Possible to fill in porcelain sink depression so that water stops pooling?
Sure, you could fill it with an epoxy, or you could tile it.
Note: Only since you don't care about aesthetics, since I suspect either will look awful in short order, if not right out of the gate. My mind's eye makes a beautiful tile sink bottom (like a shower floor) but the real world experience part of my mind doubts that the detailing at the edges could look good, not to mention mold potential.
Additional Note: First check if you may have a &quot;lifetime warranty&quot; or other applicable warranty on the sink, since some makers do have that sort of policy, but it's out the window if you attempt to fix it yourself, normally.