Abstract
This paper is concerned with the regularity criterion of weak solutions to the three-dimensional Navier-Stokes equations with nonlinear damping in critical weakLqspaces. It is proved that if the weak solution satisfies∫0T∇u1Lq,∞2q/2q-3+∇u2Lq,∞2q/2q-3/1+lne+∇uL22ds<∞, q>3/2, then the weak solution of Navier-Stokes equations with nonlinear damping is regular on(0,T].
Highlights
In this study we consider the Cauchy problem of the threedimensional Navier-Stokes equations with the nonlinear damping∂tu + (u ⋅ ∇) u + ∇π + |u|r−2 u = Δu, (1)∇ ⋅ u = 0, together with the initial data u (x, 0) = u0, (2)where u = (u1(x, t), u2(x, t), u3(x, t)) and π(x, t) denote the unknown velocity fields and the unknown pressure of the fluid. |u|r−2u, r > 2 is the nonlinear damping
The main purpose of this paper is to investigate the regularity criteria of weak solutions with the aid of two components of velocity fields in critical weak Lq space
The main idea in the proof of Theorem 2 is borrowing from the argument of previous results on classic Navier-Stokes equations [16] and together with energy methods
Summary
Since Navier-Stokes equations with the nonlinear damping (1) are a modification of the classic Navier-Stokes equations, it is necessary to mention some regularity criteria of weak solutions for Navier-Stokes equations and related fluid models [8, 9]. As for this direction, the first result of NavierStokes equations is studied by He [10] and improved by Dong and Zhang [11], Pokory [12, 13], and Zhou [14]. The main idea in the proof of Theorem 2 is borrowing from the argument of previous results on classic Navier-Stokes equations [16] and together with energy methods
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
