MTH-220.4 PDEs on Incompressible Flows

Fundamental equations for fluid dynamics equations - Eulerian and Lagrangian descriptions. Incompressible Navier-Stokes equations, Incompressible Euler equations, The Transport Theorem, Mass Conservation, Linear Momentum equation, Energy equations, Vorticity formulation.

Incompressible Navier Stokes equations - Short-time global existence and uniqueness of smooth solutions for NSE in any dimensions, Leray-Hopf global-in-time weak solutions - existence in any dimensions and uniqueness in 2D (in 3D uniqueness is open, related to one of the Clay millennium problem), Weak-strong uniqueness, Prodi-Serrin conditions.

Incompressible Euler equations - Short-time global existence and uniqueness of smooth solutions for Euler in any dimensions, Convex integration schemes for incompressible Euler equations, global-in-time weak solutions, non-uniqueness of Holder solutions, Onsager conjecture/theorem.