Abstract:
I will present a nonlinear version of the open mapping principle which
applies to constant-coefficient PDEs which are scaling-invariant and weak*
stable. An example of such a PDE is the Jacobian equation. I will discuss
the consequences of such a result for the Jacobian and its relevance towards
an answer to a long-standing problem due to Coifman, Lions, Meyer and
Semmes. As a further application I show that, for any p < ∞, the set
of initial data for which there are dissipative weak solutions in L^p_t tL^2_x to
the incompressible Euler equations is meagre in the space of solenoidal L2
fields. This is based on joint work with A. Guerra (Oxford) and S. Lindberg
(Aalto).