**Author:**

Godoi, Valdir M.S.

**Category:**

Research Papers

**Sub-Category:**

Mathematics and Applied Mathematics

**Date Published:**

March 6, 2016

**Keywords:**

Navier-Stokes equations, continuity equation, breakdown, existence, smoothness, physically reasonable solutions, gradient field, conservative field, velocity, pressure, external force, unbounded energy, millenium problem, uniqueness, non uniqueness

**Abstract:**

A solution to the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy. We have proved that there are initial velocities u¨o(x) and forces f(x,t) such that there is no physically reasonable solution to the Navier-Stokes equations for t > 0, which corresponds to the case (C) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute. Three examples are given.

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