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 $$\xi_i(x,t):= \int_{ \mathbb{ R}^3}^{ }{ \alpha(x-y,\dfrac{ t}{ \nu})v^k_i(x,0)}dy, ~~~~~ v^k_i(x,0) = \biggl(\textbf{ curl}^k(\textbf{ u}(x,0))\biggr)_i, ~~~~~ 1\le i\le 3$$ 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.
Categories: math.GM