# Get e-book Navier—Stokes Equations and Related Nonlinear Problems

On the boundary degrees of freedoms of the disk we overwrote the equation by prescribing the homogeneous boundary conditions.

## On a Modified Form of Navier-Stokes Equations for Three-Dimensional Flows

The equations related to these dofs describe the force im balance at this boundary. Testing the residual functional with the characteristic function on that boundary in the x- or y-direction we obtain the integrated stresses in the x- or y-direction :. NGS-Py 6. Getting started 2. Advanced Topics 3. Time-dependent and non-linear problems 3. Geometric modeling and mesh generation 5. AddRectangle 0 , 0 , 2 , 0. The inlet and outlet fluid pressures are known. Since the channels are at most 0.

Because there are no external forces gravity is neglected , the force term 4 is also equal to zero. The flow is driven by a higher pressure at the inlet than at the outlet.

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These results show the balance between the pressure force 2 and the viscous forces 3 in the NS equations. Along the thinner channels, the impact of viscous diffusion is larger, which leads to higher pressure drops. In engineering applications where the Reynolds number is very high, the inertial forces 1 are much larger than the viscous forces 3. Such turbulent flow problems are transient in nature; a mesh that is fine enough to resolve the size of the smallest eddies in the flow needs to be used.

## Wolf : On the local pressure of the Navier-Stokes equations and related systems

Running such simulations using the NS equations is often beyond the computational power of most of today's computers and supercomputers. These time-averaged equations can then be computed in a stationary way on a relatively coarse mesh, thus drastically reducing the computing power and time required for such simulations typically a few minutes for two-dimensional flow and a few minutes to a few days for three-dimensional flow.

Here, U and P are the time-averaged velocity and pressure, respectively. This model is often used in industrial applications because it is both robust and computationally inexpensive.

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To illustrate this flow regime, let us look at the flow in a much larger geometry than the por-scale flow: a typical ozone purification reactor. The reactor is about 40 meters long and looks like a maze with partial walls or baffles that divide the space into room-sized compartments. Based on the inlet velocity and diameter, which in this case correspond to 0. The flow compressibility is measured by the Mach number. All the previous examples are weakly compressible, meaning that the Mach number is lower than 0.

When the Mach number is very low, it is OK to assume that the flow is incompressible. This is often a good approximation for liquids, which are much less compressible than gases. The creeping flow example showing water flowing at a low speed through the porous media is a good example of incompressible flow. In some cases, the flow velocity is large enough to introduce significant changes in the density and temperature of the fluid.

The energy equation predicts the temperature in the fluid, which is needed to compute its temperature-dependent material properties. Compressible flow can be laminar or turbulent. In the next example, we look at a high-speed turbulent gas flow in a diffuser a converging and diverging nozzle. The results in these three plots show strong similarities, which confirms the strong coupling between the velocity, pressure, and temperature fields. This set-up has been studied in a number of experiments and numerical simulations by M.

Combining this with the scale-invariance 9 , we see that for fixed , we may organise the structure constants for dyadic into sextuples which sum to zero including some degenerate tuples of order less than six. This will automatically guarantee the cancellation 8 required for a steady state energy distribution, provided that. Thus we are led to the heuristic conclusion that the most stable power law distribution for the energies is the law.

Given that frequency interactions tend to cascade from low frequencies to high if only because there are so many more high frequencies than low ones , the above analysis predicts a stablising effect around this power law: scales at which a law 6 holds for some are likely to lose energy in the near-term, while scales at which a law 6 hold for some are conversely expected to gain energy, thus nudging the exponent of power law towards.

We can solve for in terms of energy dissipation as follows. If we let be the frequency scale demarcating the transition from the energy flow regime 5 to the dissipation regime 4 , we have. On the other hand, if we let be the energy dissipation at this scale which we expect to be the dominant scale of energy dissipation , we have.

Some simple algebra then lets us solve for and as. This is done by constructing a modification of the Navier-Stokes equations with a nonlinearity that obeys essentially all of the function space estimates that the true Navier-Stokes nonlinearity does, and which also obeys the energy identity, but for which one can construct solutions that blow up in finite time. Results of this type had been previously established by Montgomery-Smith , Gallagher-Paicu , and Li-Sinai for variants of the Navier-Stokes equation without the energy identity, and by Katz-Pavlovic and by Cheskidov for dyadic analogues of the Navier-Stokes equations in five and higher dimensions that obeyed the energy identity see also the work of Plechac and Sverak and of Hou and Lei that also suggest blowup for other Navier-Stokes type models obeying the energy identity in five and higher dimensions , but to my knowledge this is the first blowup result for a Navier-Stokes type equation in three dimensions that also obeys the energy identity.

### Classical Mechanics

Intriguingly, the method of proof in fact hints at a possible route to establishing blowup for the true Navier-Stokes equations, which I am now increasingly inclined to believe is the case albeit for a very small set of initial data. To state the results more precisely, recall that the Navier-Stokes equations can be written in the form.

We will work in the non-periodic setting, so the spatial domain is , and for sake of exposition I will not discuss matters of regularity or decay of the solution but we will always be working with strong notions of solution here rather than weak ones. Applying the Leray projection to divergence-free vector fields to this equation, we can eliminate the pressure, and obtain an evolution equation. The global regularity problem for Navier-Stokes is then equivalent to the global regularity problem for the evolution equation 1.

An important feature of the bilinear operator appearing in 1 is the cancellation law. This identity and its consequences provide essentially the only known a priori bound on solutions to the Navier-Stokes equations from large data and arbitrary times. Unfortunately, as discussed in this previous post , the quantities controlled by the energy identity are supercritical with respect to scaling, which is the fundamental obstacle that has defeated all attempts to solve the global regularity problem for Navier-Stokes without any additional assumptions on the data or solution e.

Theorem 1 There exists an averaged version of the bilinear operator , of the form. There are some integrability conditions on the Fourier multipliers required in the above theorem in order for the conclusion to be non-trivial, but I am omitting them here for sake of exposition. Because spatial rotations and Fourier multipliers of order are bounded on most function spaces, automatically obeys almost all of the upper bound estimates that does.

Thus, this theorem blocks any attempt to prove global regularity for the true Navier-Stokes equations which relies purely on the energy identity and on upper bound estimates for the nonlinearity; one must use some additional structure of the nonlinear operator which is not shared by an averaged version. Such additional structure certainly exists — for instance, the Navier-Stokes equation has a vorticity formulation involving only differential operators rather than pseudodifferential ones, whereas a general equation of the form 2 does not. It turns out that the particular averaged bilinear operator that we will use will be a finite linear combination of local cascade operators , which take the form.

Such operators were essentially introduced by Katz and Pavlovic as dyadic models for ; they have the essentially the same scaling property as except that one can only scale along powers of , rather than over all positive reals , and in fact they can be expressed as an average of in the sense of the above theorem, as can be shown after a somewhat tedious amount of Fourier-analytic symbol manipulations. The precise ODE that shows up depends on what precise combination of local cascade operators one is using.

Actually, Katz-Pavlovic worked with a technical variant of this particular equation, but the differences are not so important for this current discussion.

### See a Problem?

Note that the quadratic terms on the RHS carry a higher exponent of than the dissipation term; this reflects the supercritical nature of this evolution the energy is monotone decreasing in this flow, so the natural size of given the control on the energy is. There is a slight technical issue with the dissipation if one wishes to embed 3 into an equation of the form 2 , but it is minor and I will not discuss it further here. In principle, if the mode has size comparable to at some time , then energy should flow from to at a rate comparable to , so that by time or so, most of the energy of should have drained into the mode with hardly any energy dissipated.

Since the series is summable, this suggests finite time blowup for this ODE as the energy races ever more quickly to higher and higher modes. Such a scenario was indeed established by Katz and Pavlovic and refined by Cheskidov if the dissipation strength was weakened somewhat the exponent has to be lowered to be less than.

As mentioned above, this is enough to give a version of Theorem 1 in five and higher dimensions. On the other hand, it was shown a few years ago by Barbato, Morandin, and Romito that 3 in fact admits global smooth solutions at least in the dyadic case , and assuming non-negative initial data. Roughly speaking, the problem is that as energy is being transferred from to , energy is also simultaneously being transferred from to , and as such the solution races off to higher modes a bit too prematurely, without absorbing all of the energy from lower modes.

This weakens the strength of the blowup to the point where the moderately strong dissipation in 3 is enough to kill the high frequency cascade before a true singularity occurs. Because of this, the original Katz-Pavlovic model cannot quite be used to establish Theorem 1 in three dimensions. Actually, the original Katz-Pavlovic model had some additional dispersive features which allowed for another proof of global smooth solutions, which is an unpublished result of Nazarov.

To do this, I needed to insert a delay in the cascade process so that after energy was dumped into scale , it would take some time before the energy would start to transfer to scale , but the process also needed to be abrupt once the process of energy transfer started, it needed to conclude very quickly, before the delayed transfer for the next scale kicked in. The coupling constants here range widely from being very large to very small; in practice, this makes the and modes absorb very little energy, but exert a sizeable influence on the remaining modes.

If a lot of energy is suddenly dumped into , what happens next is roughly as follows: for a moderate period of time, nothing much happens other than a trickle of energy into , which in turn causes a rapid exponential growth of from a very low base. After this delay, suddenly crosses a certain threshold, at which point it causes and to exchange energy back and forth with extreme speed.

The energy from then rapidly drains into , and the process begins again with a slight loss in energy due to the dissipation. If one plots the total energy as a function of time, it looks schematically like this:. As in the previous heuristic discussion, the time between cascades from one frequency scale to the next decay exponentially, leading to blowup at some finite time. There is a real but remote possibility that this sort of construction can be adapted to the true Navier-Stokes equations.

The basic blowup mechanism in the averaged equation is that of a von Neumann machine , or more precisely a construct built within the laws of the inviscid evolution that, after some time delay, manages to suddenly create a replica of itself at a finer scale and to largely erase its original instantiation in the process.

In principle, such a von Neumann machine could also be built out of the laws of the inviscid form of the Navier-Stokes equations i. In physical terms, one would have to build the machine purely out of an ideal fluid i. The key thing missing in this program in both senses of the word to establish blowup for Navier-Stokes is to construct the logic gates within the laws of ideal fluids.

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As it turns out, I was somewhat impatient to finalise the paper and move on to other things, and the original preprint was still somewhat rough in places contradicting my own advice on this matter , with a number of typos of minor to moderate severity. But a bit more seriously, I discovered on a further proofreading that there was a subtle error in a component of the argument that I had believed to be routine — namely the persistence of higher regularity for mild solutions.