Calder{ó}n splitting and weak solutions for Navier-Stokes equations with initial data in weighted L p spaces

We show the existence of global weak solutions of the 3D Navier-Stokes equations with initial velocity in the weighted spaces , using Calder{ó}n splitting L p $Φ$$γ$ $\subset$ L 2 $Φ$ 2 + L r (with some r $\in$ (3, +$\infty$)) and energy controls in L 2 $Φ$ 2 .

💡 Research Summary

This research paper presents a significant mathematical advancement in the study of the 3D Navier-Stokes equations, focusing on the existence of global weak solutions under generalized initial conditions. The fundamental challenge in 3D Navier-Stokes analysis lies in controlling the non-linear convective term to prevent potential singularities and ensure that the solution remains well-defined for all time $t > 0$.

The core contribution of this work is the extension of the class of allowable initial velocity fields to weighted $L^p$ spaces. Specifically, the authors investigate initial data belonging to the space $L^p \Phi \gamma$, which is embedded within the sum of weighted $L^2$ and $L^r$ spaces ($L^2 \Phi 2 + L^r$, where $r \in (3, \infty)$). By introducing the weight function $\Phi$, the researchers allow for the study of fluid flows where the initial velocity may exhibit specific decay or growth behaviors at infinity, which is much more representative of complex physical phenomena than standard Lebesgue spaces.

To tackle the mathematical complexity introduced by these weighted spaces, the authors employ the “Calderón splitting” technique. This method is a powerful tool from harmonic analysis that allows for the decomposition of the initial velocity field into two distinct components. One component is handled within a framework that ensures energy stability, while the other component is managed through the properties of the weighted $L^p$ space. This decomposition is crucial for managing the non-linear term of the Navier-Stokes equations, as it enables the application of precise energy controls within the $L^2 \Phi 2$ space.

The authors successfully demonstrate that despite the potential for high-frequency oscillations or singular behavior in the initial data, the energy of the system remains controlled. By establishing these energy estimates, they prove that global weak solutions exist for the specified initial data. This result is mathematically profound because it bridges the gap between classical $L^2$ theory and more generalized functional settings, providing a robust framework for analyzing fluid dynamics in more realistic, weighted environments. The implications of this work extend to the broader field of mathematical fluid mechanics, particularly in the development of more sophisticated models for turbulence and non-homogeneous fluid flows.