Applied Mathematics Colloquium: Dr. Zachary Bradshaw

Title: Regularity of solutions to the 3D Navier-Stokes equations

In 1934 Jean Leray gave the first construction of a solution to the Navier-Stokes equations.  83 years later, the regularity, i.e., smoothness and boundedness of Leray's solutions remains an open question.  Presently, only conditional regularity criteria are available.  In this talk we introduce the Ladyzhenskaya-Prodi-Serrin regularity criteria, a classical conditional regularity criteria for Leray's weak solutions to the Navier-Stokes equations. In our discussion, special attention is paid the roles of critical versus supercritical norms in regularity issues, and how these relate to the difficulty of solving the problem of global regularity for the Navier-Stokes equations (which is one of the Millennium prizes).  We also present a recent refinement of the Ladyzhenskaya-Prodi-Serrin criteria highlighting which frequencies play an essential role in singularity formation.