satisfy these conditions: Just as in the previous case, condition 3 implies that the functions are smooth and globally defined and condition 4 means that the kinetic energy of the solution is globally bounded. The periodic Millennium Prize theorems (C) Existence and smoothness of the Navier–Stokes solutions in Let . For any initial condition satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector and a pressure satisfying conditions 3 and 4 above. (D) Breakdown of the Navier–Stokes solutions in There exists an initial condition and an external force such that there exists no solutions and satisfying conditions 3 and 4 above. Partial results The Navier–Stokes problem in two dimensions was solved by the 1960s: there exist smooth and globally defined solutions. If the initial velocity is sufficiently small then the statement is true: there are smooth and globally defined solutions to the Navier–Stokes equations. Given an initial velocity there exists a finite time T, depending on such that the Navier–Stokes equations on have smooth solutions and . It is not known if the solutions exist beyond that "blowup time" T. Jean Leray in 1934 proved the existence of so-called weak solutions to the Navier–Stokes equations, satisfying the equations in mean value, not pointwise. Terence Tao in 2016 published a finite time blowup result for an averaged version of the 3-dimensional Navier–Stokes equation. He writes that the result formalizes a "supercriticality barrier" for the global regularity problem for the true Navier–Stokes equations, and claims that the method of proof hints at a possible route to establishing blowup for the true equations. In popular culture Unsolved problems have been used to indicate a rare mathematical talent in fiction. The Navier-Stokes problem features in The Mathematician's Shiva (2014), a book about a prestigious, deceased, fictional mathematician named Rachela Karnokovitch taking the proof to her grave in protest of academia. The movie Gifted (2017) referenced the Millennium Prize problems and dealt with the potential for a 7-year-old girl and her deceased mathematician mother for solving the Navier–Stokes problem. See also List of unsolved problems in mathematics List of unsolved problems in physics Notes References Further reading External links Contributed by: Yakov Sinai The Clay Mathematics Institute's Navier–Stokes equation prize Why global regularity for Navier–Stokes is hard — Possible routes to resolution are scrutinized by Terence Tao. Navier–Stokes existence and smoothness (Millennium Prize