University of Heidelberg, **Applied Analysis Seminar**

Lukas Koch (U. Oxford)

A nonlinear open mapping principle, with applications to the Jacobian determinant and to the incompressible Euler equations

*Online via HeiCONF*

A nonlinear open mapping principle, with applications to the Jacobian determinant and to the incompressible Euler equations

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).

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