Without trying to be too far off, my ancestor Sgt. J. D Pearce was with the 8th Arkansas CSA. They went east early on in the Civil War. Very few of those men made it back as they were in so many big battles and suffered such great losses.
If that is whom you are speaking of, it is my guess that less than 10
men lived to come home though I have heard others says fewer than 100. There is even a re-enactor group in Michigan that calls itself the 8th Arkansas being named that out of repect for the tough fighting ablility of the 8th.
To be honest, I am the only person that I can recall who says someone came home in the 8 years that I have been searching.
To say Brown was at Cornith and then died in May of 1865 is actually knowing a lot on some of these men.
You may fare better to check out the Union sites of who they fought against to learn more.
Also, as far as not finding a grave, times were so hard in Arkansas at the time of his death in 1865 that there was only money for the living, and not much of that either. Marking a grave with more than a rock was a luxury that folks could not afford.
Good luck with your search.
Linda