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@PHDTHESIS{Chugreeva:679916,
author = {Chugreeva, Olga},
othercontributors = {Melcher, Christof and Westdickenberg, Maria Gabrielle},
title = {{S}tochastics meets applied analysis : stochastic
{G}inzburg-{L}andau vortices and stochastic
{L}andau-{L}ifshitz-{G}ilbert equation},
school = {RWTH Aachen University},
type = {Dissertation},
address = {Aachen},
reportid = {RWTH-2016-11479},
pages = {1 Online-Ressource (xii,130 Seiten) : Illustrationen},
year = {2016},
note = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
University 2017; Dissertation, RWTH Aachen University, 2016},
abstract = {AbstractThis work belongs to the fields of applied analysis
and stochastic partial differential equations. We study
stochastic versions of two well-known nonlinear partial
differential equations, the Landau-Lifshitz-Gilbert and the
Ginzburg-Landau equation.The deterministic prototypes of our
equations have many formal features in common and are used
to describe similar physical phenomena. We see the mixed
Ginzburg-Landau equation as the “light version” of the
Landau-Lifshitz-Gilbert equation. However, already in the
deterministic setting, the two equations are very different
in terms of the well- posedness. The Ginzburg-Landau
equation has a unique regular global in time solution. For
the Landau-Lifshitz-Gilbert equation, a regular solution
exists only locally in time, and weak solutions are not
unique. This difference is related to the fact that the
Ginzburg- Landau equation is semilinear, and the
Landau-Lifshitz-Gilbert is only quasilinear. Due to the
difference in the analytic properties, the questions that we
address for the two equations in the stochastic framework
are also very different.For the stochastic
Landau-Lifshitz-Gilbert equation, already well-posedness is
a challenging problem. The known techniques yield a solution
that is not unique and both analytically and stochastically
weak. In Chapter 2, we deal at the same time with the
non-uniquness of solution and the stochastic sense of
solvability. We propose a regular- ization of the stochastic
Landau-Lifshitz-Gilbert equation that is admissible from the
physical point of view. We show that the solution of the
regularized equation exists in the stochastically strong
sense and is unique. This follows from an argu- ment of the
Yamada-Watanabe type: For S(P)DE, solvability in the
stochastically weak sense and uniqueness in the
stochastically strong sense implies solvability in the
stochasti- cally strong sense. Accordingly, we first
construct a stochastically weak solution and then show that
it is unique in the stochastically strong sense.For the
Ginzburg-Landau equation, we focus on a more particular
question. We are interested in the dynamics of the point
singularities of the solution in the presence of random
forcing. To the best of our knowledge, we are the first to
investigate this topic. As a preparation, we consider in
Chapter 3 the mixed Ginzburg-Landau equation with
deterministic forcing of convective form. For this equation,
we derive the effective motion law. The standard toolbox
developed for equations of Ginzburg-Landau type suffices at
that point. This way, we make sure that the convective
forcing impacts the effective dynamics but does not destroy
it.In Chapter 4, we study the stochastic parabolic
Ginzburg-Landau equation with a multiplicative noise. The
noise is again of the convective form. For this equation,
existence and uniqueness of a stochastically strong regular
solution is obtained rather easily. Our main effort is
therefore devoted to the description of the point
singularities of the solution. This amounts to finding the
correct stochastic counterparts of the tools used for this
purpose in the deterministic setting. Consequently, our main
result concerns the Jacobians of the solution. We prove that
the Jacobians are tight and do correctly locate the set of
point singularities, as in the deterministic case. In
addition, we consider two special cases. For the vanishing
noise, we show that the rescaled energy densities are tight
on a space of time-dependent functions. Their limit set
corresponds to the trajectories of the point singularities
rather than to their positions at fixed time. For a
spatially uniform noise, we establish the effective motion
law, which is given by a system of stochastic differential
equations.},
cin = {113110 / 110000},
ddc = {510},
cid = {$I:(DE-82)113110_20140620$ / $I:(DE-82)110000_20140620$},
typ = {PUB:(DE-HGF)11},
urn = {urn:nbn:de:hbz:82-rwth-2016-114797},
url = {https://publications.rwth-aachen.de/record/679916},
}
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