László Székelyhidi (U Leipzig):
Convex integration for fluid dynamics
In the past decade convex integration has been established as a powerful
and versatile technique for the construction of weak solutions of
various nonlinear systems of partial differential equations arising in
fluid dynamics, including the 3D Euler and Navier-Stokes equations. The
existence theorems obtained in this way come at a high price: solutions
are highly irregular, non-differentiable, and very much non-unique as
there is usually infinitely many of them. Therefore this technique has
often been thought of as a way to obtain mathematical counterexamples in
the spirit of Weierstrass’ non-differentiable function, rather than
advancing physical theory; “pathological”, “wild”, “paradoxical”,
“counterintuitive” are some of the adjectives usually associated with
solutions obtained via convex integration. In this lecture I would like
to draw on some recent examples to show that there are many more sides
to the story, and that, with proper usage and interpretation, the convex
integration toolbox can indeed provide useful insights for problems in
mathematical hydrodynamics.
