The Robinson-Trautman (RT) spacetime is the simplest solution of General
Relativity (GR) describing a compact source surrounded by gravitational
waves. As an initial value problem, the RT spacetime evolution is a
well-posed mathematical problem. The pertinent dynamical equations are
equivalent to the so-called Calabi flow, and regular initial data evolve
smoothly towards a final state corresponding to a remnant Schwarzschild
black hole. Extensions of RT spacetimes for higher dimensions (D > 4)
were recently proposed, and the essence of the RT evolution is
unchanged: regular initial data evolve towards a final
higher-dimensional Schwarzschild black hole. The situation for D=3 is
quite different due to some peculiarities of low-dimensional GR. We will
present a D=3 RT flow mimicking the essential properties of the Calabi
flow. In particular, regular initial data evolve towards a final remnant
BTZ black hole, and any possible asymmetry in the initial data is
expelled as a radiation fluid.