Jump dynamics due to jump datum of compressible viscous Navier–Stokes flows in a bounded plane domain

In this paper, when the initial density has a jump across an interior curve in a bounded domain, we show unique existence, piecewise regularity and jump discontinuity dynamics for the density and the velocity vector governed by the Navier–Stokes equations of compressible viscous barotropic flows. A critical difficulty is in controlling the gradient of the pressure across the jump curve. This is resolved by constructing a vector function associated with the pressure jump value on the convecting curve and extending it to the whole domain.