The nonlinearities of the Navier-Stokes equations make them very different from the Laplace or other well known equation of mathematical physics.

Kozachok A.A., Kiev, Ukraine

Formulated by Clay Mathematics Institute the sixth Millennium Problems about existence and smoothness of solutions of the Navier – Stokes equations periodically was discussed at numerous forums (http://grani.ru/Society/Science/m.112524.html). On recognition of some commentators the complete presentation of problem’s solution can demand about thousand pages for mathematical formulas (http://lib.mexmat.ru/forum/viewtopic.php?t=4289). The author of Official Problem Description–Charles Fefferman has set the task about demonstration of existence and smoothness of the solution, instead of solution’s obtaining. However, the Navier-Stokes equations can be reduced correctly to more simple classical equations of mathematical physics . The problem of an existence proof of solutions of such equations is not so actual.
It is known, and are identical to infinitesimal magnitude and velocity of the relative modification of a volume element of the strained medium. Therefore divergency of acceleration , probably, there is a magnitude, identical to a acceleration of the relative modification of the same volume. In that case for incompressible liquid alongside with requirements it is necessary to accept .
The requirement for incompressible liquid is formulated by analogy and proved. Operation will convert the Navier – Stokes equations to the three-dimensional Laplace equation for pressure at some limitations of a mass force vector. The time enters into Laplace equation as parameter.
Laplacian of the Navier – Stokes equations (pressure – harmonic function) and a change of a variable (velocity on acceleration) allow to gain system of conventionally independent integro-differential equations with acceleration’s components . In that case components of the acceleration for ideal incompressible liquid are harmonic functions too. The change of a variable allows to use boundary conditions of an adhesion of a fluid absolutely correctly. According to this requirement vectors of acceleration on firm immobile boundary line are equal to null.
Conversion of the Navier-Stokes equations to more simple equations has actually removed a problem of an existence proof and smooth finish of their solution on a background. Such demonstration in view of that one of required variables is a harmonic function, it is possible to not fulfil. It is possible to use known effects about properties of harmonic functions or representation of the common decision of the Laplace equation(http://continuum-paradoxes.narod.ru, the link « Manual, p.1 », p. 58).
More in detail on a site http://continuum-paradoxes.narod.ru, the link “Russian pages”, “Sixth Millennium Problems (NAVIER-STOKES equations) is solvable by classical methods”.

Yours faithfully, Alexandr Kozachok

In two dimensions, ( with periodic bounary conditions) it was done some years ago by Dowker. I have generalized it to fluids on a 2-sphere with viscosity and coriolis force to model the atmosphere. The idea is to produce an effective theory of large scale structures in the atmosphere like hurricanes and the jet stream. I have done some simple numerical work also, which confirms the general ideas. With some help I may be able to turn this into a more realistic model of the atmosphere, including effects of convection ( which I have not yet included).

The idea is to average over small scale fluid motion to get an effective theory for larg scale structures. This is analogous to the averaging over the moleculr motion to get fluid mechanics, but carried one more step. The main point is that this second average involves dynamical variables with a non-commutative algebra (matrices in the simplest case of two dimension) rather than just functions or vector fields. Still, it is a very practical way of describing large scale objects because there are very few variables in the effective theory.

I hope to write more about this in another post.

M

It is my impression still that the papers you referred to are all about the two dimensional discrete non-commutative sphere, applied to regularize quantum field theory. For the current paper about three dimensional fluids in the continuum, it remains an open question whether any of the effects you talk about can happen. Actually it is possibly equivalent to the question I raise at the end. Does this really regularize the potential classical divergence in Fluid mechanics? That is, does the regularized equation have smooth solutions? I don’t know the answer. Let us keep talking.

I am also writing another paper on fluid mechanics on a two dimensional discrete sphere where many of the issues you raise are more directly relevant. But there I know that the equations do have smooth solutions for all time with the regularization: there is no analogue for a non-commutative anomaly in the application of fuzzy methods to this classical theory.

I have to read and understand your papers before I can write a detailed response. Couple of small points though. I am using the three dimensional sphere and not the two-sphere. Also, I am not using a discrete version of the sphere; for example not only the standard . Does that make a difference?

Moreover, my fluid mechanics remains classical even with the non-commutativity of the regularization.

Thanks for a thoughtful comment.

M

Here’s how to circumvent the problems you address.

Any time you replace a differential operator by a difference operator or an integral operator, you are introducing a coupling between points that are not infinitesimally close. What I mean by non-locality is that points that are not infinitesimally close are coupled together in the evolution equation. Violations of causality are a bit different, when the time co-ordinate is also affected. These can happen if you modify ( or discretize) relativistic theories. Since incompressible flow anyway has signals propagating infinite speed, this is not the issue in Navier-Stokes or Euler equations.

The more important point is that you would like to keep regularity when you introduce a cutoff. If the cutoff procedure itself introduces discontinuities, it will be even harder to study the regularity of solutions this way.

So, what is the proper definition of non-locality? Regardless, what you say about discontinuous point-to-point jumps in field variables is undoubtedly true in any discretized space.

When you discretize an equation, usually a derivative is replaced a difference: , a being the lattice length. This means you lose locality over a small distance scale .
Also you lose conservation of momenta larger than .

For example, it doesn’t make sense to ask for the value of the function in between two lattice points. You can do some kind of interpolation. But there will be a discontinuity either of the function itself or its derivative of sufficiently high order.

I am writing a paper with more detailed mathematics. Thanks for your interest.

“When studying a partial differential equation it is often useful to impose such a smallest possible length scale, at the cost of introducing some non-locality in the problem. This happens if we replace space by a lattice to discretize the PDE to solve it numerically.”

How exactly is non-locality introduced through spatial discretization? I must preface this by saying that I have practically no knowledge of the purely mathematical approach to the Navier Stokes equations, so my question likely exposes my accompanying ignorance. If my understanding of non-locality is correct, I don’t understand how spatial discretization would introduce such non-locality if none was there in the continuous case. Could you explain? Thanks, and good post, by the way