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Critical Thresholds in Eulerian Dynamics

We study the questions of global regularity vs. finite time breakdown in Eulerian dynamics, ut + u · ∇u = ∇F , which shows up in different contexts dictated by different modeling of F ’s. To adders these questions, we propose the notion Critical Threshold (CT), where a conditional finite time singularity depends on whether the initial configuration crosses an intrinsic, O(1) critical threshold. Our approach is based on a main new tool of spectral dynamics, where the eigenvalues, λ := λ(∇u), and eigenpairs (`, r), are traced by the forced Riccati equation λt + u · ∇λ + λ = 〈`, DFr〉. We shall outline three prototype cases. We begin with the n-dimensional Restricted Euler equations, obtaining [n/2]+1 global invariants which precisely characterize the local topology at breakdown time. Next we introduce the corresponding n-dimensional Restricted EulerPoisson (REP) system, identifying a set of [n/2] global invariants, which yield (i) sufficient conditions for finite time breakdown, and (ii) a remarkable characterization of two-dimensional initial REP configurations with global smooth solutions. And finally, we show that rotation prevents finite-time breakdown. Our study reveals the dependence of the CT phenomenon on the initial spectral gap, λ2(0) − λ1(0).