The Cauchy problem for the pressureless Euler/isentropic Navier-Stokes equations

PDE Seminar
Tuesday, December 1, 2015 - 3:05pm
1 hour (actually 50 minutes)
Skiles 006
Imperial College London
The interactions between particles and fluid have received a bulk of attention due to a number of their applications in the field of, for example, biotechnology, medicine, and in the study of sedimentation phenomenon, compressibility of droplets of the spray, cooling tower plumes, and diesel engines, etc. In this talk, we present coupled hydrodynamic equations which can formally be derived from Vlasov-Boltzmann/Navier-Stokes equations. More precisely, our proposed equations consist of the compressible pressureless Euler equations and the isentropic compressible Navier-Stokes equations. For the coupled system, we establish the global existence of classical solutions when the domain is periodic, and its large-time behavior which shows the exponential alignment between two fluid velocities. We also remark on blow-up of classical solutions in the whole space.