Navier-Stokes Equations in Gas Dynamics: Green's Function, Singularity, and Well-Posedness

The purpose of the present article is to study weak solutions of viscous conservation laws in physics. We are interested in the well-posedness theory and the propagation of singularity in the weak solutions for the initial value problem. Our approach is to convert the differential equations into integral equations on the level of weak solutions. This depends on exact analysis of the associated linear equations and their Green's functions. We carry out our approach for the Navier-Stokes equations in gas dynamics. Local in time as well as time-asymptotic behaviors of weak solutions, and the continuous dependence of the solutions on their initial data are established. (c) 2022 Wiley Periodicals LLC.

Automated summary

The purpose of this article is to study weak solutions of viscous conservation laws in physics. We convert the differential equations into integral equations on the level of weak solutions, relying on exact analysis of the associated linear equations and their Green's functions. We apply our approach to the Navier-Stokes equations in gas dynamics and establish the local in time as well as time-asymptotic behaviors of weak solutions, and the continuous dependence of the solutions on their initial data.