On the local pressure of the Navier-Stokes equations and related systems

Abstract

In the study of local regularity of weak solutions to systems related to incompressible viscous fluids local energy estimates serve as important ingredients. However, this requires certain information on the pressure. This fact has been used by V. Scheffer in the notion of a suitable weak solution to the Navier-Stokes equations, and in the proof of the partial regularity due to Caffarelli, Kohn and Nirenberg. In general domains, or in case of complex viscous fluid models a global pressure does not necessarily exist. To overcome this problem, in the present paper we construct a local pressure distribution by showing that every distribution $ \partial _t \boldsymbol u +\boldsymbol F $, which vanishes on the set of smooth solenoidal vector fields can be represented by a distribution $ \partial _t \nabla p_h +\nabla p_0 $, where $\nabla p_h \sim \boldsymbol u $ and $ \nabla p_0 \sim \boldsymbol F$.