Peristatcally induced MHD slip flow in a porous medium due to a surface acoustic wavy wall

The influences of Hall current and slip condition on the MHD flow induced by sinusoidal peristaltic wavy wall in two dimensional viscous fluid through a porous medium for moderately large Reynolds number is considered on the basis of boundary layer theory in the case where the thickness of the boundary layer is larger than the amplitude of the wavy wall. Solutions are obtained in terms of a series expansion with respect to small amplitude by a regular perturbation method. Graphs of velocity components, both for the outer and inner flows for various values of the Reynolds number, slip parameter, Hall and magnetic parameters are drawn. The inner and outer solutions are matched by the matching process. An interesting application of the present results to mechanical engineering may be the possibility of the fluid transportation without an external pressure.     

Dispersive blow-up for nonlinear Schrödinger equations revisited

The possibility of finite-time, dispersive blow-up for nonlinear equations of Schrödinger type is revisited. This mathematical phenomena is one of the conceivable explanations for oceanic and optical rogue waves. In dimension one, the fact that dispersive blow up does occur for nonlinear Schrödinger equations already appears in [9]. In the present work, the existing results are extended in several ways. In one direction, the theory is broadened to include the Davey–Stewartson and Gross–Pitaevskii equations. In another, dispersive blow up is shown to obtain for nonlinear Schrödinger equations in spatial dimensions larger than one and for more general power-law nonlinearities. As a by-product of our analysis, a sharp global smoothing estimate for the integral term appearing in Duhamel's formula is obtained.     

Existence of capillary-gravity water waves with piecewise constant vorticity

In this paper we construct small-amplitude periodic capillary-gravity water waves with a piecewise constant vorticity distribution. They describe water waves traveling on superposed linearly sheared currents that have different vorticities. This is achieved by associating to the height function formulation of the water wave problem a diffraction problem where we impose suitable transmission conditions on each line where the vorticity function has a jump. The solutions of the diffraction problem, found by using local bifurcation theory, are the desired solutions of the hydrodynamical problem.     

ADDITIVE FAMILIES OF INVARIANTS OF SKEWSYMMETRIC TENSORS

Abstract

Let d be a positive integer and F be a field of characteristic 0. Suppose that for each positive integer n, I _n is a polynomial invariant of the usual action of GL_n (F) on Λ^d(Fⁿ), such that for t ? Λ^d(F ^k) and s ? Λ^d(F ^l), I _{k + l} (t l s) = I _k(t)I _t (s), where t ⊥ s is defined in §1.4. Then we say that {I_n} is an additive family of invariants of the skewsymmetric tensors of degree d, or, briefly, an additive family of invariants. If not all the I_n are constant we say that the family is non-trivial. We show that in each even degree d there is a non-trivial additive family of invariants, but that this is not so for any odd d. These results are analogous to those in our paper [3] for symmetric tensors. Our proofs rely on the symbolic method for representing invariants of skewsymmetric tensors. To keep this paper self-contained we expound some of that theory, but for the proofs we refer to the book [2] of Grosshans, Rota and Stein.