Bibliographic Information: Abidi, H., Gui, G., & Zhang, P. (2024). GLOBAL REFINED FUJITA-KATO SOLUTION OF 3−D INHOMOGENEOUS INCOMPRESSIBLE NAVIER-STOKES EQUATIONS WITH LARGE DENSITY. arXiv preprint arXiv:2410.09386v1.
Research Objective: The paper investigates the existence and uniqueness of a global solution, specifically the "Fujita-Kato" solution, for the 3D inhomogeneous incompressible Navier-Stokes equations in critical function spaces. The research aims to relax the traditional requirement of small initial density for such solutions.
Methodology: The authors employ techniques from harmonic analysis, particularly Littlewood-Paley theory, and utilize properties of Besov and Lorentz spaces to analyze the equations. They derive a priori estimates for solutions of the linearized equations and leverage these estimates to prove the existence and uniqueness of the Fujita-Kato solution.
Key Findings: The paper demonstrates that a global unique Fujita-Kato solution exists for the 3D inhomogeneous incompressible Navier-Stokes equations even when the initial density is not restricted to be small. This finding is achieved under the condition that the initial velocity is sufficiently small in the critical Besov space Ḃ^{1/2}_{2,∞}(ℝ^{3}). The authors further establish regularity properties of this solution, including uniform-in-time estimates, which improve upon previously known exponential-in-time growth bounds.
Main Conclusions: The research significantly contributes to the understanding of the Navier-Stokes equations, particularly in the context of fluid mixtures with different densities. By removing the smallness assumption on the initial density, the study broadens the applicability of the Fujita-Kato solution framework. The uniform-in-time estimates obtained for the solution provide valuable insights into the long-term behavior of such fluid systems.
Significance: This paper makes a notable contribution to the mathematical study of fluid dynamics. The results have implications for both theoretical analysis and numerical simulations of fluid flows, particularly those involving mixing and turbulence.
Limitations and Future Research: The study focuses on the incompressible Navier-Stokes equations. Exploring similar results for the compressible case or investigating the stability of the Fujita-Kato solution under perturbations could be potential avenues for future research.
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by Hammadi Abid... at arxiv.org 10-15-2024
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