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We prove the existence of a compact global attractor for the Navier-Stokes equations of compressible flow in one space dimension, and we estimate the number of its determining modes in terms of the force and a scale invariant of the system. Of particular interest is the construction of the underlying phase space, particularly for the fluid density: we consider densities ρ for which the Lebesgue decomposition of the distribution derivative ρx has singular part with countable support and AC part in L . Remarkably, the solution operator respects this decomposition sufficiently to permit an analysis of the time–evolution of the separate components. We also show that, in the attractor, densities are in H and velocities are in H, so that, in a very precise sense, singularities in the density and in the velocity gradient disappear in the time–asymptotic limit. These results are based on a new existence and uniqueness theory for these equations in which we obtain time–independent estimates for solutions with large, discontinuous initial data, and large external forces. A crucial point in this existence theory is a new, uncontingent estimate for the viscous flux, reminiscent of the Oleinik entropy condition for scalar conservation laws. ∗ This research was supported in part by the NSF under grant No. DMS-9703703. §

The global attractor and finite determining modes for the Navier-Stokes equations of compressible flow with singular initial data