The Navier-Stokes Equation and Helmholtz Decomposition

Abstract

This work explores Navier-Stokes equation with no gravitational forces. In short, it shows that any smooth solution that decays quickly must take the form $$\text{u}(x,t)-\frac{1}{4\pi }\text{Curl}\left({\int }_{{\mathbb{R}}^3}^{}\frac{\text{Curl}(\text{u}(x^{\prime },t))}{|x-x^{\prime }|}d{V}^{\prime }\right)=-{\int }_0^t\frac{1}{\rho }\text{Grad}(\Gamma (x,s))ds.$$ Consequently, any curl free solution must be written as $$\text{u}(x,t)=-\frac{1}{\rho }\text{Grad}({\int }_0^t\Gamma (x,s)ds)$$ with $\Gamma$ a known function which is related to the heat equation. Even further it shows if there exist a value $k\in \mathbb{N}$ such that $${\text{curl}}^k((\text{u}\cdot \nabla )\text{u})(x,t)=\text{0}$$ for all $t^{\prime }\le t$ then $$\text{u}(x,t)={\text{H}}^{k+1}({\xi }_1,{\xi }_2,{\xi }_3,t)-{\int }_0^t\frac{1}{\rho }\text{Grad}(\Gamma (x,s))ds,t\in [t^{\prime },\infty )$$ with $${\xi }_i(x,t):={\int }_{{\mathbb{R}}^3}^{}\alpha (x-y,\frac{t}{\nu })v_i^k(x,0)dy,v_i^k(x,0)=({\text{curl}}^k(\text{u}(x,0)){)}_i,1\le i\le 3$$ and ${\text{H}}^k$ the $k^{th}$ application of Helmholtz operator. Hence, if there is another solution where the non-linear term is infinitly curlable then the solution is not unique. If the solution is unique, then this is the only possible solution.

arXiv commentI would like to give special thanks to all my professors especially Dr. Enakousta for helping me anytime I needed it