A drop of honey spreads slowly across a table, while water spreads quickly. The resistance to flow, viscosity, differs from one fluid to another. But what if the viscosity changes on the fly, as the substance compresses or expands? Many gases behave exactly like that, and the equations that describe their motion, the Navier–Stokes equations, become almost unbearably complex.
The authors took the case where viscosity depends on gas density by a power law: the denser the gas, the more viscous it is (the exponent is greater than 1/2 but less than 1). They considered a ball of gas that simply vanishes at infinity — the density drops to zero. Such a 'distant vacuum' is common for planetary atmospheres or interstellar gas clouds. But for a mathematician, it's a nightmare: the equations degenerate, and it's unclear whether they even have a solution.
It turned out that even with very large initial perturbations (for instance, when the gas ball is sharply compressed) the solution remains smooth and exists for arbitrarily long times. This works both in the plane and in three-dimensional space. The main trick was to circumvent three traps at once: degeneracy at the vacuum, the singularity at the center of the ball (where coordinates sort of collapse), and the nonlinear viscosity. The mathematical tool they devised for this also works for the case where the density never drops to zero: that is, for a gas in a closed container.
The Navier–Stokes equations are so difficult that a prize of one million dollars has been offered for a proof of the existence and smoothness of their solutions in the three-dimensional case: it is one of the seven Millennium Prize Problems.
In Stanislaw Lem's novel 'Solaris', the planet's ocean behaves like a fluid with variable viscosity, obeying unknown equations.