Navier-Stokes equations: Unpredictability even without butterflies?


Mathematically, the motion of a fluid is described by the so-called Navier-Stokes equations. In the spirit of Newtonian mechanics, these equations should determine the future motion of the fluid out of its initial state. However, despite the significant effort made for more than a century, this determinism has not yet been mathematically proved nor disproved. This paper offers a general perspective on the Navier-Stokes equations, the fourth millennium problem.


Navier-Stokes equations; fluid mechanics; meteorology; Newtonian determinism; millennium problems

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Mora, X. (2008). Les equacions de Navier-Stokes. Un repte al determinisme newtonià. Butlletí de la Societat Catalana de Matemàtiques, 23, 53–120. doi: 10.2436/20.2002.01.12

Seregin, G., & Šverák, V. (2017). On global weak solutions to the Cauchy problem for the Navier-Stokes equations with large L3-initial data. Nonlinear Analysis, 154, 269–296. doi: 10.1016/

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