Uniqueness of weak solutions of the Navier-Stokes equations of multidimensional, compressible flow

We prove uniqueness and continuous dependence on initial data of weak solutions of the Navier-Stokes equations of compressible flow in two and three space dimensions. The solutions we consider may display codimension-one discontinuities in density, pressure, and velocity gradient, and consequently are the generic singular solutions of this system. The key point of the analysis is that solutions with minimal regularity are best compared in a Lagrangean framework; that is, we compare the instantaneous states of corresponding fluid particles in two different solutions rather than the states of different fluid particles instantaneously occupying the same point of space-time. Estimates for H-1 differences in densities and L-2 differences in velocities are obtained by duality from bounds for the corresponding adjoint system.