Discontinuous solutions for the Navier–Stokes equations with density-dependent viscosity

Abstract

We prove the existence of a unique global-in-time weak solution of the Navier–Stokes equations that govern the motion of a compressible viscous fluid with density-dependent viscosity in two-dimensional space. The initial velocity belongs to the Sobolev space $H^1({\mathbb{R}}^2)$, and the initial fluid density is $\alpha$-Hölder continuous on both sides of a ${\mathcal{C}}^{1+\alpha }$-regular interface with some geometrical assumption. We prove that this configuration persists over time: the initial interface is transported by the flow to an interface that maintains the same regularity as the initial one. Our result generalizes a previous known result of Hoff [Comm. Pure Appl. Math. 55 (2002), 1365–140] and Hoff and Santos [Arch. Ration. Mech. Anal. 188 (2008), 509–543] concerning the propagation of regularity for discontinuity surfaces by allowing a more general non-linear pressure law and density-dependent viscosity. Moreover, it supplements the work by Danchin, Fanelli, and Paicu [Anal. PDE 13 (2020), 275–316] with global-in-time well-posedness, even for density-dependent viscosity, and we achieve uniqueness in a large space.