223 research outputs found
Stochastic Cahn-Hilliard equation with double singular nonlinearities and two reflections
We consider a stochastic partial differential equation with two logarithmic
nonlinearities, with two reflections at 1 and -1 and with a constraint of
conservation of the space average. The equation, driven by the derivative in
space of a space-time white noise, contains a bi-Laplacian in the drift. The
lack of the maximum principle for the bi-Laplacian generates difficulties for
the classical penalization method, which uses a crucial monotonicity property.
Being inspired by the works of Debussche, Gouden\`ege and Zambotti, we obtain
existence and uniqueness of solution for initial conditions in the interval
. Finally, we prove that the unique invariant measure is ergodic, and
we give a result of exponential mixing
Non elliptic SPDEs and ambit fields: existence of densities
Relying on the method developed in [debusscheromito2014], we prove the
existence of a density for two different examples of random fields indexed by
(t,x)\in(0,T]\times \Rd. The first example consists of SPDEs with Lipschitz
continuous coefficients driven by a Gaussian noise white in time and with a
stationary spatial covariance, in the setting of [dalang1999]. The density
exists on the set where the nonlinearity of the noise does not vanish.
This complements the results in [sanzsuess2015] where is assumed to be
bounded away from zero. The second example is an ambit field with a stochastic
integral term having as integrator a L\'evy basis of pure-jump, stable-like
type.Comment: 23 page
Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes
We present an abstract framework for analyzing the weak error of fully
discrete approximation schemes for linear evolution equations driven by
additive Gaussian noise. First, an abstract representation formula is derived
for sufficiently smooth test functions. The formula is then applied to the wave
equation, where the spatial approximation is done via the standard continuous
finite element method and the time discretization via an I-stable rational
approximation to the exponential function. It is found that the rate of weak
convergence is twice that of strong convergence. Furthermore, in contrast to
the parabolic case, higher order schemes in time, such as the Crank-Nicolson
scheme, are worthwhile to use if the solution is not very regular. Finally we
apply the theory to parabolic equations and detail a weak error estimate for
the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic
heat equation
Stochastic attractors for shell phenomenological models of turbulence
Recently, it has been proposed that the Navier-Stokes equations and a
relevant linear advection model have the same long-time statistical properties,
in particular, they have the same scaling exponents of their structure
functions. This assertion has been investigate rigorously in the context of
certain nonlinear deterministic phenomenological shell model, the Sabra shell
model, of turbulence and its corresponding linear advection counterpart model.
This relationship has been established through a "homotopy-like" coefficient
which bridges continuously between the two systems. That is, for
one obtains the full nonlinear model, and the corresponding linear
advection model is achieved for . In this paper, we investigate the
validity of this assertion for certain stochastic phenomenological shell models
of turbulence driven by an additive noise. We prove the continuous dependence
of the solutions with respect to the parameter . Moreover, we show the
existence of a finite-dimensional random attractor for each value of
and establish the upper semicontinuity property of this random attractors, with
respect to the parameter . This property is proved by a pathwise
argument. Our study aims toward the development of basic results and techniques
that may contribute to the understanding of the relation between the long-time
statistical properties of the nonlinear and linear models
On Nonlinear Stochastic Balance Laws
We are concerned with multidimensional stochastic balance laws. We identify a
class of nonlinear balance laws for which uniform spatial bounds for
vanishing viscosity approximations can be achieved. Moreover, we establish
temporal equicontinuity in of the approximations, uniformly in the
viscosity coefficient. Using these estimates, we supply a multidimensional
existence theory of stochastic entropy solutions. In addition, we establish an
error estimate for the stochastic viscosity method, as well as an explicit
estimate for the continuous dependence of stochastic entropy solutions on the
flux and random source functions. Various further generalizations of the
results are discussed
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
FAD binding, cobinamide binding and active site communication in the corrin reductase (CobR)
Adenosylcobalamin, the coenzyme form of vitamin B12, is one Nature's most complex coenzyme whose de novo biogenesis proceeds along either an anaerobic or aerobic metabolic pathway. The aerobic synthesis involves reduction of the centrally chelated cobalt metal ion of the corrin ring from Co(II) to Co(I) before adenosylation can take place. A corrin reductase (CobR) enzyme has been identified as the likely agent to catalyse this reduction of the metal ion. Herein, we reveal how Brucella melitensis CobR binds its coenzyme FAD (flavin dinucleotide) and we also show that the enzyme can bind a corrin substrate consistent with its role in reduction of the cobalt of the corrin ring. Stopped-flow kinetics and EPR reveal a mechanistic asymmetry in CobR dimer that provides a potential link between the two electron reduction by NADH to the single electron reduction of Co(II) to Co(I)
Longtime behavior of nonlocal Cahn-Hilliard equations
Here we consider the nonlocal Cahn-Hilliard equation with constant mobility
in a bounded domain. We prove that the associated dynamical system has an
exponential attractor, provided that the potential is regular. In order to do
that a crucial step is showing the eventual boundedness of the order parameter
uniformly with respect to the initial datum. This is obtained through an
Alikakos-Moser type argument. We establish a similar result for the viscous
nonlocal Cahn-Hilliard equation with singular (e.g., logarithmic) potential. In
this case the validity of the so-called separation property is crucial. We also
discuss the convergence of a solution to a single stationary state. The
separation property in the nonviscous case is known to hold when the mobility
degenerates at the pure phases in a proper way and the potential is of
logarithmic type. Thus, the existence of an exponential attractor can be proven
in this case as well
Stochastic Reaction-diffusion Equations Driven by Jump Processes
We establish the existence of weak martingale solutions to a class of second
order parabolic stochastic partial differential equations. The equations are
driven by multiplicative jump type noise, with a non-Lipschitz multiplicative
functional. The drift in the equations contains a dissipative nonlinearity of
polynomial growth.Comment: See journal reference for teh final published versio
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