An anonymous reader writes: While Hilbert’s broader aim of axiomatizing physics remains unfulfilled, recent research has yielded an unexpected answer to the particle-fluid question. The Boltzmann equation does not translate into the Navier-Stokes equations in all cases, because the Navier-Stokes equations — despite being exceptionally useful for modeling the weather, ocean currents, pipes, cars, airplane wings and other hydrodynamic systems, and despite the million-dollar prize offered for their exact solutions — are incomplete. The evidence suggests that truer equations of fluid dynamics can be found in a little-known, relatively unheralded theory developed by the Dutch mathematician and physicist Diederik Korteweg in the early 1900s. And yet, for some gases, even the Korteweg equations fall short, and there is no fluid picture at all.
“Navier-Stokes makes very good predictions for the air in the room,” said Slemrod, who presented the evidence last month in the journal Mathematical Modelling of Natural Phenomena. But at high altitudes, and in other near-vacuum situations, “the equations become less and less accurate.”
