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I just want to share some thoughts to demonstrate how modern PDEs work.
Navier-Stokes system describes incomprssible viscous liquid \begin{gather} \left\{\begin{aligned} & \boldsymbol{v}_{t} + (\boldsymbol{v}\cdot \nabla ) \boldsymbol{v} -\nu \Delta \boldsymbol{v}=-\nabla p,\\ &\nabla \cdot \boldsymbol{v} = 0, \end{aligned}\right. \label{eq-12.A.1} \end{gather} where $\boldsymbol{v}$ is a velocity, $p$ is pressure, $\rho=1$ a density, $\nu>0$ coefficient of viscosity. Without nonlinear terms and without $p$ it would be a heat equation.
Consider it in $\mathbb{R}_x^n\times\mathbb{R}_t^+$ with initial condition \begin{gather} \boldsymbol{v}(0)= \boldsymbol{v}_0(\boldsymbol{x}) \label{eq-12.A.2} \end{gather} with smooth initial velocity $\boldsymbol{v}_0(\boldsymbol{x})$.
Olga A. Ladyzhenskaya proved global well-posedness of the Cauchy (initial value) problem for the 2D incompressible Navier–Stokes equations—existence, uniqueness, and continuous dependence on initial data for sufficiently regular data.
At that moment (about 65 y.a.) it was application of revilutionary methods of Real Analysis, mostly of Sergey L. Sobolev's embedding theorems. Here it was important that $n=2$ ($n=1$ is a trivial case).
For $n=3$ these methods almost work here, it is a critical case. What does it mean? If we slightly reduce power in the nonlinear term or if we apply embedding theorems with slightly larger powers (sorry, but they do not hold) this proof would work.
What else is known?
For 3D case Jean Leray proved 90 y.a. local existence and uniqueness of smooth solutions and global existence of weak solutions. For small (in the proper sense) initial velocity $\boldsymbol{v}_0(\boldsymbol{x})$ Hiroshi Fujita and Tosio Kato (1964) proved existence of the global smooth solutions.
This problem has been formulated as one of the Millenium problems, but after many years of efforts by the best specialists in non-linear PDEs they reached consensunce that our current methods are insufficient and basically gave up. Still freeks (attracted by promise of glory and 1 million USD) continue trying.
How to prove global existence of weak solutions?
Considering such equation with extra "superviscosity": \begin{gather} \left\{\begin{aligned} &\rho\boldsymbol{v}_{t} + (\boldsymbol{v}\cdot \nabla ) \rho\boldsymbol{v} -\nu \Delta \boldsymbol{v}+\varepsilon\Delta^2\boldsymbol{v} =-\nabla p,\\ &\nabla \cdot \boldsymbol{v} = 0, \end{aligned}\right. \label{eq-12.A.3} \end{gather} with $\varepsilon>0$ both existence, uniqueness and regularity is easy to prove in 3D case.
Moreover, there is a compactness: there exist sequences $\varepsilon=\varepsilon_k\to +0$ and $\boldsymbol(v)_k$, $p_k$ which have weak limits $\boldsymbol(v)$, $p$ solving original problem. But we cannot conclude that this solution is regular or unique.
So we have a problem: among regular solutions there is uniqueness but we do not know about existence of such solutions.
What if?
Disclaimer.