Remarks on the complex Euler equations

Dallas Albritton; W. Jacob Ogden

doi:10.3934/cpaa.2023110

Article Contents

2024, Volume 23, Issue 10: 1407-1422. Doi: 10.3934/cpaa.2023110

This issue Previous Article Remarks on Type Ⅱ blowups of solutions to the Navier-Stokes equations Next Article A new type of stable shock formation in gas dynamics

Remarks on the complex Euler equations

Dallas Albritton^1, , and
W. Jacob Ogden^2,

1.
Department of Mathematics, University of Wisconsin–Madison, USA
2.
Department of Mathematics, University of Washington, USA

^*Corresponding author: Dallas Albritton
^*Corresponding author: Dallas Albritton

To Vladimír Šverák, on the occasion of his 65th birthday, with gratitude and admiration

Received: July 2023

Early access: October 2023

Published: October 2024

DA was supported by National Science Foundation Postdoctoral Fellowship Grant No. 2002023. WJO was supported by National Science Foundation Graduate Research Fellowship Grant No. DGE-2140004.

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Abstract

We consider a complexification of the Euler equations introduced by Šverák in [35] 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.

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Mathematics Subject Classification: Primary: 35Q31, 35Q35; Secondary: 37L40, 37D10.

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