Mathematics Colloquium

The incompressible Navier-Stokes and Euler equations are fundamental for understanding basic phenomena in fluid dynamics, but many fundamental aspects of these equations still need to be fully understood. In particular, the question of well-posedness and uniqueness within physically relevant classes of solutions is not entirely resolved. Of particular importance are the class of Leray solutions to the Navier-Stokes equations and the class of solutions to the 2d Euler equations with L^p vorticity. This talk outlines recent progress toward their understanding and discusses newly discovered nonunique solutions.