Existence and Smoothness of Solution of Navier-Stokes Equation on R 3

O. Vukovic
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引用次数: 1

Abstract

Navier-Stokes equation has for a long time been considered as one of the greatest unsolved problems in three and more dimensions. This paper proposes a solution to the aforementioned equation on R3. It introduces results from the previous literature and it proves the existence and uniqueness of smooth solution. Firstly, the concept of turbulent solution is defined. It is proved that turbulent solutions become strong solutions after some time in Navier-Stokes set of equations. However, in order to define the turbulent solution, the decay or blow-up time of solution must be examined. Differential inequality is defined and it is proved that solution of Navier-Stokes equation exists in a finite time although it exhibits blow-up solutions. The equation is introduced that establishes the distance between the strong solutions of Navier-Stokes equation and heat equation. As it is demonstrated, as the time goes to infinity, the distance decreases to zero and the solution of heat equation is identical to the solution of N-S equation. As the solution of heat equation is defined in the heat-sphere, after its analysis, it is proved that as the time goes to infinity, solution converges to the stationary state. The solution has a finite τ time and it exists when τ → ∞ that implies that it exists and it is periodic. The aforementioned statement proves the existence and smoothness of solution of Navier-Stokes equation on R3 and represents a major breakthrough in fluid dynamics and turbulence analysis.
r3上Navier-Stokes方程解的存在性与光滑性
长期以来,Navier-Stokes方程一直被认为是三维及多维空间中最大的未解问题之一。本文给出了上述方程在R3上的一个解。引入了前人的研究结果,证明了光滑解的存在唯一性。首先,定义了湍流解的概念。证明了在Navier-Stokes方程组中,湍流解在一段时间后成为强解。然而,为了定义湍流溶液,必须考察溶液的衰减或爆破时间。定义了微分不等式,并证明了Navier-Stokes方程的解在有限时间内存在,尽管它具有爆破解。介绍了建立Navier-Stokes方程强解与热方程强解之间距离的方程。结果表明,当时间趋于无穷时,距离减小到零,热方程的解与N-S方程的解相同。由于热方程的解是在热球中定义的,通过对其分析,证明了当时间趋于无穷时,解收敛于定态。解有一个有限的τ时间并且当τ→∞时它存在这意味着它存在并且它是周期性的。上述陈述证明了R3上Navier-Stokes方程解的存在性和光滑性,是流体动力学和湍流分析的重大突破。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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