A pressure control device for sealing about a tubular at a wellsite can include a rotatable member, at least two radial seals that sealingly contact the rotatable member, at least two fluid chambers, one chamber being exposed to the rotatable member between the radial seals, and a pump that pumps fluid between the chambers in response to rotation of the rotatable member. A method of operating a pressure control device at a wellsite can include providing at least two chambers in a bearing assembly of the pressure control device, and regulating pressures in the chambers via a valve system in communication with both of the chambers.