# Innovative Integration of Visual Foundation Model with a Robotic Arm on a
Mobile Platform

Shimian Zhang1 Qiuhong Lu2 1Shimian Zhang is with Shanghai Hrstek Co., Ltd
<EMAIL_ADDRESS>Lu is with Shanghai Hrstek Co., Ltd
<EMAIL_ADDRESS>
###### Abstract

In the rapidly advancing field of robotics, the fusion of state-of-the-art
visual technologies with mobile robotic arms has emerged as a critical
integration. This paper introduces a novel system that combines the Segment
Anything model (SAM) — a transformer-based visual foundation model — with a
robotic arm on a mobile platform. The design of integrating a depth camera on
the robotic arm’s end-effector ensures continuous object tracking,
significantly mitigating environmental uncertainties. By deploying on a mobile
platform, our grasping system has an enhanced mobility, playing a key role in
dynamic environments where adaptability are critical. This synthesis enables
dynamic object segmentation, tracking, and grasping. It also elevates user
interaction, allowing the robot to intuitively respond to various modalities
such as clicks, drawings, or voice commands, beyond traditional robotic
systems. Empirical assessments in both simulated and real-world demonstrate
the system’s capabilities. This configuration opens avenues for wide-ranging
applications, from industrial settings, agriculture, and household tasks, to
specialized assignments and beyond.

##  I Introduction

Figure 1: System Overview: Our integrated mobile robotic grasping system
comprises two core modules: the Visual Interpretation Module (VIM) and the
Motion Control Module (MCM). VIM, utilizing a depth camera, captures a live
scene and, through the SAM visual foundation model, segments the user-
indicated object for grasping. It then computes the object’s 3D coordinates,
relaying this data to MCM. Based on this localization, MCM plans the motion,
determining if the platform needs relocation for optimal grasping. The robotic
arm’s movement leverages inverse kinematics for precision, with a closed-loop
control anchored on continuous object tracking via the ”eye-in-hand” system.
The process ends with controlled grasping using force feedback from the arm’s
end effector.

The integration of vision systems into robotic arms has redefined the scope
and efficiency of automation across numerous sectors [6, 8, 13]. These systems
have become a cornerstone in industries where precision and repeatability are
paramount. However, traditional vision-embedded robotic arms often grapple
with challenges when faced with unrecognized objects, due to the limitation of
object detection models [9, 11]. Their performance, though exceptional for
familiar entities, deteriorates significantly with novel items. This
constraint is further exacerbated by the inherent uncertainty in object
detection performance, which often requires extensive fine-tuning.
Additionally, many of these systems are not capable to process and response
instructions delivered in natural human language, which hinders their
adaptability and user-friendliness.

In light of these challenges, we introduce a design that seamlessly integrates
the state-of-the-art visual foundation model, the Segment Anything Model (SAM)
[10], into robotic arms. When combined with a mobile platform, this design has
the potential to revolutionize robotic arm functionalities. Our innovative
approach boasts a wide range of advantages:

  * •

Universal Object Recognition: The system’s adeptness at segmenting and
identifying a diverse set of objects negates the need for continual retraining
and minimizes associated costs.

  * •

Enhanced Human Interaction: Operators are presented with a spectrum of
interaction modalities, ranging from clicks and spatial drawings to natural
language prompts, fostering an intuitive user experience.

  * •

”Eye-in-Hand” System: With the vision system integral to the arm’s design,
precise closed-loop control is achieved. Real-time object tracking further
augments this, ensuring continuous object localization and optimal grasping.

  * •

Mobile Platform Integration: By housing the robotic arm on a mobile platform,
its operational scope is significantly broadened. Vision techniques augment
traditional tasks and furnish a comprehensive understanding of dynamic
environments, facilitating both enhanced navigation and strategic grasping.

The versatility of our mobile robotic arm design paves the way for
applications spanning multiple domains, not limited to:

  * •

Industrial Manufacturing: It promises to revolutionize processes by adeptly
identifying and managing a myriad of components, thus eliminating the
requirement for continual system retraining. Further, the integrated mobile
platform magnifies the robot’s operational breadth, marking it as an
indispensable tool within expansive industrial terrains.

  * •

Consumer Environments: This design’s potential materializes prominently in
public and service sectors. Picture these robotic entities operating in
settings like grocery stores, restaurants, and hotels, fulfilling tasks based
on directives from customers or staff.

  * •

Specialized Scenarios: Where our design truly stands unparalleled is in high-
stakes, critical situations, such as taking care of disables, risky item
removal. The blend of precision and adaptability, coupled with the enhanced
coverage of the mobile platform, makes this system a choice for diverse,
especially high-risk environments.

##  II Related Works

###  II-A Visual Foundation Models

A paradigm in the field of visual foundation models is the Segment Anything
Model (SAM) by Meta AI Research [10]. SAM has emerged as a vision foundation
model [18] recognized for its unparalleled proficiency in segmenting any
object within an image. Characterized by its transformer-based structure, SAM
is trained on a large segmentation dataset with over a billion masks.
Different from the other segmentation models, SAM performs promptable
segmentation without the reliance on labeled regions. Instead, it operates on
varied prompts, ranging from point prompts via clicks, region prompts through
drawing, to text prompts in natural language. This versatility is underscored
by its impressive zero-shot transfer performance, obviating any requirement
for finetuning. Furthermore, the model’s adaptability has catalyzed a myriad
of applications across diverse scenarios like scene understanding [3],
inpainting [16], object tracking [4], and pose estimation [7].

In this work, we introduce a robotic grasping system based on SAM, envisioning
to improve the quality of traditional robotic systems through this cutting-
edge vision technology. At the same time, our research also includes an
evaluation of SAM derivatives, especially Mobile SAM [17]. This streamlined
variant, achieved through knowledge distillation, results in a model that is
approximately 60 times more compact than its predecessor, while still
performing on par with the original SAM.

###  II-B Language-driven Robotic Manipulation

With the emergence of large language models (LLMs), research of embedding
language-driven technology into robot operations has been proposed [8, 19,
13]. [8] recently introduced an innovative robotic manipulation approach named
VoxPoser. This method seamlessly integrates a large language model (LLM) with
a visual-language model (VLM). Central to their approach is the LLM’s
capability to interpret free-form language instructions, which in tandem with
its code-generation capacity, interfaces with the VLM. This collaborative
interaction translates 3D value maps into spatially grounded knowledge to the
agent, which is subsequently utilized in model-based planning frameworks to
guide the agent’s movements.

While their methodology differs from ours, both approaches utilize foundation
models to enhance traditional robotic systems, making them more responsive to
human natural language directives. Nevertheless, their system is not without
its challenges. However, their system has notable limitations. Its
environmental understanding is constrained, particularly when representing
complex objects for grasping, such as irregular packages, complex assemblies,
etc. Although the LLMs are good at natural language interpretation and code
generation, they have a potential to issue erroneous commands due to its
hallucination phenomena, as echoed in [13, 2, 1]

Our work aims to bridge these identified gaps. It favors visual foundation
models for a more representative environmental understanding and employs
strategies with depth cameras, inverse kinematics, and motion planning, that
already institutionalized in the industry, for robotic arm manipulation. Such
a methodology significantly diminishes uncertainties common in LLMs.

###  II-C Integrating SAM into Robotic Grasping

[15] recently explored the incorporation of SAM into robotic grasping
paradigms, which presents a noteworthy advancement in robotic grasping.
However, it has several remarkable limitations. First, their design utilizes
dual fixed cameras, in contrast to our “eye-in-hand” system. An ”eye-in-hand”
setup is more adapt to real-world scenarios, particularly when the robotic arm
is stationed in the wild environment. In dynamic environments, fixed camera
positions are often impractical, emphasizing the importance of a robot-
oriented vision system. Second, their choice to compile a dataset for training
grasping strategies somewhat deviates from the essence of using SAM as a
“zero-shot” foundation model without the need for specialized training or
fine-tuning. Relying on such a dataset potentially limits the system’s
adaptability across diverse real-world tasks and settings.

In contrast to their approach, we’ve integrated an ”eye-in-hand” system with
our robotic arm on a mobile platform, enhancing its adaptability in diverse
environments. Bypassing the need for grasp strategy training, we utilize the
Denavit-Hartenberg (D-H) configurations [5], inverse kinematic estimations and
continuous object tracking for precise movements. Our methodology not only
eliminates the need for task-specific training datasets, avoiding potential
data biases, but also capitalizes on the power of foundational models,
presenting a cost-effective solution for both industrial and consumer use.

##  III Methodology

Input: Live video stream from the depth camera, User prompts for object of
interest

// Object Segmentation

frame, depth = captureStream();

segment, mask = SAM(frame, user_prompt);

// Depth Estimation & Positioning

$\bar{d}$ = avgDepth(mask, depth);

$\mathbf{P}_{cam}$ = computeCamCoords($\bar{d}$, mask);

$\mathbf{P}_{arm}$ = transformCoords($\mathbf{P}_{cam}$);

// Platform Movement

if  _$\mathbf{P}_{arm}$ is out of reach_ then

movePlatform($\mathbf{P}_{arm}$);

end if

// Inverse Kinematics

$\mathbf{\Theta}$ = computeIK($\mathbf{P}_{arm}$);

$\mathbf{\pi_{\theta}}$ = motionPlan($\mathbf{\Theta}$);

executeArmMotion($\mathbf{\pi_{\theta}}$);

// Feedback Loop for Tracking

while  _Arm is in motion_ do

frame, depth = captureStream();

$\mathbf{P}^{\prime}_{arm}$ = trackAndPos(segment, frame, depth);

$\mathbf{\Theta}^{\prime}$ = computeIK($\mathbf{P}^{\prime}_{arm}$);

adjustArmMotion($\mathbf{\Theta}^{\prime}$);

end while

// Grasping

executeGrip($I_{gripper}$, PID);

Algorithm 1 Mobile grasping system pseudo-algorithm

###  III-A System Overview

The proposed robotic system integrates vision capabilities with motion control
and enhanced mobility. This system is composed of two principal modules: the
Visual Interpretation Module (VIM) and the Motion Control Module (MCM), as
Fig. 1 shows. These modules are materially supported by two integral devices:
a 6 degree-of-freedom (DoF) robotic arm with an end effector, and a versatile
mobile platform (see details in Sec. IV-A). Alg. 1 shows a comprehensive
algorithm of the grasping procedure.

###  III-B Visual Interpretation Module (VIM)

The VIM serves as the system’s sensory and interpretative cornerstone. It
encompasses a depth-sensing camera and a suite of vision foundation models,
with SAM and its variants.

####  III-B1 Visual Foundation Model Integration

Central to the VIM is the integration of SAM-like visual foundation models
(see the “Object Segmentation” stage in Alg. 1) User interaction is
facilitated through diverse modalities: point-and-clicks, region drawings, or
spoken commands. Reacting to these cues, the VIM identifies and showcases the
target object, overlaying its segmented outline on the live video feed, as
shown in Fig. 1.

####  III-B2 Depth Estimation and Object Positioning

After obtaining user confirmation, the VIM derives the three-dimensional
coordinates of the designated object (see the “Depth Measurement and
Positioning” stage in Alg. 1). This crucial data is then dispatched to the
Motion Control Module, as shown in Fig. 1.

###  III-C Motion Control Module (MCM)

The MCM acts as the robotic system’s action executor, translating visual cues
into precise movements and ensuring effective object grasping, as shown in
Fig. 1 and Alg. 1.

####  III-C1 Platform Mobility

One advantage of our system is its mobile capability to relocate when an
object is beyond the robotic arm’s reach. Specifically, the algorithm dictates
that if the target’s 3D coordinate $\mathbf{P}_{arm}$ is outside the arm’s
operational range. Such a strategy ensures that the object is always
accessible, providing a foundational prerequisite for subsequent grasping
actions. This mobility is facilitated by an object tracking-based navigation
system, allowing for dynamic adjustments based on the real-time location of
the target.

####  III-C2 Motion Planning

Central to the robot’s grasping action is its ability to plan and execute
motions. This involves a two-step process: the calculation of desired joint
angles through inverse kinematics and then determining the arm’s trajectory
for movement. As described in Alg. 1, where $\mathbf{\Theta}$ represents the
desired joint angles for the robotic arm, and $\mathbf{\pi_{\theta}}$
illustrates the planned movement trajectory. The process ensures that the end
effector can effectively approach the target object in an optimized and
collision-free manner. The arm’s kinematic behavior is modeled mathematically
using the Denavit-Hartenberg (D-H) representation [5], ensuring a systematic
approach to finding its potential configurations.

####  III-C3 Tracking Feedback

Maintaining the arm’s trajectory accuracy is paramount, especially when
accounting for potential errors arising from depth perception or servo
movements. This continuous adjustment mechanism ensures the system remains
aligned with the target even amidst unforeseen discrepancies.

####  III-C4 Grasping Mechanism

Once the arm is in the desired position, the final step is the actual act of
grasping. The system utilizes a grip force modulated by a PID controller[14],
which ensures adaptability across objects of varying sizes, shapes, and
fragility. The term $I_{gripper}$ represents the current or force reading from
the gripper, facilitating feedback-driven grip adjustments.

(a)

(b)

Figure 2: Mobile Grasping Platform overview from different viewpoints

(a) Raw Image

(b) Depth Map

(c) SAM [10]

(d) Mobile SAM [17]

Figure 3: Comparison of SAM [10] and Mobile-SAM [17] performance on both
indoor and outdoor scenarios. Zoom-in for the best view.

##  IV Experiments and Results

###  IV-A Platform Configuration Overview

Our platform is a comprehensive assembly of a 6-DoF robotic arm and a mobile
platform, specifically an all-terrain crawler AGV, as shown in Fig. 2 shows.

####  IV-A1 Camera Selection

For depth perception, our experimental setup employed the Intel Realsense-D415
depth camera. Positioned at the pinnacle of the arm, adjacent to the end-
effector, this camera supports the “eye-in-hand” grasping approach.

####  IV-A2 Robotic Arm Joints

Our robotic arm, with its six degrees of freedom, is articulated through:

  * •

Joint-0: Base Horizontal Rotation

  * •

Joint-1: Upper Arm Pitch

  * •

Joint-2: Elbow Pitch

  * •

Joint-3: Wrist Pitch

  * •

Joint-4: Wrist Radial Rotation

  * •

Joint-5: Gripper Open/Close

####  IV-A3 Mobile Platform Configuration

In our experiment, the mobile platform employs the Hokuyo UTM-30LX LIDAR. This
choice complements the depth camera by complimenting its view blind spots,
supporting a more holistic sensing of the environment. Certain SLAM and
navigation algorithms [20, 12] are employed to support the movements.

###  IV-B Performance Comparison: Original SAM vs. Mobile SAM

For efficient, lightweight visual interpretation, we conducted a comparative
analysis between the original SAM and Mobile SAM. Demonstrative examples are
presented in Fig. 3. Across diverse environments (both indoor and outdoor),
Mobile SAM matches the performance of the original SAM given identical
prompts. Notably, Mobile SAM’s model size is 60 times smaller than its
predecessor, facilitating deployment on standard home-grade graphics cards.
This means it’s well-suited for economical industrial computers with GPU
support. Based on our tests, Mobile SAM can consistently deliver rapid
responses in around 50ms, with a single NVIDIA 3060 graphics card.

Fig. 3 also showcases various applications for robotic grasping. The module
adeptly segments a spectrum of objects, from daily items to more specialized
targets such as suspicious packages removal.

###  IV-C Simulated Grasping Results

Our preliminary testing process began with simulating the grasping sequence,
depicted in Fig. 4. In this simulation, an analogue camera captures the
object’s initial state. This video feed is subsequently relayed to our visual
interpretation module, facilitating segmentation and derivation of the 3D
object coordinates for the grasping action. Desired joint angles are then
determined using our motion control module, which leverages inverse kinematic
estimation coupled with the robotic arm’s D-H model configuration. To ensure
safety and assess feasibility, joint movements are simulated before execution.
Importantly, our actual robotic arm is interfaced with this simulation,
enabling synchronized movements between the two. This simulation framework
incorporates both our robotic arm model and the depth camera, highlighting the
advantages of our integrated eye-in-hand system.

(a) Initial pose

(b) Midway pose

(c) Final pose

Figure 4: The simulated grasping process is delineated across three primary
phases: initial, midway, and final poses. The analogue camera’s viewpoint is
presented on the top-left, while corresponding joint angles are displayed on
the top-right. Zoom-in for the best view.

###  IV-D Real-World Grasping Results

Progressing from simulation, we transitioned our algorithms to the actual
robotic system. A salient feature of our system, afforded by the mobile
platform, is its ability to bridge spatial gaps. If a target is beyond the
arm’s immediate reach, the robot autonomously navigates closer, guided by
object tracking. Once close, it executes the grasping procedure. Importantly,
the system demonstrated its versatility by operating proficiently in both
indoor and outdoor settings, as Fig. 5 demonstrates.

(a)

(b)

Figure 5: Our mobile grasping platform performs grasping in different
scenarios.

##  V Conclusion & Future Work

The integration of visual foundation model with robotic arms placed on mobile
platforms represents a new stage in the field of robotics. This combination
ensures a more intuitive user experience, as individuals can effortlessly
guide the robot by clicking, drawing, or speaking, directing it towards the
desired object for grasping. The system’s adaptability allows it to segment
and grasp a wide variety of items, expanding its applicability across numerous
domains. Furthermore, leveraging the visual foundation model as a ”zero-shot”
detector, our system avoids the traditional, tedious training and testing
cycles associated with grasping robotic arms in traditional industrial
settings.

In terms of future prospects and enhancements, we aim to further refine the
grasping algorithm by incorporating the detailed contours provided by the
visual foundation models. Understanding an object’s contour can not only
improve the grasping gesture but also minimize any potential damage to
sensitive items. This becomes especially crucial in delicate operations, such
as grasping a fragile cup for disabled people, where any deformation might
cause unintended consequences.

While the mobile SAM variant offers a significant reduction in size and
increased processing speed compared to its predecessor, visual foundation
models remain computationally demanding. Our ongoing ambition is to achieve
real-time segmentation with diminished GPU reliance, possibly even eliminating
it. Approaches such as knowledge distillation and model quantification
techniques stand as promising avenues to achieve this goal.

## References

  * [1] Yejin Bang, Samuel Cahyawijaya, Nayeon Lee, Wenliang Dai, Dan Su, Bryan Wilie, Holy Lovenia, Ziwei Ji, Tiezheng Yu, Willy Chung, et al.  A multitask, multilingual, multimodal evaluation of chatgpt on reasoning, hallucination, and interactivity.  arXiv preprint arXiv:2302.04023, 2023.
  * [2] Sébastien Bubeck, Varun Chandrasekaran, Ronen Eldan, Johannes Gehrke, Eric Horvitz, Ece Kamar, Peter Lee, Yin Tat Lee, Yuanzhi Li, Scott Lundberg, et al.  Sparks of artificial general intelligence: Early experiments with gpt-4.  arXiv preprint arXiv:2303.12712, 2023.
  * [3] Zhimin Chen and Bing Li.  Bridging the domain gap: Self-supervised 3d scene understanding with foundation models.  arXiv preprint arXiv:2305.08776, 2023.
  * [4] Yangming Cheng, Liulei Li, Yuanyou Xu, Xiaodi Li, Zongxin Yang, Wenguan Wang, and Yi Yang.  Segment and track anything.  arXiv preprint arXiv:2305.06558, 2023.
  * [5] Jacques Denavit and Richard S Hartenberg.  A kinematic notation for lower-pair mechanisms based on matrices.  1955\.
  * [6] Guoguang Du, Kai Wang, Shiguo Lian, and Kaiyong Zhao.  Vision-based robotic grasping from object localization, object pose estimation to grasp estimation for parallel grippers: a review.  Artificial Intelligence Review, 54(3):1677–1734, 2021.
  * [7] Zhiwen Fan, Panwang Pan, Peihao Wang, Yifan Jiang, Dejia Xu, Hanwen Jiang, and Zhangyang Wang.  Pope: 6-dof promptable pose estimation of any object, in any scene, with one reference.  arXiv preprint arXiv:2305.15727, 2023.
  * [8] Wenlong Huang, Chen Wang, Ruohan Zhang, Yunzhu Li, Jiajun Wu, and Li Fei-Fei.  Voxposer: Composable 3d value maps for robotic manipulation with language models.  arXiv preprint arXiv:2307.05973, 2023.
  * [9] Peiyuan Jiang, Daji Ergu, Fangyao Liu, Ying Cai, and Bo Ma.  A review of yolo algorithm developments.  Procedia Computer Science, 199:1066–1073, 2022.
  * [10] Alexander Kirillov, Eric Mintun, Nikhila Ravi, Hanzi Mao, Chloe Rolland, Laura Gustafson, Tete Xiao, Spencer Whitehead, Alexander C Berg, Wan-Yen Lo, et al.  Segment anything.  arXiv preprint arXiv:2304.02643, 2023.
  * [11] Wei Liu, Dragomir Anguelov, Dumitru Erhan, Christian Szegedy, Scott Reed, Cheng-Yang Fu, and Alexander C Berg.  Ssd: Single shot multibox detector.  In Computer Vision–ECCV 2016: 14th European Conference, Amsterdam, The Netherlands, October 11–14, 2016, Proceedings, Part I 14, pages 21–37. Springer, 2016.
  * [12] Steve Macenski and Ivona Jambrecic.  Slam toolbox: Slam for the dynamic world.  Journal of Open Source Software, 6(61):2783, 2021.
  * [13] Sai Vemprala, Rogerio Bonatti, Arthur Bucker, and Ashish Kapoor.  Chatgpt for robotics: Design principles and model abilities.  Microsoft Auton. Syst. Robot. Res, 2:20, 2023.
  * [14] Antonio Visioli.  Practical PID control.  Springer Science & Business Media, 2006.
  * [15] Jiange Yang, Wenhui Tan, Chuhao Jin, Bei Liu, Jianlong Fu, Ruihua Song, and Limin Wang.  Pave the way to grasp anything: Transferring foundation models for universal pick-place robots.  arXiv preprint arXiv:2306.05716, 2023.
  * [16] Tao Yu, Runseng Feng, Ruoyu Feng, Jinming Liu, Xin Jin, Wenjun Zeng, and Zhibo Chen.  Inpaint anything: Segment anything meets image inpainting.  arXiv preprint arXiv:2304.06790, 2023.
  * [17] Chaoning Zhang, Dongshen Han, Yu Qiao, Jung Uk Kim, Sung-Ho Bae, Seungkyu Lee, and Choong Seon Hong.  Faster segment anything: Towards lightweight sam for mobile applications.  arXiv preprint arXiv:2306.14289, 2023.
  * [18] Chaoning Zhang, Sheng Zheng, Chenghao Li, Yu Qiao, Taegoo Kang, Xinru Shan, Chenshuang Zhang, Caiyan Qin, Francois Rameau, Sung-Ho Bae, et al.  A survey on segment anything model (sam): Vision foundation model meets prompt engineering.  arXiv preprint arXiv:2306.06211, 2023.
  * [19] Shimian Zhang.  Bridging intelligence and instinct: A new control paradigm for autonomous robots.  arXiv preprint arXiv:2307.10690, 2023.
  * [20] Shimian Zhang and Qiuhong Lu.  A hierarchical searching approach for laser slam re-localization on mobile robot platform.  In 2019 IEEE International Conference on Real-time Computing and Robotics (RCAR), pages 464–467. IEEE, 2019.


(6.9) $\|\nabla\Pi_{\varepsilon}\|_{B^{1}_{\infty,1}}\leq
C\left(\left(1+\varepsilon\|\nabla
a_{\varepsilon}\|_{B^{0}_{\infty,1}}^{\lambda}\right)\|\nabla\Pi_{\varepsilon}\|_{L^{2}}+\varepsilon\,\|\varrho_{\varepsilon}\,{\rm
div}\,(\boldsymbol{u}_{\varepsilon}\cdot\nabla\boldsymbol{u}_{\varepsilon})\|_{B^{0}_{\infty,1}}+\|\varrho_{\varepsilon}\,{\rm
div}\,\boldsymbol{u}_{\varepsilon}^{\perp}\|_{B^{0}_{\infty,1}}\right)\,.$

The $L^{2}$ estimate for the pressure term follows in a similar way to one
performed in (3.11), i.e.

(6.10) $\|\nabla\Pi_{\varepsilon}\|_{L^{2}}\leq
C\,\varepsilon\,\|\boldsymbol{u}_{\varepsilon}\|_{L^{2}}\|\nabla\boldsymbol{u}_{\varepsilon}\|_{L^{\infty}}+\|\boldsymbol{u}_{\varepsilon}^{\perp}\|_{L^{2}}\,.$

Next, as showed above in the bound for
$\|\partial_{i}\boldsymbol{u}_{\varepsilon}\cdot\nabla
a_{\varepsilon}\|_{B^{0}_{\infty,1}}$, combining Bony’s decomposition with the
fact that ${\rm
div}\,(\boldsymbol{u}_{\varepsilon}\cdot\nabla\boldsymbol{u}_{\varepsilon})=\nabla\boldsymbol{u}_{\varepsilon}:\nabla\boldsymbol{u}_{\varepsilon}$,
we may infer:

$\|{\rm
div}\,(\boldsymbol{u}_{\varepsilon}\cdot\nabla\boldsymbol{u}_{\varepsilon})\|_{B^{0}_{\infty,1}}\leq
C\,\|\boldsymbol{u}_{\varepsilon}\|^{2}_{B^{1}_{\infty,1}}\,.$

Now, from Proposition 3 of [16], we can estimate the $B^{1}_{\infty,1}$ norm
of the density in the following way:

$\|\varrho_{\varepsilon}\|_{B^{1}_{\infty,1}}\leq
C\,\left(\overline{\varrho}+\varepsilon\,\|\nabla
a_{\varepsilon}\|_{B^{0}_{\infty,1}}\right)\,.$

Finally, plugging the $L^{2}$ estimate (6.10) and all the above inequalities
in (6.9), one may conclude that

(6.11) $\begin{split}\|\nabla\Pi_{\varepsilon}\|_{B^{1}_{\infty,1}}&\leq
C\left(1+\varepsilon\,\|\nabla
a_{\varepsilon}\|_{B^{0}_{\infty,1}}^{\lambda}\right)\left(\varepsilon\,\|\boldsymbol{u}_{\varepsilon}\|_{L^{2}}\|\nabla\boldsymbol{u}_{\varepsilon}\|_{B^{0}_{\infty,1}}+\|\boldsymbol{u}_{\varepsilon}\|_{L^{2}}\right)\\\
&+C\left(1+\varepsilon\,\|\nabla
a_{\varepsilon}\|_{B^{0}_{\infty,1}}\right)\left(\varepsilon\,\|\boldsymbol{u}_{\varepsilon}\|_{B^{1}_{\infty,1}}^{2}+\|\boldsymbol{u}_{\varepsilon}\|_{B^{1}_{\infty,1}}\right)\\\
&\leq
C(1+\varepsilon\,\mathcal{A}_{\varepsilon}^{\lambda})(\varepsilon\,E_{\varepsilon}^{2}+E_{\varepsilon})+C(1+\varepsilon\mathcal{A}_{\varepsilon})(\varepsilon\,E_{\varepsilon}^{2}+E_{\varepsilon})\\\
&\leq
C(\varepsilon\,E_{\varepsilon}^{2}+E_{\varepsilon})(1+\varepsilon\,\mathcal{A}_{\varepsilon}+\varepsilon\,\mathcal{A}_{\varepsilon}^{\lambda})\\\
&\leq
C(\varepsilon\,E_{\varepsilon}^{2}+E_{\varepsilon})(1+\varepsilon\,\mathcal{A}_{\varepsilon}^{\lambda})\,.\end{split}$

We insert now in (6.7) the estimates found in (6.11) and in (6.8), deducing
that

(6.12) $E_{\varepsilon}(t)\leq
C\left(E_{\varepsilon}(0)+\mathcal{B}_{\varepsilon}(0)\int_{0}^{t}\exp\left(C\int_{0}^{\tau}E_{\varepsilon}(s)\,\,{\rm
d}s\right)\left(\varepsilon\,E_{\varepsilon}^{2}(\tau)+E_{\varepsilon}(\tau)\right)\,\,{\rm
d}\tau\right)\left(1+\int_{0}^{t}E_{\varepsilon}(\tau)\,{\rm d}\tau\right),$

where we have set
$\mathcal{B}_{\varepsilon}(0):=\mathcal{A}_{\varepsilon}(0)+\varepsilon\,\mathcal{A}_{\varepsilon}(0)^{\lambda+1}$.

At this point, we define $T_{\varepsilon}^{\ast}>0$ such that

(6.13)
$T_{\varepsilon}^{\ast}:=\sup\left\\{t>0:\,\mathcal{B}_{\varepsilon}(0)\int_{0}^{t}\exp\left(C\int_{0}^{\tau}E_{\varepsilon}(s)\,\,{\rm
d}s\right)\left(\varepsilon\,E_{\varepsilon}^{2}(\tau)+E_{\varepsilon}(\tau)\right)\,\,{\rm
d}\tau\leq E_{\varepsilon}(0)\right\\}\,.$

So, from (6.12) and using Grönwall’s inequality, we obtain that

$E_{\varepsilon}(t)\leq C\,E_{\varepsilon}(0)e^{CtE_{\varepsilon}(0)}\,,$

for all $t\in[0,T_{\varepsilon}^{\ast}]$.

The previous estimate implies that, for all $t\in[0,T_{\varepsilon}^{\ast}]$,
one has

$\int_{0}^{t}E_{\varepsilon}(\tau)\,{\rm d}\tau\leq
e^{CtE_{\varepsilon}(0)}-1\,.$

Analogously to inequality (5.8) in Subsection 5.3, we can argue that

$\mathcal{B}_{\varepsilon}(0)\int_{0}^{t}\exp\left(C\int_{0}^{\tau}E_{\varepsilon}(s)\,\,{\rm
d}s\right)E_{\varepsilon}(\tau)\,\,{\rm d}\tau\leq
C\mathcal{B}_{\varepsilon}(0)\left(\exp\left(e^{CtE_{\varepsilon}(0)}-1\right)-1\right)\,.$

Then, it remains to control

$\varepsilon\mathcal{B}_{\varepsilon}(0)\int_{0}^{t}\exp\left(C\int_{0}^{\tau}E_{\varepsilon}(s)\,\,{\rm
d}s\right)E_{\varepsilon}^{2}(\tau)\,\,{\rm d}\tau\,.$

For this term, we may infer that

$\begin{split}\varepsilon\mathcal{B}_{\varepsilon}(0)\int_{0}^{t}\exp\left(C\int_{0}^{\tau}E_{\varepsilon}(s)\,\,{\rm
d}s\right)E_{\varepsilon}^{2}(\tau)\,\,{\rm d}\tau&\leq
C\,\varepsilon\,\mathcal{B}_{\varepsilon}(0)\int_{0}^{t}E_{\varepsilon}^{2}(0)e^{CtE_{\varepsilon}(0)}\,\exp\left(e^{CtE_{\varepsilon}(0)}-1\right)\,\,{\rm
d}\tau\\\ &\leq
C\,\varepsilon\,\mathcal{B}_{\varepsilon}(0)E_{\varepsilon}(0)\left(\exp\left(e^{CtE_{\varepsilon}(0)}-1\right)-1\right)\,.\end{split}$

In the end, by definition (6.13) of $T^{\ast}_{\varepsilon}$, we deduce

(6.14)
$T^{\ast}_{\varepsilon}\geq\frac{C}{E_{\varepsilon}(0)}\log\left(\log\left(\frac{CE_{\varepsilon}(0)}{\max\\{\mathcal{B}_{\varepsilon}(0),\,\varepsilon\,\mathcal{B}_{\varepsilon}(0)E_{\varepsilon}(0)\\}}+1\right)+1\right),$

for a suitable constant $C>0$. This concludes the proof of Theorem 2.1.

##  Appendix A Appendix - A few tools from Fourier analysis

This appendix is devoted to present some tools from Fourier analysis, which we
have exploited in our analysis. We recall some definitions and properties of
Littlewood-Paley theory, Besov spaces and paradifferential calculus. Moreover,
using those notions, we will focus on the study of transport equations in
Besov spaces.

###  A.1 Littlewood-Paley theory and Besov spaces

In this paragraph, we recall the main results concerning the Littlewood-Paley
theory. We refer e.g. to Chapter 2 of [4] for details. For simplicity of
exposition, let us deal with the $\mathbb{R}^{d}$ case, with $d\geq 1$;
however, the whole construction can be adapted also to the $d$-dimensional
torus $\mathbb{T}^{d}$, and to the “hybrid” case
$\mathbb{R}^{d_{1}}\times\mathbb{T}^{d_{2}}$.

First of all, let us introduce the _Littlewood-Paley decomposition_. For this
we fix a smooth radial function $\chi$ such that ${\rm Supp}\,\chi\subset
B(0,2)$, $\chi\equiv 1$ in a neighborhood of $B(0,1)$ and the map
$r\mapsto\chi(r\,e)$ is non-increasing over $\mathbb{R}_{+}$ for all unitary
vectors $e\in\mathbb{R}^{d}$. Set
$\varphi\left(\xi\right)=\chi\left(\xi\right)-\chi\left(2\xi\right)$ and
$\varphi_{j}(\xi):=\varphi(2^{-j}\xi)$ for all $j\geq 0$. The dyadic blocks
$(\Delta_{j})_{j\in\mathbb{Z}}$ are defined by111We agree that $f(D)$ stands
for the pseudo-differential operator
$u\mapsto\mathcal{F}^{-1}[f(\xi)\,\widehat{u}(\xi)]$.

$\Delta_{j}\,:=\,0\quad\mbox{ if
}\;j\leq-2,\qquad\Delta_{-1}\,:=\,\chi(D)\qquad\mbox{ and
}\qquad\Delta_{j}\,:=\,\varphi(2^{-j}D)\quad\mbox{ if }\;j\geq 0\,.$

For any $j\geq 0$ fixed, we also introduce the _low frequency cut-off
operator_

(A.1) $S_{j}\,:=\,\chi(2^{-j}D)\,=\,\sum_{k\leq j-1}\Delta_{k}\,.$

Note that $S_{j}$ is a convolution operator. More precisely, after defining

$K_{0}\,:=\,\mathcal{F}^{-1}\chi\qquad\qquad\mbox{ and }\qquad\qquad
K_{j}(x)\,:=\,\mathcal{F}^{-1}[\chi(2^{-j}\cdot)](x)=2^{jd}K_{0}(2^{j}x)\,,$

for all $j\in\mathbb{N}$ and all tempered distributions
$u\in\mathcal{S}^{\prime}$, we have that $S_{j}u\,=\,K_{j}\,*\,u$. Thus the
$L^{1}$ norm of $K_{j}$ is independent of $j\geq 0$, hence $S_{j}$ maps
continuously $L^{p}$ into itself, for any $1\leq p\leq+\infty$.

The following classical property holds true: for any
$u\in\mathcal{S}^{\prime}$, then one has the equality $u=\sum_{j}\Delta_{j}u$
in the sense of $\mathcal{S}^{\prime}$. Let us also recall the so-called
_Bernstein inequalities_.

######  Lemma A.1.

Let $0<r<R$. A constant $C$ exists so that, for any non-negative integer $k$,
any couple $(p,q)$ in $[1,+\infty]^{2}$, with $p\leq q$, and any function
$u\in L^{p}$, we have, for all $\lambda>0$,

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle{{\rm Supp}\,}\,\widehat{u}\subset B(0,\lambda
R)\quad\Longrightarrow\quad\|\nabla^{k}u\|_{L^{q}}\,\leq\,C^{k+1}\,\lambda^{k+d\left(\frac{1}{p}-\frac{1}{q}\right)}\,\|u\|_{L^{p}}\;;\cr
0.0pt{\hfil$\displaystyle{{\rm
Supp}\,}\,\widehat{u}\subset\\{\xi\in\mathbb{R}^{d}\,:\,\lambda
r\leq|\xi|\leq\lambda R\\}\quad\Longrightarrow\quad
C^{-k-1}\,\lambda^{k}\|u\|_{L^{p}}\,\leq\,\|\nabla^{k}u\|_{L^{p}}\,\leq\,C^{k+1}\,\lambda^{k}\|u\|_{L^{p}}\,.\crcr}}}$

By use of Littlewood-Paley decomposition, we can define the class of Besov
spaces.

######  Definition A.2.

Let $s\in\mathbb{R}$ and $1\leq p,r\leq+\infty$. The _non-homogeneous Besov
space_ $B^{s}_{p,r}$ is defined as the subset of tempered distributions $u$
for which

$\|u\|_{B^{s}_{p,r}}\,:=\,\left\|\left(2^{js}\,\|\Delta_{j}u\|_{L^{p}}\right)_{j\geq-1}\right\|_{\ell^{r}}\,<\,+\infty\,.$

Besov spaces are interpolation spaces between Sobolev spaces. In fact, for any
$k\in\mathbb{N}$ and $p\in[1,+\infty]$ we have the chain of continuous
embeddings $B^{k}_{p,1}\hookrightarrow W^{k,p}\hookrightarrow
B^{k}_{p,\infty}$, which, when $1<p<+\infty$, can be refined to

$B^{k}_{p,\min(p,2)}\hookrightarrow W^{k,p}\hookrightarrow
B^{k}_{p,\max(p,2)}\,.$

In particular, for all $s\in\mathbb{R}$ we deduce that $B^{s}_{2,2}\equiv
H^{s}$, with equivalence of norms:

(A.2)
$\|f\|_{H^{s}}\,\sim\,\left(\sum_{j\geq-1}2^{2js}\,\|\Delta_{j}f\|^{2}_{L^{2}}\right)^{\\!\\!1/2}\,.$

As an immediate consequence of the first Bernstein inequality, one gets the
following embedding result, which generalises the Sobolev embeddings.

######  Proposition A.3.

The space $B^{s_{1}}_{p_{1},r_{1}}$ is continuously embedded in the space
$B^{s_{2}}_{p_{2},r_{2}}$ for all indices satisfying $p_{1}\,\leq\,p_{2}$ and
either $s_{2}\,<\,s_{1}-d\big{(}1/p_{1}-1/p_{2}\big{)}$, or
$s_{2}\,=\,s_{1}-d\big{(}1/p_{1}-1/p_{2}\big{)}$ and $r_{1}\leq r_{2}$.

In particular, we get the following chain of continuous embeddings:

$B^{s}_{p,r}\hookrightarrow B^{s-d/p}_{\infty,r}\hookrightarrow
B^{0}_{\infty,1}\hookrightarrow L^{\infty}\,,$

whenever the triplet $(s,p,r)\in\mathbb{R}\times[1,+\infty]^{2}$ satisfies

(A.3) $s>\frac{d}{p}\quad\quad\quad\text{or}\quad\quad\quad
s=\frac{d}{p}\quad\text{and}\quad r=1\,.$

###  A.2 Paradifferential calculus

Let us now introduce the Bony decomposition (see [6]). Once again, we refer to
Chapter 2 of [4] for details. Formally, the product of two tempered
distributions $u$ and $v$ can be decomposed into

$u\,v=\mathcal{T}_{u}v+\mathcal{T}_{v}u+\mathcal{R}(u,v)\,,$

where the paraproduct $\mathcal{T}$ and the remainder $\mathcal{R}$ are
defined as follows:

$\mathcal{T}_{u}v=\sum_{j}S_{j-1}u\,\Delta_{j}v\quad\text{and}\quad\mathcal{R}(u,v):=\sum_{j}\sum_{|k-j|}\Delta_{j}u\,\Delta_{k}v\,.$

The paraproduct and remainder operators have nice continuity properties. The
following ones will be of constant use in the paper.

######  Proposition A.4.

For any $(s,p,r)\in\mathbb{R}\times[1,+\infty]^{2}$ and $t>0$, the paraproduct
operator $\mathcal{T}$ maps continuously $L^{\infty}\times B^{s}_{p,r}$ into
$B^{s}_{p,r}$ and $B^{-t}_{\infty,\infty}\times B^{s}_{p,r}$ into
$B^{s-t}_{p,r}$. Moreover, we have the following estimates

$\|\mathcal{T}_{u}v\|_{B^{s}_{p,r}}\leq C\|u\|_{L^{\infty}}\|\nabla
v\|_{B^{s-1}_{p,r}}\quad\text{and}\quad\|\mathcal{T}_{u}v\|_{B^{s-t}_{p,r}}\leq
C\|u\|_{B^{-t}_{\infty,\infty}}\|\nabla v\|_{B^{s-1}_{p,r}}\,.$

For any $(s_{1},p_{1},r_{1})$ and $(s_{2},p_{2},r_{2})$ in
$\mathbb{R}\times[1,+\infty]^{2}$ such that $s_{1}+s_{2}>0$,
$1/p:=1/p_{1}+1/p_{2}\leq 1$ and $1/r:=1/r_{1}+1/r_{2}\leq 1$, the remainder
operator $\mathcal{R}$ maps continuously $B^{s_{1}}_{p_{1},r_{1}}\times
B^{s_{2}}_{p_{2},r_{2}}$ into $B^{s_{1}+s_{2}}_{p,r}$. In the case when
$s_{1}+s_{2}=0$ and $1/r_{1}+1/r_{2}=1$, the operator $\mathcal{R}$ is
continuous from $B^{s_{1}}_{p_{1},r_{1}}\times B^{s_{2}}_{p_{2},r_{2}}$ to
$B^{0}_{p,\infty}$.

As a consequence of the Proposition A.4, the spaces $B^{s}_{p,r}$ are Banach
algebras provided condition (A.3) holds for $s>0$. Moreover, in that case, we
have the so-called tame estimates.

######  Corollary A.5.

Let $(s,p,r)\in\mathbb{R}\times[1,+\infty]^{2}$. Then, for every $f,g\in
L^{\infty}\cap B^{s}_{p,r}$, one has

$\|fg\|_{B^{s}_{p,r}}\leq
C\left(\|f\|_{L^{\infty}}\|g\|_{B^{s}_{p,r}}+\|f\|_{B^{s}_{p,r}}\|g\|_{L^{\infty}}\right)\,.$

######  Remark A.6.

The space $B^{0}_{\infty,1}$ is not an algebra. If $f,g\in B^{0}_{\infty,1}$,
applying Proposition A.4, one can bound the paraproducts $\mathcal{T}_{f}g$
and $\mathcal{T}_{g}f$ but not the remainder $\mathcal{R}(f,g)$.

To end this paragraph, we present a fine estimate for products in which one of
the two functions is only bounded but its gradient belongs to the Besov space
$B^{s-1}_{p,r}$.

######  Proposition A.7.

Let $(s,p,r)\in\;]0,+\infty[\;\times\,[1,+\infty]^{2}$ satisfies condition
(A.3). Assume that $g\in L^{\infty}\cap B^{s}_{p,r}$ and $f$ is a bounded
function such that $\nabla f\in B^{s-1}_{p,r}$. Then, the product $fg$ belongs
to $L^{\infty}\cap B^{s}_{p,r}$ and one has the following estimate:

$\|fg\|_{B^{s}_{p,r}}\leq
C\left(\|f\|_{L^{\infty}}\|g\|_{B^{s}_{p,r}}+\|\nabla
f\|_{B^{s-1}_{p,r}}\|g\|_{L^{\infty}}\right)\,.$

###### Proof.

Taking advantage of Bony decomposition, one can write

$fg=\mathcal{T}_{f}g+\mathcal{T}_{g}f+\mathcal{R}(f,g)$

and employing Proposition A.4, we get

$\begin{split}\|\mathcal{T}_{f}g\|_{B^{s}_{p,r}}&\leq
C\|f\|_{L^{\infty}}\|g\|_{B^{s}_{p,r}}\\\
\|\mathcal{T}_{g}\,f\|_{B^{s}_{p,r}}&\leq C\|g\|_{L^{\infty}}\|\nabla
f\|_{B^{s-1}_{p,r}}\\\ \|\mathcal{R}(f,g)\|_{B^{s}_{p,r}}&\leq
C\|f\|_{B^{0}_{\infty,\infty}}\|g\|_{B^{s}_{p,r}}\leq
C\|f\|_{L^{\infty}}\|g\|_{B^{s}_{p,r}}\,.\end{split}$

This completes the proof of the proposition. ∎

###  A.3 Commutator estimates

In this paragraph, we recall the main commutator estimates widely employed
throughout the paper (we refer to Chapter 2 of [4] for full details).

######  Definition A.8.

We say that the triplet $(s,p,r)\in\mathbb{R}\times[1,+\infty]^{2}$ satisfies
the Lipschitz condition if

(A.4) $s>1+d/p\quad\text{and}\quad
r\in[1,+\infty]\quad\quad\quad\text{or}\quad\quad\quad
s=1+d/p\quad\text{and}\quad r=1\,.$

The proof of the following Lemma A.9 can be found in [15] by Danchin.

######  Lemma A.9.

Let $(s,p,r)$ satisfy condition (A.4) and $\sigma$ be in $]-1,\,s-1]$. Assume
that $w\in B^{\sigma}_{p,r}$ and $A$ is a bounded function on $\mathbb{R}^{d}$
such that $\nabla A\in B^{s-1}_{p,r}$. Then, there exists a constant
$C=C(s,p,r,\sigma,d)$ such that for all $i\in\\{1,\dots,d\\}$, we have:

$\|\partial_{i}[A,\Delta_{j}]w\|_{L^{p}}\leq C\,c_{j}\,2^{-j\sigma}\|\nabla
A\|_{B^{s-1}_{p,r}}\|w\|_{B^{\sigma}_{p,r}}\quad\text{for all }j\geq-1\,,$

with $\|(c_{j})_{j\geq-1}\|_{\ell^{r}}=1$.

The next statement concerns a standard commutator estimate between the
transport operator and the frequency localisation operator.

######  Lemma A.10.

Assume that $v\in B^{s}_{p,r}$ with $(s,p,r)$ satisfying the Lipschitz
condition (A.4). Denote by
$[v\cdot\nabla,\Delta_{j}]f=(v\cdot\nabla)\Delta_{j}f-\Delta_{j}(v\cdot\nabla)f$
the commutator between the transport operator $v\cdot\nabla$ and the frequency
localisation operator $\Delta_{j}$. Then, for every $f\in B^{s}_{p,r}$,

$\left\|\left(2^{js}\|[v\cdot\nabla,\Delta_{j}]f\|_{L^{p}}\right)_{j}\right\|_{\ell^{r}}\leq
C\left(\|\nabla v\|_{L^{\infty}}\|f\|_{B^{s}_{p,r}}+\|\nabla
v\|_{B^{s-1}_{p,r}}\|\nabla f\|_{L^{\infty}}\right)$

and also, for every $f\in B^{s-1}_{p,r}$,

$\left\|\left(2^{j(s-1)}\|[v\cdot\nabla,\Delta_{j}]f\|_{L^{p}}\right)_{j}\right\|_{\ell^{r}}\leq
C\left(\|\nabla v\|_{L^{\infty}}\|f\|_{B^{s-1}_{p,r}}+\|\nabla
v\|_{B^{s-1}_{p,r}}\|f\|_{L^{\infty}}\right)\,,$

for some constant $C=C(d,s,p)>0$.

Finally, the next result deals with commutators between paraproduct operators
and Fourier multipliers.

######  Lemma A.11.

Let $\kappa$ be a smooth function on $\mathbb{R}^{d}$, which is homogeneous of
degree $m$ away from a neighborhood of $0$. Take
$(s,p,r)\in\mathbb{R}\times[1,+\infty]^{2}$ and $v$ a vector field such that
$\nabla v\in L^{\infty}$. Then, for every $f\in B^{s}_{p,r}$, one has

$\left\|[T_{v},\kappa(D)]f\right\|_{B^{s-m+1}_{p,r}}\leq C(d,s)\,\|\nabla
v\|_{L^{\infty}}\|f\|_{B^{s}_{p,r}}\,.$

###  A.4 Transport equations

In this paragraph, we deal with the transport equations in non-homogeneous
Besov spaces. We refer to Chapter 3 of [4] for additional details. We study,
in $\mathbb{R}_{+}\times\mathbb{R}^{d}$, the initial value problem

(A.5) $\begin{cases}\partial_{t}f+v\cdot\nabla f=g\\\
f_{|t=0}=f_{0}\,.\end{cases}$

We always assume the velocity field $v=v(t,x)$ to be a Lipschitz divergence-
free function. In the case when the Lipschitz condition (A.4) is satisfied, we
have the embedding $B^{s}_{p,r}\hookrightarrow W^{1,\infty}$.

We state now the main well-posedness result concerning problem (A.5) in Besov
spaces. We point out also that the notation $C^{0}_{w}([0,T];X)$, with $X$ a
Banach space, refers to the space of functions which are continuous in time
with values in $X$ endowed with its weak topology.

######  Theorem A.12.

Let $(s,p,r)\in\mathbb{R}\times[1,+\infty]^{2}$ satisfies the Lipschitz
condition (A.4). Given $T>0$, take $g\in L^{1}_{T}(B^{s}_{p,r})$. Assume that
$v\in L^{1}_{T}(B^{s}_{p,r})$ and that there exist two real numbers $q>1$ and
$\sigma>0$ such that $v\in L^{q}_{T}(B^{-\sigma}_{\infty,\infty})$. Finally,
let $f_{0}\in B^{s}_{p,r}$ be the initial datum. Then, the transport equation
(A.5) has a unique solution $f$ in:

  * •

the space $C^{0}\left([0,T];B^{s}_{p,r}\right)$ if $r<+\infty$;

  * •

the space
$\left(\displaystyle\bigcap_{s^{\prime}<s}C^{0}\left([0,T];B^{s^{\prime}}_{p,\infty}\right)\right)\cap
C^{0}_{w}\left([0,T];B^{s}_{p,\infty}\right)$ if $r=+\infty$.

Moreover, the unique solution satisfies the following estimate:

$\|f\|_{L^{\infty}_{T}(B^{s}_{p,r})}\leq\exp\left(C\int_{0}^{T}\|\nabla
v\|_{B^{s-1}_{p,r}}\,\,{\rm
d}\tau\right)\left(\|f_{0}\|_{B^{s}_{p,r}}+\int_{0}^{T}\exp\left(-C\int_{0}^{t}\|\nabla
v\|_{B^{s-1}_{p,r}}\,\,{\rm d}\tau\right)\|g(t)\|_{B^{s}_{p,r}}\,\,{\rm
d}t\right),$

for some constant $C=C(d,s,p,r)>0$.

To conclude this paragraph, let us show a refinement of Theorem A.12, proved
by Vishik in [43] and in a different way by Hmidi and Keraani (see [34]). It
states that, if ${\rm div}\,v=0$ and the Besov regularity index is $s=0$, the
estimate in Theorem A.12 can be replaced by an inequality which is linear with
respect to $\|\nabla v\|_{L^{1}_{T}(L^{\infty})}$.

######  Theorem A.13.

Given $T>0$, assume that $v$ is a divergence-free vector field such that
$\nabla v\in L^{1}_{T}(L^{\infty})$ and let $g\in
L^{1}_{T}(B^{0}_{\infty,r})$. Take $r\in[1,+\infty]$ and $f_{0}\in
L^{1}_{T}(B^{0}_{\infty,r})$. Then, there exists a constant $C=C(d)$ such
that, for any solution $f$ to problem (A.5) in $C^{0}([0,T];B^{0}_{\infty,r})$
(or with the usual modification of $C^{0}$ into $C^{0}_{w}$ if $r=+\infty$),
we have

(A.6) $\|f\|_{L^{\infty}_{T}(B^{0}_{\infty,r})}\leq
C\left(\|f_{0}\|_{B^{0}_{\infty,r}}+\|g\|_{L^{1}_{T}(B^{0}_{\infty,r})}\right)\left(1+\int_{0}^{T}\|\nabla
v(\tau)\|_{L^{\infty}}\,\,{\rm d}\tau\right)\,.$

## References

  * [1] A. Babin, A. Mahalov, B. Nicolaenko: Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids. European J. Mech. B/Fluids, 15 (1996), n. 3, 291-300.
  * [2] A. Babin, A. Mahalov, B. Nicolaenko: Regularity and integrability of 3D Euler and Navier-Stokes equations for rotating fluids. Asymptot. Anal., 15 (1997), n. 2, 103-150.
  * [3] H. Bae, W. Lee, J. Shin: A blow-up criterion for the inhomogeneous incompressible Euler equations. Nonlinear Anal., 196 (2020), 111774.
  * [4] H. Bahouri, J.-Y. Chemin, R. Danchin: “Fourier Analysis and Nonlinear Partial Differential Equations”. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 343, Springer, Heidelberg (2011).
  * [5] J. Beale, T. Kato, A. Majda: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys., 94 (1984), n. 1, 61-66.
  * [6] J.-M. Bony: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup., 14 (1981), n. 2, 209-246.
  * [7] D. Bresch, B. Desjardins: Existence of global weak solutions for a 2D viscous shallow water equation and convergence to the quasi-geostrophic model. Comm. Math. Phys., 238 (2003), n. 1-2, 211-223.
  * [8] F. Charve: Global Well-Posedness and Asymptotics for a Geophysical Fluid System. Comm. Partial Differential Equations, 29 (2004), n. 11-12, 1919-1940.
  * [9] J.-Y. Chemin, B. Desjardins, I. Gallagher, E. Grenier: “Mathematical geophysics. An introduction to rotating fluids and the Navier-Stokes equations”. Oxford Lecture Series in Mathematics and its Applications, 32, Oxford University Press, Oxford (2006).
  * [10] D. Cobb, F. Fanelli: On the fast rotation asymptotics of a non-homogeneous incompressible MHD system. Nonlinearity, 34 (2021), n. 4, 2483.
  * [11] D. Cobb, F. Fanelli: Rigorous derivation and well-posedness of a quasi-homogeneous ideal MHD system. Nonlinear Anal. Real World Appl., 60 (2021), 103284.
  * [12] D. Cobb, F. Fanelli: Elsässer formulation of the ideal MHD and improved lifespan in two space dimensions. Submitted (2020).
  * [13] B. Cushman-Roisin:“Introduction to Geophysical Fluid Dynamics”. Prentice-Hall, Englewood Cliffs (1994).
  * [14] R. Danchin: The inviscid limit for density dependent incompressible fluids. Ann. Fac. Sci. Toulouse Math., 15 (2006), n. 4, 637-688.
  * [15] R. Danchin: On the well-posedness of the incompressible density-dependent Euler equations in the $L^{p}$ framework. J. Differential Equations, 248 (2010), n. 8, 2130-2170.
  * [16] R. Danchin, F. Fanelli: The well-posedness issue for the density-dependent Euler equations in endpoint Besov spaces. J. Math. Pures Appl. (9), 96 (2011), n. 3, 253-278.
  * [17] A. Dutrifoy: Fast averaging for long- and short-wave scaled equatiorial shallow water equations with Coriolis parameter deviating from linearity. Arch. Ration. Mech. Anal., 216 (2015), 261-312.
  * [18] A. Dutrifoy: Examples of dispersive effects in non-viscous rotating fluids. J. Math. Pures Appl. (9), 84 (2005), n. 3, 331-356.
  * [19] A. Dutrifoy, A. Majda: The dynamics of equatorial long waves: a singular limit with fast variable coefficients. Commun. Math. Sci., 4 (2006), 375-397.
  * [20] A. Dutrifoy, A. Majda, S. Schochet: A simple justification of the singular limit for equatorial shallow-water dynamics. Comm. Pure Appl. Math., 62 (2009), 322-333.
  * [21] F. Fanelli, I. Gallagher: Asymptotics of fast rotating density-dependent fluids in two space dimensions. Rev. Mat. Iberoam., 35 (2019), n. 6, 1763-1807.
  * [22] E. Feireisl, I. Gallagher, D. Gérard-Varet, A. Novotný: Multi-scale analysis of compressible viscous and rotating fluids. Comm. Math. Phys., 314 (2012), n. 3, 641-670.
  * [23] E. Feireisl, I. Gallagher, A. Novotný: A singular limit for compressible rotating fluids. SIAM J. Math. Anal., 44 (2012), n. 1, 192-205.
  * [24] E. Feireisl, B. J. Jin, A. Novotný: Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system. J. Math. Fluid Mech., 14 (2012), n. 4, 717-730.
  * [25] E. Feireisl, A. Novotný: “Singular limits in thermodynamics of viscous fluids”. Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel (2009).
  * [26] E. Feireisl, A. Novotný: Scale interactions in compressible rotating fluids. Ann. Mat. Pura Appl., 193 (2014), n. 6, 1703-1725.
  * [27] E. Feireisl, A. Novotný: Multiple scales and singular limits for compressible rotating fluids with general initial data. Comm. Partial Differential Equations, 39 (2014), n. 6, 1104-1127.
  * [28] E. Feireisl, A. Novotný, Y. Sun: Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids. Indiana Univ. Math. J., 60 (2011), n. 2, 611-631.
  * [29] I. Gallagher: A mathematical review of the analysis of the betaplane model and equatorial waves. Discrete Contin. Dyn. Syst. Ser. S, 1 (2008), n. 3, 461-480.
  * [30] I. Gallagher: The Tridimensional Navier-Stokes Equations with Almost Bidimensional Data: Stability; Uniqueness; and Life Span. Int. Math. Research Notices, (1997), n. 18, 919-935.
  * [31] I. Gallagher, L. Saint-Raymond: Weak convergence results for inhomogeneous rotating fluid equations. J. Anal. Math., 99 (2006), 1-34.
  * [32] I. Gallagher, L. Saint-Raymond: Mathematical study of the betaplane model: equatorial waves and convergence results. Mém. Soc. Math. Fr., 107 (2006).
  * [33] P. Germain: Weak-strong uniqueness for the isentropic compressible Navier-Stokes system. J. Math. Fluid Mech., 13 (2011), n. 1, 137-146.
  * [34] T. Hmidi, S. Keraani: Incompressible viscous flows in borderline Besov spaces. Arch. Ration. Mech. Anal., 189 (2008), n. 2, 283-300.
  * [35] P.-L. Lions, N. Masmoudi: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl., 77 (1998), n. 6, 585-627.
  * [36] A. Majda: “Introduction to PDEs and waves for the atmosphere and ocean”. Courant Lecture Notes in Mathematics, 9, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2003).
  * [37] V.-S. Ngo, S. Scrobogna: Dispersive effects of weakly compressible and fast rotating inviscid fluids. Discrete Contin. Dyn. Syst., 38 (2018), n. 2, 749-789.
  * [38] J. Pedlosky: “Geophysical fluid dynamics”. Springer-Verlag, New-York (1987).
  * [39] G. Prodi: Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. (4), 48 (1959), 173-182.
  * [40] S. Scrobogna: Highly rotating fluids with vertical stratification for periodic data and vanishing vertical viscosity. Rev. Mat. Iberoam., 34 (2018), n. 1, 1-58.
  * [41] J. Serrin: The initial value problem for the Navier-Stokes equations. Nonlinear Problems (Proc. Sympos., Madison, Wis., 1962), Univ. of Wisconsin Press, Madison, Wis. (1963), 69-98.
  * [42] G. K. Vallis: “Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation”. Cambridge University Press, Cambridge (2006).
  * [43] M. Vishik: Hydrodynamics in Besov spaces. Arch. Ration. Mech. Anal., 145 (1998), n. 3, 197-214.