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| Content Provider | Springer Nature Link |
|---|---|
| Author | Veiga, H. Beirão |
| Copyright Year | 2006 |
| 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. |
| Ending Page | 516 |
| Page Count | 11 |
| Starting Page | 506 |
| File Format | |
| ISSN | 14226928 |
| e-ISSN | 14226952 |
| Journal | Journal of Mathematical Fluid Mechanics |
| Issue Number | 4 |
| Volume Number | 9 |
| Language | English |
| Publisher | SP Birkhäuser Verlag Basel |
| Publisher Date | 2006-09-19 |
| Publisher Place | Basel |
| Access Restriction | One Nation One Subscription (ONOS) |
| Subject Keyword | Navier–Stokes equations Mechanics, Fluids, Thermodynamics Smoothness and regularity of solutions Initial-boundary value problems for second-order parabolic equations Mathematical Methods in Physics Fluids vorticity and regularity boundary value problems Existence, uniqueness, and regularity theory Navier-Stokes equations |
| Content Type | Text |
| Resource Type | Article |
| Subject | Applied Mathematics Mathematical Physics Condensed Matter Physics Computational Mathematics |
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