Stochastics meets applied analysis : stochastic Ginzburg-Landau vortices and stochastic Landau-Lifshitz-Gilbert equation

  • Stochastik trifft angewandte Analysis : stochastische Ginzburg-Landau Wirbel und stochastische Landau-Lifshitz-Gilbert Gleichung

Chugreeva, Olga; Melcher, Christof Erich (Thesis advisor); Westdickenberg, Maria Gabrielle (Thesis advisor)

Aachen (2016, 2017)
Dissertation / PhD Thesis

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.