# Is the Navier-Stokes equation valid in $d=2$ spatial dimensions?

In this article, the authors study the time behaviour of the velocity-velocity correlation function of a particle in a gas. If the gas is immersed in $$d$$ spatial dimensions, they find that

$$C(t)=\frac{\langle v(0)_i v(t)_i \rangle}{\langle v(0)_i^2 \rangle} \sim t^{-d/2}$$

where the average refers to an equilibrium ensemble. In two dimensions ($$d=2$$), the integral over a long time of $$1/t$$ doesn't exist. On this basis, they conclude that the self-diffusion coefficient $$D$$ does not exist in two dimensions, so that "conventional hydrodynamics does not exist in two dimensions". This is because thanks to the Green-Kubo formula

$$D \propto \int_0^{\infty} C(t) \, dt$$

the diffusion coefficient $$D$$ is logarithmically divergent for $$d=2$$.

Is this a well-known result? Does it imply that we can not apply Navier stokes for a gas in two dimensions? (i.e. interacting particles constrained on a surface can not be described in the long-wavelength limit by Navier-Stokes hydrodynamics).

As the authors of this other paper comment ($$d=2$$): Physically the long-time tails $$~1/t$$ of the correlation functions are caused by the slowly decaying hydrodynamic modes. Kinetically this is due to the possibility of recollisions, i. e. , collisions between two particles that have collided before. They lead to a much slower decay of the initial state of a particle than if they are excluded since they can still "remind" the particle of its initial state after many collisions have taken place.

Those two papers are from the '70s, which is the situation today? Can we really apply Navier Stokes (or the diffusion equation) when $$d=2$$ and $$d=1$$?

Note: clearly we can assume that Navier Stokes works in $$d=2$$, and we can numerically solve it (see this answer), my question is if it can be justified from a more fundamental kinetic approach: for sure in $$d>2$$ we can derive it from kinetic theory, but how about $$d<3$$?

Note: interestingly, not even the Boltzmann equation seems to be good for some 2D fluids (and Navier stokes may be obtained from Boltzmann), see this answer.

• I am not sure what is meant by "conventional hydrodynamics" but the Navier-Stokes equation is perfectly valid in 2D. What they may be referring to is that things like Kelvin-Helmholtz vortices can last for a really long time in 2D simulations but decay/damp very quickly in 3D. Just a thought... Commented Oct 15, 2021 at 12:34
• Jos Stam, Stable Fluids Commented Oct 16, 2021 at 1:49
• @honeste_vivere by standard/conventional hydro I just mean Navier Stokes equations. Clearly NS equations exist in 2d as a phenomenological model. However the papers that I linked argue that this phenomenological model is not valid when you try to derive it from kinetics (because of problems with diffusion in 2d) Commented Oct 16, 2021 at 16:05

There has actually been a fair amount of activity in this area recently, see, for example, this set of lecture notes.

I would argue that the answer to your question is, "yes", if properly understood.

1. The basic issue is not special to $$d=2$$. Fluid dynamics is an EFT of the low energy, low momentum behavior of correlations in a many-body system. The theory is expected to be valid as long as the system can locally equilibrate.

2. The theory is implemented by writing down the conservation laws for the conserved charges (possibly supplemented by Goldstone bosons), and expanding the currents in gradients of the thermodynamic variables. The order $$O(\nabla)$$ truncation of the stress tensor is called the Navier-Stokes equation.

3. This approach does not reproduce the correlation functions at arbitrary order in $$(\omega,k)$$. The issue is that any theory that operates with coarse grained variable like temperature $$T$$ must take into account the coarse graining scale $$l$$. General theorems in thermodynamics require that thermodynamic variables fluctuate, and that fluctuations depend on $$l$$. Indeed, by the fluctuation-dissipation relation, any theory (like Navier-Stokes) that takes into account dissipation must have fluctuations.

1. Fluctuations lead to non-analytic terms (logarithms and fractional powers) in correlation functions, and the importance of these terms is enhanced in lower dimension. While in $$d=3$$ fluctuations appear in between the order $$\nabla$$ and $$\nabla^2$$ terms, in $$d=2$$ fluctuations lead to a logarithmic divergence at the level of the Navier-Stokes equation.

2. Historically, this was first studied using kinetic approaches, but kinetic theory also does not include fluctuations in a systematic manner. Indeed, non-analyticities can be studied directly in fluid dynamics, by supplementing the theory with stochastic forces. The strength of these forces is completely fixed (at leading order) by fluctuation-dissipation relations. This approach was initiated by Landau and Lifschitz and is explained in the second volume of their book on Statistical Physics, and in the lecture notes cited above.

3. What does all of this mean for the real world? In $$d=3$$ we can work with the Navier-Stokes equation, but we have to keep in mind that the next correction is related to fluctuations. In $$d=2$$ we have to work with stochastic Navier-Stokes. This theory predicts that the shear viscosity diverges logarithmic at infinite time and infinite volume. This effect is real (it has been seen in numerical simulations, for example Leo Kadanoff et al., Phys Rev A 40 4527 (1989)), but it is hard to see in experiment.

4. In practice, a logarithmic divergence is quite weak, and two-dimensional fluids that can be studied experimentally (liquid helium films, $$2d$$ trapped atomic gases) are described quite well by the Navier-Stokes equation.

5. Final remark: We know that there is nothing intrinsically wrong with a divergent shear viscosity. Transport coefficients diverge near second order phase transitions (such as the liquid-gas phase transition) even in $$d=3$$. This is described in the famous Hohenberg-Halperin review, and these effects have been verified experimentally.

• Question about 7. If one can still use Navier-Stokes equation, what viscosity should one take to calculate a flow around a cylinder? Commented Oct 16, 2021 at 14:30
• This is like logarithmic running in field theory. The bare value of viscosity is controlled by the renormalization group. If you do a stochastic simulation you use a bare viscosity and a resolution scale (you solve the equations on a grid, for example). Then you have to logarithmically run the bare viscosity as you make the resolution finer to get physical results (for the force on the disk, for example) to remian fixed. Commented Oct 16, 2021 at 14:59
• If you just use ordinary Navier-Stokes, then you must stay with a finite system of size L, and you understand that the log divergence was cut off at the scale L. This may not be important, because the log is multplied by something like a factor 1/(4\pi) (a loop factor, in the language of field theory). Commented Oct 16, 2021 at 15:01
• "Then you have to logarithmically run the bare viscosity... " - I can be missing your point, but Navier-Stokes equations have a perfectly valid continuous solution in 2D for a flow past a cylinder. There are no divergences as the mesh size decreases. Logarithmically increasing/decreasing viscosity as you change the mesh size would just lead to zero/infinite drag. If one wants to get a physical answer for a drag, they have to converge viscosity to something finite Commented Oct 16, 2021 at 16:19
• The correct description of nature is stochastic Navier-Stokes, which requires a regulator because of the delta function correlated noise. Commented Oct 16, 2021 at 16:47

Great question. From what I was able to find, the problem is well-known under the name of "2D long-time tail problem" (for instance, mentioned in these and these proceedings). There is extensive literature on the subject with regular and numerical experiments, as well as some theoretical approaches (like the paper you mention). The research is usually concerned with the validity of the slow decay of the correlation function $$C(t)$$. You can check a recent paper "Nature of self-diffusion in two-dimensional fluids" for theory and references. The paper utilizes a self-consistent approach to the correlation function $$C(t)$$ with time-dependent diffusion and shear viscosity coefficients. The conclusion of the paper is that while the diffusion coefficient diverges at later times, the viscosity coefficient approaches a finite value. Linearized Navier-Stokes with time-dependent coefficients is used in the theory part of this paper, as well as in many other papers.

From my impression, I am not sure, whether the approach with Navier-Stokes equations with time-dependent coefficients is justified: it works numerically in the particular case that the studies consider, but I am not sure how to adopt it to the regular hydrodynamic setups in 2D, where everything is static.

• I'm not sure about this recent paper which disputes the log divergence of viscosity in 2d. The theoretical case for the log surely seems very robust. You have to be very careful with numerical simulation, because the order of limits in the Kubo relation is crucial: You have to take $k\to 0$ before you take $\omega\to 0$. This means that in a finite system you should not study arbitrarily large times. Commented Oct 16, 2021 at 13:43
• @Thomas This paper states the viscosity remains finite. Fig. 2 does appear to consider correct limits. The theoretical case in the paper does not appear particularly robust to me, because their equations for regular diffusion and vorticity diffusion are not time independent, which one almost always wants for a model to be physical. Commented Oct 16, 2021 at 17:36
• I looked again at the paper. They don't actually measure the shear stress correlator that defines viscosity. The closest they come is the particle velocity autocorrelation function (in Fig. 3), and then they have a theory that ties it to the shear viscosity. I'm not sure. If they want to make strong statements about the viscosity they really need to maesure shear stress. Commented Oct 17, 2021 at 13:01
• I don't see the details of calculation of viscosity in the paper, but I do not think they would make a claim about viscosity without actually calculating it. They mention, they were going to report the viscosity results elsewhere, which, I presume, is this paper: aip.scitation.org/doi/pdf/10.1063/1.5035119 . In the paper, on 2nd page, they mention a number of papers which resulted in finite viscosity for a 2D fluid with Leonard-Jones interacting potential. Commented Oct 19, 2021 at 15:32
• That paper is about 3d. The only comment about 2d is on page 2: " .. the self- diffusion coefficient of a two-dimensional fluid, known to diverge logarithmically .." Commented Oct 19, 2021 at 16:35