Notes on INS (16): Random thoughts

Infinite speed is no big deal ?

There may be some very small regions in space-time where there are some points with infinite speed. The flow will sill be continuous. From the “physical” point of view, the flow map \mathbb{R}_+ \times \Omega \to \Omega is more important than the velocity field ? The flow maps may have a few non-differentiable points (corresponding to infinite speed), no big deal.

Remark: Volume-preserving homeomorphisms in 3 dimensions can be approximated (in compact open topology) by volume-preserving diffeomorphims (Oxtoby ?)

Question: any theory about regularity of 3D volume-preserving homeomorphisms ?

Anyway, its easy to construct examples of volume-preserving homeomorphisms with a few singular (non-differentiable) points.

How do singularities look like ?

Nobody knows.

However, due to the “epsilon-regularity theory”, we must have a phenomenon of concentration of energy, speed, and vorticity near singular points, in every scale.

Let’s say the scales are \mu^n (where \mu < 1 is some small fixed positive number). At scale n (i.e. size \mu^n) we must have a “box” of radius of the size comparable to \mu^{n} in space, and \mu^{2n} in time. In this box, the speed (in a significant part) must be comparable to \mu^{-n} or greater.

Imagine in this box a solid torus of size \mu^{n} and thickness also comparable to \mu^{n} multiplied by a constant, and the the vector field is “almost circular” in this solid torus, and inside the solid torus the speed is comparable to \mu^{-n}

The singular point will not be in the center of the box, but lies somewhere inside the solid torus instead.

The idea is: look at torus-like boxes instead of ball-like boxes. Each subsequent solid torus (at smaller scale) lies inside the previous one (and can flow inside the previous one by “external rotation”).

This construction is not self-similar in the sense of Leray (too naive, and won’t work). Instead, it’s self-similar in a more subtle sense.

In the above construction, all the known necessary conditions (for the existence of a singular point) are satisfied: high local energy, high local enstrophy, high local vorticity in every direction, etc., at every scale ? Moreover, high BMO-1 norm ?

Twisted knots ?

In the above “solid-torus construction”, each solid torus may be represented by a knot (at the heart of the torus). Then we will hvae a sequence of knot, each one twisting around the previous one. This twsiting increases vorticity, speed, etc.

Will topological knot theory be of any help here ?! (Ref: Arnold & Khesin, Topological methods of hydrodynamics ?)

Zucon solutions

Construction of hypothetical singular solutions to the INS equation, which I will call “zucon solutions”:

The time interval will be [-1,0] (so the initial time will be -1, and the time of the singular point will be 0).

A zucon is a solution whose initial value u(-1,x) can be written as

u(-1,x) = \sum_{n=1}^\infty u_n(-1,x)

with the following properties:

1) Each u_n(-1,x) is smooth and has support contained in a compact domain \Omega_n of size \mu^n, where \mu < 1 is a constant to be determined later. The  u_n(-1,x)‘s are small enough, so that the series converges and is smooth.

2) The INS solution u_n(t,x) with the initial condition equal to u_n(-1) is smooth for t \in [-1,0]. Moreover, for t \in [-1/\mu^{2n},1], u_n(t,x) is “almost equal” to a rotation vector field of angular speed of the order 1/\mu^{2n} inside a solid torus T_n in \Omega_n. The solid torus T_n contains \Omega_{n+1}, and the size of  T_n is comparable to the size of \Omega_n divided by some constant (independent of n). The speed |u_n(t,x)| of the solution u_n(t,x) inside the solid torus T_n is comparable to 1/\mu^n for t > - \mu^{2n}.

3) We have a “near equality”

\Phi_u(t,x) = \hdots \circ \Phi_{u_3}(t,x) \circ \Phi_{u_2}(t,x) \circ \Phi_{u_1}(t,x) for all t \in [0,1]

where \Phi_u(t,x) means the flow generated by u(t,x). Do we want some kind of “almost commutativity among the vector fields u_n(t,x) ?

Put x_0 = \cap_{n=1}^n \Omega_n. Then the point (0, \Phi_u(0,x_0)) will be the singular point of the solution u(t,x) at time t= 0 (if the construction can be realized).

Question: How to construct each u_n ? The answer may lies in rotational solutions ?

Rotational (axisymmetric) solutions ?

We need to understand axisymmetric rotational solutions (with smooth initial conditions with support contained in a compact domain). I have not found any general reference about rotational solutions so far. I’ll have to find some references, and maybe to work out this problem myself. Don’t expect any singularity in rotational solution. What I want to see is the existence of rotational solutions with very small initial values, but which after a long period of time generates a high-speed rotational vector field in a small region near the “origin”. How is it possible for the rotational flow to accelerate its angular speed in some small domain ?!

Actually, there are (at least) 2 kinds of remarkable axisymmetric solutions ?

1) those with tangential speed = 0 (i.e. no rotation around the symmetric axis)

2) those whose velocity vector field is tangential (i.e. rotational flow): but is it possible ?!

The first kind can also be rotational, not around the axis, but around a horizontal circle. Question: which of these 2 kinds of flow admits speed acceleration ?

Refs:

Ladyzhenskaya O. A.: On the unique global solvability of the Cauchy problem for the Navier-Stokes equations in the presence of the axial symmetry. Zap. Nauch. Sem. LOMI 7 (1968), 155–177. (Russian) MR 0241833

Uchovskii M. R., Yudovich B. I.: Axially symmetric flows of an ideal and viscous fluid. J. Appl. Math. Mech. 32 (1968), 52–61. MR 0239293

Leonardi S., Málek J., Nečas J., Pokorný M.: On axially symmetric flows in ${\mathbb{R}}^3$. Z. Anal. Anwend. 18 (1999), 639–649. MR 1718156

Neustupa, Jiří ; Pokorný, Milan, Axisymmetric flow of Navier-Stokes fluid in the whole space with non-zero angular velocity component. (English). Mathematica Bohemica, vol. 126 (2001), issue 2, pp. 469-481 (they say that it’s NOT known if axisymmetric solutions are smooth/unique in general ! The case of zero angular momentum is proved by Ladyzhenskaya & other people)

Chae, … (2002): some improved estimates ? However the regularity question for axisymmetric solutions is still open.

Planar axisymmetric: this is the case considered by Ladyzhenskaya, Yukovich & Uchovskii. Each plane through the axis of symmetry is invariant. So this case should be very similar to the 2D case, with the same kind of estimates. Hence the proof of regularity should be ralatively simple (?). Not surprising (?).

Advance in science = lots of epsilon steps !

INS in cylindrical coordinates

(taken from wikipedia; simplified by assuming external force = 0)

The case with no rotational component:

\rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial  u_r}{\partial r} + u_z \frac{\partial u_r}{\partial z}\right) = -\frac{\partial  p}{\partial r} + \mu \left[\frac{1}{r}\frac{\partial}{\partial  r}\left(r \frac{\partial u_r}{\partial r}\right) + \frac{\partial^2  u_r}{\partial z^2} - \frac{u_r}{r^2}\right];

\rho \left(\frac{\partial u_z}{\partial t} +  u_r \frac{\partial u_z}{\partial r} + u_z \frac{\partial u_z}{\partial  z}\right) = -\frac{\partial p}{\partial z} + \mu  \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial  u_z}{\partial r}\right) + \frac{\partial^2 u_z}{\partial z^2}\right];

\frac{1}{r}\frac{\partial}{\partial r}\left(r  u_r\right) + \frac{\partial u_z}{\partial z} = 0

The rotation only case: ??? Is it possible to be rotational only ??

Hou-Li solutions

These might be just the ones I need: Dynamic Stability of the Three-Dimensional Axisymmetric Navier-Stokes Equations with Swirl, by HOU Thomas Y. (1) ; CONGMING LI (2), Communications on pure and applied mathematics, 2008, vol. 61,  no5,  pp. 661-697.

(1) California Institute of Technology, ETATS-UNIS
(2) University of Colorado, Boulder, ETATS-UNIS

Abstract: In this paper, we study the dynamic stability of the three-dimensional axisymmetric Navier-Stokes Equations with swirl. To this purpose, we propose a new one-dimensional model that approximates the Navier-Stokes equations along the symmetry axis. An important property of this one-dimensional model is that one can construct from its solutions a family of exact solutions of the three-dimensionaFinal Navier-Stokes equations. The nonlinear structure of the one-dimensional model has some very interesting properties. On one hand, it can lead to tremendous dynamic growth of the solution within a short time. On the other hand, it has a surprising dynamic depletion mechanism that prevents the solution from blowing up in finite time. By exploiting this special nonlinear structure, we prove the global regularity of the three-dimensional Navier-Stokes equations for a family of initial data, whose solutions can lead to large dynamic growth, but yet have global smooth solutions.

Vortex Stretching ?

Apparently, vortex stretching is the main mechanism of acceleration of particles in the INS flow. Work by Hou et al. showed how the flow can develop very high speed in a short time (but is still smooth, because the convection term has a flattening/vortex-depletion effect).

The key to understand singularities is the study of vorticity, vortex solutions, … ?

Ref: Majda & Bertozzi, Vorticity and Incompressible Flows, 2002.

Why singular solutions are hard to construct ?

There have been attempts to construct or  (show evidence of the existence of) singular solutions to Navier Stokes or Euler equations. However, they all failed. Why ?

Conjecture  1: Any symmmetry will make the solution regular. (e.g. axisymmetric solutions can’t be singular).

Self-similar solutions are those which are invariant w.r.t.  a scaling action. We know that self-similar singular solutions don’t exist.

Assume that conjecture 1 holds (and I think that it’s very likely that it holds). Then singular solutions that we are looking for can’t be symmetric. They must be obtained by a limiting process, and not by exact formulas ?!

Look at the space of solutions as a (maybe non-complete) metrizable space.

Introduce distance, watch distance grow with time … .

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