Decay of solutions to the three-dimensional generalized Navier-Stokes equations

TL;DR

This paper establishes optimal decay rates for solutions and their derivatives to the three-dimensional generalized Navier-Stokes equations, extending previous results and providing insights into the long-term behavior of these fluid dynamics models.

Contribution

It provides the first optimal decay estimates for solutions and derivatives of the generalized Navier-Stokes equations with small initial data.

Findings

Decay rates match those of the generalized heat equation

Results improve upon previous decay estimates for classical Navier-Stokes

Decay estimates apply to higher order derivatives of smooth solutions

Abstract

In this paper, we first obtain the temporal decay estimates for weak solutions to the three dimensional generalized Navier-Stokes equations. Then, with these estimates at disposal, we obtain the temporal decay estimates for higher order derivatives of the smooth solution with small initial data. The decay rates are optimal in the sense that they coincides with ones of the corresponding generalized heat equation. These results improve the previous known results to the classical Navier-Stokes equations.