Deformation and Symmetry in the Inviscid SQG and the 3D Euler Equations

The global regularity problem concerning the inviscid SQG and the 3D Euler equations remains outstandingly open. This paper presents several geometric observations on solutions of these equations. One observation stems from a relation between what we call Eulerian and Lagrangian deformations and reflects the alignment of the stretching directions of these deformations and the tangent direction of the level curves for the SQG equation. Various spatial symmetries in solutions to the 3D Euler equations are exploited. In addition, two observations on the curvature of the level curves of the SQG equation are also included.