Java Database Connectivity (JDBC)

I'm re-evaluating my companies choice of a DBMS, and quite frankly I'm pretty overwhelmed with the choices available.

Here's what we're trying to accomplish:

1) Writing a pure Java standalone application (although the database doesn't necessarily have to be written in Java).
2) Platform independent (we'll have platform specific installations so as long as the database has a version for Windows and Unix it should be adequate).
3) The database will need to run on the same machine as the application. (i.e. each user will have their own personal database.)
4) JDBC compliant since everything that has been done so far is with JDBC.

So far we've been using HyperSonicSQL (or hsqldb v1.60), but we're concerned about its limitations and memory resident issues. Other databases we are trying to consider at this point are MySql and PostgreSql. Any info on why one would be better than the other or links to information so I could determine myself would be greatly appreciated.

Thanks,
Josh