Harmonic Analysis and Differential Equations Student Seminar: Some New Prodi-Serrin Type Regularity Criteria for the in-Compressible Navier-Stokes Equations

Seminar | November 19 | 3:40-5 p.m. | 740 Evans Hall

 Benjamin Pineau, Berkeley

 Department of Mathematics

It is a classical result of Leray from the 1930s, that for appropriate initial data and domain, there exists a global weak solution (now known as a Leray-Hopf solution) to the n-dimensional, incompressible Navier-Stokes equations. For n ≥ 3, the question of uniqueness, and regularity of Leray-Hopf solutions remains open. On the other hand, by imposing certain “integrability” conditions on a weak solution, one can often establish global regularity using energy-type arguments. These types of conditions are often referred to as Prodi-Serrin type criteria. In this talk, I will present a relatively simple method for establishing global regularity of a weak solution, provided a certain quantity (e.g. velocity, pressure, etc.) satisfies a particular weak-Lebesgue “integrability” condition. This allows one to generalize several regularity criteria in the literature.