Local estimates of Hölder exponents in turbulent vector fields

Area of Science:

• Fluid dynamics
• Partial differential equations
• Mathematical physics

Background:

• The existence and smoothness of solutions to the Navier-Stokes equations remain a fundamental unsolved problem in mathematics and physics.
• A key question is whether singularities can develop from smooth initial conditions, specifically concerning the Hölder continuity of the velocity field at small scales.
• Existing methods like inertial dissipation and multifractal analysis offer global insights but lack local precision.

Purpose of the Study:

• To develop and apply a local statistical analysis for estimating the Hölder continuity of velocity fields in Navier-Stokes solutions.
• To compare the findings of this local analysis with established methods, particularly inertial energy dissipation.
• To investigate the spatial distribution of local Hölder exponents and its relationship with energy dissipation.

Main Methods:

• Development of a novel local statistical analysis to quantify local Hölder continuity.
• Estimation of local Hölder exponents for velocity fields.
• Comparison of local Hölder exponent distributions with the inertial energy dissipation (DℓI).

Main Results:

• The local statistical analysis provides estimates of local Hölder continuity, addressing limitations of global methods.
• A correlation was observed: increased inertial dissipation corresponds to less regular velocity fields based on local estimates.
• The spatial distribution of local Hölder exponents exhibits complex behavior and does not perfectly align with the distribution of inertial dissipation.

Conclusions:

• The developed local statistical analysis offers a more granular approach to understanding velocity field regularity in fluid dynamics.
• While inertial dissipation serves as an indicator of irregularity, it does not fully capture the localized spatial variations in Hölder continuity.
• Further research is needed to fully elucidate the complex interplay between local regularity and energy dissipation in Navier-Stokes flows.