11,542 research outputs found
Existence of positive solutions to stochastic thin-film equations
We construct martingale solutions to stochastic thin-film equations by introducing a (spatial) semidiscretization and establishing convergence. The discrete scheme allows for variants of the energy and entropy estimates in the continuous setting as long as the discrete energy does not exceed certain threshold values depending on the spatial grid size . Using a stopping time argument to prolongate high-energy paths constant in time, arbitrary moments of coupled energy/entropy functionals can be controlled. Having established Hölder regularity of approximate solutions, the convergence proof is then based on compactness arguments---in particular on Jakubowski's generalization of Skorokhod's theorem---weak convergence methods, and recent tools on martingale convergence
The stochastic thin-film equation: existence of nonnegative martingale solutions
We consider the stochastic thin-film equation with colored Gaussian
Stratonovich noise in one space dimension and establish the existence of
nonnegative weak (martingale) solutions. The construction is based on a
Trotter-Kato-type decomposition into a deterministic and a stochastic
evolution, which yields an easy to implement numerical algorithm. Compared to
previous work, no interface potential has to be included, the initial data and
the solution can have de-wetted regions of positive measure, and the
Trotter-Kato scheme allows for a simpler proof of existence than in case of
It\^o noise.Comment: 38 pages, revised version, nonnegativity proof changed, details to
time regularity and interpolation of operators adde
Improved moment estimates for invariant measures of semilinear diffusions in Hilbert spaces and applications
We study regularity properties for invariant measures of semilinear
diffusions in a separable Hilbert space. Based on a pathwise estimate for the
underlying stochastic convolution, we prove a priori estimates on such
invariant measures. As an application, we combine such estimates with a new
technique to prove the -uniqueness of the induced Kolmogorov operator,
defined on a space of cylindrical functions. Finally, examples of stochastic
Burgers equations and thin-film growth models are given to illustrate our
abstract result.Comment: 19 page
An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation
In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation is
solved numerically by using the finite difference method in combination with a
convex splitting technique of the energy functional. For the non-stochastic
case, we develop an unconditionally energy stable difference scheme which is
proved to be uniquely solvable. For the stochastic case, by adopting the same
splitting of the energy functional, we construct a similar and uniquely
solvable difference scheme with the discretized stochastic term. The resulted
schemes are nonlinear and solved by Newton iteration. For the long time
simulation, an adaptive time stepping strategy is developed based on both
first- and second-order derivatives of the energy. Numerical experiments are
carried out to verify the energy stability, the efficiency of the adaptive time
stepping and the effect of the stochastic term.Comment: This paper has been accepted for publication in SCIENCE CHINA
Mathematic
Wave energy localization by self-focusing in large molecular structures: a damped stochastic discrete nonlinear Schroedinger equation model
Wave self-focusing in molecular systems subject to thermal effects, such as
thin molecular films and long biomolecules, can be modeled by stochastic
versions of the Discrete Self-Trapping equation of Eilbeck, Lomdahl and Scott,
and this can be approximated by continuum limits in the form of stochastic
nonlinear Schroedinger equations.
Previous studies directed at the SNLS approximations have indicated that the
self-focusing of wave energy to highly localized states can be inhibited by
phase noise (modeling thermal effects) and can be restored by phase damping
(modeling heat radiation).
We show that the continuum limit is probably ill-posed in the presence of
spatially uncorrelated noise, at least with little or no damping, so that
discrete models need to be addressed directly. Also, as has been noted by other
authors, omission of damping produces highly unphysical results.
Numerical results are presented for the first time for the discrete models
including the highly nonlinear damping term, and new numerical methods are
introduced for this purpose. Previous conjectures are in general confirmed, and
the damping is shown to strongly stabilize the highly localized states of the
discrete models. It appears that the previously noted inhibition of nonlinear
wave phenomena by noise is an artifact of modeling that includes the effects of
heat, but not of heat loss.Comment: 22 pages, 13 figures, revision of talk at FPU+50 conference in Rouen,
June 200
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