Rotation Prevents Finite-Time Breakdown

Abstract

We consider a two-dimensional convection model augmented with the rotational Coriolis forcing, $U_t + U\cdot\nabla_x U = 2k U^\perp$, with a fixed $2k$ being the inverse Rossby number. We ask whether the action of dispersive rotational forcing alone, $U^\perp$, prevents the generic finite time breakdown of the free nonlinear convection. The answer provided in this work is a conditional yes. Namely, we show that the rotating Euler equations admit global smooth solutions for a subset of generic initial configurations. With other configurations, however, finite time breakdown of solutions may and actually does occur. Thus, global regularity depends on whether the initial configuration crosses an intrinsic, ${\mathcal O}(1)$ critical threshold, which is quantified in terms of the initial vorticity, $ω_0=\nabla \times U_0$, and the initial spectral gap associated with the $2\times 2$ initial velocity gradient, $η_0:=λ_2(0)-λ_1(0), λ_j(0)= λ_j(\nabla U_0)$. Specifically, global regularity of the rotational Euler equation is ensured if and only if $4k ω_0(α) +η^2_0(α) <4k^2, \forall α\in \R^2$ . We also prove that the velocity field remains smooth if and only if it is periodic. We observe yet another remarkable periodic behavior exhibited by the {\em gradient} of the velocity field. The spectral dynamics of the Eulerian formulation reveals that the vorticity and the eigenvalues (and hence the divergence) of the flow evolve with their own path-dependent period. We conclude with a kinetic formulation of the rotating Euler equation.