Mathematics

Convex integration is a technique for the construction of “strange" solutions to certain nonlinear systems of partial differential equations. The technique originates in the work of John Nash (1954) on C1 isometric embeddings and has been developed into a powerful general method for certain problems in differential geometry and in the calculus of variations.

In the last years new versions of this technique have been developed primarily for applications in fluid mechanics. Most notable achievements are the non-uniqueness of weak solutions to the incompressible Euler system and to the p-system of compressible ideal flows, the Onsager’s conjecture on anomalous dissipation in the context of the K41 theory of turbulence, the non-uniqueness of distributional solutions to the Navier-Stokes equations and of distributional solutions to the linear transport equation with Sobolev vector fields.

The course aims to provide an introduction to some aspects of this theory.