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For a nonlinear operator $T$ satisfying certain structural assumptions, ourmain theorem states that the following claims are equivalent: i) $T$ issurjective, ii) $T$ is open at zero, and iii) $T$ has a bounded right inverse.The theorem applies to numerous scale-invariant PDEs in regularity regimeswhere the equations are stable under weak$^*$ convergence. Two particularexamples we explore are the Jacobian equation and the equations ofincompressible fluid flow.  For the Jacobian, it is a long standing open problem to decide whether it isonto between the critical Sobolev space and the Hardy space. Towards a negativeanswer, we show that, if the Jacobian is onto, then it suffices to rule out theexistence of surprisingly well-behaved solutions.  For the incompressible Euler equations, we show that, for any $p<\infty$, theset of initial data for which there are dissipative weak solutions in $L^p_tL^2_x$ is meagre in the space of solenoidal $L^2$ fields. Similar results holdfor other equations of incompressible fluid dynamics.

Nonlinear open mapping principles, with applications to the Jacobian equation and other scale-invariant PDEs