Mathematics
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Item LONG-TIME EXISTENCE OF SMOOTH SOLUTIONS FOR THLong time existence of smooth solutions for the rapidly rotating shallow-water and Euler equationsE(Copyright: Society for Industrial and Applied Mathematics, 2008) CHENG, BIN; TADMOR, EITANWe 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 [Phys. D, 188 (2004), pp. 262–276] Liu and Tadmor have shown that the pressureless version of these equations admit a global smooth solution for a large set of subcritical initial configurations. In the present work we prove that when rotational force dominates the pressure, it prolongs the lifespan of smooth solutions for t ≲ ln(δ^−1); here δ ≪ 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 δ 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 a smooth, “approximate periodic” solution for a time period of days, which is the relevant time period found in NIO observations.Item Long time stability of rotational Euler dynamics(2007-05-16) Cheng, Bin; Tadmor, Eitan; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We study the stabilizing effect of rotational forcing in the nonlinear setting of two-dimensional shallow-water and Euler equations. We prove that when rotational force dominates the pressure, it prolongs the life-span of smooth solutions for $t\lesssim 1+\ln(\delta^{-1})$ where $\delta\ll 1$ is the ratio of the (inverse of) squared Froude number measuring the amplitude of pressure, relative to the (inverse of) Rossby number, measuring the dominant rotational force. The strong rotation also imposes certain periodicity to the flow in the sense that there exists a ``nearby'' periodic-in-time approximation of the exact solution. In the opposite regime of large $\delta$'s, the flow is dispersive so that the divergence field substantially decays in finite time and therefore periodicity is not retained.