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 $$ \textbf{u}(x,t)- \dfrac{1}{4\pi}\textbf{Curl}\Biggl( \int_{\mathbb{R}^{3}{}{\dfrac{\textbf{Curl}} (\textbf{u} (x^{\prime,t))}{|x-x\prime|}}dV\prime\Biggr)} = -\int_{0}^{{t}{\dfrac{1}{\rho}} \textbf{Grad}\big(\Gamma(x,s)\big)}ds.$$ Consequently, any curl free solution must be written as $$\textbf{u}(x,t) = -\dfrac{1}{\rho} \textbf{Grad}\biggl(\int_{0}^{{t}{\Gamma(x,s)} ds}\biggr)$$ 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 $$\textbf{curl}^{k\biggl((\textbf{u}\cdot} \nabla )\textbf{u}\biggr)(x,t)=\textbf{0}$$ for all $t^\prime\le t$ then $$\textbf{u}(x,t) = \textbf{H}^{{k+1}(\xi_1,\xi_2,\xi_3,t)} -\int_{0}^{{t}{\dfrac{1}{\rho}} \textbf{Grad}\big(\Gamma(x,s)\big)}ds, ~~~~~ t\in [t^{\prime,\infty)$$} with $$\xi_i(x,t):= \int_{\mathbb{R}^{3}{}{\alpha(x-y,\dfrac{t}{\nu})vk_i(x,0)}dy,} ~~~~~ v^k_i(x,0) = \biggl(\textbf{curl}^{k(\textbf{u}(x,0))\biggr)_i,} ~~~~~ 1\le i\le 3$$ and $\textbf{H}^k$ 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.