LONG-TIME EXISTENCE OF SMOOTH SOLUTIONS FOR THLong time existence of smooth solutions for the rapidly rotating shallow-water and Euler equationsE

Abstract

We study the stabilizing effect of rotational forcing in the nonlinear setting of twodimensionalshallow-water and more general models of compressible Euler equations. In [Phys. D,188 (2004), pp. 262–276] Liu and Tadmor have shown that the pressureless version of these equationsadmit a global smooth solution for a large set of subcritical initial configurations. In the presentwork we prove that when rotational force dominates the pressure, it prolongs the lifespan of smoothsolutions for t <∼ln(δ−1); here δ 1 is the ratio of the pressure gradient measured by the inversesquared Froude number, relative to the dominant rotational forces measured by the inverseRossby number. Our study reveals a “nearby” periodic-in-time approximate solution in the small δregime, upon which hinges the long-time existence of the exact smooth solution. These results arein 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 a smooth, “approximateperiodic” solution for a time period of days, which is the relevant time period found inNIO obesrvations.