Propagation of Rough Initial Data for Navier–Stokes Equation

Abstract

The purpose of this paper is to study the quantitative structure of a weak solution for an initial value problem of the compressible Navier–Stokes equation in the class of BV functions. In [T.-P. Liu and S.-H. Yu, Comm. Pure Appl. Math., 75 (2022), pp. 223–348] the precise pointwise structures of Green’s functions for various linearized problems are established, which serve as basic tools to establish the global existence of weak solutions to the Navier–Stokes equations with initial data as a small perturbation of a constant state in the BV class. In this paper, we obtain the time asymptotic pointwise structure of the weak solutions. It thus completes the initial motivation for the earlier paper on viewing the solution behavior as part of the well-posedness theory.