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The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré conjecture at the ...
In mathematics, the Navier–Stokes equations are a system of nonlinear partial differential equations for abstract vector fields of any size. In physics and engineering, they are a system of equations that model the motion of liquids or non-rarefied gases (in which the mean free path is short enough so that it can be thought of as a continuum mean instead of a collection of particles) using ...
Pages in category "Millennium Prize Problems" The following 8 pages are in this category, out of 8 total. ... Navier–Stokes existence and smoothness; P.
Millennium Prize Problems; Birch and Swinnerton-Dyer conjecture; Hodge conjecture; Navier–Stokes existence and smoothness; P versus NP problem; Poincaré conjecture (solved) Riemann hypothesis; Yang–Mills existence and mass gap
For nonlinear equations these questions are in general very hard: for example, the hardest part of Yau's solution of the Calabi conjecture was the proof of existence for a Monge–Ampere equation. The open problem of existence (and smoothness) of solutions to the Navier–Stokes equations is one of the seven Millennium Prize problems in ...
In the above equation stoke assume that at, non-stationary Navier Stokes problem converge towards the solution of the correspondent stationary problem. This solution will not depend upon the function . If this is used for the above equation consisting of Navier stokes equation and continuity equations with time derivative of pressure, then the ...
Under what conditions do smooth solutions exist for the Navier–Stokes equations, which are the equations that describe the flow of a viscous fluid? This problem, for an incompressible fluid in three dimensions, is also one of the Millennium Prize Problems in mathematics. [66]
Burgers vortex layer or Burgers vortex sheet is a strained shear layer, which is a two-dimensional analogue of Burgers vortex. This is also an exact solution of the Navier–Stokes equations, first described by Albert A. Townsend in 1951. [8] The velocity field (,,) expressed in the Cartesian coordinates are