Large Time Behavior for Weak Solutions of the 3D Globally Modified Navier-Stokes Equations

Abstract

This paper is concerned with the large time behavior of the weak solutions for three-dimensional globally modified Navier-Stokes equations. With the aid of energy methods and auxiliary decay estimates together with $L^p-L^q$ estimates of heat semigroup, we derive the optimal upper and lower decay estimates of the weak solutions for the globally modified Navier-Stokes equations as $C_{\mathrm{1}}(\mathrm{1}+t{)}^{-\mathrm{3}/\mathrm{4}}\le {∥u∥}_{L^2}\le C_{\mathrm{2}}(\mathrm{1}+t{)}^{-\mathrm{3}/\mathrm{4}},\mathrm{}t>\mathrm{1}.$ The decay rate is optimal since it coincides with that of heat equation.