TWO EXAMPLES OF WELL-POSEDNESS OF WEAK SOLUTIONS FOR QUASILINEAR EVOLUTIONARY PARTIAL DIFFERENTIAL EQUATIONS (Mathematical Analysis in Fluid and Gas Dynamics)

Abstract

To establish a well-posedness theory for weak solutions of quasilinear evolutionary partial differential equations is a difficult task. There is little control over the time evolution of weak solutions constructed by compactness methods. The purpose of the present article is to explain the construction procedures of weak solutions for hyperbolic and viscous conservation laws. These procedures allow for the establishment of well-posedness theory. For system of hyperbolic conservation laws, the weak solutions are constructed using the Riemann solutions as building blocks. For the compressible Navier-Stokes equations, one uses the Green's function approach to construct the weak solutions by solving integral equations. Within these construction procedures, the traditional Hadamard well-posedness criteria are satisfied, and the regularity and time-asymptotic behaviors of the weak solution can be studied.