Mathematical modeling and qualitative analysis of viscoelastic conductive fluids

We use an energetic variational approach to model the transport of compressible viscoelastic conductive fluids. Such a model can be called the three-dimensional compressible viscoelastic Navier–Stokes–Poisson equations. The global unique smooth solution to the Cauchy problem is obtained. In particular, we obtain the optimal time-decay rates of the solution and its higher-order spatial derivatives by using a pure energy method.

On some singular limits in damped radiation hydrodynamics

We consider the Cauchy problem for the 3D Euler system with damping coupled to radiation through two singular limits. Assuming suitable smallness hypotheses for the data, we prove that each of these two problems admits a unique smooth solution.

THE ONE-DIMENSIONAL PERIODIC BLOCH-POISSON EQUATION

We analyze the Bloch-Poisson model describing quantum steady states of electrons in thermodynamical equilibrium. The problem is set in a one-dimensional periodic geometry. The existence of a unique smooth solution for every positive temperature is proved, a convergent iterative procedure useful for the numerical simulation is obtained, and the classical limit is analyzed.