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

Abstract

For a nonlinear operator T satisfying certain structural assumptions, our main theorem states that the following claims are equivalent: i) T is surjective, ii) T is open at zero, and iii) T has a bounded right inverse. The theorem applies to numerous scale-invariant PDEs in regularity regimes where the equations are stable under weak⁎ convergence. Two particular examples we explore are the Jacobian equation and the equations of incompressible fluid flow.For the Jacobian, it is a long standing open problem to decide whether it is onto between the critical Sobolev space and the Hardy space. Towards a negative answer, we show that, if the Jacobian is onto, then it suffices to rule out the existence of surprisingly well-behaved solutions.For the incompressible Euler equations, we show that, for any p<∞, the set of initial data for which there are dissipative weak solutions in LtpLx2 is meagre in the space of solenoidal L2 fields. Similar results hold for other equations of incompressible fluid dynamics.