I know that this is an old thread, but I have the same issue on two of my droplets so I'd like to share my progress thus far.
dmesg hints that mysqld went down due to lack of available memory.
The mysql.log and mysql.err has nothing.
Found this thread when searching for people with a similar issue on digitalocean. Apparently, this is a common problem beyond digitalocean.
People have suggested on some blogs to optimize the my.cnf, which for me is in /etc/mysql/my.cnf
I'm at a stage were I'm trying to figure out which variables to alter in the my.cnf in order resolve the case.
I'll update everyone if I come up with a solution.