Vanishing viscosity limit for the Navier-Stokes equations

PDE Seminar
Tuesday, November 30, 2010 - 15:00
1 hour (actually 50 minutes)
Skiles 255
University of Houston
In this talk we will discuss the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. We will follow the approach of R.DiPerna (1983) and reduce the problem to the study of a measure-valued solution of the Euler equations, obtained as a limit of a sequence of the vanishing viscosity solutions. For a fixed pair (x,t), the (Young) measure representing the solution encodes the oscillations of the vanishing viscosity solutions near (x,t). The Tartar-Murat commutator relation with respect to two pairs of weak entropy-entropy flux kernels is used to show that the solution takes only Dirac mass values and thus it is a weak solution of the Euler equations in the usual sense. In DiPerna's paper and the follow-up papers by other authors this approach was implemented for the system of the Euler equations with the artificial viscosity. The extension of this technique to the system of the Navier-Stokes equations is complicated because of the lack of uniform (with respect to the vanishing viscosity), pointwise estimates for the solutions. We will discuss how to obtain the Tartar-Murat commutator relation and to work out the reduction argument using only the standard energy estimates. This is a joint work with Gui-Qiang Chen (Oxford University and Northwestern University).