You ask about science after civilization has broken down, but then also mention after democracy and peace are gone. If you're thinking of a lawless Mad Max sty

You ask about science after civilization has broken down, but then also mention after democracy and peace are gone. If you're thinking of a lawless Mad Max style apocalypse, most likely sciences will be regulated to discovering things that were already known but then lost to humanity. You could see it like the unrest after the Roman Empire fell (the Dark Ages) where lots of knowledge was lost and had to be learned again from scratch. It is entirely possible that things will still be recognizable as a civilization even if democratic institutions are gone and even if a nation is at war. There are pursuits of sciences under a monarchy and even authoritarian dictatorships. During time of conflict, science might focus on military technologies (which often have later civilian applications) or on technology to help pacify the population (agriculture, comfort goods, surveillance technologies).