  The Lagrangian theory of structure formation in cosmological fluids,
restricted to the matter model ``dust'', provides successful models of
large-scale structure in the Universe in the laminar regime, i.e., where the
fluid flow is single-streamed and ``dust''-shells are smooth. Beyond the epoch
of shell-crossing a qualitatively different behavior is expected, since in
general anisotropic stresses powered by multi-stream forces arise in
collisionless matter. In this paper we provide the basic framework for the
modeling of pressure-supported fluids, thus restricting attention to isotropic
stresses and to the cases where pressure can be given as a function of the
density. We derive the governing set of Lagrangian evolution equations and
study the resulting system using Lagrangian perturbation theory. We discuss the
first-order equations and compare them to the Eulerian theory of gravitational
instability, as well as to the case of plane-symmetric collapse. We obtain a
construction rule that allows to derive first-order solutions of the Lagrangian
theory from known first-order solutions of the Eulerian theory and so extend
Zel'dovich's extrapolation idea into the multi-streamed regime. These solutions
can be used to generalize current structure formation models in the spirit of
the ``adhesion approximation''.
