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The equation in the title of this question can be a relativistic analogue of the Navier-Stokes equation (in the sense that, in the low-velocity limit, it reduces to Euler's equation when $T^{ab}$ is the stress-energy tensor of a perfect fluid).

I imagine that this equation is difficult to analyze in general relativity where the covariant derivative depends on the stress-energy tensor itself.

But in special relativity, are solutions known to exist and be unique?

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    $\begingroup$ Solutions with what boundary conditions? $\endgroup$ – Ben Crowell Jan 19 '18 at 14:47
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    $\begingroup$ Solutions to what? $\nabla^aT_{ab}$ is an expression, not an equation. Do you perhaps mean $\nabla^aT_{ab}=0$? $\endgroup$ – AccidentalFourierTransform Jan 19 '18 at 21:52
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    $\begingroup$ Existence and uniqueness of Navier-Stokes is an unsolved problem even in flat space and in the non-relativistic limit. Full GR is not going to male the situation any simpler. $\endgroup$ – Thomas Jan 25 '18 at 2:07
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First, I think you mean the equation $\nabla^a T_{ab} = 0$. This is not the generalised Navier-Stokes equation, it is just a version of the $T_{ab}$ on-shell conservation equation in curved spacetime. For instance, if you write $T_{ab}$ for a point-particle you obtain the geodesic equation ($u^b\nabla_b u^a = 0$), while if you write $T_{ab}$ for electromagnetic fields, you get Maxwell's equations ($\nabla^a F_{ab} = 0$). It is true that if you write $T_{ab}$ for a general fluid you obtain a GR version of Navier Stokes equations.

The full set of classical equations you need to solve in GR is $G_{ab} = 8\pi G\,T_{ab}$ (if you take covariant derivatives you get $\nabla^a T_{ab} = 0$). These are a hardcore version of Navier-Stokes in the sense that there are $10$ extremely non-linear equations with some redundancy (choice of coordinates), plus the equations of motions for the matter/radiation, which involve the metric. There is not known general solution, and the usual procedure is to use the symmetries of the problem to guess the form of $g_{ab}$ and $T_{ab}$ and then introduce them to the equations.

In special relativity you can take an inertial frame so that the equation is $\partial^aT_{ab} = 0$, which is just the on-shell conservation of energy and momentum. For electromagnetic fields these are just Maxwell's equations, and a general solution to these is known because they are linear. For a point-particle the solution is the geodesic equation, which has the solution of the particle traveling at constant 4-velocity $u^a$. For the fluid, you just have a relativistic version of Navier-Stokes, which is non-linear and no general solutions are known.

Existence of solutions for every initial conditions is not guaranteed I believe, but if they are smooth enough (and for a physicist they always are) you expect the solution to exist. Uniqueness of solutions is guaranteed in GR if the initial data is specified on a Cauchy surface, though I believe that this is proven for matter/radiation $T_{ab}$ that satisfies some conditions, plus the initial data to be smooth enough.

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