caa a22 4500 463171055 CHVBK 20180406164813.0 cr unu---uuuuu 170326e20071201xx s 000 0 eng 10.1007/s00021-005-0210-6 doi (NATIONALLICENCE)springer-10.1007/s00021-005-0210-6 da Veiga H. Pisa, Italy aut Vorticity and Regularity for Viscous Incompressible Flows under the Dirichlet Boundary Condition. Results and Related Open Problems [Elektronische Daten] [H. da Veiga] Abstract.: In reference [7] it is proved that the solution of the evolution Navier-Stokes equations in the whole of R 3 must be smooth if the direction of the vorticity is Lipschitz continuous with respect to the space variables. In reference [5] the authors improve the above result by showing that Lipschitz continuity may be replaced by 1/2-Hölder continuity. A central point in the proofs is to estimate the integral of the term (ω · ∇)u · ω, where u is the velocity and ω = ∇ × u is the vorticity. In reference [4] we extend the main estimates on the above integral term to solutions under the slip boundary condition in the half-space R + 3 . This allows an immediate extension to this problem of the 1/2-Hölder sufficient condition. The aim of these notes is to show that under the non-slip boundary condition the above integral term may be estimated as well in a similar, even simpler, way. Nevertheless, without further hypotheses, we are not able now to extend to the non slip (or adherence) boundary condition the 1/2-Hölder sufficient condition. This is not due to the "nonlinear" term (ω · ∇)u · ω but to a boundary integral which is due to the combination of viscosity and adherence to the boundary. On the other hand, by appealing to the properties of Green functions, we are able to consider here a regular, arbitrary open set Ω. Birkhaueser, 2006 Navier-Stokes equations nationallicence boundary value problems nationallicence vorticity and regularity nationallicence Journal of Mathematical Fluid Mechanics SP Birkhäuser Verlag Basel 9/4(2007-12-01), 506-516 1422-6928 9:4<506 2007 9 21 https://doi.org/10.1007/s00021-005-0210-6 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00021-005-0210-6 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- da Veiga H. Pisa, Italy aut NATIONALLICENCE 773 0- Journal of Mathematical Fluid Mechanics SP Birkhäuser Verlag Basel 9/4(2007-12-01), 506-516 1422-6928 9:4<506 2007 9 21 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer