It's not a good idea to mix jQuery UI with jQuery Mobile. They
aren't meant to be compatible. For one, they have conflicting CSS
styles.
Compatibility is slated for a future release, but that is far off
in the future - JQuery Mobile version 2.0.
That said, some developers have gotten some jQuery UI widgets to
work with jQuery Mobile. You will have to research what others have
done. In the case of third-party widgets, they often will work with
either JQUI or JQM, if they have been specifically designed to work
with both. You will need to check documentation for the widget. There
may be some specific instructions or work-around(s).
jQuery Mobile has it's own enhancement of <select>, though
it is (ironically) nor normally used on mobile devices, because they
all have their own HTML5 <select> handling
in the browser.