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

TL;DR

This paper establishes a nonlinear open mapping principle linking surjectivity, openness, and bounded right inverses for certain operators, with applications to scale-invariant PDEs like the Jacobian and fluid flow equations.

Contribution

It introduces a unifying theorem for nonlinear operators satisfying structural assumptions, connecting fundamental properties and applying it to longstanding open problems in PDEs.

Findings

The Jacobian operator's surjectivity implies the non-existence of well-behaved solutions.

For the Euler equations, the set of initial data with dissipative solutions is meagre.

The theorem applies to PDEs stable under weak* convergence, providing new insights into their solvability.

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 $p<\infty$, 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. Similar results hold for other equations of incompressible fluid dynamics.