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 withLp-Lqestimates of heat semigroup, we derive the optimal upper and lower decay estimates of the weak solutions for the globally modified Navier-Stokes equations asC1(1+t)-3/4≤uL2≤C2(1+t)-3/4, t>1.The decay rate is optimal since it coincides with that of heat equation.
Highlights
It is well known that the motion of the viscous incompressible fluids is governed by the following classic Navier-Stokes equations [1]:∂tu + (u ⋅ ∇u) − Δu + ∇p = 0, (1) ∇ ⋅ u = 0.Here u and π denote the unknown velocity and pressure of the fluid motion, respectively
We denote by C a generic positive constant which may vary from line to line
To state the main results of this paper, we first give the definition of the weak solutions of the three-dimensional globally modified Navier-Stokes equations (3) [5]
Summary
It is well known that the motion of the viscous incompressible fluids is governed by the following classic Navier-Stokes equations [1]:. It is interesting to consider the time decay issue of this model which largely depended on the effect of low frequency of the solutions. Abstract and Applied Analysis of weak solutions for the modified Navier-Stokes equations (3). To carry out this issue, it is necessary to recall some classic time decay results of the fluid dynamical models. L2 decay of weak solutions for the Navier-Stokes equations was first studied by Schonbek [8] (see [9]). Motivated by the upper and lower decay estimates of nonlinear fluid models [14], in this study, we will develop another technique to deal with the time decay problem of the weak solutions for the globally modified Navier-Stokes equations (3). We can get the optimal time decay rate, since it coincides with that of linear equations
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