Dynamics of Singularity Surfaces for Compressible Navier-Stokes Flows in Two Space Dimensions

We prove the global existence of solutions of the Navier-Stokes equations of compressible, barotropic flow in two space dimensions with piecewise smooth initial data. These solutions remain piecewise smooth for all time, retaining simple jump discontinuities in the density and in the divergence of the velocity across a smooth curve, which is convected with the flow. The strengths of these discontinuities are shown to decay exponentially in time, more rapidly for larger acoustic speeds and smaller viscosities. The Navier-Stokes equations describe the conservation of mass and the balance of momentum: ρt + div(ρu) = 0 (1) (ρu)t + div(ρu u) + P (ρ)xj = ε∆u j + λdiv uxj . (2) Here t ≥ 0 is time, x ∈ R is the spatial coordinate, and ρ(x, t), P = P (ρ), and u(x, t) = (u(x, t), u(x, t)) are the fluid density, pressure, and velocity. ε > 0 and λ ≥ 0 are viscosity constants, and div and ∆ are the usual spatial divergence and Laplace operators. Specifically, we fix a positive, constant reference density ρ̃, and we assume that Cauchy data (ρ0, u0) is given for which ρ0 − ρ̃ is small in L ∩ L∞, u0 is small in H for some arbitrary but positive β (the L-norms must be weighted slightly), and that ρ0 is piecewise C (0 < α < β), having simple jump discontinuities across a C curve C(0). We then show that there is a global weak solution (ρ, u) for which ρ(·, t) and div u(·, t) are piecewise C, having simple jump discontinuities across a C curve C(t), which is the transport of C(0) by the velocity field u, and that certain other features of the solution concerning its singularities, readily obtainable from heuristic jump conditions, hold in a strict, pointwise sense. This research was supported in part by the NSF under Grant No. DMS-9986658. MSC 2000 : 35Q30, 76N10.