Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,852 answers , 20,624 comments
1,470 users with positive rep
501 active unimported users
More ...

Why is a smooth weak solution strong for stationary linear Stokes problem with zero-traction boundary condition?

+ 3 like - 0 dislike
124 views

Can anyone provide me with a reference giving details on how smooth generalized solutions of the stationary linear Stokes problem can be shown to be classical solutions when a zero-traction boundary condition is present? That is, given a smooth generalized solution of

$-\nu \bigtriangleup v + \bigtriangledown q = f$ on $\Omega \subset \mathbb{R}^3$

$\bigtriangledown \cdot v = 0$ on $\Omega$

$S(v,q) = 0$ on $\partial \Omega$ where $S_i(v,q) =q n_i - \nu \sum_{j=1}^3 (\partial_i v_j + \partial_j v_i)n_j$ for $i=1,2,3$

how can it be shown that the zero-traction boundary condition is met? It's not difficult to show that the first two equations are satisfied on $\Omega$ and using the relevant Green's formula one can obtain

$\int_{\partial \Omega} S(v,q) \cdot \phi = 0$

for all solenoidal $\phi \in H^1$. However, I can't quite figure out why this necessarily leads to $S(v,q)=0$.

This post imported from StackExchange MathOverflow at 2015-04-07 13:20 (UTC), posted by SE-user Navier_Stoked
asked Aug 31, 2010 in Resources and References by Navier_Stoked (0 points) [ no revision ]
retagged Jan 14, 2016
+1: Good question, great username.

This post imported from StackExchange MathOverflow at 2015-04-07 13:20 (UTC), posted by SE-user Cam McLeman
What do "smooth weak" and "smooth generalized" mean?

This post imported from StackExchange MathOverflow at 2015-04-07 13:20 (UTC), posted by SE-user Bob Terrell
What are the assumptions on $\Omega$ and $\partial \Omega$?

This post imported from StackExchange MathOverflow at 2015-04-07 13:20 (UTC), posted by SE-user Nilima Nigam

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsO$\varnothing$erflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...