In 5e, I have not seen any such provision, but there is precedent in earlier versions. As far back as the original DMG in AD&D 1e, it says this about potions:
Unless otherwise stated, the effects of a potion will last for 4 complete turns, plus 1-4 additional turns (d4). If half a potion is quaffed, the effects will last one-half as long in some cases.
Note that at the time, a "turn" was 10 minutes, and a "round" was 1 minute. With our current 6-second rounds, the equivalent would be for a potion to last 5-8 minutes, or half that for half a potion. Most of the 5e potion descriptions, OTOH, give a duration for a full potion of an hour.
A few potions were done a bit differently, for example:
Extra-Healing: This potion restores 6-27 (3d8+3) hit points of damage when wholly consumed, or 1-8 hit points of damage for each one-third potion.
In 5e, I have ruled that a standard healing potion (2d4+2) is too small to divide up, but that the stronger healing potions can be subdivided into an equivalent number of doses of standard healing potions.
With as rare as magic items are in 5e, I'm inclined to not allow dividing them into subdoses for shorter duration. "Here, everyone take a sip of this Frost Giant Strength potion, and we'll all be 23 Str for a few minutes, and let's go kick the bleep out of that dragon." = Recipe for Disaster