1,929 research outputs found
Existence and stability of solitons for the nonlinear Schr\"odinger equation on hyperbolic space
We study the existence and stability of ground state solutions or solitons to
a nonlinear stationary equation on hyperbolic space. The method of
concentration compactness applies and shows that the results correlate strongly
to those of Euclidean space.Comment: New: As noted in Banica-Duyckaerts (arXiv:1411.0846), Section 5
should read that for sufficiently large mass, sub-critical problems can be
solved via energy minimization for all d \geq 2 and as a result
Cazenave-Lions results can be applied in Section 6 with the same restriction.
These requirements were addressed by the subsequent work with Metcalfe and
Taylor in arXiv:1203.361
Phase Space Models for Stochastic Nonlinear Parabolic Waves: Wave Spread and Singularity
We derive several kinetic equations to model the large scale, low Fresnel
number behavior of the nonlinear Schrodinger (NLS) equation with a rapidly
fluctuating random potential. There are three types of kinetic equations the
longitudinal, the transverse and the longitudinal with friction. For these
nonlinear kinetic equations we address two problems: the rate of dispersion and
the singularity formation.
For the problem of dispersion, we show that the kinetic equations of the
longitudinal type produce the cubic-in-time law, that the transverse type
produce the quadratic-in-time law and that the one with friction produces the
linear-in-time law for the variance prior to any singularity.
For the problem of singularity, we show that the singularity and blow-up
conditions in the transverse case remain the same as those for the homogeneous
NLS equation with critical or supercritical self-focusing nonlinearity, but
they have changed in the longitudinal case and in the frictional case due to
the evolution of the Hamiltonian
On the propagation of an optical wave in a photorefractive medium
The aim of this paper is first to review the derivation of a model describing
the propagation of an optical wave in a photorefractive medium and to present
various mathematical results on this model: Cauchy problem, solitary waves
Vanishing viscosity limits for the degenerate lake equations with Navier boundary conditions
The paper is concerned with the vanishing viscosity limit of the
two-dimensional degenerate viscous lake equations when the Navier slip
conditions are prescribed on the impermeable boundary of a simply connected
bounded regular domain. When the initial vorticity is in the Lebesgue space
with , we show the degenerate viscous lake equations
possess a unique global solution and the solution converges to a corresponding
weak solution of the inviscid lake equations. In the special case when the
vorticity is in , an explicit convergence rate is obtained
Quantitative lower bounds for the full Boltzmann equation, Part I: Periodic boundary conditions
We prove the appearance of an explicit lower bound on the solution to the
full Boltzmann equation in the torus for a broad family of collision kernels
including in particular long-range interaction models, under the assumption of
some uniform bounds on some hydrodynamic quantities. This lower bound is
independent of time and space. When the collision kernel satisfies Grad's
cutoff assumption, the lower bound is a global Maxwellian and its asymptotic
behavior in velocity is optimal, whereas for non-cutoff collision kernels the
lower bound we obtain decreases exponentially but faster than the Maxwellian.
Our results cover solutions constructed in a spatially homogeneous setting, as
well as small-time or close-to-equilibrium solutions to the full Boltzmann
equation in the torus. The constants are explicit and depend on the a priori
bounds on the solution.Comment: 37 page
Nonlinear Schr\"odinger Equation with a White-Noise Potential: Phase-space Approach to Spread and Singularity
We propose a phase-space formulation for the nonlinear Schr\"odinger equation
with a white-noise potential in order to shed light on two issues: the rate of
spread and the singularity formation in the average sense. Our main tools are
the energy law and the variance identity. The method is completely elementary.
For the problem of wave spread, we show that the ensemble-averaged dispersion
in the critical or defocusing case follows the cubic-in-time law while in the
supercritical and subcritical focusing cases the cubic law becomes an upper and
lower bounds respectively.
We have also found that in the critical and supercritical focusing cases the
presence of a white-noise random potential results in different conditions for
singularity-with-positive-probability from the homogeneous case but does not
prevent singularity formation. We show that in the supercritical focusing case
the ensemble-averaged self-interaction energy and the momentum variance can
exceed any fixed level in a finite time with positive probability
Necessary Optimality Conditions for a Dead Oil Isotherm Optimal Control Problem
We study a system of nonlinear partial differential equations resulting from
the traditional modelling of oil engineering within the framework of the
mechanics of a continuous medium. Recent results on the problem provide
existence, uniqueness and regularity of the optimal solution. Here we obtain
the first necessary optimality conditions.Comment: 9 page
Global Existence and Regularity for the 3D Stochastic Primitive Equations of the Ocean and Atmosphere with Multiplicative White Noise
The Primitive Equations are a basic model in the study of large scale Oceanic
and Atmospheric dynamics. These systems form the analytical core of the most
advanced General Circulation Models. For this reason and due to their
challenging nonlinear and anisotropic structure the Primitive Equations have
recently received considerable attention from the mathematical community.
In view of the complex multi-scale nature of the earth's climate system, many
uncertainties appear that should be accounted for in the basic dynamical models
of atmospheric and oceanic processes. In the climate community stochastic
methods have come into extensive use in this connection. For this reason there
has appeared a need to further develop the foundations of nonlinear stochastic
partial differential equations in connection with the Primitive Equations and
more generally.
In this work we study a stochastic version of the Primitive Equations. We
establish the global existence of strong, pathwise solutions for these
equations in dimension 3 for the case of a nonlinear multiplicative noise. The
proof makes use of anisotropic estimates, estimates on the
pressure and stopping time arguments.Comment: To appear in Nonlinearit
Long-time Behavior of a Two-layer Model of Baroclinic Quasi-geostrophic Turbulence
We study a viscous two-layer quasi-geostrophic beta-plane model that is
forced by imposition of a spatially uniform vertical shear in the eastward
(zonal) component of the layer flows, or equivalently a spatially uniform
north-south temperature gradient. We prove that the model is linearly unstable,
but that non-linear solutions are bounded in time by a bound which is
independent of the initial data and is determined only by the physical
parameters of the model. We further prove, using arguments first presented in
the study of the Kuramoto-Sivashinsky equation, the existence of an absorbing
ball in appropriate function spaces, and in fact the existence of a compact
finite-dimensional attractor, and provide upper bounds for the fractal and
Hausdorff dimensions of the attractor. Finally, we show the existence of an
inertial manifold for the dynamical system generated by the model's solution
operator. Our results provide rigorous justification for observations made by
Panetta based on long-time numerical integrations of the model equations
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