The Failure of Continuous Dependence on Initial data for the Navier–Stokes equations of Compressible Flow

It is shown that physical solutions of the Navier–Stokes equations for one-dimensional, compressible flow need not depend continuously on their initial data, at least when vacuum states are allowed. Specifically, two fluid regions initially separated by a third region of very low density 6 are considered. It is shown that, as δ → 0, the (unique) solutions corresponding toδ > 0 do not in fact converge to a physical solution, but rather to a nonphysical weak solution in which the two fluids cannot collide, independent of their initial velocities, and whose separate momenta need not be conserved. A particular consequence is that solutions of the cavity problem δ = 0 are not unique.