  We develop a hydrodynamic approach to non-equilibrium conformal field theory.
We study non-equilibrium steady states in the context of one-dimensional
conformal field theory perturbed by the $T\bar T$ irrelevant operator. By
direct quantum computation, we show, to first order in the coupling, that a
relativistic hydrodynamic emerges, which is a simple modification of
one-dimensional conformal fluids. We show that it describes the steady state
and its approach, and we provide the main characteristics of the steady state,
which lies between two shock waves. The velocities of these shocks are modified
by the perturbation and equal the sound velocities of the asymptotic baths.
Pushing further this approach, we are led to conjecture that the approach to
the steady state is generically controlled by the power law $t^{-1/2}$, and
that the widths of the shocks increase with time according to $t^{1/3}$.
