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This research paper establishes the global well-posedness of the 3D incompressible Navier-Stokes equations and the 3D parabolic-elliptic Keller-Segel system in the framework of Fourier-Besov spaces with variable regularity and integrability indices.

Abstract

**Bibliographic Information:**Vergara-Hermosilla, G., & Zhao, J. (2024). Global well-posedness of the Navier-Stokes equations and the Keller-Segel system in variable Fourier-Besov spaces.*arXiv preprint arXiv:2410.05293v1*.**Research Objective:**This paper aims to prove the global well-posedness of the 3D incompressible Navier-Stokes equations and the 3D parabolic-elliptic Keller-Segel system in the context of Fourier-Besov spaces with variable regularity and integrability indices.**Methodology:**The authors utilize the framework of Fourier-Besov spaces with variable exponents and employ techniques such as Bony's paraproduct decomposition, Bernstein type inequalities, and linear estimates for the heat equation in these spaces. They establish key product estimates and utilize the structure of the equations to overcome difficulties posed by the variable exponents.**Key Findings:**The paper successfully demonstrates the existence, uniqueness, and continuous dependence on initial data of global solutions for both the Navier-Stokes equations and the Keller-Segel system under specific conditions on the variable exponents and initial data.**Main Conclusions:**The research concludes that both the 3D Navier-Stokes equations and the 3D Keller-Segel system are globally well-posed for small initial data in the chosen variable Fourier-Besov spaces. This extends previous well-posedness results obtained in classical function spaces and highlights the applicability of variable exponent spaces in analyzing fluid dynamics and mathematical biology models.**Significance:**This work contributes significantly to the study of partial differential equations by extending well-posedness results to the broader framework of variable exponent Fourier-Besov spaces. This opens up new avenues for analyzing PDEs arising in fluid mechanics and chemotaxis models.**Limitations and Future Research:**The results are limited to the specific range of variable exponents considered. Future research could explore extending these results to a wider class of exponents or investigating the behavior of solutions for large initial data in this framework.

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The provided context focuses on the Cauchy problem for both the Navier-Stokes equations and the Keller-Segel system, meaning the equations are considered on the full spatial domain $\mathbb{R}^3$ without any physical boundaries. Introducing boundary conditions significantly impacts the analysis and results:
Navier-Stokes Equations:
Dirichlet Boundary Conditions (No-slip): This is the most common condition, requiring the fluid velocity to vanish at the boundary. The presence of boundaries introduces boundary layers, regions near the boundary where viscosity plays a dominant role. The analysis becomes significantly more complex, and global well-posedness for large data in 3D remains a major open problem, even in classical Sobolev spaces.
Neumann Boundary Conditions (Slip): These specify the normal derivative of the velocity at the boundary. They are less physically realistic but mathematically simpler. Global well-posedness results are more readily available for certain geometries.
Periodic Boundary Conditions: These are often used to approximate an infinite domain. The analysis simplifies, and some techniques from the Cauchy problem can be adapted.
Keller-Segel System:
Neumann Boundary Conditions (No-flux): These are commonly used to model situations where cells cannot cross the boundary. They prevent the total mass of cells from changing, which is crucial for preventing blow-up of solutions in some cases.
Dirichlet Boundary Conditions: These fix the concentration of the chemoattractant at the boundary. They can lead to different aggregation patterns compared to Neumann conditions.
Key Changes in Analysis:
Function Spaces: The choice of function spaces needs to incorporate the boundary conditions. For Dirichlet conditions, spaces requiring functions to vanish at the boundary (e.g., Sobolev spaces $H^s_0$) are used.
Estimates: Deriving energy estimates becomes more involved due to boundary terms arising from integration by parts. New techniques, such as trace theorems and Poincaré inequalities, are often needed.
Regularity: Boundary conditions can induce singularities in the solution, making regularity analysis more challenging.

The provided results establish global well-posedness for small initial data in variable Fourier-Besov spaces. Extending these results to larger initial data is a challenging problem. Here's why and what might be required:
Scaling Criticality: The spaces considered are often scaling critical or supercritical for the equations. This means that the norms are invariant or grow under the natural scaling of the equations, making it difficult to control the nonlinear terms for large data.
Potential Blow-up: Both the Navier-Stokes equations and the Keller-Segel system can exhibit finite-time blow-up for large data in some cases. This means the solution becomes unbounded in finite time.
Possible Approaches and Conditions for Larger Data:
Weaker Spaces: One could try to work in weaker function spaces that allow for larger initial data. However, this often comes at the cost of losing uniqueness or requiring additional structural assumptions on the solutions.
Conditional Well-posedness: Instead of proving global well-posedness for all large data, one could aim for conditional well-posedness results. These guarantee existence and uniqueness under additional assumptions on the solution, such as certain norms remaining bounded.
Energy Methods: For the Navier-Stokes equations, advanced energy methods, such as those based on the partial regularity theory of Caffarelli-Kohn-Nirenberg, might be applicable. These methods exploit the structure of the equations to obtain local control on the solution.
Structure of Nonlinearities: Exploiting specific structural properties of the nonlinearities in the equations might be crucial. For the Keller-Segel system, techniques like entropy methods or Lyapunov functionals could be useful.

Studying PDEs in variable exponent spaces provides valuable insights that can be applied to a wider range of physical and biological systems:
Non-homogeneous Media: Variable exponent spaces are well-suited for modeling phenomena in non-homogeneous media, where the properties of the medium (e.g., diffusivity, permeability) vary spatially. Examples include:
Fluid flow in porous media with varying porosity.
Heat conduction in materials with spatially dependent thermal conductivity.
Image processing with non-uniform smoothing properties.
Growth and Diffusion: In biological systems, variable exponent spaces can model situations where growth or diffusion rates depend on the local density or concentration. Applications include:
Tumor growth models where the proliferation rate varies within the tumor.
Population dynamics with density-dependent dispersal.
Pattern formation in reaction-diffusion systems with spatially varying reaction rates.
Adaptive Models: Variable exponent spaces can be used to develop adaptive models that adjust their parameters based on the solution's behavior. This can lead to more accurate and efficient numerical simulations.
Theoretical Tools: The techniques developed for analyzing PDEs in variable exponent spaces, such as:
Generalized Sobolev embeddings.
Variable exponent Calderón-Zygmund theory.
Nonlinear analysis tools adapted to variable exponents.
can be transferred and extended to study other nonlinear PDEs arising in various applications.
Key Advantages of Variable Exponent Spaces:
Flexibility: They offer greater flexibility in modeling spatially heterogeneous phenomena compared to classical function spaces.
Accuracy: They can lead to more accurate representations of physical and biological processes.
Mathematical Richness: They pose interesting and challenging mathematical questions, pushing the boundaries of PDE theory.

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