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Abstract:   

We consider weak solutions of the two-dimensional Euler equations with vorticity in L^p. Since Yudovich's work in 1962, it has been known that the associated Cauchy problem is well posed when p = infinity. A classical open question is whether uniqueness still holds for finite p. We show that, for p > 1 sufficiently close to 1, the problem is in fact completely flexible in the sense of

Gromov's  h-principle. In particular, we construct weak solutions exhibiting a range of non-physical behaviors. As a consequence, this gives a negative answer to the uniqueness question for p > 1 close to 1.