2021•arXiv (Cornell University)Open access

A complete solution to the incompressible Navier-Stokes equation

Yanyou Qiao

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Abstract

In this paper, we solve the Navier-Stokes equation, one of the seven Millennium Prize Problems suggested by Clay Mathematics Institute(CMI). We prove that, for any initial velocity u0, there has been a force vector f, such that there exists no smooth solution to the respective Navier-Stokes equation. By the way, exact solutions can be explicitly solved as series, where the coefficients are all known functions determined only by u0 abd f. Similar result also holds for the Euler equation.For any n>=2, similar results also hold for dimensional Navier-Stokes equation or Euler equation.

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What this paper is about

In this paper, we solve the Navier-Stokes equation, one of the seven Millennium Prize Problems suggested by Clay Mathematics Institute(CMI). We prove that, for any initial velocity u0, there has been a force vector f, such that there exists no smooth solution to the respective Navier-Stokes equation. By the way, exact solutions can be explicitly solved as series, where the coefficients are all known functions determined only by u0 abd f. Similar result also holds for the Euler equation.For any n>=2, similar results also hold for dimensional Navier-Stokes equation or Euler equation.

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Available abstract

In this paper, we solve the Navier-Stokes equation, one of the seven Millennium Prize Problems suggested by Clay Mathematics Institute(CMI). We prove that, for any initial velocity u0, there has been a force vector f, such that there exists no smooth solution to the respective Navier-Stokes equation. By the way, exact solutions can be explicitly solved as series, where the coefficients are all known functions determined only by u0 abd f. Similar result also holds for the Euler equation.For any n>=2, similar results also hold for dimensional Navier-Stokes equation or Euler equation.

Key concepts: Compressibility, Euler equations, Mathematics, Euler's formula, Mathematical analysis, Navier–Stokes equations, Series (stratigraphy), Velocity vector

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