In one space dimension, it is well known that hyperbolic conservation
laws have unique entropy-admissible solutions, depending continuously on
the initial data. Moreover, these solutions can be obtained as limits of
vanishing viscosity approximations.
For many years it was expected that similar results would hold in
several space dimensions. However, fundamental work by De Lellis,
Szekelyhidi, and other authors, has shown that multidimensional
hyperbolic Cauchy problems usually have infinitely many weak solutions.
Moreover, all known entropy criteria fail to select a single admissible one.
In the first part of this talk I shall outline this approach based on a
Baire category argument, yielding the existence of infinitely many weak
I then wish to discuss an alternative research program,
aimed at constructing multiple solutions to some specific Cauchy
problems. Starting with some numerical simulations, here the eventual
goal is to achieve rigorous, computer-aided proofs of the existence of
two distinct self-similar solutions with the same initial data.
While solutions obtained via Baire category have turbulent nature, these
self-similar solutions are smooth, with the exception of one or two
points of singularity. They are thus much easier to visualize and