Long time existence of smooth solutions for the rapidly rotating shallow-water and Euler equations

TL;DR

This paper proves that strong rotational forces can significantly extend the lifespan of smooth solutions in shallow-water and Euler equations, aligning mathematical results with observed oceanic near inertial oscillations.

Contribution

It demonstrates that dominant rotational forces prolong smooth solution existence and introduces a near periodic-in-time approximate solution in the small pressure regime.

Findings

Rotational forcing prolongs smooth solution lifespan.

Existence of a near periodic-in-time approximate solution.

Results align with oceanic near inertial oscillation observations.

Abstract

We study the stabilizing effect of rotational forcing in the nonlinear setting of two-dimensional shallow-water and more general models of compressible Euler equations. In [H. Liu and E. Tadmor, Phys. D 188 (2004), no. 3-4, 262-276] we have shown that the pressureless version of these equations admit global smooth solution for a large set of sub-critical initial configurations. In the present work we prove that when rotational force dominates the pressure, it \emph{prolongs} the life-span of smooth solutions for t < ln(1/d); here d << 1 is the ratio of the pressure gradient measured by the inverse squared Froude number, relative to the dominant rotational forces measured by the inverse Rossby number. Our study reveals a ``nearby'' periodic-in-time approximate solution in the small d-regime, upon which hinges the long time existence of the exact smooth solution. These results are in agreement with the close-to periodic dynamics observed in the ``near inertial oscillation'' (NIO) regime which follows oceanic storms. Indeed, our results indicate the existence of smooth, ``approximate periodic'' solution for a time period of \emph{days}, which is the relevant time period found in NIO obesrvations.