# Introduction to Convex Integration

# Introduction to Convex Integration

Convex integration has its roots in the work of Nash in geometry and evolved very recently to a powerful new tool that can prove non-uniqueness for many equations. We know from undergraduate PDE course that 1-D Burgers' equation can be proven to exhibit finite-time shock via characteristics; however, such an approach was very limited and could not shed much light on other complicated models, a primary example being the Navier-Stokes equations due to being vector-valued and pressure, etc. Via convex integration we now know non-uniqueness at a relatively low regularity level for many systems of equations; examples include Euler equations, Navier-Stokes equations, MHD system, Boussinesq system, active scalars such as the surface quasi-geostrophic equations and porous media equations, etc. This phenomenon has very recently spilled over to non-uniqueness in the compressible case and stochastic case, with multiple remaining open problems. The purpose of this talk is to give an introduction to this technique of convex integration; if time remains, we discuss various open problems, especially related to the stochastic case.