Open Collections

Abstract

Existence and regularity of steady and unsteady solutions of a PDE describing the motion of a prototypical incompressible fluid with shear dependent viscosity are studied. The regularity theory is approached by studying the associated elliptic operator. A summary of the classical technique of difference quotients applied to non-linear elliptic systems is given by applying it to the elliptic system associated with a vector Burgers-like system. Interior regularity is proved for a general class of Stokes-like elliptic operators using a new solenoidal test function that permits the application difference quotient methods to systems with a divergence free constraint. Existence for steady solutions of the incompressible fluid PDE is proven; interior regularity follows immediately from regularity of the Stokes-like elliptic system. Existence and interior regularity for time dependent solutions are proven.