Smoothing Effects for Navier-Stokes Equations

TL;DR

This paper investigates the smoothing effects and long-term behavior of solutions to the 3-D Navier-Stokes equations with initial data in a critical Sobolev space, providing new energy estimates and insights into asymptotic dynamics.

Contribution

It introduces new smoothing results and energy estimates for the Navier-Stokes equations with critical initial data, advancing understanding of solution regularity and asymptotic behavior.

Findings

Proved smoothing effects for solutions with initial data in H^{1/2}( ^3)

Analyzed the asymptotic behavior of global solutions as time approaches infinity

Derived a new energy estimate for the Navier-Stokes equations

Abstract

We prove some smoothing effects for the 3-D Navier-Stokes equations for initial data belonging to the critical Sobolev space $H^{1/2}(\R^3)$. Asymptotic behavior of the global solution when the time goes to infinity is studied. We also obtain a new energy estimate. Other results in this direction and with different methods can be found in \cite{C4}.