Global Well-Posedness of Compressible Navier-Stokes Equation with BV ∧ L1 Initial Data

The purpose of this paper is to study the well-posedness problem for weak solutions of Navier-Stokes equations in gas dynamics. We consider rough initial data, in BV boolean AND L-1. The well-posedness theory of Liu and Yu (Commun Pure Appl Math 75(2):223-348, 2022) for the isentropic Navier-Stokes equations is extended to the Navier-Stokes equations with an additional equation for the conservation of energy. A key step is to treat the energy equation as mainly for the dissipation of the temperature. The dissipation is analyzed through the heat kernel with BV variable coefficient constructed in Liu and Yu (2022). This step is natural from the physical point of view, but estimates for the temperature are required to be sufficiently robust for the validity of the conservation of energy in the weak sense; for this, we establish the regularity of the solutions, particularly the estimates of their time derivatives through refined estimates of the heat kernel.

Automated summary

This paper studies the well-posedness problem for weak solutions of Navier-Stokes equations in gas dynamics, considering rough initial data in BV and L-1. The well-posedness theory previously established for the isentropic Navier-Stokes equations is extended to the Navier-Stokes equations with an additional equation for the conservation of energy. The key step is to treat the energy equation as mainly for the dissipation of temperature.