## torsdag 21 april 2016

### Velocity Blow-up to Infinity for Incompressible Euler?

In an effort to solve the Clay Navier-Stokes problem as formulated by Fefferman, Terence Tao in recent work seeks to construct a solution to the incompressible Euer equations with velocities becoming infinite in finite time, but does "not quite achieve" the goal.

Let me present some evidence indicating that the goal cannot be achieved. To this end we compare the incompressible Euler equations:
• $\frac{\partial u}{\partial t}+u\cdot\nabla u+\nabla p =0$
• $\nabla\cdot u=0$
with (i) vector-Burgers as a model of very compressible flow:
• $\frac{\partial u}{\partial t}+u\cdot\nabla u=0$
and (ii):
• $\frac{\partial u}{\partial t}+u\cdot\nabla u+\nabla p =0$,
• $\delta\Delta p=\nabla\cdot u$
with $\delta >0$ a small constant, as a model of slightly compressible flow.

For Burgers equation and so for (i), velocities may become discontinuous corresponding to the development of shocks over time, but velocities do not tend to infinity.

In case (ii) solving for the pressure p gives the following equation along a streamline $x(t)$:
• $\frac{du(x(t))}{dt} + \frac{1}{\delta}\nabla\Delta^{-1}\nabla\cdot u(x(t),t)=0$
which formally gives a bound on the possible growth of velocity in terms of $\frac{1}{\delta}$ preventing blow-up to infinity.

We conclude that neither very compressible nor slightly compressible flow appears to accommodate blow-up to infinite velocity. Is it then the incompressibility which will squeeze the flow to infinite pressure driving flow velocity to infinity? Far-fetched in my view.

On the other hand, we have strong evidence that Euler solutions become turbulent with substantial turbulent dissipation from large velocity gradients, while velocity does not spike to infinity. Again, the formulation of the Clay Navier-Stokes problem without reference to turbulence, appearently leads mathematicians into meaningless dead ends.