Mathematics

Permanent URI for this communityhttp://hdl.handle.net/1903/2261

Browse

Subject: search.filters.subject.critical threshold

Search Results

Now showing 1 - 3 of 3
  • Thumbnail Image
    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, EITAN
    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 [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.
  • Thumbnail Image
    Item
    CRITICAL THRESHOLDS IN A CONVOLUTION MODEL FOR NONLINEAR CONSERVATION LAWS
    (Copyright: Society for Industrial and Applied Mathematics, 2001) LIU, HAILIANG; TADMOR, EITAN
    In this work we consider a convolution model for nonlinear conservation laws.Due to the delicate balance between the nonlinear convection and the nonlocal forcing, this model allows for narrower shock layers than those in the viscous Burgers’ equation and yet exhibits the conditional finite time breakdown as in the damped Burgers’ equation.W e show the critical threshold phenomenon by presenting a lower threshold for the breakdown of the solutions and an upper threshold for the global existence of the smooth solution.The threshold condition depends only on the relative size of the minimum slope of the initial velocity and its maximal variation.W e show the exact blow-up rate when the slope of the initial profile is below the lower threshold.W e further prove the L1 stability of the smooth shock profile, provided the slope of the initial profile is above the critical threshold.
  • Thumbnail Image
    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.