has partial overlap with the results presented in . These authors also attempt to solve for bulk geometries with prescribed boundary conditions on $\Sigma_D$ in the long wavelength regime and interpret their results in terms of a RG flow of fluid dynamics.

[*Note added in v2:*]{} In the first version of the paper the non-relativistic metrics quoted in and in were incorrect; the metrics as presented do not solve the bulk Einstein’s equations to the desired order. These are now corrected in the current version. However, the full set of terms that we need to include in order to see the Naiver-Stokes equation on the boundary is quite large. Hence in the main text we only report the results for the case where the non-relativistic fluid moves on a Ricci flat spatial manifold in and present the general results in a new appendix . We note that the results of also correct the expressions originally derived in [@Bhattacharyya:2008kq].

Dirichlet problem for probe fields {#s:dscalar}
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To set the stage for the discussion let us consider setting up the bulk Dirichlet problem for linear PDEs in an asymptotically spacetime. As a canonical example we will consider the dynamics of a probe 