Blowup of Solutions of the Hydrostatic Euler Equations

In this paper we prove that for a certain class of initial data, smooth solutions of the hydrostatic Euler equations blow up in finite time.Comment: 7 pages; added 1 reference in section 1, paraphrased lemma 2.2, but all mathematical details remain unchange

Infinite superlinear growth of the gradient for the two-dimensional Euler equation

For two-dimensional Euler equation on the torus, we prove that the uniform norm of the gradient can grow superlinearly for some infinitely smooth initial data. We also show the exponential growth of the gradient for the finite time

The L2L^2 essential spectrum of the 2D Euler operator

Even in two dimensions, the spectrum of the linearized Euler operator is notoriously hard to compute. In this paper we give a new geometric calculation of the essential spectrum for 2D flows. This generalizes existing results---which are only available when the flow has arbitrarily long periodic orbits---and clarifies the role of individual streamlines in generating essential spectra.Comment: 14 pages; hypotheses of the main theorem clarifie

On the universality of the incompressible Euler equation on compact manifolds

The incompressible Euler equations on a compact Riemannian manifold $(M,g)$ take the form \begin{align*} \partial_t u + \nabla_u u &= - \mathrm{grad}_g p \mathrm{div}_g u &= 0. \end{align*} We show that any quadratic ODE $\partial_t y = B(y,y)$, where $B : {\bf R}^n \times {\bf R}^n \to {\bf R}^n$ is a symmetric bilinear map, can be linearly embedded into the incompressible Euler equations for some manifold $M$ if and only if $B$ obeys the cancellation condition $\langle B(y,y), y \rangle = 0$ for some positive definite inner product $\langle,\rangle$ on ${\bf R}^n$. This allows one to construct explicit solutions to the Euler equations with various dynamical features, such as quasiperiodic solutions, or solutions that transition from one steady state to another, and provides evidence for the "Turing universality" of such Euler flows.Comment: 14 pages, no figures, to appear, Discrete and Continuous Dynamical Systems. This is the final versio