Remarks on the Complex Euler Equations

TL;DR

This paper investigates the complexified Euler equations, demonstrating their nonlinear ill-posedness below analytic regularity and showing solutions can lose analyticity in finite time, with implications for related nonlinear systems.

Contribution

It proves nonlinear ill-posedness and finite-time singularity for complex Euler equations and related models, introducing an unstable manifold approach.

Findings

Complex Euler equations are nonlinearly ill-posed below analytic regularity.

Solutions can lose analyticity in finite time.

Unstable manifold construction captures high frequency instability.

Abstract

We consider a complexification of the Euler equations introduced by \v{S}ver\'ak which conserves energy. We prove that these complex Euler equations are nonlinearly ill-posed below analytic regularity and, moreover, we exhibit solutions which lose analyticity in finite time. Our examples are complex shear flows and, hence, one-dimensional. This motivates us to consider fully nonlinear systems in one spatial dimension which are non-hyperbolic near a constant equilibrium. We prove nonlinear ill-posedness and finite-time singularity for these models. Our approach is to construct an infinite-dimensional unstable manifold to capture the high frequency instability at the nonlinear level.