  The effects of a randomly moving environment on a randomly growing interface
are studied by the field theoretic renormalization group analysis. The kinetic
growth of an interface (kinetic roughening) is described by the
Kardar-Parisi-Zhang stochastic differential equation while the velocity field
of the moving medium is modelled by the Navier-Stokes equation with an external
random force. It is found that the large-scale, long-time (infrared) asymptotic
behavior of the system is divided into four nonequilibrium universality classes
related to the four types of the renormalization group equations fixed points.
In addition to the previously established regimes of asymptotic behavior
(ordinary diffusion, ordinary kinetic growth process, and passively advected
scalar field), a new nontrivial regime is found. The fixed point coordinates,
their regions of stability and the critical dimensions related to the critical
exponents (e.g., roughness exponent) are calculated to the first order of the
expansion in $\epsilon=2-d$ where $d$ is a space dimension (one-loop
approximation) or exactly. The new regime possesses a feature typical to the
the Kardar-Parisi-Zhang model: the fixed point corresponding to the regime
cannot be reached from a physical starting point. Thus, physical interpretation
is elusive.

Statistical Mechanics