Pull-in instability, as an inherent nonlinear problem, continues to become an increasingly important and interesting topic in the design of electrostatic Nano/Micro-electromechanical systems (N/MEMS) devices. Generally, the pull-in instability was studied in a continuous space, but when the electronic devices work in a porous medium, they need to be analyzed in a fractal partner. In this paper, we establish a fractal model for N/MEMS, and find a pull-in stability plateau, which can be controlled by the porous structure, and the pull-in instability can be finally converted to a stable condition. As a result, the pull-in instability can be completely eliminated, realizing the transformation of pull-in instability into pull-in stability.

a b s t r a c tWe improve the recent result of Chae and Tadmor (2008) [10] proving a one-sided threshold condition which leads to a finite-time breakdown of the Euler-Poisson equations in arbitrary dimension n.

Singular limits of a class of evolutionary systems of partial differential equations having two small parameters and hence three time scales are considered. Under appropriate conditions solutions are shown to exist and remain uniformly bounded for a fixed time as the two parameters tend to zero at different rates. A simple example shows the necessity of those conditions in order for uniform bounds to hold. Under further conditions the solutions of the original system tend to solutions of a limit equation as the parameters tend to zero.

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Abstract. We study the stabilizing effect of rotational forcing in the nonlinear setting of twodimensional 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 obesrvations. 1. Introduction and statement of main results. We are concerned here with two-dimensional systems of nonlinear Eulerian equations driven by pressure and rotational forces. It is well known that in the absence of rotation, these equations experience a finite-time breakdown: for generic smooth initial conditions, the corresponding solutions will lose C 1 -smoothness due to shock formation. The presence of rotational forces, however, has a stabilizing effect. In particular, the pressureless version of these equations admit global smooth solutions for a large set of so-called subcritical initial configurations [17]. It is therefore a natural extension to investigate the balance between the regularizing effects of rotation versus the tendency of pressure to enforce finite-time breakdown (we mention in passing the recent work [21] on a similar regularizing balance of different competing forces in the one-dimensional Euler-Poisson equations). In this paper we prove the long-time existence of rapidly rotating flows characterized by "nearby" periodic flows. Thus, rotation prolongs the lifespan of smooth solutions over increasingly long time periods, which grow longer as the rotation forces become more dominant over pressure.Our model problem is the rotational shallow water (RSW) system of equations. This system models largescale geophysical motions in a thin layer of fluid under the

This article addresses a fundamental concern regarding the incompressible approximation of fluid motions, one of the most widely used approximations in fluid mechanics. Common belief is that its accuracy is OÔ Õ, where denotes the Mach number. In this article, however, we prove an OÔ 2 Õ accuracy for the incompressible approximation of the isentropic, compressible Euler equations thanks to several decoupling properties. At the initial time, the velocity field and its first time derivative are of OÔ1Õ size, but the boundary conditions can be as stringent as the solid-wall type. The fast acoustic waves are still OÔ Õ in magnitude, since the OÔ 2 Õ error is measured in the sense of Leray projection and more physically, in time-averages. We also show when a passive scalar is transported by the flow, it is OÔ 2 Õ accurate pointwise in time to use incompressible approximation for the velocity field in the transport equation.

Abstract. We prove the global-in-time existence of large-data finite-energy weak solutions to an incompressible hybrid Vlasov-magnetohydrodynamic model in three space dimensions. The model couples three essential ingredients of magnetized plasmas: a transport equation for the probability density function, which models energetic rarefied particles of one species; the incompressible Navier-Stokes system for the bulk fluid; and a parabolic evolution equation, involving magnetic diffusivity, for the magnetic field. The physical derivation of our model is given. It is also shown that the weak solution, whose existence is established, has nonincreasing total energy, and that it satisfies a number of physically relevant properties, including conservation of the total momentum, conservation of the total mass, and nonnegativity of the probability density function for the energetic particles. The proof is based on a one-level approximation scheme, which is carefully devised to avoid increase of the total energy for the sequence of approximating solutions, in conjunction with a weak compactness argument for the sequence of approximating solutions. The key technical challenges in the analysis of the mathematical model are the nondissipative nature of the Vlasov-type particle equation and passage to the weak limits in the multilinear coupling terms.