# Vorticity and Regularity for Solutions of Initial-boundary Value Problems for the Navier—Stokes Equations

• Hugo Beirão da Veiga
Conference paper
Part of the Trends in Mathematics book series (TM)

## Abstract

In reference [7], among other side results, we prove that the solution of the evolution Navier—Stokes equations (1.1) under the Navier (or slip) boundary condition (1.2) is necessarily regular if the direction of the vorticity is 1/2-Hölder continuous with respect to the space variables. In this notes we show the main steps in the proof and made some comments on the above problem under the non-slip boundary condition (3.2).

## Keywords

Weak Solution Stokes Equation Stokes System Slip Boundary Condition Side Result
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## References

1. [1]
G.J. Beavers and D.D. Joseph, Boundary conditions of a naturally permeable wall, J. Fluid Mech., 30 (1967), 197–207.
2. [2]
H. Beirão da Veiga, Vorticity and smoothness in viscous flows,in Nonlinear Problems in Mathematical Physics and Related Topics, volume in Honor of O.A. Ladyzhenskaya, International Mathematical Series, 2, Kluwer Academic, London, 2002.Google Scholar
3. [3]
H. Beirão da Veiga, Regularity of solutions to a nonhomogeneous boundary value problem for general Stokes systems in R +n, Math. Annalen, 328 (2004), 173–192.
4. [4]
H. Beirão da Veiga, Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip type boundary conditions, Advances Diff. Eq., 9 (2004), n. 9-10, 1079–1114.
5. [5]
H. Beirão da Veiga, On the regularity of flows with Ladyzhenskaya shear dependent viscosity and slip and non-slip boundary conditions, Comm. Pure Appl. Math., 58 (2005), 552–577.
6. [6]
H. Beirão da Veiga and L. C. Berselli, On the regularizing effect of the vorticity direction in incompressible viscous flows, Differ. Integral Equ. 15 (2002), 345–356.
7. [7]
H. Beirão da Veiga, Vorticity and regularity for flows under the Navier boundary condition, Comm. Pure Appl. Analysis 5 (2006), 907–918.
8. [8]
H. Beirão da Veiga, Vorticity and regularity for viscous incompressible flows under the Dirichlet boundary condition. Results and open problems, J. Math. Fluid Mech. To appear.Google Scholar
9. [9]
C. Conca, On the application of the homogenization theory to a class of problems arising in fluid mechanics, J. Math. Pures Appl., 64 (1985), 31–75.
10. [10]
P. Constantin, Geometric statistics in turbulence, SIAMRev. 36 (1994), n. 1, 73–98.
11. [11]
P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J. 42 (1993), n. 3, 775–789.
12. [12]
G.P. Galdi and W. Layton, Approximation of the larger eddies in fluid motion: A model for space filtered flow, Math. Models and Meth. in Appl. Sciences, 3 (2000), 343–350.
13. [13]
V. John, Slip with friction and penetration with resistance boundary conditions for the Navier-Stokes equations-numerical tests and aspects of the implementations, J. Comp. Appl. Math., 147 (2002), 287–300.
14. [14]
A. Liakos, Discretization of the Navier-Stokes equations with slip boundary condition, Num. Meth. for Partial Diff. Eq., 1 (2001), 1–18.
15. [15]
C. Pare’s, Existence, uniqueness and regularity of solutions of the equations of a turbulence model for incompressible fluids, Appl. Analysis, 43 (1992), 245–296.
16. [16]
J. Silliman and L.E. Scriven, Separating flow near a static contact line: slip at a wall and shape of a free surface, J.Comput. Physics, 34 (1980), 287–313.
17. [17]
V.A. Solonnikov and V.E. Scadilov, On a boundary value problem for a stationary system of Navier-Stokes equations, Proc. Steklov Inst. Math., 125 (1973), 186–199.
18. [18]
R. Verfurth, Finite element approximation of incompressible Navier-Stokes equations with slip boundary conditions, Numer. Math., 50 (1987), 697–721.