The Regularity and Uniqueness of a Global Solution to the Isentropic Navier-Stokes Equation with Rough Initial Data

A global weak solution to the isentropic Navier-Stokes equation with initial data around a constant state in the L-1 & AND; BV class was constructed in [1]. In the current paper, we will continue to study the uniqueness and regularity of the constructed solution. The key ingredients are the Holder continuity estimates of the heat kernel in both spatial and time variables. With these finer estimates, we obtain higher order regularity of the constructed solution to Navier-Stokes equation, so that all of the derivatives in the equation of conservative form are in the strong sense. Moreover, this regularity also allows us to identify a function space such that the stability of the solutions can be established there, which eventually implies the uniqueness.