Decay of Solutions to the Three-Dimensional Generalized Navier-Stokes Equation with Nonlinear Damping Term

Abstract

In this paper, we consider the three-dimensional generalized Navier-Stokes equation with a nonlinear damping term $|u|^{β−1} (β ≥ 1).$ Firstly, utilizing the Fourier splitting method, we derive decay estimates for weak solutions to the equations when $α = 0$ and $β = 1,$ as well as when $0 < α <\frac{3}{4}$ for any $β = 2.$ Secondly, for $0 < α < \frac{5}{4}$ and any $β > {\rm max}\{ \frac{4α}{3}+1, 2\},$ we obtain the same result.