We get asked this so many times, that it's time for a sticky.
Q: How can I run Adobe Flash on Yellow Dog Linux to view YouTube?
A: You can't
Q: Why not?
A: Adobe does not have a version of Flash for PowerPC Linuxes - like Yellow Dog.
This is an problem because the PS3's Cell processor has a PowerPC central core. Adobe does have a Linux version and a PowerPC version of Flash, but the Linux version is for Intel (x86) processors and the PowerPC version is for Mac OS X which, although Unix, is binary incompatible with Linux.
In addition, Adobe Flash is closed source, so we can't compile our own version for PowerPC Linux.
The only alternative is to reverse engineer Flash from Adobe's published specifications. There are two open source projects underway to do this - gnash and swfdec - but they are fairly buggy and slow - and currently they're broken with sites like YouTube, etc.
For YouTube viewing, I use a Firefox plugin called Download Helper that extracts the files from YouTube and saves them locally. Then I use Totem (built-in), VLC or mplayer to play them back. You'll need to have gstreamer-plugins-ugly and gstreamer-plugins-bad to make this work for Totem.
You can also connect stand-alone video applications like VLC and Mplayer directly to Firefox via a scripting add-in tool called Greasemonkey. There's a Greasemonkey script called "VLCTube" that appears to seamlessly interface VLC with YouTube. I've tried the Firefox+Greasemonkey+VLCTube setup to browse YouTube, and that works well - almost as well as if you were running Flash directly.
UPDATE: Lately, I've been using a new script called "YouTube Without Flash Auto". Firefox+Greasemonkey+YouTube Without Flash Auto seems to work well, and allows more precise fine tuning of video playback.
My Youtube without Flash Auto script settings are:
Default Quality: 5 Low
Player: VLC
Autoplay: On
Player Size: small
Video Url: get_video
Greasemonkey:
https://addons.mozilla.org/en-US/firefox/addon/748/
VLCTube:
http://userscripts.org/scripts/show/25318
YouTube Without Flash Auto:
http://userscripts.org/scripts/show/50771
You'll need to add VLC to your system. An easy way to do this is by adding the extra repos listed here:
Howto: Install software (the easy way) YDL 6
Q: But wait - I can play Flash videos in the GameOS browser. That's PowerPC, right? Why can't I do this in Linux?
A: Sony purchased a custom version of Flash directly from Adobe for the GameOS, which they now maintain. This isn't for Linux - it's in the proprietary PS3 binary format. In addition, it's Sony closed source, so we can't convert it to Linux.
UPDATE: 14-Feb-2011 YouTube updated their site again. Unfortunately, that broke the YouTube Without Flash Auto and VLCTube Greasemonkey scripts. The authors of those scripts haven't released any updates in a while, so they're effectively a no-go right now. Download Helper still works great.
Also - some folks are reporting success with a program called MiniTube. This is a stand-alone application that lets you browse YouTube, etc. You can either build it from source, or obtain it via YUM from ps3bodega.
Cheers,
Paul