I always feel sorry for Botzman: Encountering fierce resistance from the scientific community for his statistical theory (bedrock now but very revolutionary at that time), he committed suicide.
So much for the high school history of science, where better theories immediately replace inadequate ones. Even Einstein didn't get his Nobel for his Special Theory.
But he did get one for his equally novel explanation of the photoelectric effect, so I wouldn't feel too sorry for him.
Nobel prizes tend not to be given for theoretical work until there is a good deal of supporting experimental evidence. This takes time, and that's why better theories do not usually replace their predecessors immediately.
Some science are in the state of decay, some science progress in little steps, and some science must be revolutionized before moving on.
The state of human knowledge is not that it always improve over time, but regress, advance, and forget as time passes.
Based on technological progress and sophistication of our society, I think it is safe to say that science mostly advance, rather than regress in systematic knowledge.
Summary: they didn't really solve it, but they re-affirmed it's stability under certain conditions, and proved some interesting mathematical properties
It is the boltzmann equation of kinetic theory. It is a differential equation for a 6 dimensional function f(r,v) which describes the number of particles at location r, with velocity v. This distribution changes as the particles fly through space and collide with one another. It has important applications in the theory of hot dilute gases (e.g. rocket engines) and the theory of conductivity in solid-state physics. There are already practical approximations and numerical methods used by physicists who need practical answers to these problems so I think this is mostly of interest to mathematicians...
