Lower and upper bounds of temporal decay for solutions to [formula omitted]-dimensional hyperviscous Navier–Stokes equations

Abstract

This paper is concerned with the lower and upper bounds of rates of decay for n-dimensional Navier–Stokes equations with fractional hyperviscosity (−Δ)α when 0<α<n+24. Taking advantage of the spectral representation technique and delicate energy estimates, we show the non-uniform decay and the upper bound of rate of optimal decay for the weak solutions. Furthermore, the lower bound of rate of decay of weak solutions is also established by employing the solution of the linearized equations of the system considered here to approximate the solution of it. Moreover, we make use of the method of bootstrap argument and the properties of generalized heat operator to obtain the lower and upper bounds of rates of optimal decay for the small data global solutions and its derivatives when initial data are in a negative Sobolev space.