280 research outputs found
Local Martingale and Pathwise Solutions for an Abstract Fluids Model
We establish the existence and uniqueness of both local martingale and local
pathwise solutions of an abstract nonlinear stochastic evolution system. The
primary application of this abstract framework is to infer the local existence
of strong, pathwise solutions to the 3D primitive equations of the oceans and
atmosphere forced by a nonlinear multiplicative white noise. Instead of
developing our results specifically for the 3D primitive equations we choose to
develop them in a slightly abstract framework which covers many related forms
of these equations (atmosphere, oceans, coupled atmosphere-ocean, on the
sphere, on the {\beta}-plane approximation etc and the incompressible
Navier-Stokes equations). In applications, all of the details are given for the
{\beta}-plane approximation of the oceans equations
Gaussian invariant measures and stationary solutions of 2D Primitive Equations
We introduce a Gaussian measure formally preserved by the 2-dimensional
Primitive Equations driven by additive Gaussian noise. Under such measure the
stochastic equations under consideration are singular: we propose a solution
theory based on the techniques developed by Gubinelli and Jara in \cite{GuJa13}
for a hyperviscous version of the equations.Comment: 15 page
Time Discrete Approximation of Weak Solutions for Stochastic Equations of Geophysical Fluid Dynamics and Applications
As a first step towards the numerical analysis of the stochastic primitive
equations of the atmosphere and oceans, we study their time discretization by
an implicit Euler scheme. From deterministic viewpoint the 3D Primitive
Equations are studied with physically realistic boundary conditions. From
probabilistic viewpoint we consider a wide class of nonlinear, state dependent,
white noise forcings. The proof of convergence of the Euler scheme covers the
equations for the oceans, atmosphere, coupled oceanic-atmospheric system and
other geophysical equations. We obtain the existence of solutions weak in PDE
and probabilistic sense, a result which is new by itself to the best of our
knowledge
Navier-Stokes-Voigt equations with memory in 3D lacking instantaneous kinematic viscosity
We consider a Navier-Stokes-Voigt fluid model where the instantaneous
kinematic viscosity has been completely replaced by a memory term incorporating
hereditary effects, in presence of Ekman damping. The dissipative character of
our model is weaker than the one where hereditary and instantaneous viscosity
coexist, previously studied by Gal and Tachim-Medjo. Nevertheless, we prove the
existence of a regular exponential attractor of finite fractal dimension under
rather sharp assumptions on the memory kernel.Comment: 26 page
Interaction of a vortex induced by a rotating cylinder with a plane
In this article,we study theoretically and numerically the interaction of a
vortex induced by a rotating cylinder with a perpendicular plane. We show the
existence of weak solutions to the swirling vortex models by using the Hopf
extension method, and by an elegant contradiction argument, respectively. We
demonstrate numerically that the model could produce phenomena of swirling
vortex including boundary layer pumping and two-celled vortex that are observed
in potential line vortex interacting with a plane and in a tornado
Boundary layer analysis of the Navier-Stokes equations with Generalized Navier boundary conditions
We study the weak boundary layer phenomenon of the Navier-Stokes equations in
a 3D bounded domain with viscosity, , under generalized Navier
friction boundary conditions, in which we allow the friction coefficient to be
a (1, 1) tensor on the boundary. When the tensor is a multiple of the identity
we obtain Navier boundary conditions, and when the tensor is the shape operator
we obtain conditions in which the vorticity vanishes on the boundary. By
constructing an explicit corrector, we prove the convergence of the
Navier-Stokes solutions to the Euler solution as the viscosity vanishes. We do
this both in the natural energy norm with a rate of order as
well as uniformly in time and space with a rate of order near the boundary and in the interior,
where decrease to 0 as the regularity of the initial velocity
increases. This work simplifies an earlier work of Iftimie and Sueur, as we use
a simple and explicit corrector (which is more easily implemented in numerical
applications). It also improves a result of Masmoudi and Rousset, who obtain
convergence uniformly in time and space via a method that does not yield a
convergence rate.Comment: Additional references and several typos fixe
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