Are you asking what will happen if a shark stops swimming, do they ever stop swimming and do they need to rest ? Or are you referring to a shark species specifically and cannot remember its actual name ?
If you are referring to a specific species, the nearest I can think of is the Pacific Sharpnose shark. In either case the information is the same. When it comes to oxygen intake, sharks use one (or both) of the following methods. One is 'buccal pumping' the other is 'ram ventilation'. Buccal pumping means that the shark draws water into its mouth and pumps it out through the gills, this is done using its 'buccal' or cheek muscles. This method requires NO forward movement and the shark can therefore be stationary and still breath. Examples of sharks that employ this method regularly are Angel sharks and Nurse sharks. The other method is one most of us are familiar with (although perhaps not in name). The majority of sharks employ 'ram ventilation' as the primary method of extracting oxygen from the water. This simply means that the shark swims along with its mouth partially open, the water rushes in and out over the gills extracting oxygen in the same way. Now as I have said the majority of sharks use ram ventilation as their primary method of breathing and indeed many 'ram ventilators' CANNOT perform buccal pumping as their facial muscles are not developed or adapted for this function. With these sharks it is ESSENTIAL that either they keep moving forward OR they must position themselves into a current sufficient to maintain a suitable amount of water flow. However, this all said sharks that use either method are both still able to rest. In the case of buccal pumping sharks, this function is maintained subconciously much like we breath subconsciously ALL the time and so these sharks can remain motionless for many hours. An experiment with a spiny dogfish (an obligate ram ventilator) indicated that swimming is coordinated by the spinal cord, not by the brain, so sharks may be able to shut down their brain and rest while still swimming. Moreover, it is actually more difficult for a ram ventilator to remain still and less efficient for oxygen extraction to do so. It is also well known that if a ram ventilator needs/wants to rest it will slow to the minimum speed necessary to maintain sufficient water flow and slow down all non-essential activity, this state may be referred to as 'sleep'. Finally, if you are wondering, your Pacific Sharpnose shark is an obligate ram ventilator. Even if you do not mean this shark, whatever you are thinking of is covered in the above information.
EDIT - YOUR HEADLINE QUESTION (What happens if a Pacific Ocean shark stops swimming??) HAS BEEN ANSWERED IN THE TEXT ABOVE. What I have written above answers every ASPECT OF why NO shark gets exhausted and plunges into 'the abyss' and when they rest, the above also explains what they do in order to remain buoyant and alive !