# Navier-Stokes fluid dynamics, Einstein gravity and holography

There was some activity a while ago, like 10 years ago, string theoreists try to relate

• the fluid dynamics, for example, governed by Navier-Stokes equation,

to

• the Einstein gravity, and its relations to holography (such as AdS/CFT),

Naively, it looks that the development there is just trying to relate one problem to the other problem. For example, explicit construction that for every solution of the incompressible Navier-Stokes equation in $$p+1$$ dimensions, there is a uniquely associated "dual" solution of the vacuum Einstein equations in $$p+2$$ dimensions. See for example, studied in

"From Navier-Stokes To Einstein" 10.1007/JHEP07(2012)146 -- by Irene Bredberg, Cynthia Keeler, Vyacheslav Lysov, Andrew Strominger

and References therein. There is also a Breakthrough Prize in Fundamental Physics awarded to the developement of this direction.

My question is that have these efforts from "string theory and Einstein gravity" also affected the mathemtical study of Navier-Stokes equation and fluid dynamics? What are new progresses? What are new and solid lessons have we learned in mathemtics of Navier-Stokes equation and fluid dynamics then after this development? (i.e. the impacts on the community of mathemticians.)

The first point to make is that the fluid/gravity correspondence relates the general theory of relativity to relativistic fluid dynamics. I don't see how the usual non-relativistic Navier-Stokes equations can benefit from this correspondence. In this context, the fluid dynamics progress I am aware of was inspired by the gravity correspondence, but did not directly make use of results from string theory. A helpful recent reference is Ashok Thillaisundaram's Ph.D. thesis Aspects of fluid dynamics and the fluid/gravity correspondence (2017).

This is a bit too long for being a comment.

An important remark is that the NS equation is parabolic and therefore the velocity at which information propagates is unbounded. For instance, if the initial datum is compactly supported, the support of the solution fills immediately the whole space. This property apparently contradicts the finite speed of relativistic equations such as that of Einstein gravity.

This paradox is resolved by the change in the dimension : gravity is in dimension $$p+2$$ while NS is in dimension $$p+1$$. Therefore these differential equations may be of different type, even though the solutions of one can be transformed into solutions of the other. An obvious example of this phenomenon is the relation between the wave equation $$\partial_t^2u-c^2\Delta u=0$$ (dimension $$p+1$$, hyperbolic, finite velocity of propagation) and the Helmoltz equation $$c^2\Delta v+\omega^2 v=0$$ (dimension $$p$$, elliptic, the Cauchy problem is ill-posed). The latter describes stationnary waves $$e^{i\omega t}v(x)$$ of the former. Mind that in general, because one dimension is lost, the correspondence works only in one direction: every solution of the simplest (NS or Helmoltz) can be transformed into a solution of the more complex (Einstein or wave eq.), but the converse is false.

In the case of NS versus Einstein, of course, the time variable of NS has little in common with the time of gravity, if any. In particular, NS describes an irreversible process, while gravity is time-reversible.

• This doesn't seem to be particularly relevant to the question, which is about a certain collection of papers, written about a decade ago, relating conformal field holography (as it arose in string theory) with fluid mechanics. Your answer seems to be about the general phenomenon of relating purely classical relativistic mechanics to its static solutions. – Ben McKay Sep 14 '19 at 8:09

I completely disagree with the author of the remark. The Helmholtz equations and the wave equation are equivalent and the solution of the Helmholtz equation $$v(\omega, \vec r)$$ is associated with the solution of the wave equation $$u(t,\vec r)$$ $$u(t,\vec r)=\int_{-\infty}^{\infty}exp(i\omega t)v(\omega, \vec r)d\omega.$$ Using $$t=t_0$$ you get the initial conditions for the wave equation. In addition, it cannot be said that the Navier-Stokes equation is parabolic due to the presence of nonlinearity. The solution of the Navier-Stokes equation with nonlinearity differs significantly from the solution without nonlinearity.