I have seen both used frequently throughout the mobile web. Some examples are news sites with fixed nav bars:
Time keeps the URL bar static, which keeps their nav buttons in the same place while scrolling:
Yet IBTimes allows default browser behavior, which makes their fixed nav bar placement inconsistent, yet gives you more screen real estate.
I can see benefits to both, so I'm wondering if there are any sort of metrics or research on which method is preferred by users.