Existence and Smoothness of Solution of Navier-Stokes Equation on R3
Author(s)
Ognjen Vukovic*
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.
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.
Cite this paper
Vukovic, O. (2015) Existence and Smoothness of Solution of Navier-Stokes Equation on R3. International Journal of Modern Nonlinear Theory and Application, 4, 117-126. doi: 10.4236/ijmnta.2015.42008.
Vukovic, O. (2015) Existence and Smoothness of Solution of Navier-Stokes Equation on R3. International Journal of Modern Nonlinear Theory and Application, 4, 117-126. doi: 10.4236/ijmnta.2015.42008.
References
[1] Oliver, M. and Titi, E.S. (2000) Remark on the Rate of Decay of Higher Order Derivatives for Solutions to the Navier-Stokes Equations in Rn. Journal of Functional Analysis, 172, 1-18.
http://dx.doi.org/10.1006/jfan.1999.3550 [2] Okabe, T. (2009) Asymptotic Energy Concentration in the Phase Space of the Weak Solutions to the Navier-Stokes Equations. Journal of Differential Equations, 246, 895-908.
http://dx.doi.org/10.1016/j.jde.2008.07.037 [3] Tao, T. (2006) Perelman’s Proof of the Poincaré Conjecture: A Nonlinear PDE Perspective.
http://arxiv.org/abs/math/0610903 [4] Cannon, J.R. (1984) The One-Dimensional Heat Equation. Vol. 23. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9781139086967
[1] Oliver, M. and Titi, E.S. (2000) Remark on the Rate of Decay of Higher Order Derivatives for Solutions to the Navier-Stokes Equations in Rn. Journal of Functional Analysis, 172, 1-18.
http://dx.doi.org/10.1006/jfan.1999.3550 [2] Okabe, T. (2009) Asymptotic Energy Concentration in the Phase Space of the Weak Solutions to the Navier-Stokes Equations. Journal of Differential Equations, 246, 895-908.
http://dx.doi.org/10.1016/j.jde.2008.07.037 [3] Tao, T. (2006) Perelman’s Proof of the Poincaré Conjecture: A Nonlinear PDE Perspective.
http://arxiv.org/abs/math/0610903 [4] Cannon, J.R. (1984) The One-Dimensional Heat Equation. Vol. 23. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9781139086967