Pipe the result to split like this:
java MyAppName | split -b500k - Console.log
This will create a new file every time you go over 500k. See the man-page for split for more details and options.
Or you can use rotatelogs
rotatelogs
nohup java MyAppName 2>&1 | rotatelogs -l Console_%Y-%m-%d.log 86400 &
This will create a new file with today's date every day