Abstract
AbstractH.-O. Bae and H.J. Choe, in a 1997 paper, established a regularity criteria for the incompressible Navier-Stokes equations in the whole space ℝ3based on two velocity components. Recently, one of the present authors extended this result to the half-space case$\begin{array}{} \displaystyle \mathbb{R}^3_+ \end{array}$. Further, this author in collaboration with J. Bemelmans and J. Brand extended the result to cylindrical domains under physical slip boundary conditions. In this note we obtain a similar result in the case of smooth arbitrary boundaries, but under a distinct, apparently very similar, slip boundary condition. They coincide just on flat portions of the boundary. Otherwise, a reciprocal reduction between the two results looks not obvious, as shown in the last section below.
Highlights
The starting point of the present paper is the well known Prodi-Serrin (P-S) su cient condition for regularity of the solutions to the incompressible Navier-Stokes equations∂tu + u · ∇u − ∆u + ∇p =, in Ω × (, T], (1.1)∇·u=, in Ω × (, T] .where u = (u, u, u ) denotes the unknown velocity of the uid and p the pressure
In a 1997 paper, established a regularity criteria for the incompressible Navier-Stokes equations in the whole space R based on two velocity components
Brand extended the result to cylindrical domains under physical slip boundary conditions
Summary
Brand extended the result to cylindrical domains under physical slip boundary conditions. In this note we obtain a similar result in the case of smooth arbitrary boundaries, but under a distinct, apparently very similar, slip boundary condition. The starting point of the present paper is the well known Prodi-Serrin (P-S) su cient condition for regularity of the solutions to the incompressible Navier-Stokes equations
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