I will present a nonlinear version of the open mapping principle which applies to constant-coefficient PDEs which are both homogeneous 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. This is based on joint work with Lukas Koch and Sauli Lindberg.

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I present a general nonlinear open mapping principle suited to applications to scale-invariant PDEs in regularity regimes where the equations are stable under weak* convergence. As an application I show that, for any $p < \infty$, the set of initial data for which there are dissipative weak solutions in $L^p_t L^2_x$ is meagre in the space of solenoidal L^2 fields. This is based on joint work with A. Guerra (Oxford) and S. Lindberg (Aalto).

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