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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Navier–Stokes existence and smoothness ： ウィキペディア英語版 Navier–Stokes existence and smoothness The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, one of the pillars of fluid mechanics (such as with turbulence). These equations describe the motion of a fluid (that is, a liquid or a gas) in space. Solutions to the Navier–Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering. Even much more basic properties of the solutions to Navier–Stokes have never been proven. For the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proved that smooth solutions always exist, or that if they do exist, they have bounded energy per unit mass. This is called the ''Navier–Stokes existence and smoothness'' problem. Since understanding the Navier–Stokes equations is considered to be the first step to understanding the elusive phenomenon of turbulence, the Clay Mathematics Institute in May 2000 made this problem one of its seven Millennium Prize problems in mathematics. It offered a US$1,000,000 prize to the first person providing a solution for a specific statement of the problem:〔(Official statement of the problem ), Clay Mathematics Institute.〕 ==The Navier–Stokes equations== (詳細はnonlinear partial differential equations for abstract vector fields of any size. In physics and engineering, they are a system of equations that models 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 continuum mechanics. The equations are a statement of Newton's second law, with the forces modeled according to those in a viscous Newtonian fluid—as the sum of contributions by pressure, viscous stress and an external body force. Since the setting of the problem proposed by the Clay Mathematics Institute is in three dimensions, for an incompressible and homogeneous fluid, only that case is considered below. Let $\backslash mathbf(\backslash boldsymbol,t)$ be a 3-dimensional vector field, the velocity of the fluid, and let $p(\backslash boldsymbol,t)$ be the pressure of the fluid.〔More precisely, is the pressure divided by the fluid density, and the density is constant for this incompressible and homogeneous fluid.〕 The Navier–Stokes equations are: : $\backslash frac\; +\; (\; \backslash mathbf\backslash cdot\backslash nabla\; )\; \backslash mathbf\; =\; -\backslash nabla\; p\; +\; \backslash nu\backslash Delta\; \backslash mathbf\; +\backslash mathbf(\backslash boldsymbol,t)$ where $\backslash nu>0$ is the kinematic viscosity, $\backslash mathbf(\backslash boldsymbol,t)$ the external volumetric force, $\backslash nabla$ is the gradient operator and $\backslash displaystyle\; \backslash Delta$ is the Laplacian operator, which is also denoted by $\backslash nabla\backslash cdot\backslash nabla$ or $\backslash nabla^2$. Note that this is a vector equation, i.e. it has three scalar equations. Writing down the coordinates of the velocity and the external force : $\backslash mathbf(\backslash boldsymbol,t)=\backslash big(\backslash ,v\_1(\backslash boldsymbol,t),\backslash ,v\_2(\backslash boldsymbol,t),\backslash ,v\_3(\backslash boldsymbol,t)\backslash ,\backslash big)\backslash ,,\backslash qquad\; \backslash mathbf(\backslash boldsymbol,t)=\backslash big(\backslash ,f\_1(\backslash boldsymbol,t),\backslash ,f\_2(\backslash boldsymbol,t),\backslash ,f\_3(\backslash boldsymbol,t)\backslash ,\backslash big)$ then for each $i=1,2,3$ there is the corresponding scalar Navier–Stokes equation: : $\backslash frac\; +\backslash sum\_^v\_j\backslash frac=\; -\backslash frac\; +\; \backslash nu\backslash sum\_^\backslash frac\; +f\_i(\backslash boldsymbol,t).$ The unknowns are the velocity $\backslash mathbf(\backslash boldsymbol,t)$ and the pressure $p(\backslash boldsymbol,t)$. Since in three dimensions, there are three equations and four unknowns (three scalar velocities and the pressure), then a supplementary equation is needed. This extra equation is the continuity equation for incompressible fluids that describes the conservation of mass of the fluid: : $\backslash nabla\backslash cdot\; \backslash mathbf\; =\; 0.$ Due to this last property, the solutions for the Navier–Stokes equations are searched in the set of solenoidal ("divergence-free") functions. For this flow of a homogeneous medium, density and viscosity are constants. The pressure ''p'' can be eliminated by taking an operator rot (alternative notation curl) of both sides of the Navier–Stokes equations. In this case the Navier–Stokes equations reduce to the vorticity-transport equations. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Navier–Stokes existence and smoothness」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.