Ill-posedness of the hydrostatic Euler and singular Vlasov equations

TL;DR

This paper develops an abstract framework to demonstrate ill-posedness in certain nonlocal PDEs with unbounded unstable spectra, applying it to hydrostatic Euler, kinetic incompressible Euler, and Vlasov-Dirac-Benney systems.

Contribution

It introduces a general method for proving ill-posedness in nonlocal PDEs with unbounded spectra, applied to multiple complex equations.

Findings

Proves ill-posedness of hydrostatic Euler equations.

Establishes ill-posedness for kinetic incompressible Euler equations.

Shows ill-posedness of the Vlasov-Dirac-Benney system.

Abstract

In this paper, we develop an abstract framework to establish ill-posedness in the sense of Hadamard for some nonlocal PDEs displaying unbounded unstable spectra. We apply it to prove the ill-posedness for the hydrostatic Euler equations as well as for the kinetic incompressible Euler equations and the Vlasov-Dirac-Benney system.