$DATE$ is the way to go. It uses the (system) date at the time the search is run.
The typical way would be to use Advanced Search:
('2' = $USER$ AND '3' >= $DATE$ AND '3' < $DATE$ + 24*60*60)
Field id 2 is the Submitter-field, and 3 is the Create Date.
This all uses the Client Time Zone. If you need the Servers Time Zone, you might need to call the server and get the $DATE$ from there instead. For example using a Service-call that returned $DATE$ into a temporary field tmpServerDate.
Best Regards - Misi, RRR