4,558 research outputs found
Geodesic Flow on the Diffeomorphism Group of the circle
We show that certain right-invariant metrics endow the infinite-dimensional
Lie group of all smooth orientation-preserving diffeomorphisms of the circle
with a Riemannian structure. The study of the Riemannian exponential map allows
us to prove infinite-dimensional counterparts of results from classical
Riemannian geometry: the Riemannian exponential map is a smooth local
diffeomorphism and the length-minimizing property of the geodesics holds.Comment: 15 page
On the particle paths and the stagnation points in small-amplitude deep-water waves
In order to obtain quite precise information about the shape of the particle
paths below small-amplitude gravity waves travelling on irrotational deep
water, analytic solutions of the nonlinear differential equation system
describing the particle motion are provided. All these solutions are not closed
curves. Some particle trajectories are peakon-like, others can be expressed
with the aid of the Jacobi elliptic functions or with the aid of the
hyperelliptic functions. Remarks on the stagnation points of the
small-amplitude irrotational deep-water waves are also made.Comment: to appear in J. Math. Fluid Mech. arXiv admin note: text overlap with
arXiv:1106.382
Steady water waves with multiple critical layers: interior dynamics
We study small-amplitude steady water waves with multiple critical layers.
Those are rotational two-dimensional gravity-waves propagating over a perfect
fluid of finite depth. It is found that arbitrarily many critical layers with
cat's-eye vortices are possible, with different structure at different levels
within the fluid. The corresponding vorticity depends linearly on the stream
function.Comment: 14 pages, 3 figures. As accepted for publication in J. Math. Fluid
Mec
On periodic water waves with Coriolis effects and isobaric streamlines
In this paper we prove that solutions of the f-plane approximation for
equatorial geophysical deep water waves, which have the property that the
pressure is constant along the streamlines and do not possess stagnation
points,are Gerstner-type waves. Furthermore, for waves traveling over a flat
bed, we prove that there are only laminar flow solutions with these properties.Comment: To appear in Journal of Nonlinear Mathematical Physics; 15 page
Particle trajectories in linearized irrotational shallow water flows
We investigate the particle trajectories in an irrotational shallow water
flow over a flat bed as periodic waves propagate on the water's free surface.
Within the linear water wave theory, we show that there are no closed orbits
for the water particles beneath the irrotational shallow water waves. Depending
on the strength of underlying uniform current, we obtain that some particle
trajectories are undulating path to the right or to the left, some are looping
curves with a drift to the right and others are parabolic curves or curves
which have only one loop
Categoricity and Negation. A Note on Kripke’s Affirmativism
We argue that, if taken seriously, Kripke's view that a language for science can dispense with a negation operator is to be rejected. Part of the argument is a proof that positive logic, i.e., classical propositional logic without negation, is not categorical
On the critical dissipative quasi-geostrophic equation
The 2D quasi-geostrophic (QG) equation is a two dimensional model of the 3D
incompressible Euler equations. When dissipation is included in the model then
solutions always exist if the dissipation's wave number dependence is
super-linear. Below this critical power the dissipation appears to be
insufficient. For instance, it is not known if the critical dissipative QG
equation has global smooth solutions for arbitrary large initial data. In this
paper we prove existence and uniqueness of global classical solutions of the
critical dissipative QG equation for initial data that have small
norm. The importance of an smallness condition is due to the fact
that is a conserved norm for the non-dissipative QG equation and
is non-increasing on all solutions of the dissipative QG., irrespective of
size.Comment: 12 page
Equations of the Camassa-Holm Hierarchy
The squared eigenfunctions of the spectral problem associated with the
Camassa-Holm (CH) equation represent a complete basis of functions, which helps
to describe the inverse scattering transform for the CH hierarchy as a
generalized Fourier transform (GFT). All the fundamental properties of the CH
equation, such as the integrals of motion, the description of the equations of
the whole hierarchy, and their Hamiltonian structures, can be naturally
expressed using the completeness relation and the recursion operator, whose
eigenfunctions are the squared solutions. Using the GFT, we explicitly describe
some members of the CH hierarchy, including integrable deformations for the CH
equation. We also show that solutions of some - dimensional members of
the CH hierarchy can be constructed using results for the inverse scattering
transform for the CH equation. We give an example of the peakon solution of one
such equation.Comment: 10 page
Nonexistence of self-similar singularities for the 3D incompressible Euler equations
We prove that there exists no self-similar finite time blowing up solution to
the 3D incompressible Euler equations. By similar method we also show
nonexistence of self-similar blowing up solutions to the divergence-free
transport equation in . This result has direct applications to the
density dependent Euler equations, the Boussinesq system, and the
quasi-geostrophic equations, for which we also show nonexistence of
self-similar blowing up solutions.Comment: This version refines the previous one by relaxing the condition of
compact support for the vorticit
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