

uniformly for all $x_{0}\in\mathbb{R}$.

Then we seek to improve the constant in these bounds. The gain will come from
the fact that the $C$’s will always come paired either with extra $\epsilon$
factors, or with $T\epsilon^{6}$ factors.

To a large extent the proof largely repeats the proof of the corresponding
result in [21], so we review the steps and expand the portion where the
argument differs here.

_STEP 1:_ The proof of the energy bound (3.3). This is done by integrating the
density flux relation (2.9) for the localized mass $\mathbf{M}_{k}(u)$. The
argument in [21] applies unchanged.

_STEP 2:_ The proof of the $L^{6}$ Strichartz bound (3.3). In [21] this is
proved together with (3.5) by integrating the interaction Morawetz identity
applied to the pair $(u_{k},u_{k})$. However, in the focusing case the sign of
the $u_{k}^{6}$ contribution changes, and the same argument no longer applies.

Instead, here we will estimate the $L^{6}$ norm directly using an
interpolation argument. Precisely, for the function $v_{k}=|u_{k}|^{2}$, using
(3.7) and (3.9), we have the bounds

$\|v_{k}\|_{L^{\infty}_{t}L^{1}_{x}}\lesssim C^{2}\epsilon^{2}c_{k}^{2},$

respectively

$\|v_{k}\|_{L^{2}_{t}\dot{H}^{1}_{x}}\lesssim C^{2}\epsilon^{2}c_{k}^{2}.$

We interpolate between the two estimates in homogeneous Sobolev spaces, with
weights $5/9$ and $4/9$. We obtain

$\|v_{k}\|_{L^{\frac{9}{2}}_{t}\dot{W}^{\frac{4}{9},\frac{9}{7}}_{x}}\lesssim\|v_{k}\|_{L^{\infty}_{t}L^{1}_{x}}^{\frac{5}{9}}\|v_{k}\|_{L^{2}_{t}\dot{H}^{1}_{x}}^{\frac{4}{9}}\lesssim
C^{2}\epsilon^{2}c_{k}^{2}.$

By Sobolev embeddings $\dot{W}^{\frac{4}{9},\frac{9}{7}}$ embeds in $L^{3}$
so, using also Hölder’s inequality with respect to time, we obtain

$\|v_{k}\|_{L^{3}_{t}L_{x}^{3}}\lesssim
T^{\frac{1}{9}}\|v_{k}\|_{L^{\frac{9}{2}}L^{3}}\lesssim
T^{\frac{1}{9}}\|v_{k}\|_{L^{\frac{9}{2}}\dot{W}^{\frac{4}{9},\frac{9}{7}}}\lesssim
C^{2}T^{\frac{1}{9}}\epsilon^{2}c_{k}^{2}=C^{2}(T\epsilon^{6})^{\frac{1}{9}}\epsilon^{\frac{4}{3}}c_{k}^{2}.$

This implies the desired Strichartz bound (3.4) under the time constraint
(3.2).

_STEP 3:_ The proof of the bilinear $L^{2}$ bound (3.6). This is again exactly
as in [21], by applying the interaction Morawetz identity to the functions
$(u_{A},u_{B}(\cdot+x_{0}))$. We note that the $L^{6}$ bound is used as an
input in this proof, and the defocusing assumption (H2) is not needed. Here we
view (3.5) as a special case of (3.6), and no longer in conjunction with
(3.3).

##  4\. Long time energy estimates

The frequency envelope bounds in Theorem 4 provide us with uniform energy
bounds on the $\epsilon^{-6}$ time-scale, and so they do not suffice in order
to prove our main result in Theorem 1, which is on the $\epsilon^{-8}$ time-
scale. To fill in this gap, we will prove a direct energy estimate on the
$\epsilon^{-8}$ time-scale. Precisely, we will show the following:

######  Proposition 4.1.

Let $u$ be an $L^{2}$ solution for (1.4) in a time interval $[0,T]$. Assume
that the initial data for (1.4) satisfies

(4.1) $\|u_{0}\|_{L_{x}^{2}}\leq\epsilon\ll 1,$

and that $T\ll\epsilon^{-8}$. Then the solution $u$ satisfies

(4.2) $\|u\|_{L_{t}^{\infty}(0,T;L_{x}^{2})}\leq 4\epsilon.$

Once we have this proposition, a continuity argument based on the local well-
posedness for (1.4) in $L^{2}$ implies Theorem 1.

###### Proof.

It suffices to prove that the conclusion holds assuming that we have the
bootstrap assumption

(4.3) $\|u\|_{L^{\infty}(0,T;L^{2})}\leq 8\epsilon.$

Instead of tracking directly the mass $\mathbf{M}(u)=\|u\|_{L_{x}^{2}}^{2}$,
it is more efficient to work with the modified mass

$\mathbf{M}^{\sharp}(u)=\int_{\mathbb{R}}M^{\sharp}(u)\,dx.$

In view of the bound (2.6), we have

$\mathbf{M}(u)=\mathbf{M}^{\sharp}(u)+O(\epsilon^{4}).$

Since $\epsilon\ll 1$, we have

(4.4) $\mathbf{M}(u)(0)\leq 2\epsilon^{2},$

and it suffices to show that

(4.5) $\mathbf{M}(u)(t)\leq 4\epsilon^{2},\qquad t\in[0,T].$

In view of Proposition 2.3, the time evolution of $\mathbf{M}^{\sharp}$ is
given by

(4.6)
$\frac{d}{dt}\mathbf{M}^{\sharp}(u)=\int_{\mathbb{R}}R^{6}_{m}(u,{\bar{u}},u,{\bar{u}},u,{\bar{u}})\,dx.$

To bound its growth, we use the following

######  Lemma 4.2.

Assume that the bounds (3.3)-(3.6) for $u$ hold in a time interval $[0,T]$.
Then we have

(4.7)
$\|R^{6}_{m}\|_{L^{1}_{t,x}([0,T]\times\mathbb{R})}\lesssim\epsilon^{4}.$

This is Lemma 7.3 in [21]. We apply this lemma on time intervals of size
$\epsilon^{-6}$, where the bounds (3.3)-(3.6) hold in view of Theorem 4, and
then add up the results. Then for $T>\epsilon^{-6}$ we get

(4.8)
$\|R^{6}_{m}\|_{L^{1}_{t,x}([0,T]\times\mathbb{R})}\lesssim\epsilon^{4}(T\epsilon^{6}).$

Hence for $T\ll\epsilon^{-8}$ we arrive at

(4.9) $\|R^{6}_{m}\|_{L^{1}_{t,x}([0,T]\times\mathbb{R})}\ll\epsilon^{2}.$

This allows us to obtain (4.5) from (4.4), thereby concluding the proof of the
proposition.

∎

##  5\. Optimality remarks

As noted earlier, a key obstruction to global dispersive estimates in the
focusing case is given by the potential existence of small solitons. Here we
will restrict our attention to the simplest model, namely the focusing cubic
NLS, and test the optimality of our estimates on solitons for this model. It
is not so difficult to show that small solitons exist for our model whenever
the focusing assumption (H4) is satisfied in some frequency region.

All cubic NLS solitons are equivalent modulo scaling and Galilean
transformations. Our bounds are Galilean invariant, so we set the soliton
velocity to zero and we focus on scaling. Then the unit scale soliton has the
form

$u(x,t)=e^{it}Q(x),\qquad Q(x)=\operatorname{sech}(x).$

Rescaled to the frequency scale $\lambda$, this yields the solitons

$u_{\lambda}(x,t)=e^{it\lambda^{2}}Q_{\lambda}(x),\qquad
Q_{\lambda}(x)=\lambda Q(\lambda x),$

which has initial data size

$\|u_{0}\|_{L_{x}^{2}}^{2}=\|Q_{\lambda}\|_{L^{2}_{x}}^{2}=\int_{\mathbb{R}}\lambda^{2}Q^{2}(\lambda
x)\,dx\approx\lambda,$

so for Theorem 1 we will choose $\lambda=\epsilon^{2}$.

On the other hand,

$\|u_{\lambda}\|_{L^{6}_{t,x}(0,T,\mathbb{R})}^{6}=T\int_{\mathbb{R}}\lambda^{6}Q^{6}(\lambda
x)\,dx\approx T\lambda^{5}.$

Then it is easily seen that we have approximate equality in (1.10) and (1.13).

Similarly, we compute the bilinear Strichartz norm. Due to the bound from
below on $|I|$, we have $c\gtrsim\lambda^{-\frac{1}{2}}$ whereas $u$ is
concentrated at frequency $\lesssim\lambda$. Then

$\|\partial_{x}|u_{\lambda}|^{2}\|_{L^{2}_{t}(0,T;\dot{H}^{\frac{1}{2}}_{x}+cL^{2}_{x})}^{2}\approx
c^{-2}\|\partial_{x}|u_{\lambda}|^{2}\|_{L^{2}_{t,x}(0,T;L^{2})}^{2}=\lambda(T\lambda^{2})^{-1}T\int_{\mathbb{R}}\lambda^{6}\partial_{x}Q_{x}^{2}(\lambda
x)\,dx\approx\lambda^{2}.$

This corresponds to having equality in (1.11) and (1.14).

## References

  * [1] Thomas Alazard and Jean-Marc Delort.  Global solutions and asymptotic behavior for two dimensional gravity water waves.  Ann. Sci. Éc. Norm. Supér. (4), 48(5):1149–1238, 2015.
  * [2] Michael Borghese, Robert Jenkins, and Kenneth D. T.-R. McLaughlin.  Long time asymptotic behavior of the focusing nonlinear Schrödinger equation.  Ann. Inst. H. Poincaré C Anal. Non Linéaire, 35(4):887–920, 2018.
  * [3] Jean Bourgain.  A remark on normal forms and the “$I$-method” for periodic NLS.  J. Anal. Math., 94:125–157, 2004.
  * [4] J. Colliander, M. Grillakis, and N. Tzirakis.  Tensor products and correlation estimates with applications to nonlinear Schrödinger equations.  Comm. Pure Appl. Math., 62(7):920–968, 2009.
  * [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao.  Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation.  Math. Res. Lett., 9(5-6):659–682, 2002.
  * [6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao.  Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^{3}$.  Comm. Pure Appl. Math., 57(8):987–1014, 2004.
  * [7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao.  Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{3}$.  Ann. of Math. (2), 167(3):767–865, 2008.
  * [8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao.  Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^{2}$.  Discrete Contin. Dyn. Syst., 21(3):665–686, 2008.
  * [9] Percy Deift and Xin Zhou.  Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space.  Comm. Pure Appl. Math., 56(8):1029–1077, 2003.  Dedicated to the memory of Jürgen K. Moser.
  * [10] Jean-Marc Delort.  Semiclassical microlocal normal forms and global solutions of modified one-dimensional KG equations.  Ann. Inst. Fourier (Grenoble), 66(4):1451–1528, 2016.
  * [11] Benjamin Dodson.  Global well-posedness and scattering for the defocusing, $L^{2}$ critical, nonlinear Schrödinger equation when $d=1$.  Amer. J. Math., 138(2):531–569, 2016.
  * [12] Benjamin Dodson.  Global well-posedness and scattering for the defocusing, mass-critical generalized KdV equation.  Ann. PDE, 3(1):Paper No. 5, 35, 2017.
  * [13] Nakao Hayashi and Pavel I. Naumkin.  Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations.  Amer. J. Math., 120(2):369–389, 1998.
  * [14] Nakao Hayashi and Pavel I. Naumkin.  Large time asymptotics for the fractional nonlinear Schrödinger equation.  Adv. Differential Equations, 25(1-2):31–80, 2020.
  * [15] John K. Hunter and Mihaela Ifrim.  Enhanced life span of smooth solutions of a Burgers-Hilbert equation.  SIAM J. Math. Anal., 44(3):2039–2052, 2012.
  * [16] John K. Hunter, Mihaela Ifrim, Daniel Tataru, and Tak Kwong Wong.  Long time solutions for a Burgers-Hilbert equation via a modified energy method.  Proc. Amer. Math. Soc., 143(8):3407–3412, 2015.
  * [17] Mihaela Ifrim and Daniel Tataru.  Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension.  Nonlinearity, 28(8):2661–2675, 2015.
  * [18] Mihaela Ifrim and Daniel Tataru.  Two dimensional water waves in holomorphic coordinates II: Global solutions.  Bull. Soc. Math. France, 144(2):369–394, 2016.
  * [19] Mihaela Ifrim and Daniel Tataru.  The lifespan of small data solutions in two dimensional capillary water waves.  Arch. Ration. Mech. Anal., 225(3):1279–1346, 2017.
  * [20] Mihaela Ifrim and Daniel Tataru.  Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation.  Ann. Sci. Éc. Norm. Supér. (4), 52(2):297–335, 2019.
  * [21] Mihaela Ifrim and Daniel Tataru.  Global solutions for 1D cubic defocusing dispersive equations: Part I.  arXiv e-prints, page arXiv:2205.12212, May 2022.
  * [22] Mihaela Ifrim and Daniel Tataru.  Testing by wave packets and modified scattering in nonlinear dispersive pde’s.  arXiv e-prints, page arXiv:2204.13285, April 2022.
  * [23] Jun Kato and Fabio Pusateri.  A new proof of long-range scattering for critical nonlinear Schrödinger equations.  Differential Integral Equations, 24(9-10):923–940, 2011.
  * [24] Hans Lindblad, Jonas Lührmann, and Avy Soffer.  Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities.  Arch. Ration. Mech. Anal., 241(3):1459–1527, 2021.
  * [25] Hans Lindblad and Avy Soffer.  Scattering and small data completeness for the critical nonlinear Schrödinger equation.  Nonlinearity, 19(2):345–353, 2006.
  * [26] Fabrice Planchon and Luis Vega.  Bilinear virial identities and applications.  Ann. Sci. Éc. Norm. Supér. (4), 42(2):261–290, 2009.
  * [27] E. Ryckman and M. Visan.  Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{1+4}$.  Amer. J. Math., 129(1):1–60, 2007.
  * [28] Jalal Shatah.  Normal forms and quadratic nonlinear Klein-Gordon equations.  Comm. Pure Appl. Math., 38(5):685–696, 1985.
  * [29] Terence Tao.  Global regularity of wave maps. II. Small energy in two dimensions.  Comm. Math. Phys., 224(2):443–544, 2001.
  * [30] Terence Tao.  Global well-posedness of the Benjamin-Ono equation in $H^{1}({\bf R})$.  J. Hyperbolic Differ. Equ., 1(1):27–49, 2004.