Navier–Stokes equations remain elusive

Nature : A set of equations devised to explain the motion of fluids continue to confound mathematicians. First proposed in the 19th century by a French engineer and an Irish mathematician–physicist, the Navier–Stokes equations have been used to model the weather, ocean currents, and air flow around airplane wings. Despite the equations’ evident success in describing turbulent systems, no one has proven that solutions in three dimensions can always be found and are always finite. Because of their importance, the Navier–Stokes equations were established as one of the seven Millennium Prize Problems in 2000 by the Clay Mathematics Institute of Cambridge, Massachusetts. Although two mathematicians—Penny Smith of Lehigh University in Pennsylvania and Mukhtarbay Otelbaev of the Eurasian National University in Kazakhstan—have each claimed to have cracked the problem, both proofs have been found to contain errors. Only one of the Millennium Prize problems has been solved so far: the Poincaré conjecture by Grigory Perelman in 2002.