On the spectral dynamics of the deformation tensor and new a priori estimates for the 3D Euler equations
Dongho Chae
TL;DR
This work analyzes the spectral dynamics of the deformation tensor for solutions of the 3D incompressible Euler equations on the torus, with the aim of linking eigenvalue dynamics to vorticity growth and regularity. The authors derive an evolution equation for the eigenvalues in the $L^2$ sense and use it to obtain new a priori $L^2$ estimates for the vorticity, including a simple sufficient condition for controlling the $L^2$-norm of vorticity. They also establish decay in time estimates for ratios of the deformation-tensor eigenvalues and show that the $L^2$ dynamics of vorticity are governed by the second largest eigenvalue of the deformation tensor. The results offer insight into how spectral properties constrain potential finite-time singularities and connect to classical criteria such as Beale-Kato-Majda, highlighting the practical impact of spectral information for understanding Euler regularity.
Abstract
In this paper we study the dynamics of eigenvalues of the deformation tensor for solutions of the 3D incompressible Euler equations. Using the evolution equation of the $L^2$ norm of spectra, we deduce new a priori estimates of the $L^2$ norm of vorticity. As an immediate corollary of the estimate we obtain a new sufficient condition of $L^2$ norm control of vorticity. We also obtain decay in time estimates of the ratios of the eigenvalues. In the remarks we discuss what these estimates suggest in the study of searching initial data leading to a possible finite time singularities. We find that the dynamical behaviors of $L^2$ norm of vorticity are controlled completely by the second largest eigenvalue of the deformation tensor.